Subgroups — C₆₇×#10476 #30689727

Order 17152.

506 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#461×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#1990	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#1954	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1802	C₂	1	1	yes	maximal, characteristic
5	C₂×#335×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#332×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1296	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×SD₁₆×C₆₇	—	2	1	no	—
9	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
11	#461×C₆₇	—	2	2	no	—
12	#461×C₆₇	—	2	2	no	—
13	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×#103×C₆₇	—	2	1	no	—
17	C₂×#102×C₆₇	—	2	1	no	—
18	C₂×C₂²⋊C₈×C₆₇	—	2	1	no	—
19	C₂×C₂²⋊C₈×C₆₇	—	2	1	no	—
20	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
21	#384×C₆₇	C₂²	1	1	yes	characteristic
22	#340×C₆₇	—	2	1	no	—
23	#340×C₆₇	—	2	1	no	—
24	#340×C₆₇	—	2	1	no	—
25	#335×C₆₇	—	2	2	no	—
26	#332×C₆₇	—	2	2	no	—
27	C₂²×Q₈×C₆₇	—	2	1	no	—
28	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
29	C₂³×C₄×C₆₇	—	2	1	no	—
30	C₂×SD₁₆×C₆₇	—	2	4	no	—
31	C₂×SD₁₆×C₆₇	—	4	2	no	—
32	C₂×SD₁₆×C₆₇	—	4	2	no	—
33	C₂²×C₈×C₆₇	—	2	1	no	—
34	C₂²×C₈×C₆₇	—	2	1	no	—
35	C₂²×C₈×C₆₇	—	2	1	no	—
36	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
38	C₄⋊Q₈×C₆₇	—	2	1	no	—
39	C₄⋊Q₈×C₆₇	—	2	1	no	—
40	C₄⋊Q₈×C₆₇	—	2	1	no	—
41	C₄⋊Q₈×C₆₇	D₄	1	2	yes	—
42	C₄×D₄×C₆₇	—	2	1	no	—
43	C₄×D₄×C₆₇	—	2	1	no	—
44	C₄×D₄×C₆₇	—	2	1	no	—
45	C₄×D₄×C₆₇	—	2	1	no	—
46	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
47	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
48	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
49	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
50	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
51	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
52	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
53	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
54	C₄⋊C₈×C₆₇	—	2	1	no	—
55	C₄⋊C₈×C₆₇	—	2	1	no	—
56	C₄⋊C₈×C₆₇	—	2	1	no	—
57	C₄⋊C₈×C₆₇	—	2	1	no	—
58	C₄⋊C₈×C₆₇	—	2	1	no	—
59	C₄⋊C₈×C₆₇	D₄	1	2	yes	—
60	#103×C₆₇	—	2	4	no	—
61	#102×C₆₇	—	4	2	no	—
62	C₂²⋊C₈×C₆₇	—	4	2	no	—
63	C₂²⋊C₈×C₆₇	—	4	2	no	—
64	C₄×C₈×C₆₇	—	2	1	no	—
65	C₄×C₈×C₆₇	—	2	1	no	—
66	C₂⁴×C₆₇	—	2	1	no	—
67	C₂×Q₈×C₆₇	—	2	2	no	—
68	C₂×Q₈×C₆₇	—	4	1	no	—
69	C₂×Q₈×C₆₇	—	4	1	no	—
70	C₂×Q₈×C₆₇	—	4	1	no	—
71	C₂×Q₈×C₆₇	—	4	1	no	—
72	C₂×Q₈×C₆₇	—	4	1	no	—
73	C₂×D₄×C₆₇	—	2	1	no	—
74	C₂×D₄×C₆₇	—	2	1	no	—
75	C₂×D₄×C₆₇	—	2	1	no	—
76	C₂×D₄×C₆₇	—	2	1	no	—
77	C₂×D₄×C₆₇	—	2	1	no	—
78	C₂×D₄×C₆₇	D₈	1	2	yes	—
79	C₂²×C₄×C₆₇	—	2	1	no	—
80	C₂²×C₄×C₆₇	—	4	1	no	—
81	C₂²×C₄×C₆₇	—	4	1	no	—
82	C₂²×C₄×C₆₇	—	4	1	no	—
83	C₂²×C₄×C₆₇	—	4	1	no	—
84	C₂²×C₄×C₆₇	—	4	1	no	—
85	C₂²×C₄×C₆₇	—	4	1	no	—
86	C₂²×C₄×C₆₇	—	4	1	no	—
87	C₂²×C₄×C₆₇	—	4	1	no	—
88	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
91	SD₁₆×C₆₇	—	4	8	no	—
92	C₂×C₈×C₆₇	—	2	1	no	—
93	C₂×C₈×C₆₇	—	2	1	no	—
94	C₂×C₈×C₆₇	—	2	2	no	—
95	C₂×C₈×C₆₇	—	2	1	no	—
96	C₂×C₈×C₆₇	—	2	1	no	—
97	C₂×C₈×C₆₇	—	2	1	no	—
98	C₂×C₈×C₆₇	—	4	1	no	—
99	C₂×C₈×C₆₇	—	4	1	no	—
100	C₂×C₈×C₆₇	—	4	1	no	—
101	C₂×C₈×C₆₇	—	4	1	no	—
102	C₂×C₈×C₆₇	—	4	1	no	—
103	C₂×C₈×C₆₇	—	4	1	no	—
104	C₂×C₈×C₆₇	—	4	1	no	—
105	C₂×C₈×C₆₇	—	4	1	no	—
106	C₂×C₈×C₆₇	SD₁₆	1	2	yes	complemented
107	C₄⋊C₄×C₆₇	—	2	1	no	—
108	C₄⋊C₄×C₆₇	—	2	2	no	—
109	C₄⋊C₄×C₆₇	—	2	2	no	—
110	C₄⋊C₄×C₆₇	—	2	1	no	—
111	C₄⋊C₄×C₆₇	—	4	1	no	—
112	C₄⋊C₄×C₆₇	—	4	1	no	—
113	C₄⋊C₄×C₆₇	—	4	1	no	—
114	C₄⋊C₄×C₆₇	—	4	1	no	—
115	C₄⋊C₄×C₆₇	—	4	1	no	—
116	C₄⋊C₄×C₆₇	—	4	1	no	—
117	C₄⋊C₄×C₆₇	—	8	1	no	—
118	C₂²⋊C₄×C₆₇	—	4	1	no	—
119	C₂²⋊C₄×C₆₇	—	4	1	no	—
120	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
121	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
122	C₄²×C₆₇	#41	1	1	yes	characteristic
123	C₄²×C₆₇	#41	1	1	yes	characteristic
124	C₂³×C₆₇	—	2	2	no	—
125	C₂³×C₆₇	—	4	1	no	—
126	C₂³×C₆₇	—	4	1	no	—
127	C₂³×C₆₇	—	4	1	no	—
128	C₂³×C₆₇	—	4	1	no	—
129	C₂³×C₆₇	—	4	1	no	—
130	C₂³×C₆₇	—	4	1	no	—
131	C₂³×C₆₇	#121	1	1	yes	characteristic
132	Q₈×C₆₇	—	4	2	no	—
133	Q₈×C₆₇	—	4	2	no	—
134	Q₈×C₆₇	—	8	1	no	—
135	Q₈×C₆₇	—	8	1	no	—
136	D₄×C₆₇	—	2	2	no	—
137	D₄×C₆₇	—	2	2	no	—
138	D₄×C₆₇	—	4	1	no	—
139	D₄×C₆₇	—	4	1	no	—
140	C₂×C₄×C₆₇	—	2	1	no	—
141	C₂×C₄×C₆₇	—	2	1	no	—
142	C₂×C₄×C₆₇	—	4	1	no	—
143	C₂×C₄×C₆₇	—	4	2	no	—
144	C₂×C₄×C₆₇	—	4	1	no	—
145	C₂×C₄×C₆₇	—	4	1	no	—
146	C₂×C₄×C₆₇	—	8	1	no	—
147	C₂×C₄×C₆₇	—	8	1	no	—
148	C₂×C₄×C₆₇	—	8	1	no	—
149	C₂×C₄×C₆₇	—	8	1	no	—
150	C₂×C₄×C₆₇	—	8	1	no	—
151	C₂×C₄×C₆₇	—	8	1	no	—
152	C₂×C₄×C₆₇	—	8	1	no	—
153	C₂×C₄×C₆₇	—	8	1	no	—
154	C₂×C₄×C₆₇	—	8	1	no	—
155	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
156	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
157	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
158	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
159	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
160	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
161	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
162	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
163	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
164	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
165	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
166	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
167	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
168	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
169	C₈×C₆₇	—	2	2	no	—
170	C₈×C₆₇	—	4	2	no	—
171	C₈×C₆₇	—	4	1	no	—
172	C₈×C₆₇	—	8	1	no	—
173	C₈×C₆₇	—	8	1	no	—
174	C₂²×C₆₇	—	4	2	no	—
175	C₂²×C₆₇	—	4	2	no	—
176	C₂²×C₆₇	—	4	2	no	—
177	C₂²×C₆₇	—	8	1	no	—
178	C₂²×C₆₇	—	8	1	no	—
179	C₂²×C₆₇	—	8	1	no	—
180	C₂²×C₆₇	—	8	1	no	—
181	C₂²×C₆₇	#394	1	1	yes	characteristic
182	C₂²×C₆₇	#461	1	1	yes	characteristic
183	C₂²×C₆₇	#463	1	1	yes	characteristic
184	C₂²×C₆₇	#466	1	1	yes	characteristic
185	C₂²×C₆₇	#468	1	1	yes	characteristic
186	C₂²×C₆₇	#471	1	1	yes	characteristic
187	C₂²×C₆₇	#473	1	1	yes	characteristic
188	C₄×C₆₇	—	2	1	no	—
189	C₄×C₆₇	—	2	1	no	—
190	C₄×C₆₇	—	8	1	no	—
191	C₄×C₆₇	—	8	2	no	—
192	C₄×C₆₇	—	16	1	no	—
193	C₄×C₆₇	#447	1	1	yes	characteristic
194	C₄×C₆₇	#452	1	1	yes	characteristic
195	C₄×C₆₇	#460	1	1	yes	characteristic
196	C₄×C₆₇	#464	1	1	yes	characteristic
197	C₄×C₆₇	#465	1	1	yes	characteristic
198	C₄×C₆₇	#468	1	1	yes	characteristic
199	C₄×C₆₇	#480	1	1	yes	characteristic
200	C₄×C₆₇	#482	1	1	yes	characteristic
201	#10476	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
202	C₂×C₆₇	—	8	2	no	—
203	C₂×C₆₇	#1673	1	1	yes	characteristic
204	C₂×C₆₇	#1681	1	1	yes	characteristic
205	C₂×C₆₇	#1691	1	1	yes	characteristic
206	C₂×C₆₇	#1699	1	1	yes	characteristic
207	C₂×C₆₇	#2033	1	1	yes	characteristic
208	C₂×C₆₇	#2052	1	1	yes	characteristic
209	C₂×C₆₇	#2055	1	1	yes	characteristic
210	C₂×#461	C₂×C₆₇	1	1	yes	characteristic
211	#1990	C₂×C₆₇	1	1	yes	characteristic, complemented
212	#1954	C₂×C₆₇	1	1	yes	characteristic
213	#1802	C₂×C₆₇	1	1	yes	characteristic
214	C₂×#335	C₂×C₆₇	1	1	yes	characteristic, complemented
215	C₂×#332	C₂×C₆₇	1	1	yes	characteristic, complemented
216	#1296	C₂×C₆₇	1	1	yes	characteristic, complemented
217	C₆₇	#10476	1	1	yes	characteristic, direct product factor, Sylow
218	C₂²×SD₁₆	—	2	1	no	—
219	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
220	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
221	#461	—	2	2	no	—
222	#461	—	2	2	no	—
223	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
224	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
225	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
226	C₂×#103	—	2	1	no	—
227	C₂×#102	—	2	1	no	—
228	C₂×C₂²⋊C₈	—	2	1	no	—
229	C₂×C₂²⋊C₈	—	2	1	no	—
230	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
231	#384	C₂²×C₆₇	1	1	yes	characteristic
232	#340	—	2	1	no	—
233	#340	—	2	1	no	—
234	#340	—	2	1	no	—
235	#335	—	2	2	no	—
236	#332	—	2	2	no	—
237	C₂²×Q₈	—	2	1	no	—
238	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
239	C₂³×C₄	—	2	1	no	—
240	C₂×SD₁₆	—	2	4	no	—
241	C₂×SD₁₆	—	4	2	no	—
242	C₂×SD₁₆	—	4	2	no	—
243	C₂²×C₈	—	2	1	no	—
244	C₂²×C₈	—	2	1	no	—
245	C₂²×C₈	—	2	1	no	—
246	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
247	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
248	C₄⋊Q₈	—	2	1	no	—
249	C₄⋊Q₈	—	2	1	no	—
250	C₄⋊Q₈	—	2	1	no	—
251	C₄⋊Q₈	D₄×C₆₇	1	2	yes	—
252	C₄×D₄	—	2	1	no	—
253	C₄×D₄	—	2	1	no	—
254	C₄×D₄	—	2	1	no	—
255	C₄×D₄	—	2	1	no	—
256	C₂×C₄⋊C₄	—	2	1	no	—
257	C₂×C₄⋊C₄	—	2	1	no	—
258	C₂×C₄⋊C₄	—	2	1	no	—
259	C₂×C₄⋊C₄	—	2	1	no	—
260	C₂×C₄⋊C₄	—	2	1	no	—
261	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
262	C₂×C₂²⋊C₄	—	2	1	no	—
263	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
264	C₄⋊C₈	—	2	1	no	—
265	C₄⋊C₈	—	2	1	no	—
266	C₄⋊C₈	—	2	1	no	—
267	C₄⋊C₈	—	2	1	no	—
268	C₄⋊C₈	—	2	1	no	—
269	C₄⋊C₈	D₄×C₆₇	1	2	yes	—
270	#103	—	2	4	no	—
271	#102	—	4	2	no	—
272	C₂²⋊C₈	—	4	2	no	—
273	C₂²⋊C₈	—	4	2	no	—
274	C₄×C₈	—	2	1	no	—
275	C₄×C₈	—	2	1	no	—
276	C₂⁴	—	2	1	no	—
277	C₂×Q₈	—	2	2	no	—
278	C₂×Q₈	—	4	1	no	—
279	C₂×Q₈	—	4	1	no	—
280	C₂×Q₈	—	4	1	no	—
281	C₂×Q₈	—	4	1	no	—
282	C₂×Q₈	—	4	1	no	—
283	C₂×D₄	—	2	1	no	—
284	C₂×D₄	—	2	1	no	—
285	C₂×D₄	—	2	1	no	—
286	C₂×D₄	—	2	1	no	—
287	C₂×D₄	—	2	1	no	—
288	C₂×D₄	D₈×C₆₇	1	2	yes	—
289	C₂²×C₄	—	2	1	no	—
290	C₂²×C₄	—	4	1	no	—
291	C₂²×C₄	—	4	1	no	—
292	C₂²×C₄	—	4	1	no	—
293	C₂²×C₄	—	4	1	no	—
294	C₂²×C₄	—	4	1	no	—
295	C₂²×C₄	—	4	1	no	—
296	C₂²×C₄	—	4	1	no	—
297	C₂²×C₄	—	4	1	no	—
298	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
299	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
300	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
301	SD₁₆	—	4	8	no	—
302	C₂×C₈	—	2	1	no	—
303	C₂×C₈	—	2	1	no	—
304	C₂×C₈	—	2	2	no	—
305	C₂×C₈	—	2	1	no	—
306	C₂×C₈	—	2	1	no	—
307	C₂×C₈	—	2	1	no	—
308	C₂×C₈	—	4	1	no	—
309	C₂×C₈	—	4	1	no	—
310	C₂×C₈	—	4	1	no	—
311	C₂×C₈	—	4	1	no	—
312	C₂×C₈	—	4	1	no	—
313	C₂×C₈	—	4	1	no	—
314	C₂×C₈	—	4	1	no	—
315	C₂×C₈	—	4	1	no	—
316	C₂×C₈	SD₁₆×C₆₇	1	2	yes	complemented
317	C₄⋊C₄	—	2	1	no	—
318	C₄⋊C₄	—	2	2	no	—
319	C₄⋊C₄	—	2	2	no	—
320	C₄⋊C₄	—	2	1	no	—
321	C₄⋊C₄	—	4	1	no	—
322	C₄⋊C₄	—	4	1	no	—
323	C₄⋊C₄	—	4	1	no	—
324	C₄⋊C₄	—	4	1	no	—
325	C₄⋊C₄	—	4	1	no	—
326	C₄⋊C₄	—	4	1	no	—
327	C₄⋊C₄	—	8	1	no	—
328	C₂²⋊C₄	—	4	1	no	—
329	C₂²⋊C₄	—	4	1	no	—
330	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
331	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
332	C₄²	#41×C₆₇	1	1	yes	characteristic
333	C₄²	#41×C₆₇	1	1	yes	characteristic
334	C₂³	—	2	2	no	—
335	C₂³	—	4	1	no	—
336	C₂³	—	4	1	no	—
337	C₂³	—	4	1	no	—
338	C₂³	—	4	1	no	—
339	C₂³	—	4	1	no	—
340	C₂³	—	4	1	no	—
341	C₂³	#121×C₆₇	1	1	yes	characteristic
342	Q₈	—	4	2	no	—
343	Q₈	—	4	2	no	—
344	Q₈	—	8	1	no	—
345	Q₈	—	8	1	no	—
346	D₄	—	2	2	no	—
347	D₄	—	2	2	no	—
348	D₄	—	4	1	no	—
349	D₄	—	4	1	no	—
350	C₂×C₄	—	2	1	no	—
351	C₂×C₄	—	2	1	no	—
352	C₂×C₄	—	4	1	no	—
353	C₂×C₄	—	4	1	no	—
354	C₂×C₄	—	4	1	no	—
355	C₂×C₄	—	4	2	no	—
356	C₂×C₄	—	8	1	no	—
357	C₂×C₄	—	8	1	no	—
358	C₂×C₄	—	8	1	no	—
359	C₂×C₄	—	8	1	no	—
360	C₂×C₄	—	8	1	no	—
361	C₂×C₄	—	8	1	no	—
362	C₂×C₄	—	8	1	no	—
363	C₂×C₄	—	8	1	no	—
364	C₂×C₄	—	8	1	no	—
365	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
366	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
367	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
368	C₂×C₄	#123×C₆₇	1	1	yes	characteristic
369	C₂×C₄	#123×C₆₇	1	1	yes	characteristic
370	C₂×C₄	#124×C₆₇	1	1	yes	characteristic
371	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
372	C₂×C₄	C₂×SD₁₆×C₆₇	1	1	yes	characteristic
373	C₂×C₄	#135×C₆₇	1	1	yes	characteristic
374	C₂×C₄	#135×C₆₇	1	1	yes	characteristic
375	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
376	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
377	C₂×C₄	#137×C₆₇	1	1	yes	characteristic
378	C₂×C₄	#137×C₆₇	1	1	yes	characteristic
379	C₈	—	2	2	no	—
380	C₈	—	4	1	no	—
381	C₈	—	4	2	no	—
382	C₈	—	8	1	no	—
383	C₈	—	8	1	no	—
384	C₂²	—	4	2	no	—
385	C₂²	—	4	2	no	—
386	C₂²	—	4	2	no	—
387	C₂²	—	8	1	no	—
388	C₂²	—	8	1	no	—
389	C₂²	—	8	1	no	—
390	C₂²	—	8	1	no	—
391	C₂²	#394×C₆₇	1	1	yes	characteristic
392	C₂²	#461×C₆₇	1	1	yes	characteristic
393	C₂²	#463×C₆₇	1	1	yes	characteristic
394	C₂²	#466×C₆₇	1	1	yes	characteristic
395	C₂²	#468×C₆₇	1	1	yes	characteristic
396	C₂²	#471×C₆₇	1	1	yes	characteristic
397	C₂²	#473×C₆₇	1	1	yes	characteristic
398	C₄	—	2	1	no	—
399	C₄	—	2	1	no	—
400	C₄	—	8	2	no	—
401	C₄	—	8	1	no	—
402	C₄	—	16	1	no	—
403	C₄	#447×C₆₇	1	1	yes	characteristic
404	C₄	#452×C₆₇	1	1	yes	characteristic
405	C₄	#460×C₆₇	1	1	yes	characteristic
406	C₄	#464×C₆₇	1	1	yes	characteristic
407	C₄	#465×C₆₇	1	1	yes	characteristic
408	C₄	#468×C₆₇	1	1	yes	characteristic
409	C₄	#480×C₆₇	1	1	yes	characteristic
410	C₄	#482×C₆₇	1	1	yes	characteristic
411	C₂	—	8	2	no	—
412	C₂	C₆₇×#1673	1	1	yes	characteristic
413	C₂	C₆₇×#1681	1	1	yes	characteristic
414	C₂	C₆₇×#1691	1	1	yes	characteristic
415	C₂	C₆₇×#1699	1	1	yes	characteristic
416	C₂	C₆₇×#2033	1	1	yes	characteristic
417	C₂	C₆₇×#2052	1	1	yes	characteristic
418	C₂	C₆₇×#2055	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	#121
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₆₇
prime radical	C₂²×D₄×C₆₇	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#1990	C₂	semidirect
5. C₂×#335×C₆₇	C₂	semidirect
6. C₂×#332×C₆₇	C₂	semidirect
7. C₆₇×#1296	C₂	semidirect
106. C₂×C₈×C₆₇	SD₁₆	semidirect
201. #10476	C₆₇	direct (N × H)
211. #1990	C₂×C₆₇	semidirect
214. C₂×#335	C₂×C₆₇	semidirect
215. C₂×#332	C₂×C₆₇	semidirect
216. #1296	C₂×C₆₇	semidirect
217. C₆₇	#10476	direct (N × H)
316. C₂×C₈	SD₁₆×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#461×C₆₇	2
C₆₇×#1990	2
C₆₇×#1954	2
C₆₇×#1802	2
C₂×#335×C₆₇	2
C₂×#332×C₆₇	2
C₆₇×#1296	2
#10476	67

Subgroups — C67×#10476 #30689727

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#10476 #30689727