Subgroups — C₆₇×#33032 #30712283

Order 17152.

974 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#474×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#461×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2862	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄²×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1988	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1981	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1928	C₂	1	4	yes	maximal, complemented
8	C₆₇×#1877	C₂	1	4	yes	maximal
9	C₆₇×#1831	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×SD₁₆×C₆₇	C₂²	1	2	yes	—
11	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
14	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
15	C₂²×C₄²×C₆₇	—	2	1	no	—
16	#474×C₆₇	—	2	4	no	—
17	#461×C₆₇	—	2	4	no	—
18	C₂×#107×C₆₇	C₂²	1	2	yes	—
19	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×#103×C₆₇	C₂²	1	2	yes	—
22	C₂×#103×C₆₇	C₂²	1	4	yes	—
23	C₂×#102×C₆₇	C₂²	1	2	yes	—
24	#391×C₆₇	—	2	1	no	—
25	#391×C₆₇	—	2	1	no	—
26	#391×C₆₇	C₂²	1	4	yes	—
27	#382×C₆₇	—	2	2	no	—
28	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
29	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
30	C₄×C₂²⋊C₄×C₆₇	—	2	2	no	—
31	C₄×C₂²⋊C₄×C₆₇	—	2	1	no	—
32	C₄³×C₆₇	C₂²	1	1	yes	characteristic
33	#340×C₆₇	C₂²	1	4	yes	—
34	#336×C₆₇	C₂²	1	4	yes	—
35	C₂²×Q₈×C₆₇	—	2	1	no	—
36	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
37	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
38	C₂³×C₄×C₆₇	—	2	2	no	—
39	C₂³×C₄×C₆₇	—	2	1	no	—
40	C₂×SD₁₆×C₆₇	—	2	4	no	—
41	C₂×SD₁₆×C₆₇	—	2	8	no	—
42	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
43	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
44	C₄⋊Q₈×C₆₇	—	2	2	no	—
45	C₄⋊Q₈×C₆₇	—	2	2	no	—
46	C₄⋊Q₈×C₆₇	—	2	2	no	—
47	C₄⋊Q₈×C₆₇	—	2	2	no	—
48	C₄×D₄×C₆₇	—	2	4	no	—
49	C₄×D₄×C₆₇	—	2	4	no	—
50	C₄×D₄×C₆₇	—	2	2	no	—
51	C₄×D₄×C₆₇	D₄	1	4	yes	—
52	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
53	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
54	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
55	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
56	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
57	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
58	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
59	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
60	C₂×C₄²×C₆₇	—	2	2	no	—
61	C₂×C₄²×C₆₇	—	2	1	no	—
62	C₂×C₄²×C₆₇	—	4	4	no	—
63	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
64	C₂×C₄²×C₆₇	D₄	1	2	yes	—
65	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
66	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
67	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
68	#107×C₆₇	—	2	4	no	—
69	C₄⋊C₈×C₆₇	—	2	2	no	—
70	C₄⋊C₈×C₆₇	—	2	2	no	—
71	#103×C₆₇	—	2	4	no	—
72	#103×C₆₇	—	2	8	no	—
73	#102×C₆₇	—	2	4	no	—
74	#95×C₆₇	—	2	2	no	—
75	#95×C₆₇	—	2	1	no	—
76	#95×C₆₇	—	2	4	no	—
77	#95×C₆₇	—	2	4	no	—
78	C₂⁴×C₆₇	—	2	1	no	—
79	C₂×Q₈×C₆₇	—	2	2	no	—
80	C₂×Q₈×C₆₇	—	2	4	no	—
81	C₂×Q₈×C₆₇	—	2	2	no	—
82	C₂×Q₈×C₆₇	—	2	4	no	—
83	C₂×Q₈×C₆₇	—	4	2	no	—
84	C₂×Q₈×C₆₇	—	4	2	no	—
85	C₂×Q₈×C₆₇	—	4	2	no	—
86	C₂×D₄×C₆₇	—	2	1	no	—
87	C₂×D₄×C₆₇	—	2	2	no	—
88	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
89	C₂×D₄×C₆₇	#41	1	4	yes	—
90	C₂²×C₄×C₆₇	—	2	8	no	—
91	C₂²×C₄×C₆₇	—	2	1	no	—
92	C₂²×C₄×C₆₇	—	2	2	no	—
93	C₂²×C₄×C₆₇	—	2	4	no	—
94	C₂²×C₄×C₆₇	—	2	2	no	—
95	C₂²×C₄×C₆₇	—	2	2	no	—
96	C₂²×C₄×C₆₇	—	4	4	no	—
97	C₂²×C₄×C₆₇	—	4	2	no	—
98	C₂²×C₄×C₆₇	—	4	2	no	—
99	C₂²×C₄×C₆₇	—	4	4	no	—
100	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
101	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
103	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
105	SD₁₆×C₆₇	—	4	8	no	—
106	C₂×C₈×C₆₇	—	2	2	no	—
107	C₂×C₈×C₆₇	—	2	2	no	—
108	C₂×C₈×C₆₇	—	2	4	no	—
109	C₂×C₈×C₆₇	—	2	4	no	—
110	C₄⋊C₄×C₆₇	—	2	1	no	—
111	C₄⋊C₄×C₆₇	—	2	2	no	—
112	C₄⋊C₄×C₆₇	—	2	2	no	—
113	C₄⋊C₄×C₆₇	—	2	2	no	—
114	C₄⋊C₄×C₆₇	—	4	2	no	—
115	C₄⋊C₄×C₆₇	—	4	2	no	—
116	C₄⋊C₄×C₆₇	—	4	2	no	—
117	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
118	C₄⋊C₄×C₆₇	#41	1	4	yes	—
119	C₂²⋊C₄×C₆₇	—	2	8	no	—
120	C₂²⋊C₄×C₆₇	—	2	4	no	—
121	C₄²×C₆₇	—	2	2	no	—
122	C₄²×C₆₇	—	2	2	no	—
123	C₄²×C₆₇	—	2	4	no	—
124	C₄²×C₆₇	—	2	2	no	—
125	C₄²×C₆₇	—	2	2	no	—
126	C₄²×C₆₇	—	4	4	no	—
127	C₄²×C₆₇	—	4	2	no	—
128	C₄²×C₆₇	C₂×D₄	1	2	yes	—
129	C₄²×C₆₇	#41	1	2	yes	—
130	C₂³×C₆₇	—	2	2	no	—
131	C₂³×C₆₇	—	2	4	no	—
132	C₂³×C₆₇	—	4	1	no	—
133	C₂³×C₆₇	—	4	1	no	—
134	C₂³×C₆₇	—	4	2	no	—
135	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
136	Q₈×C₆₇	—	4	2	no	—
137	Q₈×C₆₇	—	4	4	no	—
138	Q₈×C₆₇	—	4	2	no	—
139	Q₈×C₆₇	—	8	4	no	—
140	D₄×C₆₇	—	2	4	no	—
141	D₄×C₆₇	—	2	2	no	—
142	D₄×C₆₇	#124	1	4	yes	—
143	C₂×C₄×C₆₇	—	2	2	no	—
144	C₂×C₄×C₆₇	—	2	2	no	—
145	C₂×C₄×C₆₇	—	2	2	no	—
146	C₂×C₄×C₆₇	—	2	2	no	—
147	C₂×C₄×C₆₇	—	2	2	no	—
148	C₂×C₄×C₆₇	—	2	4	no	—
149	C₂×C₄×C₆₇	—	2	2	no	—
150	C₂×C₄×C₆₇	—	2	2	no	—
151	C₂×C₄×C₆₇	—	4	16	no	—
152	C₂×C₄×C₆₇	—	4	1	no	—
153	C₂×C₄×C₆₇	—	4	2	no	—
154	C₂×C₄×C₆₇	—	4	1	no	—
155	C₂×C₄×C₆₇	—	4	2	no	—
156	C₂×C₄×C₆₇	—	4	8	no	—
157	C₂×C₄×C₆₇	—	4	2	no	—
158	C₂×C₄×C₆₇	—	4	2	no	—
159	C₂×C₄×C₆₇	—	4	4	no	—
160	C₂×C₄×C₆₇	—	4	2	no	—
161	C₂×C₄×C₆₇	—	4	2	no	—
162	C₂×C₄×C₆₇	#121	1	2	yes	—
163	C₂×C₄×C₆₇	#121	1	2	yes	—
164	C₂×C₄×C₆₇	#135	1	2	yes	—
165	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
166	C₂×C₄×C₆₇	#137	1	4	yes	—
167	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
168	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
169	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
170	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
171	C₈×C₆₇	—	4	2	no	—
172	C₈×C₆₇	—	4	2	no	—
173	C₂²×C₆₇	—	2	4	no	—
174	C₂²×C₆₇	—	4	4	no	—
175	C₂²×C₆₇	—	4	4	no	—
176	C₂²×C₆₇	—	4	2	no	—
177	C₂²×C₆₇	—	4	2	no	—
178	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
179	C₂²×C₆₇	#525	1	2	yes	—
180	C₂²×C₆₇	C₂×#135	1	1	yes	characteristic
181	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
182	C₂²×C₆₇	C₂×#137	1	2	yes	—
183	C₄×C₆₇	—	2	4	no	—
184	C₄×C₆₇	—	2	2	no	—
185	C₄×C₆₇	—	4	2	no	—
186	C₄×C₆₇	—	4	4	no	—
187	C₄×C₆₇	—	4	2	no	—
188	C₄×C₆₇	—	8	2	no	—
189	C₄×C₆₇	—	8	2	no	—
190	C₄×C₆₇	#463	1	4	yes	—
191	C₄×C₆₇	C₂×#124	1	1	yes	characteristic
192	C₄×C₆₇	#529	1	2	yes	—
193	C₄×C₆₇	#532	1	1	yes	characteristic
194	#33032	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
195	C₂×C₆₇	—	4	4	no	—
196	C₂×C₆₇	#2850	1	1	yes	characteristic
197	C₂×C₆₇	C₂×#463	1	1	yes	characteristic
198	C₂×C₆₇	#3039	1	1	yes	characteristic
199	C₂×C₆₇	#3107	1	2	yes	—
200	C₂×C₆₇	#3110	1	2	yes	—
201	C₂×#474	C₂×C₆₇	1	1	yes	characteristic
202	C₂×#461	C₂×C₆₇	1	1	yes	characteristic
203	#2862	C₂×C₆₇	1	1	yes	characteristic, complemented
204	C₄²×D₄	C₂×C₆₇	1	1	yes	characteristic
205	#1988	C₂×C₆₇	1	1	yes	characteristic, complemented
206	#1981	C₂×C₆₇	1	1	yes	characteristic, complemented
207	#1928	C₂×C₆₇	1	4	yes	complemented
208	#1877	C₂×C₆₇	1	4	yes	—
209	#1831	C₂×C₆₇	1	1	yes	characteristic, complemented
210	C₆₇	#33032	1	1	yes	characteristic, direct product factor, Sylow
211	C₂²×SD₁₆	C₂²×C₆₇	1	2	yes	—
212	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
213	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
214	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
215	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
216	C₂²×C₄²	—	2	1	no	—
217	#474	—	2	4	no	—
218	#461	—	2	4	no	—
219	C₂×#107	C₂²×C₆₇	1	2	yes	—
220	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
221	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
222	C₂×#103	C₂²×C₆₇	1	4	yes	—
223	C₂×#103	C₂²×C₆₇	1	2	yes	—
224	C₂×#102	C₂²×C₆₇	1	2	yes	—
225	#391	—	2	1	no	—
226	#391	—	2	1	no	—
227	#391	C₂²×C₆₇	1	4	yes	—
228	#382	—	2	2	no	—
229	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
230	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
231	C₄×C₂²⋊C₄	—	2	2	no	—
232	C₄×C₂²⋊C₄	—	2	1	no	—
233	C₄³	C₂²×C₆₇	1	1	yes	characteristic
234	#340	C₂²×C₆₇	1	4	yes	—
235	#336	C₂²×C₆₇	1	4	yes	—
236	C₂²×Q₈	—	2	1	no	—
237	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
238	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
239	C₂³×C₄	—	2	2	no	—
240	C₂³×C₄	—	2	1	no	—
241	C₂×SD₁₆	—	2	4	no	—
242	C₂×SD₁₆	—	2	8	no	—
243	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
244	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
245	C₄⋊Q₈	—	2	2	no	—
246	C₄⋊Q₈	—	2	2	no	—
247	C₄⋊Q₈	—	2	2	no	—
248	C₄⋊Q₈	—	2	2	no	—
249	C₄×D₄	—	2	4	no	—
250	C₄×D₄	—	2	4	no	—
251	C₄×D₄	—	2	2	no	—
252	C₄×D₄	D₄×C₆₇	1	4	yes	—
253	C₂×C₄⋊C₄	—	2	1	no	—
254	C₂×C₄⋊C₄	—	2	1	no	—
255	C₂×C₄⋊C₄	—	2	1	no	—
256	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
257	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
258	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
259	C₂×C₂²⋊C₄	—	2	2	no	—
260	C₂×C₂²⋊C₄	—	2	1	no	—
261	C₂×C₄²	—	2	2	no	—
262	C₂×C₄²	—	2	1	no	—
263	C₂×C₄²	—	4	4	no	—
264	C₂×C₄²	D₄×C₆₇	1	2	yes	—
265	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
266	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
267	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
268	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
269	#107	—	2	4	no	—
270	C₄⋊C₈	—	2	2	no	—
271	C₄⋊C₈	—	2	2	no	—
272	#103	—	2	4	no	—
273	#103	—	2	8	no	—
274	#102	—	2	4	no	—
275	#95	—	2	2	no	—
276	#95	—	2	1	no	—
277	#95	—	2	4	no	—
278	#95	—	2	4	no	—
279	C₂⁴	—	2	1	no	—
280	C₂×Q₈	—	2	2	no	—
281	C₂×Q₈	—	2	4	no	—
282	C₂×Q₈	—	2	2	no	—
283	C₂×Q₈	—	2	4	no	—
284	C₂×Q₈	—	4	2	no	—
285	C₂×Q₈	—	4	2	no	—
286	C₂×Q₈	—	4	2	no	—
287	C₂×D₄	—	2	1	no	—
288	C₂×D₄	—	2	2	no	—
289	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
290	C₂×D₄	#41×C₆₇	1	4	yes	—
291	C₂²×C₄	—	2	8	no	—
292	C₂²×C₄	—	2	1	no	—
293	C₂²×C₄	—	2	2	no	—
294	C₂²×C₄	—	2	4	no	—
295	C₂²×C₄	—	2	2	no	—
296	C₂²×C₄	—	2	2	no	—
297	C₂²×C₄	—	4	4	no	—
298	C₂²×C₄	—	4	2	no	—
299	C₂²×C₄	—	4	2	no	—
300	C₂²×C₄	—	4	4	no	—
301	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
302	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
303	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
304	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
305	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
306	SD₁₆	—	4	8	no	—
307	C₂×C₈	—	2	2	no	—
308	C₂×C₈	—	2	2	no	—
309	C₂×C₈	—	2	4	no	—
310	C₂×C₈	—	2	4	no	—
311	C₄⋊C₄	—	2	1	no	—
312	C₄⋊C₄	—	2	2	no	—
313	C₄⋊C₄	—	2	2	no	—
314	C₄⋊C₄	—	2	2	no	—
315	C₄⋊C₄	—	4	2	no	—
316	C₄⋊C₄	—	4	2	no	—
317	C₄⋊C₄	—	4	2	no	—
318	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
319	C₄⋊C₄	#41×C₆₇	1	4	yes	—
320	C₂²⋊C₄	—	2	8	no	—
321	C₂²⋊C₄	—	2	4	no	—
322	C₄²	—	2	2	no	—
323	C₄²	—	2	2	no	—
324	C₄²	—	2	4	no	—
325	C₄²	—	2	2	no	—
326	C₄²	—	2	2	no	—
327	C₄²	—	4	4	no	—
328	C₄²	—	4	2	no	—
329	C₄²	C₂×D₄×C₆₇	1	2	yes	—
330	C₄²	#41×C₆₇	1	2	yes	—
331	C₂³	—	2	2	no	—
332	C₂³	—	2	4	no	—
333	C₂³	—	4	1	no	—
334	C₂³	—	4	1	no	—
335	C₂³	—	4	2	no	—
336	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
337	Q₈	—	4	2	no	—
338	Q₈	—	4	4	no	—
339	Q₈	—	4	2	no	—
340	Q₈	—	8	4	no	—
341	D₄	—	2	4	no	—
342	D₄	—	2	2	no	—
343	D₄	#124×C₆₇	1	4	yes	—
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	2	2	no	—
346	C₂×C₄	—	2	2	no	—
347	C₂×C₄	—	2	2	no	—
348	C₂×C₄	—	2	2	no	—
349	C₂×C₄	—	2	4	no	—
350	C₂×C₄	—	2	2	no	—
351	C₂×C₄	—	2	2	no	—
352	C₂×C₄	—	4	16	no	—
353	C₂×C₄	—	4	1	no	—
354	C₂×C₄	—	4	1	no	—
355	C₂×C₄	—	4	2	no	—
356	C₂×C₄	—	4	2	no	—
357	C₂×C₄	—	4	8	no	—
358	C₂×C₄	—	4	2	no	—
359	C₂×C₄	—	4	2	no	—
360	C₂×C₄	—	4	4	no	—
361	C₂×C₄	—	4	2	no	—
362	C₂×C₄	—	4	2	no	—
363	C₂×C₄	#121×C₆₇	1	2	yes	—
364	C₂×C₄	#121×C₆₇	1	2	yes	—
365	C₂×C₄	#135×C₆₇	1	2	yes	—
366	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
367	C₂×C₄	#137×C₆₇	1	4	yes	—
368	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
369	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
370	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
371	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
372	C₈	—	4	2	no	—
373	C₈	—	4	2	no	—
374	C₂²	—	2	4	no	—
375	C₂²	—	4	2	no	—
376	C₂²	—	4	2	no	—
377	C₂²	—	4	4	no	—
378	C₂²	—	4	4	no	—
379	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
380	C₂²	#525×C₆₇	1	2	yes	—
381	C₂²	C₂×#135×C₆₇	1	1	yes	characteristic
382	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
383	C₂²	C₂×#137×C₆₇	1	2	yes	—
384	C₄	—	2	2	no	—
385	C₄	—	2	4	no	—
386	C₄	—	4	2	no	—
387	C₄	—	4	2	no	—
388	C₄	—	4	4	no	—
389	C₄	—	8	2	no	—
390	C₄	—	8	2	no	—
391	C₄	#463×C₆₇	1	4	yes	—
392	C₄	C₂×#124×C₆₇	1	1	yes	characteristic
393	C₄	#529×C₆₇	1	2	yes	—
394	C₄	#532×C₆₇	1	1	yes	characteristic
395	C₂	—	4	4	no	—
396	C₂	C₆₇×#2850	1	1	yes	characteristic
397	C₂	C₂×#463×C₆₇	1	1	yes	characteristic
398	C₂	C₆₇×#3039	1	1	yes	characteristic
399	C₂	C₆₇×#3107	1	2	yes	—
400	C₂	C₆₇×#3110	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂²×D₄×C₆₇	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₄³×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₆₇×#2862	C₂	semidirect
5. C₆₇×#1988	C₂	semidirect
6. C₆₇×#1981	C₂	semidirect
7. C₆₇×#1928	C₂	semidirect
9. C₆₇×#1831	C₂	semidirect
194. #33032	C₆₇	direct (N × H)
203. #2862	C₂×C₆₇	semidirect
205. #1988	C₂×C₆₇	semidirect
206. #1981	C₂×C₆₇	semidirect
207. #1928	C₂×C₆₇	semidirect
209. #1831	C₂×C₆₇	semidirect
210. C₆₇	#33032	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#474×C₆₇	2
C₂×#461×C₆₇	2
C₆₇×#2862	2
C₄²×D₄×C₆₇	2
C₆₇×#1988	2
C₆₇×#1981	2
C₆₇×#1928	2
C₆₇×#1877	2
C₆₇×#1831	2
#33032	67

Subgroups — C67×#33032 #30712283

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33032 #30712283