Subgroups — C₆₇×#19902 #30699153

Order 17152.

672 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3334	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₇×#3330	C₂	1	2	yes	maximal, complemented
3	C₆₇×#3283	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1707	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1699	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1673	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1669	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1654	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1654	C₂	1	2	yes	maximal, complemented
10	C₆₇×#1589	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#135×C₆₇	—	2	1	no	—
13	#547×C₆₇	C₂²	1	2	yes	—
14	#547×C₆₇	C₂²	1	2	yes	—
15	#500×C₆₇	C₂²	1	1	yes	characteristic
16	#498×C₆₇	C₂²	1	2	yes	—
17	#494×C₆₇	C₂²	1	1	yes	characteristic
18	#487×C₆₇	C₂²	1	2	yes	—
19	#484×C₆₇	—	2	2	no	—
20	#480×C₆₇	—	2	2	no	—
21	#474×C₆₇	C₂²	1	2	yes	—
22	#474×C₆₇	C₂²	1	2	yes	—
23	#468×C₆₇	—	2	2	no	—
24	#467×C₆₇	—	2	1	no	—
25	#466×C₆₇	—	2	1	no	—
26	#461×C₆₇	C₂²	1	2	yes	—
27	#461×C₆₇	C₂²	1	2	yes	—
28	#452×C₆₇	—	2	2	no	—
29	#449×C₆₇	—	2	2	no	—
30	C₄×D₈×C₆₇	C₂²	1	1	yes	characteristic
31	C₄×D₈×C₆₇	C₂²	1	1	yes	characteristic
32	#436×C₆₇	C₂²	1	2	yes	—
33	C₈×D₄×C₆₇	C₂²	1	1	yes	characteristic
34	C₈×D₄×C₆₇	C₂²	1	1	yes	characteristic
35	C₂×#107×C₆₇	—	2	1	no	—
36	#417×C₆₇	—	2	2	no	—
37	#335×C₆₇	C₂²	1	1	yes	characteristic
38	#335×C₆₇	C₂²	1	1	yes	characteristic
39	#332×C₆₇	C₂²	1	2	yes	—
40	#332×C₆₇	C₂²	1	2	yes	—
41	D₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
42	D₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
43	D₄⋊C₈×C₆₇	C₂²	1	2	yes	—
44	C₈²×C₆₇	C₂²	1	1	yes	characteristic
45	C₂×#41×C₆₇	—	2	2	no	—
46	C₂×#41×C₆₇	—	2	2	no	—
47	#135×C₆₇	—	4	4	no	—
48	C₂×Q₁₆×C₆₇	—	2	1	no	—
49	C₂×Q₁₆×C₆₇	—	2	1	no	—
50	C₂×SD₁₆×C₆₇	—	2	2	no	—
51	C₂×SD₁₆×C₆₇	—	2	2	no	—
52	C₂×D₈×C₆₇	D₄	1	1	yes	characteristic
53	C₂×D₈×C₆₇	D₄	1	1	yes	characteristic
54	C₂×M₁₆×C₆₇	—	2	2	no	—
55	C₂²×C₈×C₆₇	—	2	1	no	—
56	C₂²×C₈×C₆₇	—	2	1	no	—
57	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
58	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
59	#123×C₆₇	—	4	2	no	—
60	#123×C₆₇	—	4	2	no	—
61	C₂²⋊Q₈×C₆₇	—	2	2	no	—
62	C₂²⋊Q₈×C₆₇	—	2	2	no	—
63	#121×C₆₇	—	2	2	no	—
64	#121×C₆₇	—	2	2	no	—
65	C₄×D₄×C₆₇	C₂³	1	1	yes	characteristic
66	C₄×D₄×C₆₇	C₂³	1	2	yes	—
67	C₄×D₄×C₆₇	C₂³	1	1	yes	characteristic
68	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
69	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
70	#107×C₆₇	—	2	1	no	—
71	#107×C₆₇	—	2	1	no	—
72	#107×C₆₇	—	2	2	no	—
73	#107×C₆₇	—	2	2	no	—
74	#107×C₆₇	D₄	1	1	yes	characteristic
75	#107×C₆₇	D₄	1	1	yes	characteristic
76	#106×C₆₇	—	4	2	no	—
77	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
78	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
79	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
80	#103×C₆₇	—	2	2	no	—
81	#103×C₆₇	—	2	2	no	—
82	#103×C₆₇	—	2	2	no	—
83	#103×C₆₇	—	2	2	no	—
84	#102×C₆₇	—	2	2	no	—
85	#102×C₆₇	—	2	2	no	—
86	#102×C₆₇	D₄	1	2	yes	—
87	#102×C₆₇	D₄	1	1	yes	characteristic
88	#102×C₆₇	D₄	1	1	yes	characteristic
89	C₂²⋊C₈×C₆₇	—	2	2	no	—
90	C₂²⋊C₈×C₆₇	—	2	1	no	—
91	C₂²⋊C₈×C₆₇	—	2	1	no	—
92	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
94	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
95	#41×C₆₇	—	4	4	no	—
96	#41×C₆₇	—	4	4	no	—
97	#41×C₆₇	—	8	2	no	—
98	#41×C₆₇	—	8	2	no	—
99	C₂×Q₈×C₆₇	—	2	2	no	—
100	C₂×Q₈×C₆₇	—	2	2	no	—
101	C₂×D₄×C₆₇	—	4	2	no	—
102	C₂×D₄×C₆₇	—	4	2	no	—
103	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
104	C₂×D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
105	C₂×D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₆₇	—	2	1	no	—
107	C₂²×C₄×C₆₇	—	2	2	no	—
108	C₂²×C₄×C₆₇	—	2	1	no	—
109	C₂²×C₄×C₆₇	—	4	2	no	—
110	C₂²×C₄×C₆₇	—	4	2	no	—
111	Q₁₆×C₆₇	—	4	2	no	—
112	Q₁₆×C₆₇	—	8	1	no	—
113	SD₁₆×C₆₇	—	4	4	no	—
114	SD₁₆×C₆₇	—	8	2	no	—
115	D₈×C₆₇	—	2	2	no	—
116	D₈×C₆₇	—	2	2	no	—
117	M₁₆×C₆₇	—	8	2	no	—
118	C₂×C₈×C₆₇	—	2	1	no	—
119	C₂×C₈×C₆₇	—	2	1	no	—
120	C₂×C₈×C₆₇	—	2	2	no	—
121	C₂×C₈×C₆₇	—	2	1	no	—
122	C₂×C₈×C₆₇	—	4	2	no	—
123	C₂×C₈×C₆₇	—	4	2	no	—
124	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
125	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
126	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
127	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
128	C₄⋊C₄×C₆₇	—	2	2	no	—
129	C₄⋊C₄×C₆₇	—	2	2	no	—
130	C₄⋊C₄×C₆₇	—	2	2	no	—
131	C₄⋊C₄×C₆₇	—	2	2	no	—
132	C₄⋊C₄×C₆₇	—	4	2	no	—
133	C₄⋊C₄×C₆₇	—	4	2	no	—
134	C₄⋊C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
135	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
136	C₄⋊C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
137	C₂²⋊C₄×C₆₇	—	2	1	no	—
138	C₂²⋊C₄×C₆₇	—	2	2	no	—
139	C₂²⋊C₄×C₆₇	—	2	1	no	—
140	C₂²⋊C₄×C₆₇	—	4	2	no	—
141	C₂²⋊C₄×C₆₇	—	4	2	no	—
142	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
143	C₂³×C₆₇	—	2	1	no	—
144	C₂³×C₆₇	—	2	1	no	—
145	C₂³×C₆₇	—	2	2	no	—
146	Q₈×C₆₇	—	4	2	no	—
147	Q₈×C₆₇	—	4	2	no	—
148	Q₈×C₆₇	—	4	2	no	—
149	Q₈×C₆₇	—	4	2	no	—
150	D₄×C₆₇	—	2	1	no	—
151	D₄×C₆₇	—	2	2	no	—
152	D₄×C₆₇	—	2	1	no	—
153	D₄×C₆₇	—	2	1	no	—
154	D₄×C₆₇	—	2	2	no	—
155	D₄×C₆₇	—	8	4	no	—
156	D₄×C₆₇	—	8	4	no	—
157	D₄×C₆₇	#135	1	2	yes	complemented
158	C₂×C₄×C₆₇	—	2	1	no	—
159	C₂×C₄×C₆₇	—	2	2	no	—
160	C₂×C₄×C₆₇	—	2	1	no	—
161	C₂×C₄×C₆₇	—	4	2	no	—
162	C₂×C₄×C₆₇	—	4	1	no	—
163	C₂×C₄×C₆₇	—	4	1	no	—
164	C₂×C₄×C₆₇	—	4	2	no	—
165	C₂×C₄×C₆₇	—	4	1	no	—
166	C₂×C₄×C₆₇	—	4	1	no	—
167	C₂×C₄×C₆₇	—	4	2	no	—
168	C₂×C₄×C₆₇	—	4	2	no	—
169	C₂×C₄×C₆₇	—	8	4	no	—
170	C₂×C₄×C₆₇	—	8	4	no	—
171	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
172	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
173	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
174	C₈×C₆₇	—	2	2	no	—
175	C₈×C₆₇	—	2	1	no	—
176	C₈×C₆₇	—	2	1	no	—
177	C₈×C₆₇	—	2	1	no	—
178	C₈×C₆₇	—	4	1	no	—
179	C₈×C₆₇	—	4	1	no	—
180	C₈×C₆₇	—	4	2	no	—
181	C₈×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
182	C₂²×C₆₇	—	2	2	no	—
183	C₂²×C₆₇	—	4	1	no	—
184	C₂²×C₆₇	—	4	1	no	—
185	C₂²×C₆₇	—	4	1	no	—
186	C₂²×C₆₇	—	4	2	no	—
187	C₂²×C₆₇	—	4	2	no	—
188	C₂²×C₆₇	—	4	1	no	—
189	C₂²×C₆₇	—	4	1	no	—
190	C₂²×C₆₇	—	4	2	no	—
191	C₂²×C₆₇	D₄²	1	1	yes	characteristic
192	C₄×C₆₇	—	2	1	no	—
193	C₄×C₆₇	—	4	1	no	—
194	C₄×C₆₇	—	4	1	no	—
195	C₄×C₆₇	—	4	2	no	—
196	C₄×C₆₇	—	8	2	no	—
197	C₄×C₆₇	—	8	2	no	—
198	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
199	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
200	C₄×C₆₇	#576	1	1	yes	characteristic
201	C₄×C₆₇	#577	1	1	yes	characteristic
202	#19902	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
203	C₂×C₆₇	—	4	2	no	—
204	C₂×C₆₇	—	8	1	no	—
205	C₂×C₆₇	—	8	2	no	—
206	C₂×C₆₇	#3272	1	1	yes	characteristic
207	C₂×C₆₇	#3280	1	1	yes	characteristic
208	C₂×C₆₇	#3293	1	1	yes	characteristic
209	#3334	C₂×C₆₇	1	1	yes	characteristic, complemented
210	#3330	C₂×C₆₇	1	2	yes	complemented
211	#3283	C₂×C₆₇	1	1	yes	characteristic, complemented
212	#1707	C₂×C₆₇	1	1	yes	characteristic, complemented
213	#1699	C₂×C₆₇	1	2	yes	complemented
214	#1673	C₂×C₆₇	1	1	yes	characteristic, complemented
215	#1669	C₂×C₆₇	1	1	yes	characteristic, complemented
216	#1654	C₂×C₆₇	1	2	yes	complemented
217	#1654	C₂×C₆₇	1	2	yes	complemented
218	#1589	C₂×C₆₇	1	1	yes	characteristic, complemented
219	C₈×D₈	C₂×C₆₇	1	1	yes	characteristic, complemented
220	C₆₇	#19902	1	1	yes	characteristic, direct product factor, Sylow
221	C₂×#135	—	2	1	no	—
222	#547	C₂²×C₆₇	1	2	yes	—
223	#547	C₂²×C₆₇	1	2	yes	—
224	#500	C₂²×C₆₇	1	1	yes	characteristic
225	#498	C₂²×C₆₇	1	2	yes	—
226	#494	C₂²×C₆₇	1	1	yes	characteristic
227	#487	C₂²×C₆₇	1	2	yes	—
228	#484	—	2	2	no	—
229	#480	—	2	2	no	—
230	#474	C₂²×C₆₇	1	2	yes	—
231	#474	C₂²×C₆₇	1	2	yes	—
232	#468	—	2	2	no	—
233	#467	—	2	1	no	—
234	#466	—	2	1	no	—
235	#461	C₂²×C₆₇	1	2	yes	—
236	#461	C₂²×C₆₇	1	2	yes	—
237	#452	—	2	2	no	—
238	#449	—	2	2	no	—
239	C₄×D₈	C₂²×C₆₇	1	1	yes	characteristic
240	C₄×D₈	C₂²×C₆₇	1	1	yes	characteristic
241	#436	C₂²×C₆₇	1	2	yes	—
242	C₈×D₄	C₂²×C₆₇	1	1	yes	characteristic
243	C₈×D₄	C₂²×C₆₇	1	1	yes	characteristic
244	C₂×#107	—	2	1	no	—
245	#417	—	2	2	no	—
246	#335	C₂²×C₆₇	1	1	yes	characteristic
247	#335	C₂²×C₆₇	1	1	yes	characteristic
248	#332	C₂²×C₆₇	1	2	yes	—
249	#332	C₂²×C₆₇	1	2	yes	—
250	D₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
251	D₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
252	D₄⋊C₈	C₂²×C₆₇	1	2	yes	—
253	C₈²	C₂²×C₆₇	1	1	yes	characteristic
254	C₂×#41	—	2	2	no	—
255	C₂×#41	—	2	2	no	—
256	#135	—	4	4	no	—
257	C₂×Q₁₆	—	2	1	no	—
258	C₂×Q₁₆	—	2	1	no	—
259	C₂×SD₁₆	—	2	2	no	—
260	C₂×SD₁₆	—	2	2	no	—
261	C₂×D₈	D₄×C₆₇	1	1	yes	characteristic
262	C₂×D₈	D₄×C₆₇	1	1	yes	characteristic
263	C₂×M₁₆	—	2	2	no	—
264	C₂²×C₈	—	2	1	no	—
265	C₂²×C₈	—	2	1	no	—
266	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
267	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
268	#123	—	4	2	no	—
269	#123	—	4	2	no	—
270	C₂²⋊Q₈	—	2	2	no	—
271	C₂²⋊Q₈	—	2	2	no	—
272	#121	—	2	2	no	—
273	#121	—	2	2	no	—
274	C₄×D₄	C₂³×C₆₇	1	1	yes	characteristic
275	C₄×D₄	C₂³×C₆₇	1	1	yes	characteristic
276	C₄×D₄	C₂³×C₆₇	1	2	yes	—
277	C₂×C₄⋊C₄	—	2	2	no	—
278	C₂×C₄⋊C₄	—	2	2	no	—
279	#107	—	2	1	no	—
280	#107	—	2	1	no	—
281	#107	—	2	2	no	—
282	#107	—	2	2	no	—
283	#107	D₄×C₆₇	1	1	yes	characteristic
284	#107	D₄×C₆₇	1	1	yes	characteristic
285	#106	—	4	2	no	—
286	C₄⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
287	C₄⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
288	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
289	#103	—	2	2	no	—
290	#103	—	2	2	no	—
291	#103	—	2	2	no	—
292	#103	—	2	2	no	—
293	#102	—	2	2	no	—
294	#102	—	2	2	no	—
295	#102	D₄×C₆₇	1	1	yes	characteristic
296	#102	D₄×C₆₇	1	2	yes	—
297	#102	D₄×C₆₇	1	1	yes	characteristic
298	C₂²⋊C₈	—	2	1	no	—
299	C₂²⋊C₈	—	2	1	no	—
300	C₂²⋊C₈	—	2	2	no	—
301	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
302	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
303	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
304	#41	—	4	4	no	—
305	#41	—	4	4	no	—
306	#41	—	8	2	no	—
307	#41	—	8	2	no	—
308	C₂×Q₈	—	2	2	no	—
309	C₂×Q₈	—	2	2	no	—
310	C₂×D₄	—	4	2	no	—
311	C₂×D₄	—	4	2	no	—
312	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
313	C₂×D₄	C₂×D₄×C₆₇	1	1	yes	characteristic
314	C₂×D₄	C₂×D₄×C₆₇	1	1	yes	characteristic
315	C₂²×C₄	—	2	1	no	—
316	C₂²×C₄	—	2	2	no	—
317	C₂²×C₄	—	2	1	no	—
318	C₂²×C₄	—	4	2	no	—
319	C₂²×C₄	—	4	2	no	—
320	Q₁₆	—	4	2	no	—
321	Q₁₆	—	8	1	no	—
322	SD₁₆	—	4	4	no	—
323	SD₁₆	—	8	2	no	—
324	D₈	—	2	2	no	—
325	D₈	—	2	2	no	—
326	M₁₆	—	8	2	no	—
327	C₂×C₈	—	2	1	no	—
328	C₂×C₈	—	2	1	no	—
329	C₂×C₈	—	2	2	no	—
330	C₂×C₈	—	2	1	no	—
331	C₂×C₈	—	4	2	no	—
332	C₂×C₈	—	4	2	no	—
333	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
334	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
335	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
336	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
337	C₄⋊C₄	—	2	2	no	—
338	C₄⋊C₄	—	2	2	no	—
339	C₄⋊C₄	—	2	2	no	—
340	C₄⋊C₄	—	2	2	no	—
341	C₄⋊C₄	—	4	2	no	—
342	C₄⋊C₄	—	4	2	no	—
343	C₄⋊C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
344	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
345	C₄⋊C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
346	C₂²⋊C₄	—	2	1	no	—
347	C₂²⋊C₄	—	2	2	no	—
348	C₂²⋊C₄	—	2	1	no	—
349	C₂²⋊C₄	—	4	2	no	—
350	C₂²⋊C₄	—	4	2	no	—
351	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
352	C₂³	—	2	1	no	—
353	C₂³	—	2	1	no	—
354	C₂³	—	2	2	no	—
355	Q₈	—	4	2	no	—
356	Q₈	—	4	2	no	—
357	Q₈	—	4	2	no	—
358	Q₈	—	4	2	no	—
359	D₄	—	2	1	no	—
360	D₄	—	2	2	no	—
361	D₄	—	2	1	no	—
362	D₄	—	2	1	no	—
363	D₄	—	2	2	no	—
364	D₄	—	8	4	no	—
365	D₄	—	8	4	no	—
366	D₄	#135×C₆₇	1	2	yes	complemented
367	C₂×C₄	—	2	1	no	—
368	C₂×C₄	—	2	2	no	—
369	C₂×C₄	—	2	1	no	—
370	C₂×C₄	—	4	2	no	—
371	C₂×C₄	—	4	1	no	—
372	C₂×C₄	—	4	1	no	—
373	C₂×C₄	—	4	2	no	—
374	C₂×C₄	—	4	1	no	—
375	C₂×C₄	—	4	1	no	—
376	C₂×C₄	—	4	2	no	—
377	C₂×C₄	—	4	2	no	—
378	C₂×C₄	—	8	4	no	—
379	C₂×C₄	—	8	4	no	—
380	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
381	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
382	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
383	C₈	—	2	2	no	—
384	C₈	—	2	1	no	—
385	C₈	—	2	1	no	—
386	C₈	—	2	1	no	—
387	C₈	—	4	1	no	—
388	C₈	—	4	1	no	—
389	C₈	—	4	2	no	—
390	C₈	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
391	C₂²	—	2	2	no	—
392	C₂²	—	4	1	no	—
393	C₂²	—	4	1	no	—
394	C₂²	—	4	1	no	—
395	C₂²	—	4	2	no	—
396	C₂²	—	4	2	no	—
397	C₂²	—	4	1	no	—
398	C₂²	—	4	1	no	—
399	C₂²	—	4	2	no	—
400	C₂²	D₄²×C₆₇	1	1	yes	characteristic
401	C₄	—	2	1	no	—
402	C₄	—	4	1	no	—
403	C₄	—	4	1	no	—
404	C₄	—	4	2	no	—
405	C₄	—	8	2	no	—
406	C₄	—	8	2	no	—
407	C₄	C₂×#135×C₆₇	1	1	yes	characteristic
408	C₄	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
409	C₄	#576×C₆₇	1	1	yes	characteristic
410	C₄	#577×C₆₇	1	1	yes	characteristic
411	C₂	—	4	2	no	—
412	C₂	—	8	1	no	—
413	C₂	—	8	2	no	—
414	C₂	C₆₇×#3272	1	1	yes	characteristic
415	C₂	C₆₇×#3280	1	1	yes	characteristic
416	C₂	C₆₇×#3293	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₇
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#3334	C₂	semidirect
2. C₆₇×#3330	C₂	semidirect
3. C₆₇×#3283	C₂	semidirect
4. C₆₇×#1707	C₂	semidirect
5. C₆₇×#1699	C₂	semidirect
6. C₆₇×#1673	C₂	semidirect
7. C₆₇×#1669	C₂	semidirect
8. C₆₇×#1654	C₂	semidirect
9. C₆₇×#1654	C₂	semidirect
10. C₆₇×#1589	C₂	semidirect
11. C₈×D₈×C₆₇	C₂	semidirect
157. D₄×C₆₇	#135	semidirect
202. #19902	C₆₇	direct (N × H)
209. #3334	C₂×C₆₇	semidirect
210. #3330	C₂×C₆₇	semidirect
211. #3283	C₂×C₆₇	semidirect
212. #1707	C₂×C₆₇	semidirect
213. #1699	C₂×C₆₇	semidirect
214. #1673	C₂×C₆₇	semidirect
215. #1669	C₂×C₆₇	semidirect
216. #1654	C₂×C₆₇	semidirect
217. #1654	C₂×C₆₇	semidirect
218. #1589	C₂×C₆₇	semidirect
219. C₈×D₈	C₂×C₆₇	semidirect
220. C₆₇	#19902	direct (N × H)
366. D₄	#135×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3334	2
C₆₇×#3330	2
C₆₇×#3283	2
C₆₇×#1707	2
C₆₇×#1699	2
C₆₇×#1673	2
C₆₇×#1669	2
C₆₇×#1654	2
C₆₇×#1589	2
C₈×D₈×C₆₇	2
#19902	67

Subgroups — C67×#19902 #30699153

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#19902 #30699153