Subgroups — C₆₇×#17990 #30697241

Order 17152.

584 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	C₆₇×#3167	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2985	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2981	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1588	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1585	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1583	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1582	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1568	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1497	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#1496	C₂	1	1	yes	maximal, characteristic
11	C₆₇×#1452	C₂	1	1	yes	maximal, characteristic, complemented
12	#533×C₆₇	C₂²	1	1	yes	characteristic
13	#532×C₆₇	C₂²	1	1	yes	characteristic
14	#498×C₆₇	C₄	1	1	yes	characteristic
15	#492×C₆₇	C₄	1	1	yes	characteristic
16	#487×C₆₇	C₄	1	1	yes	characteristic
17	#486×C₆₇	C₄	1	1	yes	characteristic
18	#465×C₆₇	C₄	1	2	yes	—
19	#465×C₆₇	C₄	1	2	yes	—
20	#446×C₆₇	—	2	2	no	—
21	#446×C₆₇	C₂²	1	2	yes	—
22	C₈×Q₈×C₆₇	C₂²	1	2	yes	—
23	C₄×SD₁₆×C₆₇	C₂²	1	2	yes	—
24	C₄×SD₁₆×C₆₇	C₂²	1	2	yes	—
25	#436×C₆₇	C₂²	1	2	yes	—
26	#435×C₆₇	—	2	2	no	—
27	C₈×D₄×C₆₇	C₂²	1	2	yes	—
28	#433×C₆₇	—	2	2	no	—
29	#432×C₆₇	C₂²	1	1	yes	characteristic
30	#432×C₆₇	C₂²	1	1	yes	characteristic
31	#431×C₆₇	C₂²	1	1	yes	characteristic
32	#431×C₆₇	C₂²	1	1	yes	characteristic
33	#408×C₆₇	—	2	2	no	—
34	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
35	#335×C₆₇	C₂²	1	2	yes	—
36	#334×C₆₇	C₂²	1	2	yes	—
37	Q₈⋊C₈×C₆₇	C₂²	1	2	yes	—
38	Q₈⋊C₈×C₆₇	C₂²	1	2	yes	—
39	D₄⋊C₈×C₆₇	C₂²	1	2	yes	—
40	D₄⋊C₈×C₆₇	C₂²	1	2	yes	—
41	#322×C₆₇	C₂²	1	1	yes	characteristic
42	#322×C₆₇	C₂²	1	1	yes	characteristic
43	#322×C₆₇	C₂²	1	1	yes	characteristic
44	#322×C₆₇	C₂²	1	1	yes	characteristic
45	C₂×#41×C₆₇	—	2	1	no	—
46	C₂×SD₁₆×C₆₇	C₂×C₄	1	2	yes	—
47	C₂×SD₁₆×C₆₇	C₂×C₄	1	2	yes	—
48	C₂×M₁₆×C₆₇	—	2	2	no	—
49	C₂²×C₈×C₆₇	—	2	2	no	—
50	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
51	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
52	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
53	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
54	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
55	#127×C₆₇	C₂×C₄	1	1	yes	characteristic
56	#125×C₆₇	—	2	1	no	—
57	#124×C₆₇	—	2	1	no	—
58	C₂²⋊Q₈×C₆₇	—	2	1	no	—
59	C₂²⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
60	#121×C₆₇	—	2	1	no	—
61	#121×C₆₇	C₂×C₄	1	2	yes	—
62	C₄×Q₈×C₆₇	—	2	1	no	—
63	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
64	C₄×D₄×C₆₇	—	2	1	no	—
65	C₄×D₄×C₆₇	C₂³	1	2	yes	—
66	#117×C₆₇	—	2	1	no	—
67	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
68	#106×C₆₇	C₂×C₄	1	2	yes	—
69	#106×C₆₇	C₂×C₄	1	2	yes	—
70	C₄⋊C₈×C₆₇	—	2	2	no	—
71	C₄⋊C₈×C₆₇	—	2	2	no	—
72	C₄⋊C₈×C₆₇	—	2	1	no	—
73	C₄⋊C₈×C₆₇	D₄	1	2	yes	—
74	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
75	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
76	#103×C₆₇	C₂×C₄	1	2	yes	—
77	#103×C₆₇	C₂×C₄	1	2	yes	—
78	#102×C₆₇	C₂×C₄	1	2	yes	—
79	#102×C₆₇	C₂×C₄	1	2	yes	—
80	C₂²⋊C₈×C₆₇	—	2	1	no	—
81	C₂²⋊C₈×C₆₇	—	2	1	no	—
82	C₂²⋊C₈×C₆₇	—	2	1	no	—
83	C₂²⋊C₈×C₆₇	—	2	2	no	—
84	C₂²⋊C₈×C₆₇	—	2	2	no	—
85	C₂²⋊C₈×C₆₇	D₄	1	2	yes	—
86	#97×C₆₇	—	2	2	no	—
87	#97×C₆₇	—	2	1	no	—
88	#97×C₆₇	—	2	1	no	—
89	C₄×C₈×C₆₇	—	2	2	no	—
90	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
91	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
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	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
96	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
97	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
98	#41×C₆₇	—	8	2	no	—
99	C₂×Q₈×C₆₇	—	2	1	no	—
100	C₂×Q₈×C₆₇	—	2	1	no	—
101	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
102	C₂×D₄×C₆₇	—	2	2	no	—
103	C₂×D₄×C₆₇	—	2	1	no	—
104	C₂×D₄×C₆₇	—	2	1	no	—
105	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
106	C₂²×C₄×C₆₇	—	2	2	no	—
107	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
108	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
110	SD₁₆×C₆₇	—	4	2	no	—
111	SD₁₆×C₆₇	—	4	2	no	—
112	M₁₆×C₆₇	—	8	2	no	—
113	C₂×C₈×C₆₇	—	2	2	no	—
114	C₂×C₈×C₆₇	—	2	2	no	—
115	C₂×C₈×C₆₇	—	2	1	no	—
116	C₂×C₈×C₆₇	—	2	1	no	—
117	C₂×C₈×C₆₇	—	4	1	no	—
118	C₂×C₈×C₆₇	—	4	1	no	—
119	C₂×C₈×C₆₇	—	8	2	no	—
120	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
121	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
122	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
123	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
124	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
125	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
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	2	no	—
131	C₄⋊C₄×C₆₇	—	2	2	no	—
132	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
133	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
134	C₂²⋊C₄×C₆₇	—	2	2	no	—
135	C₂²⋊C₄×C₆₇	—	2	2	no	—
136	C₂²⋊C₄×C₆₇	—	2	2	no	—
137	C₄²×C₆₇	—	2	2	no	—
138	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
139	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
140	C₄²×C₆₇	#41	1	1	yes	characteristic
141	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
142	C₂³×C₆₇	—	2	2	no	—
143	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
144	Q₈×C₆₇	—	2	2	no	—
145	Q₈×C₆₇	—	2	2	no	—
146	Q₈×C₆₇	—	8	1	no	—
147	Q₈×C₆₇	—	8	1	no	—
148	D₄×C₆₇	—	2	2	no	—
149	D₄×C₆₇	—	2	2	no	—
150	D₄×C₆₇	—	8	1	no	—
151	D₄×C₆₇	—	8	1	no	—
152	D₄×C₆₇	—	8	2	no	—
153	C₂×C₄×C₆₇	—	2	1	no	—
154	C₂×C₄×C₆₇	—	2	1	no	—
155	C₂×C₄×C₆₇	—	2	2	no	—
156	C₂×C₄×C₆₇	—	2	2	no	—
157	C₂×C₄×C₆₇	—	2	2	no	—
158	C₂×C₄×C₆₇	—	2	2	no	—
159	C₂×C₄×C₆₇	—	4	1	no	—
160	C₂×C₄×C₆₇	—	4	2	no	—
161	C₂×C₄×C₆₇	—	4	2	no	—
162	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
163	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
164	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
165	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
166	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
167	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
168	C₈×C₆₇	—	2	1	no	—
169	C₈×C₆₇	—	2	1	no	—
170	C₈×C₆₇	—	2	1	no	—
171	C₈×C₆₇	—	2	1	no	—
172	C₈×C₆₇	—	2	2	no	—
173	C₈×C₆₇	—	2	2	no	—
174	C₈×C₆₇	—	4	1	no	—
175	C₈×C₆₇	—	4	1	no	—
176	C₈×C₆₇	—	4	2	no	—
177	C₈×C₆₇	—	4	2	no	—
178	C₂²×C₆₇	—	2	1	no	—
179	C₂²×C₆₇	—	2	1	no	—
180	C₂²×C₆₇	—	2	1	no	—
181	C₂²×C₆₇	—	4	2	no	—
182	C₂²×C₆₇	—	4	2	no	—
183	C₂²×C₆₇	—	4	2	no	—
184	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
185	C₄×C₆₇	—	2	1	no	—
186	C₄×C₆₇	—	2	1	no	—
187	C₄×C₆₇	—	2	1	no	—
188	C₄×C₆₇	—	2	1	no	—
189	C₄×C₆₇	—	4	2	no	—
190	C₄×C₆₇	—	4	2	no	—
191	C₄×C₆₇	—	4	2	no	—
192	C₄×C₆₇	#568	1	1	yes	characteristic
193	C₄×C₆₇	#568	1	1	yes	characteristic
194	C₄×C₆₇	#577	1	2	yes	—
195	#17990	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
196	C₂×C₆₇	—	4	1	no	—
197	C₂×C₆₇	—	8	2	no	—
198	C₂×C₆₇	#2932	1	1	yes	characteristic
199	C₂×C₆₇	#2949	1	1	yes	characteristic
200	C₂×C₆₇	#2957	1	1	yes	characteristic
201	#3167	C₂×C₆₇	1	1	yes	characteristic
202	#2985	C₂×C₆₇	1	1	yes	characteristic, complemented
203	#2981	C₂×C₆₇	1	1	yes	characteristic
204	#1588	C₂×C₆₇	1	2	yes	complemented
205	#1585	C₂×C₆₇	1	2	yes	complemented
206	#1583	C₂×C₆₇	1	2	yes	complemented
207	#1582	C₂×C₆₇	1	2	yes	complemented
208	#1568	C₂×C₆₇	1	1	yes	characteristic, complemented
209	#1497	C₂×C₆₇	1	1	yes	characteristic, complemented
210	#1496	C₂×C₆₇	1	1	yes	characteristic
211	#1452	C₂×C₆₇	1	1	yes	characteristic, complemented
212	C₆₇	#17990	1	1	yes	characteristic, direct product factor, Sylow
213	#533	C₂²×C₆₇	1	1	yes	characteristic
214	#532	C₂²×C₆₇	1	1	yes	characteristic
215	#498	C₄×C₆₇	1	1	yes	characteristic
216	#492	C₄×C₆₇	1	1	yes	characteristic
217	#487	C₄×C₆₇	1	1	yes	characteristic
218	#486	C₄×C₆₇	1	1	yes	characteristic
219	#465	C₄×C₆₇	1	2	yes	—
220	#465	C₄×C₆₇	1	2	yes	—
221	#446	—	2	2	no	—
222	#446	C₂²×C₆₇	1	2	yes	—
223	C₈×Q₈	C₂²×C₆₇	1	2	yes	—
224	C₄×SD₁₆	C₂²×C₆₇	1	2	yes	—
225	C₄×SD₁₆	C₂²×C₆₇	1	2	yes	—
226	#436	C₂²×C₆₇	1	2	yes	—
227	#435	—	2	2	no	—
228	C₈×D₄	C₂²×C₆₇	1	2	yes	—
229	#433	—	2	2	no	—
230	#432	C₂²×C₆₇	1	1	yes	characteristic
231	#432	C₂²×C₆₇	1	1	yes	characteristic
232	#431	C₂²×C₆₇	1	1	yes	characteristic
233	#431	C₂²×C₆₇	1	1	yes	characteristic
234	#408	—	2	2	no	—
235	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
236	#335	C₂²×C₆₇	1	2	yes	—
237	#334	C₂²×C₆₇	1	2	yes	—
238	Q₈⋊C₈	C₂²×C₆₇	1	2	yes	—
239	Q₈⋊C₈	C₂²×C₆₇	1	2	yes	—
240	D₄⋊C₈	C₂²×C₆₇	1	2	yes	—
241	D₄⋊C₈	C₂²×C₆₇	1	2	yes	—
242	#322	C₂²×C₆₇	1	1	yes	characteristic
243	#322	C₂²×C₆₇	1	1	yes	characteristic
244	#322	C₂²×C₆₇	1	1	yes	characteristic
245	#322	C₂²×C₆₇	1	1	yes	characteristic
246	C₂×#41	—	2	1	no	—
247	C₂×SD₁₆	C₂×C₄×C₆₇	1	2	yes	—
248	C₂×SD₁₆	C₂×C₄×C₆₇	1	2	yes	—
249	C₂×M₁₆	—	2	2	no	—
250	C₂²×C₈	—	2	2	no	—
251	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
252	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
253	C₄⋊Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
254	C₄⋊Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
255	C₄⋊Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
256	#127	C₂×C₄×C₆₇	1	1	yes	characteristic
257	#125	—	2	1	no	—
258	#124	—	2	1	no	—
259	C₂²⋊Q₈	—	2	1	no	—
260	C₂²⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
261	#121	—	2	1	no	—
262	#121	C₂×C₄×C₆₇	1	2	yes	—
263	C₄×Q₈	—	2	1	no	—
264	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
265	C₄×D₄	—	2	1	no	—
266	C₄×D₄	C₂³×C₆₇	1	2	yes	—
267	#117	—	2	1	no	—
268	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
269	#106	C₂×C₄×C₆₇	1	2	yes	—
270	#106	C₂×C₄×C₆₇	1	2	yes	—
271	C₄⋊C₈	—	2	2	no	—
272	C₄⋊C₈	—	2	2	no	—
273	C₄⋊C₈	—	2	1	no	—
274	C₄⋊C₈	D₄×C₆₇	1	2	yes	—
275	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
276	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
277	#103	C₂×C₄×C₆₇	1	2	yes	—
278	#103	C₂×C₄×C₆₇	1	2	yes	—
279	#102	C₂×C₄×C₆₇	1	2	yes	—
280	#102	C₂×C₄×C₆₇	1	2	yes	—
281	C₂²⋊C₈	—	2	1	no	—
282	C₂²⋊C₈	—	2	1	no	—
283	C₂²⋊C₈	—	2	1	no	—
284	C₂²⋊C₈	—	2	2	no	—
285	C₂²⋊C₈	—	2	2	no	—
286	C₂²⋊C₈	D₄×C₆₇	1	2	yes	—
287	#97	—	2	2	no	—
288	#97	—	2	1	no	—
289	#97	—	2	1	no	—
290	C₄×C₈	—	2	2	no	—
291	C₄×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
292	C₄×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
293	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
294	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
295	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
296	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
297	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
298	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
299	#41	—	8	2	no	—
300	C₂×Q₈	—	2	1	no	—
301	C₂×Q₈	—	2	1	no	—
302	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
303	C₂×D₄	—	2	2	no	—
304	C₂×D₄	—	2	1	no	—
305	C₂×D₄	—	2	1	no	—
306	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
307	C₂²×C₄	—	2	2	no	—
308	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
309	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
310	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
311	SD₁₆	—	4	2	no	—
312	SD₁₆	—	4	2	no	—
313	M₁₆	—	8	2	no	—
314	C₂×C₈	—	2	2	no	—
315	C₂×C₈	—	2	2	no	—
316	C₂×C₈	—	2	1	no	—
317	C₂×C₈	—	2	1	no	—
318	C₂×C₈	—	4	1	no	—
319	C₂×C₈	—	4	1	no	—
320	C₂×C₈	—	8	2	no	—
321	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
322	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
323	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
324	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
325	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
326	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
327	C₄⋊C₄	—	2	1	no	—
328	C₄⋊C₄	—	2	1	no	—
329	C₄⋊C₄	—	2	1	no	—
330	C₄⋊C₄	—	2	1	no	—
331	C₄⋊C₄	—	2	2	no	—
332	C₄⋊C₄	—	2	2	no	—
333	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
334	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
335	C₂²⋊C₄	—	2	2	no	—
336	C₂²⋊C₄	—	2	2	no	—
337	C₂²⋊C₄	—	2	2	no	—
338	C₄²	—	2	2	no	—
339	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
340	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
341	C₄²	#41×C₆₇	1	1	yes	characteristic
342	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
343	C₂³	—	2	2	no	—
344	C₂³	C₄×D₄×C₆₇	1	1	yes	characteristic
345	Q₈	—	2	2	no	—
346	Q₈	—	2	2	no	—
347	Q₈	—	8	1	no	—
348	Q₈	—	8	1	no	—
349	D₄	—	2	2	no	—
350	D₄	—	2	2	no	—
351	D₄	—	8	1	no	—
352	D₄	—	8	1	no	—
353	D₄	—	8	2	no	—
354	C₂×C₄	—	2	1	no	—
355	C₂×C₄	—	2	1	no	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	2	2	no	—
358	C₂×C₄	—	2	2	no	—
359	C₂×C₄	—	2	2	no	—
360	C₂×C₄	—	4	1	no	—
361	C₂×C₄	—	4	2	no	—
362	C₂×C₄	—	4	2	no	—
363	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
364	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
365	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
366	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
367	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
368	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
369	C₈	—	2	1	no	—
370	C₈	—	2	1	no	—
371	C₈	—	2	1	no	—
372	C₈	—	2	1	no	—
373	C₈	—	2	2	no	—
374	C₈	—	2	2	no	—
375	C₈	—	4	1	no	—
376	C₈	—	4	1	no	—
377	C₈	—	4	2	no	—
378	C₈	—	4	2	no	—
379	C₂²	—	2	1	no	—
380	C₂²	—	2	1	no	—
381	C₂²	—	2	1	no	—
382	C₂²	—	4	2	no	—
383	C₂²	—	4	2	no	—
384	C₂²	—	4	2	no	—
385	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
386	C₄	—	2	1	no	—
387	C₄	—	2	1	no	—
388	C₄	—	2	1	no	—
389	C₄	—	2	1	no	—
390	C₄	—	4	2	no	—
391	C₄	—	4	2	no	—
392	C₄	—	4	2	no	—
393	C₄	#568×C₆₇	1	1	yes	characteristic
394	C₄	#568×C₆₇	1	1	yes	characteristic
395	C₄	#577×C₆₇	1	2	yes	—
396	C₂	—	4	1	no	—
397	C₂	—	8	2	no	—
398	C₂	C₆₇×#2932	1	1	yes	characteristic
399	C₂	C₆₇×#2949	1	1	yes	characteristic
400	C₂	C₆₇×#2957	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₄²	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	#532×C₆₇	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ #431×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#2985	C₂	semidirect
4. C₆₇×#1588	C₂	semidirect
5. C₆₇×#1585	C₂	semidirect
6. C₆₇×#1583	C₂	semidirect
7. C₆₇×#1582	C₂	semidirect
8. C₆₇×#1568	C₂	semidirect
9. C₆₇×#1497	C₂	semidirect
11. C₆₇×#1452	C₂	semidirect
195. #17990	C₆₇	direct (N × H)
202. #2985	C₂×C₆₇	semidirect
204. #1588	C₂×C₆₇	semidirect
205. #1585	C₂×C₆₇	semidirect
206. #1583	C₂×C₆₇	semidirect
207. #1582	C₂×C₆₇	semidirect
208. #1568	C₂×C₆₇	semidirect
209. #1497	C₂×C₆₇	semidirect
211. #1452	C₂×C₆₇	semidirect
212. C₆₇	#17990	direct (N × H)

Maximal subgroups

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

subgroup	index
C₆₇×#3167	2
C₆₇×#2985	2
C₆₇×#2981	2
C₆₇×#1588	2
C₆₇×#1585	2
C₆₇×#1583	2
C₆₇×#1582	2
C₆₇×#1568	2
C₆₇×#1497	2
C₆₇×#1496	2
C₆₇×#1452	2
#17990	67

Subgroups — C67×#17990 #30697241

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#17990 #30697241