Subgroups — C₆₇×#33165 #30712416

Order 17152.

1126 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#464×C₆₇	C₂	1	4	yes	maximal, complemented
2	C₂×#462×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#460×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#2858	C₂	1	1	yes	maximal, characteristic
5	C₄²×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1983	C₂	1	4	yes	maximal, complemented
7	C₆₇×#1982	C₂	1	1	yes	maximal, characteristic
8	C₆₇×#1981	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1842	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×Q₁₆×C₆₇	C₂²	1	2	yes	—
11	C₂²×SD₁₆×C₆₇	C₂²	1	2	yes	—
12	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×#124×C₆₇	C₂²	1	4	yes	—
15	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
17	#464×C₆₇	—	2	16	no	—
18	#462×C₆₇	—	2	4	no	—
19	#460×C₆₇	—	2	4	no	—
20	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₈×C₆₇	C₂²	1	4	yes	—
23	C₂×#103×C₆₇	C₂²	1	2	yes	—
24	C₂×#103×C₆₇	C₂²	1	4	yes	—
25	C₂×#102×C₆₇	C₂²	1	2	yes	—
26	C₂×#102×C₆₇	C₂²	1	4	yes	—
27	#390×C₆₇	—	2	1	no	—
28	#390×C₆₇	—	2	1	no	—
29	#390×C₆₇	—	2	2	no	—
30	C₄×C₄⋊C₄×C₆₇	—	2	1	no	—
31	C₄×C₄⋊C₄×C₆₇	—	2	2	no	—
32	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
33	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
34	C₄³×C₆₇	—	2	1	no	—
35	C₄³×C₆₇	C₂²	1	1	yes	characteristic
36	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
37	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
38	C₂²×D₄×C₆₇	—	2	1	no	—
39	C₂²×D₄×C₆₇	—	2	1	no	—
40	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
41	C₂×Q₁₆×C₆₇	—	2	4	no	—
42	C₂×Q₁₆×C₆₇	—	2	8	no	—
43	C₂×SD₁₆×C₆₇	—	2	4	no	—
44	C₂×SD₁₆×C₆₇	—	2	8	no	—
45	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
46	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
47	C₄⋊Q₈×C₆₇	—	2	2	no	—
48	C₄⋊Q₈×C₆₇	—	2	2	no	—
49	#127×C₆₇	—	2	2	no	—
50	#127×C₆₇	—	2	2	no	—
51	#124×C₆₇	—	2	8	no	—
52	#124×C₆₇	—	2	8	no	—
53	C₄×Q₈×C₆₇	—	2	2	no	—
54	C₄×Q₈×C₆₇	—	2	4	no	—
55	C₄×Q₈×C₆₇	D₄	1	4	yes	—
56	C₄×Q₈×C₆₇	D₄	1	8	yes	complemented
57	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
58	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
59	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
60	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
61	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
62	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
63	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
64	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
65	C₂×C₄²×C₆₇	—	2	1	no	—
66	C₂×C₄²×C₆₇	—	2	2	no	—
67	C₂×C₄²×C₆₇	—	2	2	no	—
68	C₂×C₄²×C₆₇	—	2	1	no	—
69	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
70	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
71	C₂×C₄²×C₆₇	D₄	1	2	yes	—
72	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
73	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
74	C₄⋊C₈×C₆₇	—	2	2	no	—
75	C₄⋊C₈×C₆₇	—	2	2	no	—
76	C₄⋊C₈×C₆₇	—	2	8	no	—
77	#103×C₆₇	—	2	4	no	—
78	#103×C₆₇	—	2	8	no	—
79	#102×C₆₇	—	2	4	no	—
80	#102×C₆₇	—	2	8	no	—
81	#95×C₆₇	—	2	2	no	—
82	#95×C₆₇	—	2	1	no	—
83	C₂⁴×C₆₇	—	2	2	no	—
84	C₂×Q₈×C₆₇	—	2	1	no	—
85	C₂×Q₈×C₆₇	—	2	2	no	—
86	C₂×Q₈×C₆₇	—	2	4	no	—
87	C₂×Q₈×C₆₇	—	2	2	no	—
88	C₂×Q₈×C₆₇	—	2	4	no	—
89	C₂×Q₈×C₆₇	—	2	2	no	—
90	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
91	C₂×Q₈×C₆₇	C₂×D₄	1	4	yes	—
92	C₂×D₄×C₆₇	—	2	4	no	—
93	C₂×D₄×C₆₇	—	2	2	no	—
94	C₂×D₄×C₆₇	—	2	4	no	—
95	C₂×D₄×C₆₇	—	2	2	no	—
96	C₂×D₄×C₆₇	—	4	4	no	—
97	C₂×D₄×C₆₇	—	4	2	no	—
98	C₂×D₄×C₆₇	—	4	2	no	—
99	C₂×D₄×C₆₇	—	4	2	no	—
100	C₂×D₄×C₆₇	—	4	2	no	—
101	C₂²×C₄×C₆₇	—	2	1	no	—
102	C₂²×C₄×C₆₇	—	2	2	no	—
103	C₂²×C₄×C₆₇	—	2	1	no	—
104	C₂²×C₄×C₆₇	—	2	2	no	—
105	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
106	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
107	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
108	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
110	Q₁₆×C₆₇	—	4	8	no	—
111	SD₁₆×C₆₇	—	4	8	no	—
112	C₂×C₈×C₆₇	—	2	2	no	—
113	C₂×C₈×C₆₇	—	2	4	no	—
114	C₂×C₈×C₆₇	—	2	4	no	—
115	C₂×C₈×C₆₇	—	2	2	no	—
116	C₄⋊C₄×C₆₇	—	2	4	no	—
117	C₄⋊C₄×C₆₇	—	2	8	no	—
118	C₄⋊C₄×C₆₇	—	2	2	no	—
119	C₄⋊C₄×C₆₇	—	2	1	no	—
120	C₄⋊C₄×C₆₇	—	4	2	no	—
121	C₄⋊C₄×C₆₇	—	4	2	no	—
122	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
123	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
124	C₂²⋊C₄×C₆₇	—	4	8	no	—
125	C₂²⋊C₄×C₆₇	—	4	8	no	—
126	C₄²×C₆₇	—	2	4	no	—
127	C₄²×C₆₇	—	2	8	no	—
128	C₄²×C₆₇	—	2	2	no	—
129	C₄²×C₆₇	—	2	2	no	—
130	C₄²×C₆₇	—	4	2	no	—
131	C₄²×C₆₇	—	4	4	no	—
132	C₄²×C₆₇	—	4	2	no	—
133	C₄²×C₆₇	C₂×D₄	1	2	yes	—
134	C₄²×C₆₇	C₂×D₄	1	4	yes	—
135	C₄²×C₆₇	#41	1	4	yes	—
136	C₄²×C₆₇	#41	1	2	yes	—
137	C₂³×C₆₇	—	4	4	no	—
138	C₂³×C₆₇	—	4	2	no	—
139	C₂³×C₆₇	—	4	2	no	—
140	C₂³×C₆₇	—	4	2	no	—
141	C₂³×C₆₇	—	4	4	no	—
142	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
143	Q₈×C₆₇	—	2	2	no	—
144	Q₈×C₆₇	—	2	4	no	—
145	Q₈×C₆₇	—	4	2	no	—
146	Q₈×C₆₇	—	4	4	no	—
147	Q₈×C₆₇	—	4	2	no	—
148	Q₈×C₆₇	#127	1	4	yes	—
149	D₄×C₆₇	—	4	2	no	—
150	D₄×C₆₇	—	4	4	no	—
151	D₄×C₆₇	—	4	2	no	—
152	D₄×C₆₇	—	8	4	no	—
153	D₄×C₆₇	—	8	4	no	—
154	C₂×C₄×C₆₇	—	2	2	no	—
155	C₂×C₄×C₆₇	—	2	4	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	—	2	2	no	—
160	C₂×C₄×C₆₇	—	2	2	no	—
161	C₂×C₄×C₆₇	—	2	4	no	—
162	C₂×C₄×C₆₇	—	2	2	no	—
163	C₂×C₄×C₆₇	—	4	2	no	—
164	C₂×C₄×C₆₇	—	4	2	no	—
165	C₂×C₄×C₆₇	—	4	2	no	—
166	C₂×C₄×C₆₇	—	4	4	no	—
167	C₂×C₄×C₆₇	—	4	2	no	—
168	C₂×C₄×C₆₇	#121	1	4	yes	—
169	C₂×C₄×C₆₇	#121	1	4	yes	—
170	C₂×C₄×C₆₇	#121	1	2	yes	—
171	C₂×C₄×C₆₇	#121	1	2	yes	—
172	C₂×C₄×C₆₇	#135	1	2	yes	—
173	C₂×C₄×C₆₇	#137	1	2	yes	—
174	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
175	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
176	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
177	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
178	C₈×C₆₇	—	4	2	no	—
179	C₈×C₆₇	—	4	2	no	—
180	C₂²×C₆₇	—	8	2	no	—
181	C₂²×C₆₇	—	8	2	no	—
182	C₂²×C₆₇	—	8	4	no	—
183	C₂²×C₆₇	—	8	4	no	—
184	C₂²×C₆₇	—	8	2	no	—
185	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
186	C₂²×C₆₇	C₂×#121	1	2	yes	—
187	C₂²×C₆₇	C₂×#135	1	1	yes	characteristic
188	C₂²×C₆₇	C₂×#137	1	1	yes	characteristic
189	C₂²×C₆₇	#575	1	2	yes	—
190	C₄×C₆₇	—	2	4	no	—
191	C₄×C₆₇	—	2	4	no	—
192	C₄×C₆₇	—	2	4	no	—
193	C₄×C₆₇	—	2	2	no	—
194	C₄×C₆₇	—	4	4	no	—
195	C₄×C₆₇	—	4	2	no	—
196	C₄×C₆₇	—	8	2	no	—
197	C₄×C₆₇	#464	1	4	yes	complemented
198	C₄×C₆₇	C₂×#127	1	1	yes	characteristic
199	C₄×C₆₇	#532	1	1	yes	characteristic
200	C₄×C₆₇	#532	1	2	yes	—
201	#33165	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
202	C₂×C₆₇	—	16	2	no	—
203	C₂×C₆₇	#2856	1	1	yes	characteristic
204	C₂×C₆₇	C₂×#464	1	1	yes	characteristic
205	C₂×C₆₇	#3035	1	2	yes	—
206	C₂×C₆₇	#3037	1	2	yes	—
207	C₂×C₆₇	#3040	1	1	yes	characteristic
208	C₂×#464	C₂×C₆₇	1	4	yes	complemented
209	C₂×#462	C₂×C₆₇	1	1	yes	characteristic, complemented
210	C₂×#460	C₂×C₆₇	1	1	yes	characteristic
211	#2858	C₂×C₆₇	1	1	yes	characteristic
212	C₄²×Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
213	#1983	C₂×C₆₇	1	4	yes	complemented
214	#1982	C₂×C₆₇	1	1	yes	characteristic
215	#1981	C₂×C₆₇	1	1	yes	characteristic, complemented
216	#1842	C₂×C₆₇	1	1	yes	characteristic, complemented
217	C₆₇	#33165	1	1	yes	characteristic, direct product factor, Sylow
218	C₂²×Q₁₆	C₂²×C₆₇	1	2	yes	—
219	C₂²×SD₁₆	C₂²×C₆₇	1	2	yes	—
220	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
221	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
222	C₂×#124	C₂²×C₆₇	1	4	yes	—
223	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
224	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
225	#464	—	2	16	no	—
226	#462	—	2	4	no	—
227	#460	—	2	4	no	—
228	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
229	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
230	C₂×C₄⋊C₈	C₂²×C₆₇	1	4	yes	—
231	C₂×#103	C₂²×C₆₇	1	4	yes	—
232	C₂×#103	C₂²×C₆₇	1	2	yes	—
233	C₂×#102	C₂²×C₆₇	1	4	yes	—
234	C₂×#102	C₂²×C₆₇	1	2	yes	—
235	#390	—	2	1	no	—
236	#390	—	2	1	no	—
237	#390	—	2	2	no	—
238	C₄×C₄⋊C₄	—	2	1	no	—
239	C₄×C₄⋊C₄	—	2	2	no	—
240	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
241	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
242	C₄³	—	2	1	no	—
243	C₄³	C₂²×C₆₇	1	1	yes	characteristic
244	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
245	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
246	C₂²×D₄	—	2	1	no	—
247	C₂²×D₄	—	2	1	no	—
248	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
249	C₂×Q₁₆	—	2	4	no	—
250	C₂×Q₁₆	—	2	8	no	—
251	C₂×SD₁₆	—	2	4	no	—
252	C₂×SD₁₆	—	2	8	no	—
253	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
254	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
255	C₄⋊Q₈	—	2	2	no	—
256	C₄⋊Q₈	—	2	2	no	—
257	#127	—	2	2	no	—
258	#127	—	2	2	no	—
259	#124	—	2	8	no	—
260	#124	—	2	8	no	—
261	C₄×Q₈	—	2	2	no	—
262	C₄×Q₈	—	2	4	no	—
263	C₄×Q₈	D₄×C₆₇	1	4	yes	—
264	C₄×Q₈	D₄×C₆₇	1	8	yes	complemented
265	C₂×C₄⋊C₄	—	2	1	no	—
266	C₂×C₄⋊C₄	—	2	2	no	—
267	C₂×C₄⋊C₄	—	2	1	no	—
268	C₂×C₄⋊C₄	—	2	1	no	—
269	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
270	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
271	C₂×C₂²⋊C₄	—	2	4	no	—
272	C₂×C₂²⋊C₄	—	2	4	no	—
273	C₂×C₄²	—	2	1	no	—
274	C₂×C₄²	—	2	2	no	—
275	C₂×C₄²	—	2	2	no	—
276	C₂×C₄²	—	2	1	no	—
277	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
278	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
279	C₂×C₄²	D₄×C₆₇	1	2	yes	—
280	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
281	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
282	C₄⋊C₈	—	2	2	no	—
283	C₄⋊C₈	—	2	2	no	—
284	C₄⋊C₈	—	2	8	no	—
285	#103	—	2	4	no	—
286	#103	—	2	8	no	—
287	#102	—	2	4	no	—
288	#102	—	2	8	no	—
289	#95	—	2	2	no	—
290	#95	—	2	1	no	—
291	C₂⁴	—	2	2	no	—
292	C₂×Q₈	—	2	1	no	—
293	C₂×Q₈	—	2	2	no	—
294	C₂×Q₈	—	2	4	no	—
295	C₂×Q₈	—	2	2	no	—
296	C₂×Q₈	—	2	4	no	—
297	C₂×Q₈	—	2	2	no	—
298	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
299	C₂×Q₈	C₂×D₄×C₆₇	1	4	yes	—
300	C₂×D₄	—	2	4	no	—
301	C₂×D₄	—	2	2	no	—
302	C₂×D₄	—	2	4	no	—
303	C₂×D₄	—	2	2	no	—
304	C₂×D₄	—	4	4	no	—
305	C₂×D₄	—	4	2	no	—
306	C₂×D₄	—	4	2	no	—
307	C₂×D₄	—	4	2	no	—
308	C₂×D₄	—	4	2	no	—
309	C₂²×C₄	—	2	1	no	—
310	C₂²×C₄	—	2	2	no	—
311	C₂²×C₄	—	2	1	no	—
312	C₂²×C₄	—	2	2	no	—
313	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
314	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
315	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
316	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
317	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
318	Q₁₆	—	4	8	no	—
319	SD₁₆	—	4	8	no	—
320	C₂×C₈	—	2	2	no	—
321	C₂×C₈	—	2	4	no	—
322	C₂×C₈	—	2	4	no	—
323	C₂×C₈	—	2	2	no	—
324	C₄⋊C₄	—	2	4	no	—
325	C₄⋊C₄	—	2	8	no	—
326	C₄⋊C₄	—	2	2	no	—
327	C₄⋊C₄	—	2	1	no	—
328	C₄⋊C₄	—	4	2	no	—
329	C₄⋊C₄	—	4	2	no	—
330	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
331	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
332	C₂²⋊C₄	—	4	8	no	—
333	C₂²⋊C₄	—	4	8	no	—
334	C₄²	—	2	4	no	—
335	C₄²	—	2	8	no	—
336	C₄²	—	2	2	no	—
337	C₄²	—	2	2	no	—
338	C₄²	—	4	2	no	—
339	C₄²	—	4	4	no	—
340	C₄²	—	4	2	no	—
341	C₄²	C₂×D₄×C₆₇	1	2	yes	—
342	C₄²	C₂×D₄×C₆₇	1	4	yes	—
343	C₄²	#41×C₆₇	1	4	yes	—
344	C₄²	#41×C₆₇	1	2	yes	—
345	C₂³	—	4	4	no	—
346	C₂³	—	4	2	no	—
347	C₂³	—	4	2	no	—
348	C₂³	—	4	2	no	—
349	C₂³	—	4	4	no	—
350	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
351	Q₈	—	2	2	no	—
352	Q₈	—	2	4	no	—
353	Q₈	—	4	2	no	—
354	Q₈	—	4	4	no	—
355	Q₈	—	4	2	no	—
356	Q₈	#127×C₆₇	1	4	yes	—
357	D₄	—	4	2	no	—
358	D₄	—	4	4	no	—
359	D₄	—	4	2	no	—
360	D₄	—	8	4	no	—
361	D₄	—	8	4	no	—
362	C₂×C₄	—	2	2	no	—
363	C₂×C₄	—	2	4	no	—
364	C₂×C₄	—	2	2	no	—
365	C₂×C₄	—	2	2	no	—
366	C₂×C₄	—	2	2	no	—
367	C₂×C₄	—	2	2	no	—
368	C₂×C₄	—	2	2	no	—
369	C₂×C₄	—	2	4	no	—
370	C₂×C₄	—	2	2	no	—
371	C₂×C₄	—	4	2	no	—
372	C₂×C₄	—	4	2	no	—
373	C₂×C₄	—	4	2	no	—
374	C₂×C₄	—	4	4	no	—
375	C₂×C₄	—	4	2	no	—
376	C₂×C₄	#121×C₆₇	1	4	yes	—
377	C₂×C₄	#121×C₆₇	1	4	yes	—
378	C₂×C₄	#121×C₆₇	1	2	yes	—
379	C₂×C₄	#121×C₆₇	1	2	yes	—
380	C₂×C₄	#135×C₆₇	1	2	yes	—
381	C₂×C₄	#137×C₆₇	1	2	yes	—
382	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
383	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
384	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
385	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
386	C₈	—	4	2	no	—
387	C₈	—	4	2	no	—
388	C₂²	—	8	2	no	—
389	C₂²	—	8	4	no	—
390	C₂²	—	8	4	no	—
391	C₂²	—	8	2	no	—
392	C₂²	—	8	2	no	—
393	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
394	C₂²	C₂×#121×C₆₇	1	2	yes	—
395	C₂²	C₂×#135×C₆₇	1	1	yes	characteristic
396	C₂²	C₂×#137×C₆₇	1	1	yes	characteristic
397	C₂²	#575×C₆₇	1	2	yes	—
398	C₄	—	2	4	no	—
399	C₄	—	2	4	no	—
400	C₄	—	2	2	no	—
401	C₄	—	2	4	no	—
402	C₄	—	4	4	no	—
403	C₄	—	4	2	no	—
404	C₄	—	8	2	no	—
405	C₄	#464×C₆₇	1	4	yes	complemented
406	C₄	C₂×#127×C₆₇	1	1	yes	characteristic
407	C₄	#532×C₆₇	1	1	yes	characteristic
408	C₄	#532×C₆₇	1	2	yes	—
409	C₂	—	16	2	no	—
410	C₂	C₆₇×#2856	1	1	yes	characteristic
411	C₂	C₂×#464×C₆₇	1	1	yes	characteristic
412	C₂	C₆₇×#3035	1	2	yes	—
413	C₂	C₆₇×#3037	1	2	yes	—
414	C₂	C₆₇×#3040	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂×#127×C₆₇	C₂²
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
1. C₂×#464×C₆₇	C₂	semidirect
2. C₂×#462×C₆₇	C₂	semidirect
5. C₄²×Q₈×C₆₇	C₂	semidirect
6. C₆₇×#1983	C₂	semidirect
8. C₆₇×#1981	C₂	semidirect
9. C₆₇×#1842	C₂	semidirect
56. C₄×Q₈×C₆₇	D₄	semidirect
197. C₄×C₆₇	#464	semidirect
201. #33165	C₆₇	direct (N × H)
208. C₂×#464	C₂×C₆₇	semidirect
209. C₂×#462	C₂×C₆₇	semidirect
212. C₄²×Q₈	C₂×C₆₇	semidirect
213. #1983	C₂×C₆₇	semidirect
215. #1981	C₂×C₆₇	semidirect
216. #1842	C₂×C₆₇	semidirect
217. C₆₇	#33165	direct (N × H)
264. C₄×Q₈	D₄×C₆₇	semidirect
405. C₄	#464×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#464×C₆₇	2
C₂×#462×C₆₇	2
C₂×#460×C₆₇	2
C₆₇×#2858	2
C₄²×Q₈×C₆₇	2
C₆₇×#1983	2
C₆₇×#1982	2
C₆₇×#1981	2
C₆₇×#1842	2
#33165	67

Subgroups — C67×#33165 #30712416

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33165 #30712416