Subgroups — C₆₇×#32944 #30712195

Order 17152.

1320 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#568×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#471×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₂×#468×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×#1816	C₂	1	2	yes	maximal, complemented
11	C₆₇×#1811	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	#568×C₆₇	—	2	8	no	—
17	C₂×#131×C₆₇	—	2	2	no	—
18	C₂×#131×C₆₇	C₂²	1	2	yes	—
19	C₂×#131×C₆₇	C₂²	1	2	yes	—
20	C₂²×M₁₆×C₆₇	—	2	2	no	—
21	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
22	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×#121×C₆₇	C₂²	1	1	yes	characteristic
25	#471×C₆₇	—	2	8	no	—
26	#468×C₆₇	—	2	2	no	—
27	#468×C₆₇	—	2	2	no	—
28	#452×C₆₇	—	2	2	no	—
29	#451×C₆₇	—	2	1	no	—
30	#450×C₆₇	—	2	1	no	—
31	#449×C₆₇	—	2	2	no	—
32	#448×C₆₇	—	2	1	no	—
33	#447×C₆₇	—	2	1	no	—
34	C₂×#108×C₆₇	C₂²	1	2	yes	complemented
35	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic, complemented
36	#417×C₆₇	C₂²	1	2	yes	—
37	#417×C₆₇	C₂²	1	2	yes	—
38	C₂×#103×C₆₇	C₂²	1	1	yes	characteristic, complemented
39	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic, complemented
40	C₂×#101×C₆₇	C₂²	1	2	yes	complemented
41	C₂×#100×C₆₇	C₂²	1	2	yes	complemented
42	#407×C₆₇	C₂²	1	1	yes	characteristic
43	#407×C₆₇	C₂²	1	1	yes	characteristic
44	#343×C₆₇	C₂²	1	2	yes	complemented
45	#343×C₆₇	C₂²	1	2	yes	complemented
46	#341×C₆₇	C₂²	1	2	yes	complemented
47	C₂×#41×C₆₇	—	2	4	no	—
48	C₂×#41×C₆₇	—	2	4	no	—
49	C₂×#41×C₆₇	—	2	1	no	—
50	C₂×#41×C₆₇	—	2	1	no	—
51	C₂×#41×C₆₇	D₄	1	2	yes	—
52	C₂×#41×C₆₇	C₂³	1	2	yes	—
53	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
54	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
55	C₂²×D₄×C₆₇	—	2	1	no	—
56	C₂²×D₄×C₆₇	—	2	1	no	—
57	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
58	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
59	C₂³×C₄×C₆₇	—	2	1	no	—
60	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
61	#137×C₆₇	—	2	4	no	—
62	#137×C₆₇	—	2	4	no	—
63	AGL(1,ℤ/8)×C₆₇	—	2	4	no	—
64	AGL(1,ℤ/8)×C₆₇	—	2	4	no	—
65	C₂×Q₁₆×C₆₇	—	2	2	no	—
66	C₂×SD₁₆×C₆₇	—	2	2	no	—
67	C₂×SD₁₆×C₆₇	—	2	2	no	—
68	C₂×SD₁₆×C₆₇	—	2	2	no	—
69	C₂×SD₁₆×C₆₇	—	2	4	no	—
70	C₂×D₈×C₆₇	—	2	2	no	—
71	#131×C₆₇	—	2	16	no	—
72	#131×C₆₇	—	2	4	no	—
73	#131×C₆₇	—	2	4	no	—
74	#131×C₆₇	—	2	4	no	—
75	#131×C₆₇	D₄	1	8	yes	complemented
76	C₂×M₁₆×C₆₇	—	2	16	no	—
77	C₂×M₁₆×C₆₇	—	2	1	no	—
78	C₂×M₁₆×C₆₇	—	2	4	no	—
79	C₂×M₁₆×C₆₇	—	2	2	no	—
80	C₂×M₁₆×C₆₇	—	2	2	no	—
81	C₂×M₁₆×C₆₇	—	2	4	no	—
82	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
83	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
84	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
85	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
86	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
87	C₂²×C₈×C₆₇	—	2	1	no	—
88	C₂²×C₈×C₆₇	—	2	2	no	—
89	C₂²×C₈×C₆₇	—	2	2	no	—
90	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
91	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
92	C₂²⋊Q₈×C₆₇	—	2	1	no	—
93	C₂²⋊Q₈×C₆₇	—	2	1	no	—
94	C₂²⋊Q₈×C₆₇	—	2	1	no	—
95	C₂²⋊Q₈×C₆₇	—	2	1	no	—
96	#121×C₆₇	—	2	1	no	—
97	#121×C₆₇	—	2	1	no	—
98	#121×C₆₇	—	2	1	no	—
99	#121×C₆₇	—	2	1	no	—
100	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
101	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
102	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
103	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
104	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
105	#108×C₆₇	—	2	4	no	—
106	#107×C₆₇	—	2	2	no	—
107	#103×C₆₇	—	2	2	no	—
108	#103×C₆₇	—	2	2	no	—
109	#103×C₆₇	—	2	2	no	—
110	#102×C₆₇	—	2	2	no	—
111	#102×C₆₇	—	2	2	no	—
112	#102×C₆₇	—	2	2	no	—
113	#101×C₆₇	—	2	4	no	—
114	#100×C₆₇	—	2	4	no	—
115	C₂²⋊C₈×C₆₇	—	2	1	no	—
116	C₂²⋊C₈×C₆₇	—	2	1	no	—
117	C₂²⋊C₈×C₆₇	—	2	1	no	—
118	C₂²⋊C₈×C₆₇	—	2	1	no	—
119	C₂⁴×C₆₇	—	2	1	no	—
120	C₂⁴×C₆₇	—	2	1	no	—
121	C₂⁴×C₆₇	#41	1	1	yes	characteristic
122	#41×C₆₇	—	2	4	no	—
123	#41×C₆₇	—	2	8	no	—
124	#41×C₆₇	—	2	4	no	—
125	#41×C₆₇	—	2	4	no	—
126	#41×C₆₇	—	2	4	no	—
127	#41×C₆₇	—	2	2	no	—
128	#41×C₆₇	—	2	2	no	—
129	#41×C₆₇	C₂×D₄	1	4	yes	—
130	#41×C₆₇	#41	1	4	yes	—
131	C₂×Q₈×C₆₇	—	2	4	no	—
132	C₂×Q₈×C₆₇	—	2	1	no	—
133	C₂×Q₈×C₆₇	—	2	2	no	—
134	C₂×Q₈×C₆₇	—	2	1	no	—
135	C₂×Q₈×C₆₇	—	2	1	no	—
136	C₂×Q₈×C₆₇	—	2	2	no	—
137	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
138	C₂×D₄×C₆₇	—	2	8	no	—
139	C₂×D₄×C₆₇	—	2	2	no	—
140	C₂×D₄×C₆₇	—	2	2	no	—
141	C₂×D₄×C₆₇	—	2	4	no	—
142	C₂×D₄×C₆₇	—	2	1	no	—
143	C₂×D₄×C₆₇	—	2	2	no	—
144	C₂×D₄×C₆₇	—	2	1	no	—
145	C₂×D₄×C₆₇	—	2	1	no	—
146	C₂×D₄×C₆₇	—	2	2	no	—
147	C₂×D₄×C₆₇	—	4	1	no	—
148	C₂×D₄×C₆₇	—	4	1	no	—
149	C₂×D₄×C₆₇	—	8	2	no	—
150	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
151	C₂²×C₄×C₆₇	—	2	4	no	—
152	C₂²×C₄×C₆₇	—	2	4	no	—
153	C₂²×C₄×C₆₇	—	2	1	no	—
154	C₂²×C₄×C₆₇	—	2	2	no	—
155	C₂²×C₄×C₆₇	—	2	2	no	—
156	C₂²×C₄×C₆₇	—	2	2	no	—
157	C₂²×C₄×C₆₇	—	2	2	no	—
158	C₂²×C₄×C₆₇	—	2	1	no	—
159	C₂²×C₄×C₆₇	—	2	1	no	—
160	C₂²×C₄×C₆₇	—	2	1	no	—
161	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
162	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
163	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
164	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
165	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
166	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
167	Q₁₆×C₆₇	—	4	4	no	—
168	SD₁₆×C₆₇	—	4	4	no	—
169	SD₁₆×C₆₇	—	4	4	no	—
170	SD₁₆×C₆₇	—	4	4	no	—
171	D₈×C₆₇	—	4	4	no	—
172	M₁₆×C₆₇	—	2	4	no	—
173	M₁₆×C₆₇	—	2	8	no	—
174	M₁₆×C₆₇	—	2	2	no	—
175	M₁₆×C₆₇	—	2	4	no	—
176	M₁₆×C₆₇	—	2	8	no	—
177	M₁₆×C₆₇	—	2	4	no	—
178	M₁₆×C₆₇	—	2	2	no	—
179	M₁₆×C₆₇	—	2	2	no	—
180	M₁₆×C₆₇	—	2	2	no	—
181	M₁₆×C₆₇	—	2	2	no	—
182	M₁₆×C₆₇	C₂×D₄	1	4	yes	—
183	C₂×C₈×C₆₇	—	2	4	no	—
184	C₂×C₈×C₆₇	—	2	8	no	—
185	C₂×C₈×C₆₇	—	2	2	no	—
186	C₂×C₈×C₆₇	—	2	2	no	—
187	C₂×C₈×C₆₇	—	2	8	no	—
188	C₂×C₈×C₆₇	—	2	2	no	—
189	C₂×C₈×C₆₇	—	2	2	no	—
190	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
191	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
192	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
193	C₄⋊C₄×C₆₇	—	2	1	no	—
194	C₄⋊C₄×C₆₇	—	2	1	no	—
195	C₄⋊C₄×C₆₇	—	2	1	no	—
196	C₄⋊C₄×C₆₇	—	2	1	no	—
197	C₄⋊C₄×C₆₇	—	4	1	no	—
198	C₄⋊C₄×C₆₇	—	4	1	no	—
199	C₂²⋊C₄×C₆₇	—	4	1	no	—
200	C₂²⋊C₄×C₆₇	—	4	1	no	—
201	C₂²⋊C₄×C₆₇	—	4	1	no	—
202	C₂²⋊C₄×C₆₇	—	4	1	no	—
203	C₂³×C₆₇	—	2	4	no	—
204	C₂³×C₆₇	—	2	2	no	—
205	C₂³×C₆₇	—	2	2	no	—
206	C₂³×C₆₇	—	4	1	no	—
207	C₂³×C₆₇	—	4	1	no	—
208	C₂³×C₆₇	—	4	1	no	—
209	C₂³×C₆₇	—	4	2	no	—
210	C₂³×C₆₇	—	8	1	no	—
211	C₂³×C₆₇	—	8	1	no	—
212	C₂³×C₆₇	—	8	1	no	—
213	C₂³×C₆₇	—	8	1	no	—
214	C₂³×C₆₇	#121	1	1	yes	characteristic
215	C₂³×C₆₇	#121	1	1	yes	characteristic
216	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
217	Q₈×C₆₇	—	2	4	no	—
218	Q₈×C₆₇	—	2	2	no	—
219	Q₈×C₆₇	—	4	1	no	—
220	Q₈×C₆₇	—	4	2	no	—
221	Q₈×C₆₇	—	4	1	no	—
222	Q₈×C₆₇	#121	1	4	yes	—
223	D₄×C₆₇	—	2	8	no	—
224	D₄×C₆₇	—	2	4	no	—
225	D₄×C₆₇	—	2	4	no	—
226	D₄×C₆₇	—	2	4	no	—
227	D₄×C₆₇	—	2	2	no	—
228	D₄×C₆₇	—	4	1	no	—
229	D₄×C₆₇	—	4	2	no	—
230	D₄×C₆₇	—	4	1	no	—
231	D₄×C₆₇	—	8	2	no	—
232	D₄×C₆₇	—	8	2	no	—
233	D₄×C₆₇	#121	1	4	yes	—
234	C₂×C₄×C₆₇	—	2	4	no	—
235	C₂×C₄×C₆₇	—	2	8	no	—
236	C₂×C₄×C₆₇	—	2	2	no	—
237	C₂×C₄×C₆₇	—	2	2	no	—
238	C₂×C₄×C₆₇	—	2	4	no	—
239	C₂×C₄×C₆₇	—	2	1	no	—
240	C₂×C₄×C₆₇	—	2	1	no	—
241	C₂×C₄×C₆₇	—	2	2	no	—
242	C₂×C₄×C₆₇	—	2	2	no	—
243	C₂×C₄×C₆₇	—	2	1	no	—
244	C₂×C₄×C₆₇	—	2	4	no	—
245	C₂×C₄×C₆₇	—	2	1	no	—
246	C₂×C₄×C₆₇	—	4	1	no	—
247	C₂×C₄×C₆₇	—	4	1	no	—
248	C₂×C₄×C₆₇	—	4	1	no	—
249	C₂×C₄×C₆₇	—	4	2	no	—
250	C₂×C₄×C₆₇	—	8	1	no	—
251	C₂×C₄×C₆₇	—	8	1	no	—
252	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
253	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
254	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
255	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
256	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
257	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
258	C₈×C₆₇	—	2	4	no	—
259	C₈×C₆₇	—	2	2	no	—
260	C₈×C₆₇	—	4	2	no	—
261	C₈×C₆₇	—	4	2	no	—
262	C₈×C₆₇	#120	1	4	yes	—
263	C₂²×C₆₇	—	2	1	no	—
264	C₂²×C₆₇	—	2	1	no	—
265	C₂²×C₆₇	—	2	1	no	—
266	C₂²×C₆₇	—	2	1	no	—
267	C₂²×C₆₇	—	2	4	no	—
268	C₂²×C₆₇	—	2	1	no	—
269	C₂²×C₆₇	—	2	1	no	—
270	C₂²×C₆₇	—	2	1	no	—
271	C₂²×C₆₇	—	2	1	no	—
272	C₂²×C₆₇	—	4	2	no	—
273	C₂²×C₆₇	—	4	2	no	—
274	C₂²×C₆₇	—	8	2	no	—
275	C₂²×C₆₇	—	8	4	no	—
276	C₂²×C₆₇	—	8	1	no	—
277	C₂²×C₆₇	—	8	1	no	—
278	C₂²×C₆₇	—	8	1	no	—
279	C₂²×C₆₇	—	16	2	no	—
280	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
281	C₂²×C₆₇	#525	1	2	yes	—
282	C₄×C₆₇	—	2	1	no	—
283	C₄×C₆₇	—	2	4	no	—
284	C₄×C₆₇	—	2	1	no	—
285	C₄×C₆₇	—	8	1	no	—
286	C₄×C₆₇	—	8	1	no	—
287	C₄×C₆₇	—	8	1	no	—
288	C₄×C₆₇	C₂×#120	1	1	yes	characteristic
289	C₄×C₆₇	C₂×#121	1	2	yes	—
290	C₄×C₆₇	#525	1	1	yes	characteristic
291	#32944	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
292	C₂×C₆₇	—	2	2	no	—
293	C₂×C₆₇	—	4	1	no	—
294	C₂×C₆₇	—	4	1	no	—
295	C₂×C₆₇	—	4	4	no	—
296	C₂×C₆₇	—	16	1	no	—
297	C₂×C₆₇	#2847	1	1	yes	characteristic
298	C₂×C₆₇	#3068	1	1	yes	characteristic
299	C₂×C₆₇	#3069	1	1	yes	characteristic
300	C₂×#568	C₂×C₆₇	1	1	yes	characteristic, complemented
301	C₂×#471	C₂×C₆₇	1	2	yes	complemented
302	C₂×#468	C₂×C₆₇	1	1	yes	characteristic, complemented
303	#3007	C₂×C₆₇	1	1	yes	characteristic, complemented
304	#3006	C₂×C₆₇	1	1	yes	characteristic
305	#1908	C₂×C₆₇	1	2	yes	complemented
306	#1907	C₂×C₆₇	1	2	yes	complemented
307	#1901	C₂×C₆₇	1	1	yes	characteristic, complemented
308	#1900	C₂×C₆₇	1	1	yes	characteristic, complemented
309	#1816	C₂×C₆₇	1	2	yes	complemented
310	#1811	C₂×C₆₇	1	1	yes	characteristic, complemented
311	C₆₇	#32944	1	1	yes	characteristic, direct product factor, Sylow
312	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
313	C₂×#137	C₂²×C₆₇	1	2	yes	—
314	C₂×AGL(1,ℤ/8)	C₂²×C₆₇	1	2	yes	—
315	C₂²×SD₁₆	C₂²×C₆₇	1	1	yes	characteristic, complemented
316	#568	—	2	8	no	—
317	C₂×#131	—	2	2	no	—
318	C₂×#131	C₂²×C₆₇	1	2	yes	—
319	C₂×#131	C₂²×C₆₇	1	2	yes	—
320	C₂²×M₁₆	—	2	2	no	—
321	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
322	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
323	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
324	C₂×#121	C₂²×C₆₇	1	1	yes	characteristic
325	#471	—	2	8	no	—
326	#468	—	2	2	no	—
327	#468	—	2	2	no	—
328	#452	—	2	2	no	—
329	#451	—	2	1	no	—
330	#450	—	2	1	no	—
331	#449	—	2	2	no	—
332	#448	—	2	1	no	—
333	#447	—	2	1	no	—
334	C₂×#108	C₂²×C₆₇	1	2	yes	complemented
335	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic, complemented
336	#417	C₂²×C₆₇	1	2	yes	—
337	#417	C₂²×C₆₇	1	2	yes	—
338	C₂×#103	C₂²×C₆₇	1	1	yes	characteristic, complemented
339	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic, complemented
340	C₂×#101	C₂²×C₆₇	1	2	yes	complemented
341	C₂×#100	C₂²×C₆₇	1	2	yes	complemented
342	#407	C₂²×C₆₇	1	1	yes	characteristic
343	#407	C₂²×C₆₇	1	1	yes	characteristic
344	#343	C₂²×C₆₇	1	2	yes	complemented
345	#343	C₂²×C₆₇	1	2	yes	complemented
346	#341	C₂²×C₆₇	1	2	yes	complemented
347	C₂×#41	—	2	4	no	—
348	C₂×#41	—	2	4	no	—
349	C₂×#41	—	2	1	no	—
350	C₂×#41	—	2	1	no	—
351	C₂×#41	D₄×C₆₇	1	2	yes	—
352	C₂×#41	C₂³×C₆₇	1	2	yes	—
353	C₂²×Q₈	D₄×C₆₇	1	1	yes	characteristic
354	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
355	C₂²×D₄	—	2	1	no	—
356	C₂²×D₄	—	2	1	no	—
357	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
358	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
359	C₂³×C₄	—	2	1	no	—
360	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
361	#137	—	2	4	no	—
362	#137	—	2	4	no	—
363	AGL(1,ℤ/8)	—	2	4	no	—
364	AGL(1,ℤ/8)	—	2	4	no	—
365	C₂×Q₁₆	—	2	2	no	—
366	C₂×SD₁₆	—	2	2	no	—
367	C₂×SD₁₆	—	2	2	no	—
368	C₂×SD₁₆	—	2	2	no	—
369	C₂×SD₁₆	—	2	4	no	—
370	C₂×D₈	—	2	2	no	—
371	#131	—	2	16	no	—
372	#131	—	2	4	no	—
373	#131	—	2	4	no	—
374	#131	—	2	4	no	—
375	#131	D₄×C₆₇	1	8	yes	complemented
376	C₂×M₁₆	—	2	16	no	—
377	C₂×M₁₆	—	2	1	no	—
378	C₂×M₁₆	—	2	4	no	—
379	C₂×M₁₆	—	2	2	no	—
380	C₂×M₁₆	—	2	2	no	—
381	C₂×M₁₆	—	2	4	no	—
382	C₂×M₁₆	D₄×C₆₇	1	2	yes	—
383	C₂×M₁₆	D₄×C₆₇	1	2	yes	—
384	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
385	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
386	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
387	C₂²×C₈	—	2	1	no	—
388	C₂²×C₈	—	2	2	no	—
389	C₂²×C₈	—	2	2	no	—
390	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
391	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
392	C₂²⋊Q₈	—	2	1	no	—
393	C₂²⋊Q₈	—	2	1	no	—
394	C₂²⋊Q₈	—	2	1	no	—
395	C₂²⋊Q₈	—	2	1	no	—
396	#121	—	2	1	no	—
397	#121	—	2	1	no	—
398	#121	—	2	1	no	—
399	#121	—	2	1	no	—
400	C₂×C₄⋊C₄	—	2	1	no	—
401	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
402	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
403	C₂×C₂²⋊C₄	—	2	1	no	—
404	C₂×C₂²⋊C₄	—	2	1	no	—
405	#108	—	2	4	no	—
406	#107	—	2	2	no	—
407	#103	—	2	2	no	—
408	#103	—	2	2	no	—
409	#103	—	2	2	no	—
410	#102	—	2	2	no	—
411	#102	—	2	2	no	—
412	#102	—	2	2	no	—
413	#101	—	2	4	no	—
414	#100	—	2	4	no	—
415	C₂²⋊C₈	—	2	1	no	—
416	C₂²⋊C₈	—	2	1	no	—
417	C₂²⋊C₈	—	2	1	no	—
418	C₂²⋊C₈	—	2	1	no	—
419	C₂⁴	—	2	1	no	—
420	C₂⁴	—	2	1	no	—
421	C₂⁴	#41×C₆₇	1	1	yes	characteristic
422	#41	—	2	4	no	—
423	#41	—	2	8	no	—
424	#41	—	2	4	no	—
425	#41	—	2	4	no	—
426	#41	—	2	4	no	—
427	#41	—	2	2	no	—
428	#41	—	2	2	no	—
429	#41	C₂×D₄×C₆₇	1	4	yes	—
430	#41	#41×C₆₇	1	4	yes	—
431	C₂×Q₈	—	2	4	no	—
432	C₂×Q₈	—	2	1	no	—
433	C₂×Q₈	—	2	2	no	—
434	C₂×Q₈	—	2	1	no	—
435	C₂×Q₈	—	2	1	no	—
436	C₂×Q₈	—	2	2	no	—
437	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
438	C₂×D₄	—	2	8	no	—
439	C₂×D₄	—	2	2	no	—
440	C₂×D₄	—	2	2	no	—
441	C₂×D₄	—	2	4	no	—
442	C₂×D₄	—	2	1	no	—
443	C₂×D₄	—	2	2	no	—
444	C₂×D₄	—	2	1	no	—
445	C₂×D₄	—	2	1	no	—
446	C₂×D₄	—	2	2	no	—
447	C₂×D₄	—	4	1	no	—
448	C₂×D₄	—	4	1	no	—
449	C₂×D₄	—	8	2	no	—
450	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
451	C₂²×C₄	—	2	4	no	—
452	C₂²×C₄	—	2	4	no	—
453	C₂²×C₄	—	2	1	no	—
454	C₂²×C₄	—	2	2	no	—
455	C₂²×C₄	—	2	2	no	—
456	C₂²×C₄	—	2	2	no	—
457	C₂²×C₄	—	2	2	no	—
458	C₂²×C₄	—	2	1	no	—
459	C₂²×C₄	—	2	1	no	—
460	C₂²×C₄	—	2	1	no	—
461	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
462	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
463	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
464	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
465	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
466	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
467	Q₁₆	—	4	4	no	—
468	SD₁₆	—	4	4	no	—
469	SD₁₆	—	4	4	no	—
470	SD₁₆	—	4	4	no	—
471	D₈	—	4	4	no	—
472	M₁₆	—	2	4	no	—
473	M₁₆	—	2	8	no	—
474	M₁₆	—	2	2	no	—
475	M₁₆	—	2	4	no	—
476	M₁₆	—	2	8	no	—
477	M₁₆	—	2	4	no	—
478	M₁₆	—	2	2	no	—
479	M₁₆	—	2	2	no	—
480	M₁₆	—	2	2	no	—
481	M₁₆	—	2	2	no	—
482	M₁₆	C₂×D₄×C₆₇	1	4	yes	—
483	C₂×C₈	—	2	4	no	—
484	C₂×C₈	—	2	8	no	—
485	C₂×C₈	—	2	2	no	—
486	C₂×C₈	—	2	2	no	—
487	C₂×C₈	—	2	8	no	—
488	C₂×C₈	—	2	2	no	—
489	C₂×C₈	—	2	2	no	—
490	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
491	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
492	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
493	C₄⋊C₄	—	2	1	no	—
494	C₄⋊C₄	—	2	1	no	—
495	C₄⋊C₄	—	2	1	no	—
496	C₄⋊C₄	—	2	1	no	—
497	C₄⋊C₄	—	4	1	no	—
498	C₄⋊C₄	—	4	1	no	—
499	C₂²⋊C₄	—	4	1	no	—
500	C₂²⋊C₄	—	4	1	no	—
501	C₂²⋊C₄	—	4	1	no	—
502	C₂²⋊C₄	—	4	1	no	—
503	C₂³	—	2	4	no	—
504	C₂³	—	2	2	no	—
505	C₂³	—	2	2	no	—
506	C₂³	—	4	1	no	—
507	C₂³	—	4	1	no	—
508	C₂³	—	4	1	no	—
509	C₂³	—	4	2	no	—
510	C₂³	—	8	1	no	—
511	C₂³	—	8	1	no	—
512	C₂³	—	8	1	no	—
513	C₂³	—	8	1	no	—
514	C₂³	#121×C₆₇	1	1	yes	characteristic
515	C₂³	#121×C₆₇	1	1	yes	characteristic
516	C₂³	C₂×#41×C₆₇	1	1	yes	characteristic
517	Q₈	—	2	4	no	—
518	Q₈	—	2	2	no	—
519	Q₈	—	4	1	no	—
520	Q₈	—	4	2	no	—
521	Q₈	—	4	1	no	—
522	Q₈	#121×C₆₇	1	4	yes	—
523	D₄	—	2	8	no	—
524	D₄	—	2	4	no	—
525	D₄	—	2	4	no	—
526	D₄	—	2	4	no	—
527	D₄	—	2	2	no	—
528	D₄	—	4	1	no	—
529	D₄	—	4	2	no	—
530	D₄	—	4	1	no	—
531	D₄	—	8	2	no	—
532	D₄	—	8	2	no	—
533	D₄	#121×C₆₇	1	4	yes	—
534	C₂×C₄	—	2	4	no	—
535	C₂×C₄	—	2	8	no	—
536	C₂×C₄	—	2	2	no	—
537	C₂×C₄	—	2	2	no	—
538	C₂×C₄	—	2	4	no	—
539	C₂×C₄	—	2	1	no	—
540	C₂×C₄	—	2	1	no	—
541	C₂×C₄	—	2	2	no	—
542	C₂×C₄	—	2	2	no	—
543	C₂×C₄	—	2	1	no	—
544	C₂×C₄	—	2	4	no	—
545	C₂×C₄	—	2	1	no	—
546	C₂×C₄	—	4	1	no	—
547	C₂×C₄	—	4	1	no	—
548	C₂×C₄	—	4	1	no	—
549	C₂×C₄	—	4	2	no	—
550	C₂×C₄	—	8	1	no	—
551	C₂×C₄	—	8	1	no	—
552	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
553	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
554	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
555	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
556	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
557	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
558	C₈	—	2	4	no	—
559	C₈	—	2	2	no	—
560	C₈	—	4	2	no	—
561	C₈	—	4	2	no	—
562	C₈	#120×C₆₇	1	4	yes	—
563	C₂²	—	2	1	no	—
564	C₂²	—	2	1	no	—
565	C₂²	—	2	1	no	—
566	C₂²	—	2	1	no	—
567	C₂²	—	2	1	no	—
568	C₂²	—	2	1	no	—
569	C₂²	—	2	1	no	—
570	C₂²	—	2	1	no	—
571	C₂²	—	2	4	no	—
572	C₂²	—	4	2	no	—
573	C₂²	—	4	2	no	—
574	C₂²	—	8	2	no	—
575	C₂²	—	8	4	no	—
576	C₂²	—	8	1	no	—
577	C₂²	—	8	1	no	—
578	C₂²	—	8	1	no	—
579	C₂²	—	16	2	no	—
580	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
581	C₂²	#525×C₆₇	1	2	yes	—
582	C₄	—	2	1	no	—
583	C₄	—	2	4	no	—
584	C₄	—	2	1	no	—
585	C₄	—	8	1	no	—
586	C₄	—	8	1	no	—
587	C₄	—	8	1	no	—
588	C₄	C₂×#120×C₆₇	1	1	yes	characteristic
589	C₄	C₂×#121×C₆₇	1	2	yes	—
590	C₄	#525×C₆₇	1	1	yes	characteristic
591	C₂	—	2	2	no	—
592	C₂	—	4	1	no	—
593	C₂	—	4	1	no	—
594	C₂	—	4	4	no	—
595	C₂	—	16	1	no	—
596	C₂	C₆₇×#2847	1	1	yes	characteristic
597	C₂	C₆₇×#3068	1	1	yes	characteristic
598	C₂	C₆₇×#3069	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. C₂×#568×C₆₇	C₂	semidirect
2. C₂×#471×C₆₇	C₂	semidirect
3. C₂×#468×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×#1816	C₂	semidirect
11. C₆₇×#1811	C₂	semidirect
15. C₂²×SD₁₆×C₆₇	C₂²	semidirect
34. C₂×#108×C₆₇	C₂²	semidirect
35. C₂×#107×C₆₇	C₂²	semidirect
38. C₂×#103×C₆₇	C₂²	semidirect
39. C₂×#102×C₆₇	C₂²	semidirect
40. C₂×#101×C₆₇	C₂²	semidirect
41. C₂×#100×C₆₇	C₂²	semidirect
44. #343×C₆₇	C₂²	semidirect
45. #343×C₆₇	C₂²	semidirect
46. #341×C₆₇	C₂²	semidirect
75. #131×C₆₇	D₄	semidirect
291. #32944	C₆₇	direct (N × H)
300. C₂×#568	C₂×C₆₇	semidirect
301. C₂×#471	C₂×C₆₇	semidirect
302. C₂×#468	C₂×C₆₇	semidirect
303. #3007	C₂×C₆₇	semidirect
305. #1908	C₂×C₆₇	semidirect
306. #1907	C₂×C₆₇	semidirect
307. #1901	C₂×C₆₇	semidirect
308. #1900	C₂×C₆₇	semidirect
309. #1816	C₂×C₆₇	semidirect
310. #1811	C₂×C₆₇	semidirect
311. C₆₇	#32944	direct (N × H)
315. C₂²×SD₁₆	C₂²×C₆₇	semidirect
334. C₂×#108	C₂²×C₆₇	semidirect
335. C₂×#107	C₂²×C₆₇	semidirect
338. C₂×#103	C₂²×C₆₇	semidirect
339. C₂×#102	C₂²×C₆₇	semidirect
340. C₂×#101	C₂²×C₆₇	semidirect
341. C₂×#100	C₂²×C₆₇	semidirect
344. #343	C₂²×C₆₇	semidirect
345. #343	C₂²×C₆₇	semidirect
346. #341	C₂²×C₆₇	semidirect
375. #131	D₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#568×C₆₇	2
C₂×#471×C₆₇	2
C₂×#468×C₆₇	2
C₆₇×#3007	2
C₆₇×#3006	2
C₆₇×#1908	2
C₆₇×#1907	2
C₆₇×#1901	2
C₆₇×#1900	2
C₆₇×#1816	2
C₆₇×#1811	2
#32944	67

Subgroups — C67×#32944 #30712195

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#32944 #30712195