Subgroups — C₆₇×#19395 #30698646

Order 17152.

628 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3231	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₇×#3110	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#3101	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#2972	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1683	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1679	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1673	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1671	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1572	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#1554	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1544	C₂	1	1	yes	maximal, characteristic, complemented
12	#533×C₆₇	C₂²	1	1	yes	characteristic
13	#533×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
15	#500×C₆₇	C₂²	1	1	yes	characteristic, complemented
16	#498×C₆₇	C₂²	1	1	yes	characteristic, complemented
17	#487×C₆₇	C₂²	1	1	yes	characteristic, complemented
18	#487×C₆₇	C₂²	1	1	yes	characteristic, complemented
19	#485×C₆₇	—	2	2	no	—
20	#484×C₆₇	—	2	1	no	—
21	#483×C₆₇	—	2	1	no	—
22	#479×C₆₇	—	2	2	no	—
23	#476×C₆₇	C₂²	1	2	yes	—
24	#474×C₆₇	C₂²	1	2	yes	—
25	#461×C₆₇	C₂²	1	2	yes	—
26	#461×C₆₇	C₂²	1	2	yes	—
27	#450×C₆₇	—	2	1	no	—
28	#450×C₆₇	—	2	1	no	—
29	#440×C₆₇	—	2	2	no	—
30	C₄×SD₁₆×C₆₇	—	2	2	no	—
31	C₈×D₄×C₆₇	—	2	2	no	—
32	C₈×D₄×C₆₇	C₂²	1	2	yes	—
33	C₈×D₄×C₆₇	C₂²	1	2	yes	—
34	#432×C₆₇	C₂²	1	1	yes	characteristic
35	#431×C₆₇	C₂²	1	1	yes	characteristic
36	#431×C₆₇	C₂²	1	1	yes	characteristic
37	#431×C₆₇	C₂²	1	1	yes	characteristic
38	C₂×C₂²⋊C₈×C₆₇	—	2	2	no	—
39	#335×C₆₇	C₂²	1	2	yes	complemented
40	#334×C₆₇	C₂²	1	2	yes	complemented
41	#333×C₆₇	C₂²	1	2	yes	complemented
42	#332×C₆₇	C₂²	1	2	yes	complemented
43	#332×C₆₇	C₂²	1	2	yes	complemented
44	#332×C₆₇	C₂²	1	2	yes	complemented
45	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
46	C₂³×C₄×C₆₇	—	2	1	no	—
47	C₂×SD₁₆×C₆₇	—	2	2	no	—
48	C₂×SD₁₆×C₆₇	—	2	2	no	—
49	C₂²×C₈×C₆₇	—	2	2	no	—
50	C₂²×C₈×C₆₇	—	2	2	no	—
51	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
52	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
53	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
54	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
55	#125×C₆₇	—	2	1	no	—
56	#125×C₆₇	—	2	1	no	—
57	C₂²⋊Q₈×C₆₇	—	2	1	no	—
58	C₂²⋊Q₈×C₆₇	—	2	1	no	—
59	C₂²⋊Q₈×C₆₇	—	2	1	no	—
60	C₂²⋊Q₈×C₆₇	—	2	1	no	—
61	C₄×Q₈×C₆₇	—	2	1	no	—
62	C₄×Q₈×C₆₇	—	2	1	no	—
63	C₄×Q₈×C₆₇	—	2	1	no	—
64	C₄×Q₈×C₆₇	—	2	1	no	—
65	C₄×D₄×C₆₇	—	2	1	no	—
66	C₄×D₄×C₆₇	—	2	1	no	—
67	C₄×D₄×C₆₇	D₄	1	2	yes	—
68	C₄×D₄×C₆₇	C₂³	1	2	yes	—
69	#117×C₆₇	—	2	1	no	—
70	#117×C₆₇	—	2	1	no	—
71	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
72	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
73	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
74	#107×C₆₇	—	2	2	no	—
75	#107×C₆₇	—	2	2	no	—
76	#106×C₆₇	—	2	2	no	—
77	#106×C₆₇	—	2	2	no	—
78	C₄⋊C₈×C₆₇	D₄	1	2	yes	—
79	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
80	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
81	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
82	#103×C₆₇	—	2	2	no	—
83	#103×C₆₇	—	2	2	no	—
84	#103×C₆₇	—	2	2	no	—
85	#103×C₆₇	—	2	2	no	—
86	#102×C₆₇	—	2	2	no	—
87	#102×C₆₇	—	2	2	no	—
88	C₂²⋊C₈×C₆₇	—	2	1	no	—
89	C₂²⋊C₈×C₆₇	—	2	1	no	—
90	C₂²⋊C₈×C₆₇	—	2	1	no	—
91	C₂²⋊C₈×C₆₇	—	2	2	no	—
92	C₂²⋊C₈×C₆₇	—	2	2	no	—
93	C₂²⋊C₈×C₆₇	D₄	1	2	yes	—
94	#97×C₆₇	—	2	1	no	—
95	C₄×C₈×C₆₇	—	2	1	no	—
96	C₄×C₈×C₆₇	—	2	1	no	—
97	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
98	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
99	C₂⁴×C₆₇	—	2	1	no	—
100	C₂×Q₈×C₆₇	—	2	1	no	—
101	C₂×Q₈×C₆₇	—	2	1	no	—
102	C₂×Q₈×C₆₇	—	2	1	no	—
103	C₂×Q₈×C₆₇	—	2	1	no	—
104	C₂×D₄×C₆₇	—	2	4	no	—
105	C₂×D₄×C₆₇	—	2	1	no	—
106	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
107	C₂²×C₄×C₆₇	—	2	1	no	—
108	C₂²×C₄×C₆₇	—	2	2	no	—
109	C₂²×C₄×C₆₇	—	2	2	no	—
110	C₂²×C₄×C₆₇	—	8	2	no	—
111	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
114	SD₁₆×C₆₇	—	4	4	no	—
115	SD₁₆×C₆₇	—	4	4	no	—
116	C₂×C₈×C₆₇	—	2	2	no	—
117	C₂×C₈×C₆₇	—	2	2	no	—
118	C₂×C₈×C₆₇	—	2	2	no	—
119	C₂×C₈×C₆₇	—	8	2	no	—
120	C₂×C₈×C₆₇	—	8	2	no	—
121	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
122	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
123	C₄⋊C₄×C₆₇	—	2	1	no	—
124	C₄⋊C₄×C₆₇	—	2	1	no	—
125	C₄⋊C₄×C₆₇	—	2	1	no	—
126	C₄⋊C₄×C₆₇	—	2	1	no	—
127	C₄⋊C₄×C₆₇	—	2	1	no	—
128	C₄⋊C₄×C₆₇	—	2	1	no	—
129	C₄⋊C₄×C₆₇	—	2	1	no	—
130	C₄⋊C₄×C₆₇	—	2	1	no	—
131	C₄⋊C₄×C₆₇	—	2	1	no	—
132	C₄⋊C₄×C₆₇	—	4	1	no	—
133	C₄⋊C₄×C₆₇	—	4	1	no	—
134	C₄⋊C₄×C₆₇	—	4	1	no	—
135	C₄⋊C₄×C₆₇	—	4	1	no	—
136	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
137	C₂²⋊C₄×C₆₇	—	2	2	no	—
138	C₂²⋊C₄×C₆₇	—	2	2	no	—
139	C₂²⋊C₄×C₆₇	—	4	1	no	—
140	C₂²⋊C₄×C₆₇	—	4	1	no	—
141	C₄²×C₆₇	—	4	1	no	—
142	C₄²×C₆₇	—	4	1	no	—
143	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
144	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
145	C₄²×C₆₇	#41	1	1	yes	characteristic
146	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
147	C₂³×C₆₇	—	2	4	no	—
148	C₂³×C₆₇	—	2	2	no	—
149	C₂³×C₆₇	—	8	1	no	—
150	C₂³×C₆₇	—	8	1	no	—
151	C₂³×C₆₇	#121	1	1	yes	characteristic
152	Q₈×C₆₇	—	4	2	no	—
153	Q₈×C₆₇	—	4	2	no	—
154	Q₈×C₆₇	—	4	2	no	—
155	Q₈×C₆₇	—	8	1	no	—
156	D₄×C₆₇	—	2	4	no	—
157	D₄×C₆₇	—	2	2	no	—
158	D₄×C₆₇	#135	1	4	yes	—
159	C₂×C₄×C₆₇	—	2	2	no	—
160	C₂×C₄×C₆₇	—	2	2	no	—
161	C₂×C₄×C₆₇	—	2	2	no	—
162	C₂×C₄×C₆₇	—	4	1	no	—
163	C₂×C₄×C₆₇	—	4	2	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	1	no	—
168	C₂×C₄×C₆₇	—	4	1	no	—
169	C₂×C₄×C₆₇	—	8	2	no	—
170	C₂×C₄×C₆₇	—	8	1	no	—
171	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
172	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
173	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
174	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
175	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
176	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
177	C₈×C₆₇	—	2	2	no	—
178	C₈×C₆₇	—	2	2	no	—
179	C₈×C₆₇	—	4	2	no	—
180	C₈×C₆₇	—	4	2	no	—
181	C₈×C₆₇	—	4	2	no	—
182	C₂²×C₆₇	—	2	1	no	—
183	C₂²×C₆₇	—	2	1	no	—
184	C₂²×C₆₇	—	2	4	no	—
185	C₂²×C₆₇	—	2	1	no	—
186	C₂²×C₆₇	—	4	2	no	—
187	C₂²×C₆₇	—	4	2	no	—
188	C₂²×C₆₇	—	8	4	no	—
189	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
190	C₄×C₆₇	—	2	1	no	—
191	C₄×C₆₇	—	2	1	no	—
192	C₄×C₆₇	—	2	1	no	—
193	C₄×C₆₇	—	2	1	no	—
194	C₄×C₆₇	—	4	2	no	—
195	C₄×C₆₇	—	8	1	no	—
196	C₄×C₆₇	—	8	1	no	—
197	C₄×C₆₇	—	8	1	no	—
198	C₄×C₆₇	—	8	1	no	—
199	C₄×C₆₇	C₂×#135	1	2	yes	—
200	C₄×C₆₇	#577	1	2	yes	—
201	#19395	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
202	C₂×C₆₇	—	4	1	no	—
203	C₂×C₆₇	—	4	4	no	—
204	C₂×C₆₇	#3042	1	1	yes	characteristic
205	C₂×C₆₇	#3050	1	1	yes	characteristic
206	C₂×C₆₇	#3069	1	1	yes	characteristic
207	#3231	C₂×C₆₇	1	1	yes	characteristic, complemented
208	#3110	C₂×C₆₇	1	1	yes	characteristic
209	#3101	C₂×C₆₇	1	1	yes	characteristic
210	#2972	C₂×C₆₇	1	1	yes	characteristic
211	#1683	C₂×C₆₇	1	2	yes	complemented
212	#1679	C₂×C₆₇	1	2	yes	complemented
213	#1673	C₂×C₆₇	1	2	yes	complemented
214	#1671	C₂×C₆₇	1	2	yes	complemented
215	#1572	C₂×C₆₇	1	1	yes	characteristic, complemented
216	#1554	C₂×C₆₇	1	1	yes	characteristic, complemented
217	#1544	C₂×C₆₇	1	1	yes	characteristic, complemented
218	C₆₇	#19395	1	1	yes	characteristic, direct product factor, Sylow
219	#533	C₂²×C₆₇	1	1	yes	characteristic
220	#533	C₂²×C₆₇	1	1	yes	characteristic
221	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
222	#500	C₂²×C₆₇	1	1	yes	characteristic, complemented
223	#498	C₂²×C₆₇	1	1	yes	characteristic, complemented
224	#487	C₂²×C₆₇	1	1	yes	characteristic, complemented
225	#487	C₂²×C₆₇	1	1	yes	characteristic, complemented
226	#485	—	2	2	no	—
227	#484	—	2	1	no	—
228	#483	—	2	1	no	—
229	#479	—	2	2	no	—
230	#476	C₂²×C₆₇	1	2	yes	—
231	#474	C₂²×C₆₇	1	2	yes	—
232	#461	C₂²×C₆₇	1	2	yes	—
233	#461	C₂²×C₆₇	1	2	yes	—
234	#450	—	2	1	no	—
235	#450	—	2	1	no	—
236	#440	—	2	2	no	—
237	C₄×SD₁₆	—	2	2	no	—
238	C₈×D₄	—	2	2	no	—
239	C₈×D₄	C₂²×C₆₇	1	2	yes	—
240	C₈×D₄	C₂²×C₆₇	1	2	yes	—
241	#432	C₂²×C₆₇	1	1	yes	characteristic
242	#431	C₂²×C₆₇	1	1	yes	characteristic
243	#431	C₂²×C₆₇	1	1	yes	characteristic
244	#431	C₂²×C₆₇	1	1	yes	characteristic
245	C₂×C₂²⋊C₈	—	2	2	no	—
246	#335	C₂²×C₆₇	1	2	yes	complemented
247	#334	C₂²×C₆₇	1	2	yes	complemented
248	#333	C₂²×C₆₇	1	2	yes	complemented
249	#332	C₂²×C₆₇	1	2	yes	complemented
250	#332	C₂²×C₆₇	1	2	yes	complemented
251	#332	C₂²×C₆₇	1	2	yes	complemented
252	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
253	C₂³×C₄	—	2	1	no	—
254	C₂×SD₁₆	—	2	2	no	—
255	C₂×SD₁₆	—	2	2	no	—
256	C₂²×C₈	—	2	2	no	—
257	C₂²×C₈	—	2	2	no	—
258	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
259	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
260	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
261	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
262	#125	—	2	1	no	—
263	#125	—	2	1	no	—
264	C₂²⋊Q₈	—	2	1	no	—
265	C₂²⋊Q₈	—	2	1	no	—
266	C₂²⋊Q₈	—	2	1	no	—
267	C₂²⋊Q₈	—	2	1	no	—
268	C₄×Q₈	—	2	1	no	—
269	C₄×Q₈	—	2	1	no	—
270	C₄×Q₈	—	2	1	no	—
271	C₄×Q₈	—	2	1	no	—
272	C₄×D₄	—	2	1	no	—
273	C₄×D₄	—	2	1	no	—
274	C₄×D₄	D₄×C₆₇	1	2	yes	—
275	C₄×D₄	C₂³×C₆₇	1	2	yes	—
276	#117	—	2	1	no	—
277	#117	—	2	1	no	—
278	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
279	C₂×C₂²⋊C₄	—	2	1	no	—
280	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
281	#107	—	2	2	no	—
282	#107	—	2	2	no	—
283	#106	—	2	2	no	—
284	#106	—	2	2	no	—
285	C₄⋊C₈	D₄×C₆₇	1	2	yes	—
286	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
287	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
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	C₂²⋊C₈	—	2	1	no	—
296	C₂²⋊C₈	—	2	1	no	—
297	C₂²⋊C₈	—	2	1	no	—
298	C₂²⋊C₈	—	2	2	no	—
299	C₂²⋊C₈	—	2	2	no	—
300	C₂²⋊C₈	D₄×C₆₇	1	2	yes	—
301	#97	—	2	1	no	—
302	C₄×C₈	—	2	1	no	—
303	C₄×C₈	—	2	1	no	—
304	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
305	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
306	C₂⁴	—	2	1	no	—
307	C₂×Q₈	—	2	1	no	—
308	C₂×Q₈	—	2	1	no	—
309	C₂×Q₈	—	2	1	no	—
310	C₂×Q₈	—	2	1	no	—
311	C₂×D₄	—	2	4	no	—
312	C₂×D₄	—	2	1	no	—
313	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
314	C₂²×C₄	—	2	1	no	—
315	C₂²×C₄	—	2	2	no	—
316	C₂²×C₄	—	2	2	no	—
317	C₂²×C₄	—	8	2	no	—
318	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
319	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
320	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
321	SD₁₆	—	4	4	no	—
322	SD₁₆	—	4	4	no	—
323	C₂×C₈	—	2	2	no	—
324	C₂×C₈	—	2	2	no	—
325	C₂×C₈	—	2	2	no	—
326	C₂×C₈	—	8	2	no	—
327	C₂×C₈	—	8	2	no	—
328	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
329	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
330	C₄⋊C₄	—	2	1	no	—
331	C₄⋊C₄	—	2	1	no	—
332	C₄⋊C₄	—	2	1	no	—
333	C₄⋊C₄	—	2	1	no	—
334	C₄⋊C₄	—	2	1	no	—
335	C₄⋊C₄	—	2	1	no	—
336	C₄⋊C₄	—	2	1	no	—
337	C₄⋊C₄	—	2	1	no	—
338	C₄⋊C₄	—	2	1	no	—
339	C₄⋊C₄	—	4	1	no	—
340	C₄⋊C₄	—	4	1	no	—
341	C₄⋊C₄	—	4	1	no	—
342	C₄⋊C₄	—	4	1	no	—
343	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
344	C₂²⋊C₄	—	2	2	no	—
345	C₂²⋊C₄	—	2	2	no	—
346	C₂²⋊C₄	—	4	1	no	—
347	C₂²⋊C₄	—	4	1	no	—
348	C₄²	—	4	1	no	—
349	C₄²	—	4	1	no	—
350	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
351	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
352	C₄²	#41×C₆₇	1	1	yes	characteristic
353	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
354	C₂³	—	2	4	no	—
355	C₂³	—	2	2	no	—
356	C₂³	—	8	1	no	—
357	C₂³	—	8	1	no	—
358	C₂³	#121×C₆₇	1	1	yes	characteristic
359	Q₈	—	4	2	no	—
360	Q₈	—	4	2	no	—
361	Q₈	—	4	2	no	—
362	Q₈	—	8	1	no	—
363	D₄	—	2	4	no	—
364	D₄	—	2	2	no	—
365	D₄	#135×C₆₇	1	4	yes	—
366	C₂×C₄	—	2	2	no	—
367	C₂×C₄	—	2	2	no	—
368	C₂×C₄	—	2	2	no	—
369	C₂×C₄	—	4	1	no	—
370	C₂×C₄	—	4	2	no	—
371	C₂×C₄	—	4	2	no	—
372	C₂×C₄	—	4	1	no	—
373	C₂×C₄	—	4	1	no	—
374	C₂×C₄	—	4	1	no	—
375	C₂×C₄	—	4	1	no	—
376	C₂×C₄	—	8	2	no	—
377	C₂×C₄	—	8	1	no	—
378	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
379	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
380	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
381	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
382	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
383	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
384	C₈	—	2	2	no	—
385	C₈	—	2	2	no	—
386	C₈	—	4	2	no	—
387	C₈	—	4	2	no	—
388	C₈	—	4	2	no	—
389	C₂²	—	2	1	no	—
390	C₂²	—	2	1	no	—
391	C₂²	—	2	4	no	—
392	C₂²	—	2	1	no	—
393	C₂²	—	4	2	no	—
394	C₂²	—	4	2	no	—
395	C₂²	—	8	4	no	—
396	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
397	C₄	—	2	1	no	—
398	C₄	—	2	1	no	—
399	C₄	—	2	1	no	—
400	C₄	—	2	1	no	—
401	C₄	—	4	2	no	—
402	C₄	—	8	1	no	—
403	C₄	—	8	1	no	—
404	C₄	—	8	1	no	—
405	C₄	—	8	1	no	—
406	C₄	C₂×#135×C₆₇	1	2	yes	—
407	C₄	#577×C₆₇	1	2	yes	—
408	C₂	—	4	1	no	—
409	C₂	—	4	4	no	—
410	C₂	C₆₇×#3042	1	1	yes	characteristic
411	C₂	C₆₇×#3050	1	1	yes	characteristic
412	C₂	C₆₇×#3069	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₇
prime radical	C₂²×D₄×C₆₇	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#3231	C₂	semidirect
5. C₆₇×#1683	C₂	semidirect
6. C₆₇×#1679	C₂	semidirect
7. C₆₇×#1673	C₂	semidirect
8. C₆₇×#1671	C₂	semidirect
9. C₆₇×#1572	C₂	semidirect
10. C₆₇×#1554	C₂	semidirect
11. C₆₇×#1544	C₂	semidirect
15. #500×C₆₇	C₂²	semidirect
16. #498×C₆₇	C₂²	semidirect
17. #487×C₆₇	C₂²	semidirect
18. #487×C₆₇	C₂²	semidirect
39. #335×C₆₇	C₂²	semidirect
40. #334×C₆₇	C₂²	semidirect
41. #333×C₆₇	C₂²	semidirect
42. #332×C₆₇	C₂²	semidirect
43. #332×C₆₇	C₂²	semidirect
44. #332×C₆₇	C₂²	semidirect
201. #19395	C₆₇	direct (N × H)
207. #3231	C₂×C₆₇	semidirect
211. #1683	C₂×C₆₇	semidirect
212. #1679	C₂×C₆₇	semidirect
213. #1673	C₂×C₆₇	semidirect
214. #1671	C₂×C₆₇	semidirect
215. #1572	C₂×C₆₇	semidirect
216. #1554	C₂×C₆₇	semidirect
217. #1544	C₂×C₆₇	semidirect
218. C₆₇	#19395	direct (N × H)
222. #500	C₂²×C₆₇	semidirect
223. #498	C₂²×C₆₇	semidirect
224. #487	C₂²×C₆₇	semidirect
225. #487	C₂²×C₆₇	semidirect
246. #335	C₂²×C₆₇	semidirect
247. #334	C₂²×C₆₇	semidirect
248. #333	C₂²×C₆₇	semidirect
249. #332	C₂²×C₆₇	semidirect
250. #332	C₂²×C₆₇	semidirect
251. #332	C₂²×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3231	2
C₆₇×#3110	2
C₆₇×#3101	2
C₆₇×#2972	2
C₆₇×#1683	2
C₆₇×#1679	2
C₆₇×#1673	2
C₆₇×#1671	2
C₆₇×#1572	2
C₆₇×#1554	2
C₆₇×#1544	2
#19395	67

Subgroups — C67×#19395 #30698646

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#19395 #30698646