Subgroups — C₆₇×#32943 #30712194

Order 17152.

1316 conjugacy classes of subgroups.

The subgroup lattice has 500 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
2	C₂×#473×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₂×#467×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#3007	C₂	1	2	yes	maximal
5	C₆₇×#1908	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1901	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1816	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1810	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
10	C₂×#137×C₆₇	C₂²	1	4	yes	—
11	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic, complemented
12	C₂×#131×C₆₇	—	2	8	no	—
13	C₂×#131×C₆₇	—	2	1	no	—
14	C₂×#131×C₆₇	—	2	1	no	—
15	C₂×#131×C₆₇	C₂²	1	2	yes	—
16	C₂×#131×C₆₇	C₂²	1	2	yes	—
17	C₂²×M₁₆×C₆₇	—	2	1	no	—
18	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
19	C₂³×C₈×C₆₇	—	2	1	no	—
20	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
22	#473×C₆₇	—	2	8	no	—
23	#467×C₆₇	—	2	1	no	—
24	#467×C₆₇	—	2	1	no	—
25	#467×C₆₇	—	2	1	no	—
26	#467×C₆₇	—	2	1	no	—
27	#452×C₆₇	—	2	4	no	—
28	#451×C₆₇	—	2	2	no	—
29	#450×C₆₇	—	2	2	no	—
30	C₂×#108×C₆₇	C₂²	1	2	yes	complemented
31	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic, complemented
32	#417×C₆₇	C₂²	1	4	yes	—
33	C₂×#103×C₆₇	C₂²	1	2	yes	complemented
34	C₂×#101×C₆₇	C₂²	1	4	yes	complemented
35	#407×C₆₇	C₂²	1	2	yes	—
36	#343×C₆₇	C₂²	1	4	yes	complemented
37	#341×C₆₇	C₂²	1	2	yes	complemented
38	C₂×#41×C₆₇	—	2	4	no	—
39	C₂×#41×C₆₇	—	2	4	no	—
40	C₂×#41×C₆₇	—	2	1	no	—
41	C₂×#41×C₆₇	—	2	1	no	—
42	C₂×#41×C₆₇	D₄	1	2	yes	—
43	C₂×#41×C₆₇	C₂³	1	2	yes	—
44	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
45	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
46	C₂²×D₄×C₆₇	—	2	1	no	—
47	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
48	C₂³×C₄×C₆₇	—	2	1	no	—
49	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
50	#137×C₆₇	—	2	8	no	—
51	#137×C₆₇	—	2	8	no	—
52	C₂×Q₁₆×C₆₇	—	2	4	no	—
53	C₂×Q₁₆×C₆₇	—	2	1	no	—
54	C₂×Q₁₆×C₆₇	—	2	1	no	—
55	C₂×Q₁₆×C₆₇	—	2	4	no	—
56	C₂×SD₁₆×C₆₇	—	2	4	no	—
57	#131×C₆₇	—	2	8	no	—
58	#131×C₆₇	—	2	8	no	—
59	#131×C₆₇	—	2	4	no	—
60	#131×C₆₇	—	2	4	no	—
61	#131×C₆₇	—	2	4	no	—
62	#131×C₆₇	D₄	1	8	yes	—
63	C₂×M₁₆×C₆₇	—	2	8	no	—
64	C₂×M₁₆×C₆₇	—	2	1	no	—
65	C₂×M₁₆×C₆₇	—	2	4	no	—
66	C₂×M₁₆×C₆₇	—	2	2	no	—
67	C₂×M₁₆×C₆₇	—	2	2	no	—
68	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
69	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
70	C₂²×C₈×C₆₇	—	2	8	no	—
71	C₂²×C₈×C₆₇	—	2	1	no	—
72	C₂²×C₈×C₆₇	—	2	2	no	—
73	C₂²×C₈×C₆₇	—	2	2	no	—
74	C₂²×C₈×C₆₇	—	2	4	no	—
75	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
76	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
77	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
78	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
79	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
80	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
81	C₂²⋊Q₈×C₆₇	—	2	2	no	—
82	C₂²⋊Q₈×C₆₇	—	2	2	no	—
83	C₂²⋊Q₈×C₆₇	—	2	2	no	—
84	C₂²⋊Q₈×C₆₇	—	2	2	no	—
85	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
86	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
87	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
88	#108×C₆₇	—	2	4	no	—
89	#107×C₆₇	—	2	1	no	—
90	#107×C₆₇	—	2	1	no	—
91	#103×C₆₇	—	2	4	no	—
92	#103×C₆₇	—	2	2	no	—
93	#103×C₆₇	—	2	2	no	—
94	#102×C₆₇	—	2	4	no	—
95	#101×C₆₇	—	2	8	no	—
96	C₂²⋊C₈×C₆₇	—	2	2	no	—
97	C₂²⋊C₈×C₆₇	—	2	2	no	—
98	C₂⁴×C₆₇	—	2	1	no	—
99	C₂⁴×C₆₇	#41	1	1	yes	characteristic
100	#41×C₆₇	—	2	4	no	—
101	#41×C₆₇	—	2	8	no	—
102	#41×C₆₇	—	2	4	no	—
103	#41×C₆₇	—	2	4	no	—
104	#41×C₆₇	—	2	4	no	—
105	#41×C₆₇	—	2	2	no	—
106	#41×C₆₇	—	2	2	no	—
107	#41×C₆₇	C₂×D₄	1	4	yes	—
108	#41×C₆₇	#41	1	4	yes	—
109	C₂×Q₈×C₆₇	—	2	4	no	—
110	C₂×Q₈×C₆₇	—	2	1	no	—
111	C₂×Q₈×C₆₇	—	2	4	no	—
112	C₂×Q₈×C₆₇	—	2	2	no	—
113	C₂×Q₈×C₆₇	—	2	2	no	—
114	C₂×Q₈×C₆₇	—	2	4	no	—
115	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
116	C₂×D₄×C₆₇	—	2	8	no	—
117	C₂×D₄×C₆₇	—	2	2	no	—
118	C₂×D₄×C₆₇	—	2	2	no	—
119	C₂×D₄×C₆₇	—	2	4	no	—
120	C₂×D₄×C₆₇	—	2	1	no	—
121	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
122	C₂²×C₄×C₆₇	—	2	4	no	—
123	C₂²×C₄×C₆₇	—	2	4	no	—
124	C₂²×C₄×C₆₇	—	2	1	no	—
125	C₂²×C₄×C₆₇	—	2	2	no	—
126	C₂²×C₄×C₆₇	—	2	2	no	—
127	C₂²×C₄×C₆₇	—	2	2	no	—
128	C₂²×C₄×C₆₇	—	2	2	no	—
129	C₂²×C₄×C₆₇	—	2	2	no	—
130	C₂²×C₄×C₆₇	—	2	2	no	—
131	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
132	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
133	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
134	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
135	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
136	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
137	Q₁₆×C₆₇	—	4	8	no	—
138	Q₁₆×C₆₇	—	4	2	no	—
139	Q₁₆×C₆₇	—	4	2	no	—
140	SD₁₆×C₆₇	—	4	8	no	—
141	M₁₆×C₆₇	—	2	4	no	—
142	M₁₆×C₆₇	—	2	8	no	—
143	M₁₆×C₆₇	—	2	2	no	—
144	M₁₆×C₆₇	—	2	8	no	—
145	M₁₆×C₆₇	—	2	4	no	—
146	M₁₆×C₆₇	—	2	4	no	—
147	M₁₆×C₆₇	C₂×D₄	1	4	yes	—
148	C₂×C₈×C₆₇	—	2	4	no	—
149	C₂×C₈×C₆₇	—	2	8	no	—
150	C₂×C₈×C₆₇	—	2	2	no	—
151	C₂×C₈×C₆₇	—	2	2	no	—
152	C₂×C₈×C₆₇	—	2	2	no	—
153	C₂×C₈×C₆₇	—	2	8	no	—
154	C₂×C₈×C₆₇	—	2	2	no	—
155	C₂×C₈×C₆₇	—	2	2	no	—
156	C₂×C₈×C₆₇	—	2	2	no	—
157	C₂×C₈×C₆₇	—	2	4	no	—
158	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
159	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
160	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
161	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
162	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
163	C₄⋊C₄×C₆₇	—	2	2	no	—
164	C₄⋊C₄×C₆₇	—	2	2	no	—
165	C₄⋊C₄×C₆₇	—	4	2	no	—
166	C₄⋊C₄×C₆₇	—	4	2	no	—
167	C₂²⋊C₄×C₆₇	—	4	2	no	—
168	C₂²⋊C₄×C₆₇	—	4	2	no	—
169	C₂³×C₆₇	—	2	4	no	—
170	C₂³×C₆₇	—	2	2	no	—
171	C₂³×C₆₇	—	2	2	no	—
172	C₂³×C₆₇	—	2	2	no	—
173	C₂³×C₆₇	—	4	1	no	—
174	C₂³×C₆₇	—	4	2	no	—
175	C₂³×C₆₇	—	4	2	no	—
176	C₂³×C₆₇	#121	1	1	yes	characteristic
177	C₂³×C₆₇	#121	1	1	yes	characteristic
178	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
179	Q₈×C₆₇	—	2	4	no	—
180	Q₈×C₆₇	—	2	2	no	—
181	Q₈×C₆₇	—	4	2	no	—
182	Q₈×C₆₇	—	4	4	no	—
183	Q₈×C₆₇	—	4	2	no	—
184	Q₈×C₆₇	#121	1	4	yes	—
185	D₄×C₆₇	—	2	8	no	—
186	D₄×C₆₇	—	2	4	no	—
187	D₄×C₆₇	—	2	4	no	—
188	D₄×C₆₇	—	2	4	no	—
189	D₄×C₆₇	—	2	2	no	—
190	D₄×C₆₇	#121	1	4	yes	—
191	C₂×C₄×C₆₇	—	2	4	no	—
192	C₂×C₄×C₆₇	—	2	8	no	—
193	C₂×C₄×C₆₇	—	2	2	no	—
194	C₂×C₄×C₆₇	—	2	2	no	—
195	C₂×C₄×C₆₇	—	2	1	no	—
196	C₂×C₄×C₆₇	—	2	4	no	—
197	C₂×C₄×C₆₇	—	2	1	no	—
198	C₂×C₄×C₆₇	—	2	2	no	—
199	C₂×C₄×C₆₇	—	2	2	no	—
200	C₂×C₄×C₆₇	—	2	1	no	—
201	C₂×C₄×C₆₇	—	2	4	no	—
202	C₂×C₄×C₆₇	—	2	1	no	—
203	C₂×C₄×C₆₇	—	4	2	no	—
204	C₂×C₄×C₆₇	—	4	2	no	—
205	C₂×C₄×C₆₇	—	4	4	no	—
206	C₂×C₄×C₆₇	—	8	2	no	—
207	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
208	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
209	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
210	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
211	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
212	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
213	C₈×C₆₇	—	2	1	no	—
214	C₈×C₆₇	—	2	4	no	—
215	C₈×C₆₇	—	2	1	no	—
216	C₈×C₆₇	—	4	4	no	—
217	C₈×C₆₇	#120	1	2	yes	—
218	C₈×C₆₇	#120	1	2	yes	—
219	C₂²×C₆₇	—	2	1	no	—
220	C₂²×C₆₇	—	2	1	no	—
221	C₂²×C₆₇	—	2	1	no	—
222	C₂²×C₆₇	—	2	1	no	—
223	C₂²×C₆₇	—	2	1	no	—
224	C₂²×C₆₇	—	2	4	no	—
225	C₂²×C₆₇	—	2	1	no	—
226	C₂²×C₆₇	—	2	1	no	—
227	C₂²×C₆₇	—	4	4	no	—
228	C₂²×C₆₇	—	4	8	no	—
229	C₂²×C₆₇	—	4	2	no	—
230	C₂²×C₆₇	—	4	2	no	—
231	C₂²×C₆₇	#473	1	2	yes	complemented
232	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
233	C₂²×C₆₇	#525	1	2	yes	—
234	C₄×C₆₇	—	2	1	no	—
235	C₄×C₆₇	—	2	4	no	—
236	C₄×C₆₇	—	2	1	no	—
237	C₄×C₆₇	—	8	2	no	—
238	C₄×C₆₇	—	8	2	no	—
239	C₄×C₆₇	C₂×#120	1	1	yes	characteristic
240	C₄×C₆₇	C₂×#121	1	2	yes	—
241	C₄×C₆₇	#525	1	1	yes	characteristic
242	#32943	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
243	C₂×C₆₇	—	2	2	no	—
244	C₂×C₆₇	—	2	2	no	—
245	C₂×C₆₇	—	4	1	no	—
246	C₂×C₆₇	—	4	4	no	—
247	C₂×C₆₇	#2847	1	1	yes	characteristic
248	C₂×C₆₇	C₂×#473	1	1	yes	characteristic
249	C₂×C₆₇	#3067	1	1	yes	characteristic
250	C₂²×#131	C₂×C₆₇	1	1	yes	characteristic
251	C₂×#473	C₂×C₆₇	1	2	yes	complemented
252	C₂×#467	C₂×C₆₇	1	1	yes	characteristic, complemented
253	#3007	C₂×C₆₇	1	2	yes	—
254	#1908	C₂×C₆₇	1	4	yes	complemented
255	#1901	C₂×C₆₇	1	2	yes	complemented
256	#1816	C₂×C₆₇	1	2	yes	complemented
257	#1810	C₂×C₆₇	1	1	yes	characteristic, complemented
258	C₆₇	#32943	1	1	yes	characteristic, direct product factor, Sylow
259	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
260	C₂×#137	C₂²×C₆₇	1	4	yes	—
261	C₂²×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic, complemented
262	C₂×#131	—	2	8	no	—
263	C₂×#131	—	2	1	no	—
264	C₂×#131	—	2	1	no	—
265	C₂×#131	C₂²×C₆₇	1	2	yes	—
266	C₂×#131	C₂²×C₆₇	1	2	yes	—
267	C₂²×M₁₆	—	2	1	no	—
268	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
269	C₂³×C₈	—	2	1	no	—
270	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
271	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
272	#473	—	2	8	no	—
273	#467	—	2	1	no	—
274	#467	—	2	1	no	—
275	#467	—	2	1	no	—
276	#467	—	2	1	no	—
277	#452	—	2	4	no	—
278	#451	—	2	2	no	—
279	#450	—	2	2	no	—
280	C₂×#108	C₂²×C₆₇	1	2	yes	complemented
281	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic, complemented
282	#417	C₂²×C₆₇	1	4	yes	—
283	C₂×#103	C₂²×C₆₇	1	2	yes	complemented
284	C₂×#101	C₂²×C₆₇	1	4	yes	complemented
285	#407	C₂²×C₆₇	1	2	yes	—
286	#343	C₂²×C₆₇	1	4	yes	complemented
287	#341	C₂²×C₆₇	1	2	yes	complemented
288	C₂×#41	—	2	4	no	—
289	C₂×#41	—	2	4	no	—
290	C₂×#41	—	2	1	no	—
291	C₂×#41	—	2	1	no	—
292	C₂×#41	D₄×C₆₇	1	2	yes	—
293	C₂×#41	C₂³×C₆₇	1	2	yes	—
294	C₂²×Q₈	D₄×C₆₇	1	1	yes	characteristic
295	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
296	C₂²×D₄	—	2	1	no	—
297	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
298	C₂³×C₄	—	2	1	no	—
299	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
300	#137	—	2	8	no	—
301	#137	—	2	8	no	—
302	C₂×Q₁₆	—	2	4	no	—
303	C₂×Q₁₆	—	2	1	no	—
304	C₂×Q₁₆	—	2	1	no	—
305	C₂×Q₁₆	—	2	4	no	—
306	C₂×SD₁₆	—	2	4	no	—
307	#131	—	2	8	no	—
308	#131	—	2	8	no	—
309	#131	—	2	4	no	—
310	#131	—	2	4	no	—
311	#131	—	2	4	no	—
312	#131	D₄×C₆₇	1	8	yes	—
313	C₂×M₁₆	—	2	8	no	—
314	C₂×M₁₆	—	2	1	no	—
315	C₂×M₁₆	—	2	4	no	—
316	C₂×M₁₆	—	2	2	no	—
317	C₂×M₁₆	—	2	2	no	—
318	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
319	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
320	C₂²×C₈	—	2	8	no	—
321	C₂²×C₈	—	2	1	no	—
322	C₂²×C₈	—	2	2	no	—
323	C₂²×C₈	—	2	2	no	—
324	C₂²×C₈	—	2	4	no	—
325	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
326	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
327	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
328	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
329	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
330	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
331	C₂²⋊Q₈	—	2	2	no	—
332	C₂²⋊Q₈	—	2	2	no	—
333	C₂²⋊Q₈	—	2	2	no	—
334	C₂²⋊Q₈	—	2	2	no	—
335	C₂×C₄⋊C₄	—	2	2	no	—
336	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
337	C₂×C₂²⋊C₄	—	2	2	no	—
338	#108	—	2	4	no	—
339	#107	—	2	1	no	—
340	#107	—	2	1	no	—
341	#103	—	2	4	no	—
342	#103	—	2	2	no	—
343	#103	—	2	2	no	—
344	#102	—	2	4	no	—
345	#101	—	2	8	no	—
346	C₂²⋊C₈	—	2	2	no	—
347	C₂²⋊C₈	—	2	2	no	—
348	C₂⁴	—	2	1	no	—
349	C₂⁴	#41×C₆₇	1	1	yes	characteristic
350	#41	—	2	4	no	—
351	#41	—	2	8	no	—
352	#41	—	2	4	no	—
353	#41	—	2	4	no	—
354	#41	—	2	4	no	—
355	#41	—	2	2	no	—
356	#41	—	2	2	no	—
357	#41	C₂×D₄×C₆₇	1	4	yes	—
358	#41	#41×C₆₇	1	4	yes	—
359	C₂×Q₈	—	2	4	no	—
360	C₂×Q₈	—	2	1	no	—
361	C₂×Q₈	—	2	4	no	—
362	C₂×Q₈	—	2	2	no	—
363	C₂×Q₈	—	2	2	no	—
364	C₂×Q₈	—	2	4	no	—
365	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
366	C₂×D₄	—	2	8	no	—
367	C₂×D₄	—	2	2	no	—
368	C₂×D₄	—	2	2	no	—
369	C₂×D₄	—	2	4	no	—
370	C₂×D₄	—	2	1	no	—
371	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
372	C₂²×C₄	—	2	4	no	—
373	C₂²×C₄	—	2	4	no	—
374	C₂²×C₄	—	2	1	no	—
375	C₂²×C₄	—	2	2	no	—
376	C₂²×C₄	—	2	2	no	—
377	C₂²×C₄	—	2	2	no	—
378	C₂²×C₄	—	2	2	no	—
379	C₂²×C₄	—	2	2	no	—
380	C₂²×C₄	—	2	2	no	—
381	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
382	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
383	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
384	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
385	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
386	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
387	Q₁₆	—	4	8	no	—
388	Q₁₆	—	4	2	no	—
389	Q₁₆	—	4	2	no	—
390	SD₁₆	—	4	8	no	—
391	M₁₆	—	2	4	no	—
392	M₁₆	—	2	8	no	—
393	M₁₆	—	2	2	no	—
394	M₁₆	—	2	8	no	—
395	M₁₆	—	2	4	no	—
396	M₁₆	—	2	4	no	—
397	M₁₆	C₂×D₄×C₆₇	1	4	yes	—
398	C₂×C₈	—	2	4	no	—
399	C₂×C₈	—	2	8	no	—
400	C₂×C₈	—	2	2	no	—
401	C₂×C₈	—	2	2	no	—
402	C₂×C₈	—	2	2	no	—
403	C₂×C₈	—	2	8	no	—
404	C₂×C₈	—	2	2	no	—
405	C₂×C₈	—	2	2	no	—
406	C₂×C₈	—	2	2	no	—
407	C₂×C₈	—	2	4	no	—
408	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
409	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
410	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
411	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
412	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
413	C₄⋊C₄	—	2	2	no	—
414	C₄⋊C₄	—	2	2	no	—
415	C₄⋊C₄	—	4	2	no	—
416	C₄⋊C₄	—	4	2	no	—
417	C₂²⋊C₄	—	4	2	no	—
418	C₂²⋊C₄	—	4	2	no	—
419	C₂³	—	2	4	no	—
420	C₂³	—	2	2	no	—
421	C₂³	—	2	2	no	—
422	C₂³	—	2	2	no	—
423	C₂³	—	4	1	no	—
424	C₂³	—	4	2	no	—
425	C₂³	—	4	2	no	—
426	C₂³	#121×C₆₇	1	1	yes	characteristic
427	C₂³	#121×C₆₇	1	1	yes	characteristic
428	C₂³	C₂×#41×C₆₇	1	1	yes	characteristic
429	Q₈	—	2	4	no	—
430	Q₈	—	2	2	no	—
431	Q₈	—	4	2	no	—
432	Q₈	—	4	4	no	—
433	Q₈	—	4	2	no	—
434	Q₈	#121×C₆₇	1	4	yes	—
435	D₄	—	2	8	no	—
436	D₄	—	2	4	no	—
437	D₄	—	2	4	no	—
438	D₄	—	2	4	no	—
439	D₄	—	2	2	no	—
440	D₄	#121×C₆₇	1	4	yes	—
441	C₂×C₄	—	2	4	no	—
442	C₂×C₄	—	2	8	no	—
443	C₂×C₄	—	2	2	no	—
444	C₂×C₄	—	2	2	no	—
445	C₂×C₄	—	2	1	no	—
446	C₂×C₄	—	2	4	no	—
447	C₂×C₄	—	2	1	no	—
448	C₂×C₄	—	2	2	no	—
449	C₂×C₄	—	2	2	no	—
450	C₂×C₄	—	2	1	no	—
451	C₂×C₄	—	2	4	no	—
452	C₂×C₄	—	2	1	no	—
453	C₂×C₄	—	4	2	no	—
454	C₂×C₄	—	4	2	no	—
455	C₂×C₄	—	4	4	no	—
456	C₂×C₄	—	8	2	no	—
457	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
458	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
459	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
460	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
461	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
462	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
463	C₈	—	2	1	no	—
464	C₈	—	2	4	no	—
465	C₈	—	2	1	no	—
466	C₈	—	4	4	no	—
467	C₈	#120×C₆₇	1	2	yes	—
468	C₈	#120×C₆₇	1	2	yes	—
469	C₂²	—	2	1	no	—
470	C₂²	—	2	1	no	—
471	C₂²	—	2	1	no	—
472	C₂²	—	2	1	no	—
473	C₂²	—	2	1	no	—
474	C₂²	—	2	1	no	—
475	C₂²	—	2	1	no	—
476	C₂²	—	2	4	no	—
477	C₂²	—	4	4	no	—
478	C₂²	—	4	8	no	—
479	C₂²	—	4	2	no	—
480	C₂²	—	4	2	no	—
481	C₂²	#473×C₆₇	1	2	yes	complemented
482	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
483	C₂²	#525×C₆₇	1	2	yes	—
484	C₄	—	2	1	no	—
485	C₄	—	2	4	no	—
486	C₄	—	2	1	no	—
487	C₄	—	8	2	no	—
488	C₄	—	8	2	no	—
489	C₄	C₂×#120×C₆₇	1	1	yes	characteristic
490	C₄	C₂×#121×C₆₇	1	2	yes	—
491	C₄	#525×C₆₇	1	1	yes	characteristic
492	C₂	—	2	2	no	—
493	C₂	—	2	2	no	—
494	C₂	—	4	1	no	—
495	C₂	—	4	4	no	—
496	C₂	C₆₇×#2847	1	1	yes	characteristic
497	C₂	C₂×#473×C₆₇	1	1	yes	characteristic
498	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	C₂²×#41×C₆₇	C₂²
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
2. C₂×#473×C₆₇	C₂	semidirect
3. C₂×#467×C₆₇	C₂	semidirect
5. C₆₇×#1908	C₂	semidirect
6. C₆₇×#1901	C₂	semidirect
7. C₆₇×#1816	C₂	semidirect
8. C₆₇×#1810	C₂	semidirect
11. C₂²×Q₁₆×C₆₇	C₂²	semidirect
30. C₂×#108×C₆₇	C₂²	semidirect
31. C₂×#107×C₆₇	C₂²	semidirect
33. C₂×#103×C₆₇	C₂²	semidirect
34. C₂×#101×C₆₇	C₂²	semidirect
36. #343×C₆₇	C₂²	semidirect
37. #341×C₆₇	C₂²	semidirect
231. C₂²×C₆₇	#473	semidirect
242. #32943	C₆₇	direct (N × H)
251. C₂×#473	C₂×C₆₇	semidirect
252. C₂×#467	C₂×C₆₇	semidirect
254. #1908	C₂×C₆₇	semidirect
255. #1901	C₂×C₆₇	semidirect
256. #1816	C₂×C₆₇	semidirect
257. #1810	C₂×C₆₇	semidirect
258. C₆₇	#32943	direct (N × H)
261. C₂²×Q₁₆	C₂²×C₆₇	semidirect
280. C₂×#108	C₂²×C₆₇	semidirect
281. C₂×#107	C₂²×C₆₇	semidirect
283. C₂×#103	C₂²×C₆₇	semidirect
284. C₂×#101	C₂²×C₆₇	semidirect
286. #343	C₂²×C₆₇	semidirect
287. #341	C₂²×C₆₇	semidirect
481. C₂²	#473×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×#473×C₆₇	2
C₂×#467×C₆₇	2
C₆₇×#3007	2
C₆₇×#1908	2
C₆₇×#1901	2
C₆₇×#1816	2
C₆₇×#1810	2
#32943	67

Subgroups — C67×#32943 #30712194

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#32943 #30712194