Subgroups — C₆₇×#32941 #30712192

Order 17152.

1342 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#131×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#471×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₂×#465×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#3007	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#3006	C₂	1	1	yes	maximal, characteristic
6	C₆₇×#1908	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1907	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1901	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1900	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#1815	C₂	1	2	yes	maximal, complemented
11	C₆₇×#1810	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×#137×C₆₇	C₂²	1	2	yes	—
14	C₂×AGL(1,ℤ/8)×C₆₇	C₂²	1	2	yes	—
15	C₂²×SD₁₆×C₆₇	C₂²	1	1	yes	characteristic, complemented
16	C₂×#131×C₆₇	—	2	8	no	—
17	C₂×#131×C₆₇	—	2	1	no	—
18	C₂×#131×C₆₇	—	2	1	no	—
19	C₂×#131×C₆₇	C₂²	1	2	yes	—
20	C₂×#131×C₆₇	C₂²	1	2	yes	—
21	C₂²×M₁₆×C₆₇	—	2	1	no	—
22	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
23	C₂³×C₈×C₆₇	—	2	1	no	—
24	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×#121×C₆₇	C₂²	1	1	yes	characteristic
27	#471×C₆₇	—	2	8	no	—
28	#465×C₆₇	—	2	1	no	—
29	#465×C₆₇	—	2	1	no	—
30	#465×C₆₇	—	2	1	no	—
31	#465×C₆₇	—	2	1	no	—
32	#452×C₆₇	—	2	2	no	—
33	#451×C₆₇	—	2	1	no	—
34	#450×C₆₇	—	2	1	no	—
35	#449×C₆₇	—	2	2	no	—
36	#448×C₆₇	—	2	1	no	—
37	#447×C₆₇	—	2	1	no	—
38	C₂×#108×C₆₇	C₂²	1	2	yes	complemented
39	C₂×#106×C₆₇	C₂²	1	1	yes	characteristic, complemented
40	#417×C₆₇	C₂²	1	2	yes	—
41	#417×C₆₇	C₂²	1	2	yes	—
42	C₂×#103×C₆₇	C₂²	1	1	yes	characteristic, complemented
43	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic, complemented
44	C₂×#101×C₆₇	C₂²	1	2	yes	complemented
45	C₂×#100×C₆₇	C₂²	1	2	yes	complemented
46	#407×C₆₇	C₂²	1	1	yes	characteristic
47	#407×C₆₇	C₂²	1	1	yes	characteristic
48	#343×C₆₇	C₂²	1	2	yes	complemented
49	#343×C₆₇	C₂²	1	2	yes	complemented
50	#341×C₆₇	C₂²	1	2	yes	complemented
51	C₂×#41×C₆₇	—	2	4	no	—
52	C₂×#41×C₆₇	—	2	4	no	—
53	C₂×#41×C₆₇	—	2	1	no	—
54	C₂×#41×C₆₇	—	2	1	no	—
55	C₂×#41×C₆₇	D₄	1	2	yes	—
56	C₂×#41×C₆₇	C₂³	1	2	yes	—
57	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
58	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
59	C₂²×D₄×C₆₇	—	2	1	no	—
60	C₂²×D₄×C₆₇	—	2	1	no	—
61	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
62	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
63	C₂³×C₄×C₆₇	—	2	1	no	—
64	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
65	#137×C₆₇	—	2	4	no	—
66	#137×C₆₇	—	2	4	no	—
67	AGL(1,ℤ/8)×C₆₇	—	2	4	no	—
68	AGL(1,ℤ/8)×C₆₇	—	2	4	no	—
69	C₂×Q₁₆×C₆₇	—	2	2	no	—
70	C₂×SD₁₆×C₆₇	—	2	2	no	—
71	C₂×SD₁₆×C₆₇	—	2	2	no	—
72	C₂×SD₁₆×C₆₇	—	2	1	no	—
73	C₂×SD₁₆×C₆₇	—	2	1	no	—
74	C₂×SD₁₆×C₆₇	—	2	4	no	—
75	C₂×D₈×C₆₇	—	2	2	no	—
76	#131×C₆₇	—	2	8	no	—
77	#131×C₆₇	—	2	8	no	—
78	#131×C₆₇	—	2	4	no	—
79	#131×C₆₇	—	2	4	no	—
80	#131×C₆₇	—	2	4	no	—
81	#131×C₆₇	D₄	1	8	yes	complemented
82	C₂×M₁₆×C₆₇	—	2	8	no	—
83	C₂×M₁₆×C₆₇	—	2	1	no	—
84	C₂×M₁₆×C₆₇	—	2	4	no	—
85	C₂×M₁₆×C₆₇	—	2	2	no	—
86	C₂×M₁₆×C₆₇	—	2	2	no	—
87	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
88	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
89	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
90	C₂²×C₈×C₆₇	—	2	8	no	—
91	C₂²×C₈×C₆₇	—	2	1	no	—
92	C₂²×C₈×C₆₇	—	2	2	no	—
93	C₂²×C₈×C₆₇	—	2	2	no	—
94	C₂²×C₈×C₆₇	—	2	4	no	—
95	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
96	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
97	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
98	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
99	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
100	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
101	C₂²⋊Q₈×C₆₇	—	2	1	no	—
102	C₂²⋊Q₈×C₆₇	—	2	1	no	—
103	C₂²⋊Q₈×C₆₇	—	2	1	no	—
104	C₂²⋊Q₈×C₆₇	—	2	1	no	—
105	#121×C₆₇	—	2	1	no	—
106	#121×C₆₇	—	2	1	no	—
107	#121×C₆₇	—	2	1	no	—
108	#121×C₆₇	—	2	1	no	—
109	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
110	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
111	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
112	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
113	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
114	#108×C₆₇	—	2	4	no	—
115	#106×C₆₇	—	2	1	no	—
116	#106×C₆₇	—	2	1	no	—
117	#103×C₆₇	—	2	2	no	—
118	#103×C₆₇	—	2	2	no	—
119	#103×C₆₇	—	2	1	no	—
120	#103×C₆₇	—	2	1	no	—
121	#102×C₆₇	—	2	2	no	—
122	#102×C₆₇	—	2	2	no	—
123	#102×C₆₇	—	2	1	no	—
124	#102×C₆₇	—	2	1	no	—
125	#101×C₆₇	—	2	4	no	—
126	#100×C₆₇	—	2	4	no	—
127	C₂²⋊C₈×C₆₇	—	2	1	no	—
128	C₂²⋊C₈×C₆₇	—	2	1	no	—
129	C₂²⋊C₈×C₆₇	—	2	1	no	—
130	C₂²⋊C₈×C₆₇	—	2	1	no	—
131	C₂⁴×C₆₇	—	2	1	no	—
132	C₂⁴×C₆₇	—	2	1	no	—
133	C₂⁴×C₆₇	#41	1	1	yes	characteristic
134	#41×C₆₇	—	2	4	no	—
135	#41×C₆₇	—	2	8	no	—
136	#41×C₆₇	—	2	4	no	—
137	#41×C₆₇	—	2	4	no	—
138	#41×C₆₇	—	2	4	no	—
139	#41×C₆₇	—	2	2	no	—
140	#41×C₆₇	—	2	2	no	—
141	#41×C₆₇	C₂×D₄	1	4	yes	—
142	#41×C₆₇	#41	1	4	yes	—
143	C₂×Q₈×C₆₇	—	2	4	no	—
144	C₂×Q₈×C₆₇	—	2	1	no	—
145	C₂×Q₈×C₆₇	—	2	2	no	—
146	C₂×Q₈×C₆₇	—	2	1	no	—
147	C₂×Q₈×C₆₇	—	2	1	no	—
148	C₂×Q₈×C₆₇	—	2	2	no	—
149	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
150	C₂×D₄×C₆₇	—	2	8	no	—
151	C₂×D₄×C₆₇	—	2	2	no	—
152	C₂×D₄×C₆₇	—	2	2	no	—
153	C₂×D₄×C₆₇	—	2	4	no	—
154	C₂×D₄×C₆₇	—	2	1	no	—
155	C₂×D₄×C₆₇	—	2	2	no	—
156	C₂×D₄×C₆₇	—	2	1	no	—
157	C₂×D₄×C₆₇	—	2	1	no	—
158	C₂×D₄×C₆₇	—	2	2	no	—
159	C₂×D₄×C₆₇	—	4	1	no	—
160	C₂×D₄×C₆₇	—	4	1	no	—
161	C₂×D₄×C₆₇	—	8	2	no	—
162	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
163	C₂²×C₄×C₆₇	—	2	4	no	—
164	C₂²×C₄×C₆₇	—	2	4	no	—
165	C₂²×C₄×C₆₇	—	2	1	no	—
166	C₂²×C₄×C₆₇	—	2	2	no	—
167	C₂²×C₄×C₆₇	—	2	2	no	—
168	C₂²×C₄×C₆₇	—	2	2	no	—
169	C₂²×C₄×C₆₇	—	2	2	no	—
170	C₂²×C₄×C₆₇	—	2	1	no	—
171	C₂²×C₄×C₆₇	—	2	1	no	—
172	C₂²×C₄×C₆₇	—	2	1	no	—
173	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
174	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
175	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
176	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
177	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
178	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
179	Q₁₆×C₆₇	—	4	4	no	—
180	SD₁₆×C₆₇	—	4	4	no	—
181	SD₁₆×C₆₇	—	4	4	no	—
182	SD₁₆×C₆₇	—	4	2	no	—
183	SD₁₆×C₆₇	—	4	2	no	—
184	D₈×C₆₇	—	4	4	no	—
185	M₁₆×C₆₇	—	2	4	no	—
186	M₁₆×C₆₇	—	2	8	no	—
187	M₁₆×C₆₇	—	2	2	no	—
188	M₁₆×C₆₇	—	2	8	no	—
189	M₁₆×C₆₇	—	2	2	no	—
190	M₁₆×C₆₇	—	2	2	no	—
191	M₁₆×C₆₇	—	2	2	no	—
192	M₁₆×C₆₇	—	2	2	no	—
193	M₁₆×C₆₇	C₂×D₄	1	4	yes	—
194	C₂×C₈×C₆₇	—	2	4	no	—
195	C₂×C₈×C₆₇	—	2	8	no	—
196	C₂×C₈×C₆₇	—	2	2	no	—
197	C₂×C₈×C₆₇	—	2	2	no	—
198	C₂×C₈×C₆₇	—	2	2	no	—
199	C₂×C₈×C₆₇	—	2	8	no	—
200	C₂×C₈×C₆₇	—	2	2	no	—
201	C₂×C₈×C₆₇	—	2	2	no	—
202	C₂×C₈×C₆₇	—	2	2	no	—
203	C₂×C₈×C₆₇	—	2	2	no	—
204	C₂×C₈×C₆₇	—	2	2	no	—
205	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
206	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
207	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
208	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
209	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
210	C₄⋊C₄×C₆₇	—	2	1	no	—
211	C₄⋊C₄×C₆₇	—	2	1	no	—
212	C₄⋊C₄×C₆₇	—	2	1	no	—
213	C₄⋊C₄×C₆₇	—	2	1	no	—
214	C₄⋊C₄×C₆₇	—	4	1	no	—
215	C₄⋊C₄×C₆₇	—	4	1	no	—
216	C₂²⋊C₄×C₆₇	—	4	1	no	—
217	C₂²⋊C₄×C₆₇	—	4	1	no	—
218	C₂²⋊C₄×C₆₇	—	4	1	no	—
219	C₂²⋊C₄×C₆₇	—	4	1	no	—
220	C₂³×C₆₇	—	2	4	no	—
221	C₂³×C₆₇	—	2	2	no	—
222	C₂³×C₆₇	—	2	2	no	—
223	C₂³×C₆₇	—	2	2	no	—
224	C₂³×C₆₇	—	4	1	no	—
225	C₂³×C₆₇	—	4	2	no	—
226	C₂³×C₆₇	—	4	2	no	—
227	C₂³×C₆₇	—	4	1	no	—
228	C₂³×C₆₇	—	4	2	no	—
229	C₂³×C₆₇	—	8	1	no	—
230	C₂³×C₆₇	—	8	1	no	—
231	C₂³×C₆₇	#121	1	1	yes	characteristic
232	C₂³×C₆₇	#121	1	1	yes	characteristic
233	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
234	Q₈×C₆₇	—	2	4	no	—
235	Q₈×C₆₇	—	2	2	no	—
236	Q₈×C₆₇	—	4	1	no	—
237	Q₈×C₆₇	—	4	2	no	—
238	Q₈×C₆₇	—	4	1	no	—
239	Q₈×C₆₇	#121	1	4	yes	—
240	D₄×C₆₇	—	2	8	no	—
241	D₄×C₆₇	—	2	4	no	—
242	D₄×C₆₇	—	2	4	no	—
243	D₄×C₆₇	—	2	4	no	—
244	D₄×C₆₇	—	2	2	no	—
245	D₄×C₆₇	—	4	1	no	—
246	D₄×C₆₇	—	4	2	no	—
247	D₄×C₆₇	—	4	1	no	—
248	D₄×C₆₇	—	8	2	no	—
249	D₄×C₆₇	—	8	2	no	—
250	D₄×C₆₇	#121	1	4	yes	—
251	C₂×C₄×C₆₇	—	2	4	no	—
252	C₂×C₄×C₆₇	—	2	8	no	—
253	C₂×C₄×C₆₇	—	2	2	no	—
254	C₂×C₄×C₆₇	—	2	2	no	—
255	C₂×C₄×C₆₇	—	2	1	no	—
256	C₂×C₄×C₆₇	—	2	4	no	—
257	C₂×C₄×C₆₇	—	2	1	no	—
258	C₂×C₄×C₆₇	—	2	2	no	—
259	C₂×C₄×C₆₇	—	2	2	no	—
260	C₂×C₄×C₆₇	—	2	1	no	—
261	C₂×C₄×C₆₇	—	2	4	no	—
262	C₂×C₄×C₆₇	—	2	1	no	—
263	C₂×C₄×C₆₇	—	4	1	no	—
264	C₂×C₄×C₆₇	—	4	1	no	—
265	C₂×C₄×C₆₇	—	4	1	no	—
266	C₂×C₄×C₆₇	—	4	2	no	—
267	C₂×C₄×C₆₇	—	8	1	no	—
268	C₂×C₄×C₆₇	—	8	1	no	—
269	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
270	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
271	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
272	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
273	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
274	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
275	C₈×C₆₇	—	2	1	no	—
276	C₈×C₆₇	—	2	4	no	—
277	C₈×C₆₇	—	2	1	no	—
278	C₈×C₆₇	—	4	2	no	—
279	C₈×C₆₇	—	4	2	no	—
280	C₈×C₆₇	#120	1	2	yes	—
281	C₈×C₆₇	#120	1	2	yes	—
282	C₂²×C₆₇	—	2	1	no	—
283	C₂²×C₆₇	—	2	1	no	—
284	C₂²×C₆₇	—	2	1	no	—
285	C₂²×C₆₇	—	2	1	no	—
286	C₂²×C₆₇	—	2	1	no	—
287	C₂²×C₆₇	—	2	4	no	—
288	C₂²×C₆₇	—	2	1	no	—
289	C₂²×C₆₇	—	2	1	no	—
290	C₂²×C₆₇	—	4	4	no	—
291	C₂²×C₆₇	—	4	8	no	—
292	C₂²×C₆₇	—	4	2	no	—
293	C₂²×C₆₇	—	4	2	no	—
294	C₂²×C₆₇	—	8	1	no	—
295	C₂²×C₆₇	—	8	1	no	—
296	C₂²×C₆₇	—	8	1	no	—
297	C₂²×C₆₇	—	16	2	no	—
298	C₂²×C₆₇	#471	1	2	yes	complemented
299	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
300	C₂²×C₆₇	#525	1	2	yes	—
301	C₄×C₆₇	—	2	1	no	—
302	C₄×C₆₇	—	2	4	no	—
303	C₄×C₆₇	—	2	1	no	—
304	C₄×C₆₇	—	8	1	no	—
305	C₄×C₆₇	—	8	1	no	—
306	C₄×C₆₇	—	8	1	no	—
307	C₄×C₆₇	C₂×#120	1	1	yes	characteristic
308	C₄×C₆₇	C₂×#121	1	2	yes	—
309	C₄×C₆₇	#525	1	1	yes	characteristic
310	#32941	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
311	C₂×C₆₇	—	2	2	no	—
312	C₂×C₆₇	—	2	2	no	—
313	C₂×C₆₇	—	4	1	no	—
314	C₂×C₆₇	—	4	4	no	—
315	C₂×C₆₇	—	16	1	no	—
316	C₂×C₆₇	#2847	1	1	yes	characteristic
317	C₂×C₆₇	C₂×#471	1	1	yes	characteristic
318	C₂×C₆₇	#3067	1	1	yes	characteristic
319	C₂²×#131	C₂×C₆₇	1	1	yes	characteristic, complemented
320	C₂×#471	C₂×C₆₇	1	2	yes	complemented
321	C₂×#465	C₂×C₆₇	1	1	yes	characteristic, complemented
322	#3007	C₂×C₆₇	1	1	yes	characteristic, complemented
323	#3006	C₂×C₆₇	1	1	yes	characteristic
324	#1908	C₂×C₆₇	1	2	yes	complemented
325	#1907	C₂×C₆₇	1	2	yes	complemented
326	#1901	C₂×C₆₇	1	1	yes	characteristic, complemented
327	#1900	C₂×C₆₇	1	1	yes	characteristic, complemented
328	#1815	C₂×C₆₇	1	2	yes	complemented
329	#1810	C₂×C₆₇	1	1	yes	characteristic, complemented
330	C₆₇	#32941	1	1	yes	characteristic, direct product factor, Sylow
331	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
332	C₂×#137	C₂²×C₆₇	1	2	yes	—
333	C₂×AGL(1,ℤ/8)	C₂²×C₆₇	1	2	yes	—
334	C₂²×SD₁₆	C₂²×C₆₇	1	1	yes	characteristic, complemented
335	C₂×#131	—	2	8	no	—
336	C₂×#131	—	2	1	no	—
337	C₂×#131	—	2	1	no	—
338	C₂×#131	C₂²×C₆₇	1	2	yes	—
339	C₂×#131	C₂²×C₆₇	1	2	yes	—
340	C₂²×M₁₆	—	2	1	no	—
341	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
342	C₂³×C₈	—	2	1	no	—
343	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
344	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
345	C₂×#121	C₂²×C₆₇	1	1	yes	characteristic
346	#471	—	2	8	no	—
347	#465	—	2	1	no	—
348	#465	—	2	1	no	—
349	#465	—	2	1	no	—
350	#465	—	2	1	no	—
351	#452	—	2	2	no	—
352	#451	—	2	1	no	—
353	#450	—	2	1	no	—
354	#449	—	2	2	no	—
355	#448	—	2	1	no	—
356	#447	—	2	1	no	—
357	C₂×#108	C₂²×C₆₇	1	2	yes	complemented
358	C₂×#106	C₂²×C₆₇	1	1	yes	characteristic, complemented
359	#417	C₂²×C₆₇	1	2	yes	—
360	#417	C₂²×C₆₇	1	2	yes	—
361	C₂×#103	C₂²×C₆₇	1	1	yes	characteristic, complemented
362	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic, complemented
363	C₂×#101	C₂²×C₆₇	1	2	yes	complemented
364	C₂×#100	C₂²×C₆₇	1	2	yes	complemented
365	#407	C₂²×C₆₇	1	1	yes	characteristic
366	#407	C₂²×C₆₇	1	1	yes	characteristic
367	#343	C₂²×C₆₇	1	2	yes	complemented
368	#343	C₂²×C₆₇	1	2	yes	complemented
369	#341	C₂²×C₆₇	1	2	yes	complemented
370	C₂×#41	—	2	4	no	—
371	C₂×#41	—	2	4	no	—
372	C₂×#41	—	2	1	no	—
373	C₂×#41	—	2	1	no	—
374	C₂×#41	D₄×C₆₇	1	2	yes	—
375	C₂×#41	C₂³×C₆₇	1	2	yes	—
376	C₂²×Q₈	D₄×C₆₇	1	1	yes	characteristic
377	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
378	C₂²×D₄	—	2	1	no	—
379	C₂²×D₄	—	2	1	no	—
380	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
381	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
382	C₂³×C₄	—	2	1	no	—
383	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
384	#137	—	2	4	no	—
385	#137	—	2	4	no	—
386	AGL(1,ℤ/8)	—	2	4	no	—
387	AGL(1,ℤ/8)	—	2	4	no	—
388	C₂×Q₁₆	—	2	2	no	—
389	C₂×SD₁₆	—	2	2	no	—
390	C₂×SD₁₆	—	2	2	no	—
391	C₂×SD₁₆	—	2	1	no	—
392	C₂×SD₁₆	—	2	1	no	—
393	C₂×SD₁₆	—	2	4	no	—
394	C₂×D₈	—	2	2	no	—
395	#131	—	2	8	no	—
396	#131	—	2	8	no	—
397	#131	—	2	4	no	—
398	#131	—	2	4	no	—
399	#131	—	2	4	no	—
400	#131	D₄×C₆₇	1	8	yes	complemented
401	C₂×M₁₆	—	2	8	no	—
402	C₂×M₁₆	—	2	1	no	—
403	C₂×M₁₆	—	2	4	no	—
404	C₂×M₁₆	—	2	2	no	—
405	C₂×M₁₆	—	2	2	no	—
406	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
407	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
408	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
409	C₂²×C₈	—	2	8	no	—
410	C₂²×C₈	—	2	1	no	—
411	C₂²×C₈	—	2	2	no	—
412	C₂²×C₈	—	2	2	no	—
413	C₂²×C₈	—	2	4	no	—
414	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
415	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
416	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
417	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
418	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
419	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
420	C₂²⋊Q₈	—	2	1	no	—
421	C₂²⋊Q₈	—	2	1	no	—
422	C₂²⋊Q₈	—	2	1	no	—
423	C₂²⋊Q₈	—	2	1	no	—
424	#121	—	2	1	no	—
425	#121	—	2	1	no	—
426	#121	—	2	1	no	—
427	#121	—	2	1	no	—
428	C₂×C₄⋊C₄	—	2	1	no	—
429	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
430	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
431	C₂×C₂²⋊C₄	—	2	1	no	—
432	C₂×C₂²⋊C₄	—	2	1	no	—
433	#108	—	2	4	no	—
434	#106	—	2	1	no	—
435	#106	—	2	1	no	—
436	#103	—	2	2	no	—
437	#103	—	2	2	no	—
438	#103	—	2	1	no	—
439	#103	—	2	1	no	—
440	#102	—	2	2	no	—
441	#102	—	2	2	no	—
442	#102	—	2	1	no	—
443	#102	—	2	1	no	—
444	#101	—	2	4	no	—
445	#100	—	2	4	no	—
446	C₂²⋊C₈	—	2	1	no	—
447	C₂²⋊C₈	—	2	1	no	—
448	C₂²⋊C₈	—	2	1	no	—
449	C₂²⋊C₈	—	2	1	no	—
450	C₂⁴	—	2	1	no	—
451	C₂⁴	—	2	1	no	—
452	C₂⁴	#41×C₆₇	1	1	yes	characteristic
453	#41	—	2	4	no	—
454	#41	—	2	8	no	—
455	#41	—	2	4	no	—
456	#41	—	2	4	no	—
457	#41	—	2	4	no	—
458	#41	—	2	2	no	—
459	#41	—	2	2	no	—
460	#41	C₂×D₄×C₆₇	1	4	yes	—
461	#41	#41×C₆₇	1	4	yes	—
462	C₂×Q₈	—	2	4	no	—
463	C₂×Q₈	—	2	1	no	—
464	C₂×Q₈	—	2	2	no	—
465	C₂×Q₈	—	2	1	no	—
466	C₂×Q₈	—	2	1	no	—
467	C₂×Q₈	—	2	2	no	—
468	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
469	C₂×D₄	—	2	8	no	—
470	C₂×D₄	—	2	2	no	—
471	C₂×D₄	—	2	2	no	—
472	C₂×D₄	—	2	4	no	—
473	C₂×D₄	—	2	1	no	—
474	C₂×D₄	—	2	2	no	—
475	C₂×D₄	—	2	1	no	—
476	C₂×D₄	—	2	1	no	—
477	C₂×D₄	—	2	2	no	—
478	C₂×D₄	—	4	1	no	—
479	C₂×D₄	—	4	1	no	—
480	C₂×D₄	—	8	2	no	—
481	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
482	C₂²×C₄	—	2	4	no	—
483	C₂²×C₄	—	2	4	no	—
484	C₂²×C₄	—	2	1	no	—
485	C₂²×C₄	—	2	2	no	—
486	C₂²×C₄	—	2	2	no	—
487	C₂²×C₄	—	2	2	no	—
488	C₂²×C₄	—	2	2	no	—
489	C₂²×C₄	—	2	1	no	—
490	C₂²×C₄	—	2	1	no	—
491	C₂²×C₄	—	2	1	no	—
492	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
493	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
494	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
495	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
496	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
497	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
498	Q₁₆	—	4	4	no	—
499	SD₁₆	—	4	4	no	—
500	SD₁₆	—	4	4	no	—
501	SD₁₆	—	4	2	no	—
502	SD₁₆	—	4	2	no	—
503	D₈	—	4	4	no	—
504	M₁₆	—	2	4	no	—
505	M₁₆	—	2	8	no	—
506	M₁₆	—	2	2	no	—
507	M₁₆	—	2	8	no	—
508	M₁₆	—	2	2	no	—
509	M₁₆	—	2	2	no	—
510	M₁₆	—	2	2	no	—
511	M₁₆	—	2	2	no	—
512	M₁₆	C₂×D₄×C₆₇	1	4	yes	—
513	C₂×C₈	—	2	4	no	—
514	C₂×C₈	—	2	8	no	—
515	C₂×C₈	—	2	2	no	—
516	C₂×C₈	—	2	2	no	—
517	C₂×C₈	—	2	2	no	—
518	C₂×C₈	—	2	8	no	—
519	C₂×C₈	—	2	2	no	—
520	C₂×C₈	—	2	2	no	—
521	C₂×C₈	—	2	2	no	—
522	C₂×C₈	—	2	2	no	—
523	C₂×C₈	—	2	2	no	—
524	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
525	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
526	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
527	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
528	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
529	C₄⋊C₄	—	2	1	no	—
530	C₄⋊C₄	—	2	1	no	—
531	C₄⋊C₄	—	2	1	no	—
532	C₄⋊C₄	—	2	1	no	—
533	C₄⋊C₄	—	4	1	no	—
534	C₄⋊C₄	—	4	1	no	—
535	C₂²⋊C₄	—	4	1	no	—
536	C₂²⋊C₄	—	4	1	no	—
537	C₂²⋊C₄	—	4	1	no	—
538	C₂²⋊C₄	—	4	1	no	—
539	C₂³	—	2	4	no	—
540	C₂³	—	2	2	no	—
541	C₂³	—	2	2	no	—
542	C₂³	—	2	2	no	—
543	C₂³	—	4	1	no	—
544	C₂³	—	4	2	no	—
545	C₂³	—	4	2	no	—
546	C₂³	—	4	1	no	—
547	C₂³	—	4	2	no	—
548	C₂³	—	8	1	no	—
549	C₂³	—	8	1	no	—
550	C₂³	#121×C₆₇	1	1	yes	characteristic
551	C₂³	#121×C₆₇	1	1	yes	characteristic
552	C₂³	C₂×#41×C₆₇	1	1	yes	characteristic
553	Q₈	—	2	4	no	—
554	Q₈	—	2	2	no	—
555	Q₈	—	4	1	no	—
556	Q₈	—	4	2	no	—
557	Q₈	—	4	1	no	—
558	Q₈	#121×C₆₇	1	4	yes	—
559	D₄	—	2	8	no	—
560	D₄	—	2	4	no	—
561	D₄	—	2	4	no	—
562	D₄	—	2	4	no	—
563	D₄	—	2	2	no	—
564	D₄	—	4	1	no	—
565	D₄	—	4	2	no	—
566	D₄	—	4	1	no	—
567	D₄	—	8	2	no	—
568	D₄	—	8	2	no	—
569	D₄	#121×C₆₇	1	4	yes	—
570	C₂×C₄	—	2	4	no	—
571	C₂×C₄	—	2	8	no	—
572	C₂×C₄	—	2	2	no	—
573	C₂×C₄	—	2	2	no	—
574	C₂×C₄	—	2	1	no	—
575	C₂×C₄	—	2	4	no	—
576	C₂×C₄	—	2	1	no	—
577	C₂×C₄	—	2	2	no	—
578	C₂×C₄	—	2	2	no	—
579	C₂×C₄	—	2	1	no	—
580	C₂×C₄	—	2	4	no	—
581	C₂×C₄	—	2	1	no	—
582	C₂×C₄	—	4	1	no	—
583	C₂×C₄	—	4	1	no	—
584	C₂×C₄	—	4	1	no	—
585	C₂×C₄	—	4	2	no	—
586	C₂×C₄	—	8	1	no	—
587	C₂×C₄	—	8	1	no	—
588	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
589	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
590	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
591	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
592	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
593	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
594	C₈	—	2	1	no	—
595	C₈	—	2	4	no	—
596	C₈	—	2	1	no	—
597	C₈	—	4	2	no	—
598	C₈	—	4	2	no	—
599	C₈	#120×C₆₇	1	2	yes	—
600	C₈	#120×C₆₇	1	2	yes	—
601	C₂²	—	2	1	no	—
602	C₂²	—	2	1	no	—
603	C₂²	—	2	1	no	—
604	C₂²	—	2	1	no	—
605	C₂²	—	2	1	no	—
606	C₂²	—	2	1	no	—
607	C₂²	—	2	1	no	—
608	C₂²	—	2	4	no	—
609	C₂²	—	4	4	no	—
610	C₂²	—	4	8	no	—
611	C₂²	—	4	2	no	—
612	C₂²	—	4	2	no	—
613	C₂²	—	8	1	no	—
614	C₂²	—	8	1	no	—
615	C₂²	—	8	1	no	—
616	C₂²	—	16	2	no	—
617	C₂²	#471×C₆₇	1	2	yes	complemented
618	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
619	C₂²	#525×C₆₇	1	2	yes	—
620	C₄	—	2	1	no	—
621	C₄	—	2	4	no	—
622	C₄	—	2	1	no	—
623	C₄	—	8	1	no	—
624	C₄	—	8	1	no	—
625	C₄	—	8	1	no	—
626	C₄	C₂×#120×C₆₇	1	1	yes	characteristic
627	C₄	C₂×#121×C₆₇	1	2	yes	—
628	C₄	#525×C₆₇	1	1	yes	characteristic
629	C₂	—	2	2	no	—
630	C₂	—	2	2	no	—
631	C₂	—	4	1	no	—
632	C₂	—	4	4	no	—
633	C₂	—	16	1	no	—
634	C₂	C₆₇×#2847	1	1	yes	characteristic
635	C₂	C₂×#471×C₆₇	1	1	yes	characteristic
636	C₂	C₆₇×#3067	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₆₇×#3006	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂³×C₄×C₆₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂²×#131×C₆₇	C₂	semidirect
2. C₂×#471×C₆₇	C₂	semidirect
3. C₂×#465×C₆₇	C₂	semidirect
4. C₆₇×#3007	C₂	semidirect
6. C₆₇×#1908	C₂	semidirect
7. C₆₇×#1907	C₂	semidirect
8. C₆₇×#1901	C₂	semidirect
9. C₆₇×#1900	C₂	semidirect
10. C₆₇×#1815	C₂	semidirect
11. C₆₇×#1810	C₂	semidirect
15. C₂²×SD₁₆×C₆₇	C₂²	semidirect
38. C₂×#108×C₆₇	C₂²	semidirect
39. C₂×#106×C₆₇	C₂²	semidirect
42. C₂×#103×C₆₇	C₂²	semidirect
43. C₂×#102×C₆₇	C₂²	semidirect
44. C₂×#101×C₆₇	C₂²	semidirect
45. C₂×#100×C₆₇	C₂²	semidirect
48. #343×C₆₇	C₂²	semidirect
49. #343×C₆₇	C₂²	semidirect
50. #341×C₆₇	C₂²	semidirect
81. #131×C₆₇	D₄	semidirect
298. C₂²×C₆₇	#471	semidirect
310. #32941	C₆₇	direct (N × H)
319. C₂²×#131	C₂×C₆₇	semidirect
320. C₂×#471	C₂×C₆₇	semidirect
321. C₂×#465	C₂×C₆₇	semidirect
322. #3007	C₂×C₆₇	semidirect
324. #1908	C₂×C₆₇	semidirect
325. #1907	C₂×C₆₇	semidirect
326. #1901	C₂×C₆₇	semidirect
327. #1900	C₂×C₆₇	semidirect
328. #1815	C₂×C₆₇	semidirect
329. #1810	C₂×C₆₇	semidirect
330. C₆₇	#32941	direct (N × H)
334. C₂²×SD₁₆	C₂²×C₆₇	semidirect
357. C₂×#108	C₂²×C₆₇	semidirect
358. C₂×#106	C₂²×C₆₇	semidirect
361. C₂×#103	C₂²×C₆₇	semidirect
362. C₂×#102	C₂²×C₆₇	semidirect
363. C₂×#101	C₂²×C₆₇	semidirect
364. C₂×#100	C₂²×C₆₇	semidirect
367. #343	C₂²×C₆₇	semidirect
368. #343	C₂²×C₆₇	semidirect
369. #341	C₂²×C₆₇	semidirect
400. #131	D₄×C₆₇	semidirect
617. C₂²	#471×C₆₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×#131×C₆₇	2
C₂×#471×C₆₇	2
C₂×#465×C₆₇	2
C₆₇×#3007	2
C₆₇×#3006	2
C₆₇×#1908	2
C₆₇×#1907	2
C₆₇×#1901	2
C₆₇×#1900	2
C₆₇×#1815	2
C₆₇×#1810	2
#32941	67

Subgroups — C67×#32941 #30712192

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#32941 #30712192