Subgroups — C₆₇×#60639 #30739890

Order 17152.

3314 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#575×C₆₇	C₂	1	4	yes	maximal, complemented
2	C₂×#532×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₆₇×#3153	C₂	1	8	yes	maximal, complemented
4	C₆₇×#3152	C₂	1	4	yes	maximal, complemented
5	C₆₇×#3151	C₂	1	4	yes	maximal, complemented
6	C₆₇×#3150	C₂	1	4	yes	maximal, complemented
7	C₆₇×#3147	C₂	1	4	yes	maximal, complemented
8	C₂×C₄×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#41×C₆₇	C₂²	1	4	yes	—
10	#575×C₆₇	—	2	32	no	—
11	C₂×#137×C₆₇	C₂²	1	16	yes	complemented
12	C₂×AGL(1,ℤ/8)×C₆₇	C₂²	1	16	yes	complemented
13	C₂×#135×C₆₇	C₂²	1	16	yes	complemented
14	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
15	#532×C₆₇	—	2	4	no	—
16	#532×C₆₇	—	2	4	no	—
17	#532×C₆₇	C₂²	1	4	yes	—
18	#532×C₆₇	C₂²	1	4	yes	—
19	#532×C₆₇	C₂²	1	4	yes	complemented
20	#532×C₆₇	C₂²	1	4	yes	complemented
21	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	complemented
22	C₂×#127×C₆₇	C₂²	1	2	yes	complemented
23	C₂×#124×C₆₇	C₂²	1	4	yes	complemented
24	C₂×#121×C₆₇	—	2	4	no	—
25	C₂×C₄×D₄×C₆₇	—	2	4	no	—
26	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
27	#497×C₆₇	C₂²	1	16	yes	complemented
28	#496×C₆₇	C₂²	1	16	yes	complemented
29	#495×C₆₇	C₂²	1	16	yes	complemented
30	#494×C₆₇	C₂²	1	4	yes	complemented
31	#493×C₆₇	C₂²	1	4	yes	complemented
32	#492×C₆₇	C₂²	1	8	yes	complemented
33	C₄×M₁₆×C₆₇	C₂²	1	4	yes	complemented
34	C₄×M₁₆×C₆₇	C₂²	1	4	yes	complemented
35	C₂×#97×C₆₇	C₂²	1	2	yes	complemented
36	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	complemented
37	C₂×#41×C₆₇	—	2	4	no	—
38	C₂×#41×C₆₇	—	2	4	no	—
39	C₂×#41×C₆₇	—	2	32	no	—
40	C₂×#41×C₆₇	C₂³	1	8	yes	—
41	C₂×#41×C₆₇	C₂³	1	8	yes	—
42	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
43	C₂²×D₄×C₆₇	—	2	4	no	—
44	C₂²×D₄×C₆₇	—	2	4	no	—
45	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
46	C₂³×C₄×C₆₇	—	2	4	no	—
47	C₂³×C₄×C₆₇	D₄	1	2	yes	—
48	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
49	#137×C₆₇	—	2	64	no	—
50	AGL(1,ℤ/8)×C₆₇	—	2	64	no	—
51	#135×C₆₇	—	2	64	no	—
52	C₂×Q₁₆×C₆₇	C₂³	1	16	yes	—
53	C₂×SD₁₆×C₆₇	C₂³	1	32	yes	—
54	C₂×D₈×C₆₇	C₂³	1	16	yes	—
55	C₂×M₁₆×C₆₇	D₄	1	16	yes	complemented
56	C₂×M₁₆×C₆₇	C₂³	1	8	yes	—
57	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
58	C₄⋊Q₈×C₆₇	—	2	4	no	—
59	C₄⋊Q₈×C₆₇	C₂³	1	4	yes	—
60	C₄⋊Q₈×C₆₇	C₂³	1	4	yes	—
61	#127×C₆₇	—	2	4	no	—
62	#127×C₆₇	C₂³	1	4	yes	—
63	#127×C₆₇	C₂³	1	4	yes	—
64	#124×C₆₇	—	2	8	no	—
65	#124×C₆₇	C₂³	1	8	yes	—
66	#124×C₆₇	C₂³	1	8	yes	—
67	#121×C₆₇	—	2	8	no	—
68	#121×C₆₇	—	2	8	no	—
69	#121×C₆₇	—	2	8	no	—
70	#121×C₆₇	—	2	8	no	—
71	C₄×D₄×C₆₇	—	2	8	no	—
72	C₄×D₄×C₆₇	—	2	8	no	—
73	C₄×D₄×C₆₇	—	2	8	no	—
74	C₄×D₄×C₆₇	—	2	8	no	—
75	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
76	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
77	C₂×C₄²×C₆₇	D₄	1	4	yes	—
78	C₂×C₄²×C₆₇	D₄	1	2	yes	—
79	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
80	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
81	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
82	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
83	#97×C₆₇	C₂³	1	8	yes	—
84	C₄×C₈×C₆₇	C₂³	1	4	yes	—
85	C₄×C₈×C₆₇	C₂³	1	4	yes	—
86	C₂⁴×C₆₇	—	2	4	no	—
87	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
88	#41×C₆₇	—	2	32	no	—
89	#41×C₆₇	—	2	32	no	—
90	#41×C₆₇	—	4	16	no	—
91	#41×C₆₇	—	4	16	no	—
92	C₂×Q₈×C₆₇	—	2	4	no	—
93	C₂×Q₈×C₆₇	—	2	16	no	—
94	C₂×Q₈×C₆₇	C₂⁴	1	8	yes	—
95	C₂×D₄×C₆₇	—	2	8	no	—
96	C₂×D₄×C₆₇	—	2	8	no	—
97	C₂×D₄×C₆₇	—	2	16	no	—
98	C₂×D₄×C₆₇	—	2	8	no	—
99	C₂×D₄×C₆₇	—	2	8	no	—
100	C₂×D₄×C₆₇	—	2	4	no	—
101	C₂×D₄×C₆₇	—	4	16	no	—
102	C₂×D₄×C₆₇	—	4	16	no	—
103	C₂×D₄×C₆₇	C₂⁴	1	8	yes	—
104	C₂²×C₄×C₆₇	—	2	4	no	—
105	C₂²×C₄×C₆₇	—	2	8	no	—
106	C₂²×C₄×C₆₇	—	2	8	no	—
107	C₂²×C₄×C₆₇	—	4	4	no	—
108	C₂²×C₄×C₆₇	—	4	16	no	—
109	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
110	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
111	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
112	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
113	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
114	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
115	C₂²×C₄×C₆₇	C₂×D₄	1	8	yes	—
116	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
117	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
118	Q₁₆×C₆₇	—	2	32	no	—
119	SD₁₆×C₆₇	—	2	64	no	—
120	D₈×C₆₇	—	2	32	no	—
121	M₁₆×C₆₇	C₂×D₄	1	32	yes	complemented
122	C₂×C₈×C₆₇	C₂×D₄	1	16	yes	complemented
123	C₂×C₈×C₆₇	C₂⁴	1	8	yes	—
124	C₄⋊C₄×C₆₇	—	2	8	no	—
125	C₄⋊C₄×C₆₇	—	2	8	no	—
126	C₂²⋊C₄×C₆₇	—	2	16	no	—
127	C₂²⋊C₄×C₆₇	—	2	16	no	—
128	C₄²×C₆₇	C₂×D₄	1	4	yes	—
129	C₄²×C₆₇	C₂×D₄	1	4	yes	—
130	C₄²×C₆₇	C₂×D₄	1	2	yes	—
131	C₄²×C₆₇	C₂×D₄	1	2	yes	—
132	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
133	C₄²×C₆₇	C₂⁴	1	2	yes	—
134	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
135	C₂³×C₆₇	—	2	8	no	—
136	C₂³×C₆₇	—	4	2	no	—
137	C₂³×C₆₇	—	4	16	no	—
138	C₂³×C₆₇	—	4	8	no	—
139	C₂³×C₆₇	#127	1	4	yes	—
140	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
141	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
142	Q₈×C₆₇	—	2	16	no	—
143	Q₈×C₆₇	—	4	8	no	—
144	D₄×C₆₇	—	2	16	no	—
145	D₄×C₆₇	—	4	16	no	—
146	D₄×C₆₇	—	4	16	no	—
147	D₄×C₆₇	—	4	16	no	—
148	D₄×C₆₇	—	4	8	no	—
149	D₄×C₆₇	—	4	16	no	—
150	C₂×C₄×C₆₇	—	2	4	no	—
151	C₂×C₄×C₆₇	—	2	8	no	—
152	C₂×C₄×C₆₇	—	2	8	no	—
153	C₂×C₄×C₆₇	—	4	4	no	—
154	C₂×C₄×C₆₇	—	4	2	no	—
155	C₂×C₄×C₆₇	—	4	8	no	—
156	C₂×C₄×C₆₇	—	4	16	no	—
157	C₂×C₄×C₆₇	—	4	16	no	—
158	C₂×C₄×C₆₇	#127	1	8	yes	—
159	C₂×C₄×C₆₇	#127	1	4	yes	—
160	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
161	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
162	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
163	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
164	C₂×C₄×C₆₇	C₂²×D₄	1	8	yes	—
165	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
166	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
167	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
168	C₈×C₆₇	C₂²×D₄	1	16	yes	—
169	C₂²×C₆₇	—	2	4	no	—
170	C₂²×C₆₇	—	2	2	no	—
171	C₂²×C₆₇	—	4	4	no	—
172	C₂²×C₆₇	—	4	8	no	—
173	C₂²×C₆₇	—	4	16	no	—
174	C₂²×C₆₇	—	8	16	no	—
175	C₂²×C₆₇	C₂×#127	1	4	yes	—
176	C₂²×C₆₇	C₂×#127	1	2	yes	—
177	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
178	C₄×C₆₇	—	2	2	no	—
179	C₄×C₆₇	—	2	4	no	—
180	C₄×C₆₇	—	4	8	no	—
181	C₄×C₆₇	C₂×#127	1	2	yes	—
182	C₄×C₆₇	C₂×#127	1	4	yes	—
183	C₄×C₆₇	#575	1	4	yes	complemented
184	C₄×C₆₇	C₂³×D₄	1	2	yes	—
185	#60639	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
186	C₂×C₆₇	—	2	4	no	—
187	C₂×C₆₇	—	2	2	no	—
188	C₂×C₆₇	—	8	8	no	—
189	C₂×C₆₇	C₂²×#127	1	1	yes	characteristic
190	C₂×C₆₇	C₂×#575	1	2	yes	—
191	C₂×#575	C₂×C₆₇	1	4	yes	complemented
192	C₂×#532	C₂×C₆₇	1	2	yes	complemented
193	#3153	C₂×C₆₇	1	8	yes	complemented
194	#3152	C₂×C₆₇	1	4	yes	complemented
195	#3151	C₂×C₆₇	1	4	yes	complemented
196	#3150	C₂×C₆₇	1	4	yes	complemented
197	#3147	C₂×C₆₇	1	4	yes	complemented
198	C₂×C₄×M₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
199	C₆₇	#60639	1	1	yes	characteristic, direct product factor, Sylow
200	C₂²×#41	C₂²×C₆₇	1	4	yes	—
201	#575	—	2	32	no	—
202	C₂×#137	C₂²×C₆₇	1	16	yes	complemented
203	C₂×AGL(1,ℤ/8)	C₂²×C₆₇	1	16	yes	complemented
204	C₂×#135	C₂²×C₆₇	1	16	yes	complemented
205	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
206	#532	—	2	4	no	—
207	#532	—	2	4	no	—
208	#532	C₂²×C₆₇	1	4	yes	—
209	#532	C₂²×C₆₇	1	4	yes	—
210	#532	C₂²×C₆₇	1	4	yes	complemented
211	#532	C₂²×C₆₇	1	4	yes	complemented
212	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	complemented
213	C₂×#127	C₂²×C₆₇	1	2	yes	complemented
214	C₂×#124	C₂²×C₆₇	1	4	yes	complemented
215	C₂×#121	—	2	4	no	—
216	C₂×C₄×D₄	—	2	4	no	—
217	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
218	#497	C₂²×C₆₇	1	16	yes	complemented
219	#496	C₂²×C₆₇	1	16	yes	complemented
220	#495	C₂²×C₆₇	1	16	yes	complemented
221	#494	C₂²×C₆₇	1	4	yes	complemented
222	#493	C₂²×C₆₇	1	4	yes	complemented
223	#492	C₂²×C₆₇	1	8	yes	complemented
224	C₄×M₁₆	C₂²×C₆₇	1	4	yes	complemented
225	C₄×M₁₆	C₂²×C₆₇	1	4	yes	complemented
226	C₂×#97	C₂²×C₆₇	1	2	yes	complemented
227	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	complemented
228	C₂×#41	—	2	4	no	—
229	C₂×#41	—	2	4	no	—
230	C₂×#41	—	2	32	no	—
231	C₂×#41	C₂³×C₆₇	1	8	yes	—
232	C₂×#41	C₂³×C₆₇	1	8	yes	—
233	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
234	C₂²×D₄	—	2	4	no	—
235	C₂²×D₄	—	2	4	no	—
236	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
237	C₂³×C₄	—	2	4	no	—
238	C₂³×C₄	D₄×C₆₇	1	2	yes	—
239	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
240	#137	—	2	64	no	—
241	AGL(1,ℤ/8)	—	2	64	no	—
242	#135	—	2	64	no	—
243	C₂×Q₁₆	C₂³×C₆₇	1	16	yes	—
244	C₂×SD₁₆	C₂³×C₆₇	1	32	yes	—
245	C₂×D₈	C₂³×C₆₇	1	16	yes	—
246	C₂×M₁₆	D₄×C₆₇	1	16	yes	complemented
247	C₂×M₁₆	C₂³×C₆₇	1	8	yes	—
248	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
249	C₄⋊Q₈	—	2	4	no	—
250	C₄⋊Q₈	C₂³×C₆₇	1	4	yes	—
251	C₄⋊Q₈	C₂³×C₆₇	1	4	yes	—
252	#127	—	2	4	no	—
253	#127	C₂³×C₆₇	1	4	yes	—
254	#127	C₂³×C₆₇	1	4	yes	—
255	#124	—	2	8	no	—
256	#124	C₂³×C₆₇	1	8	yes	—
257	#124	C₂³×C₆₇	1	8	yes	—
258	#121	—	2	8	no	—
259	#121	—	2	8	no	—
260	#121	—	2	8	no	—
261	#121	—	2	8	no	—
262	C₄×D₄	—	2	8	no	—
263	C₄×D₄	—	2	8	no	—
264	C₄×D₄	—	2	8	no	—
265	C₄×D₄	—	2	8	no	—
266	C₂×C₄⋊C₄	—	2	4	no	—
267	C₂×C₂²⋊C₄	—	2	8	no	—
268	C₂×C₄²	D₄×C₆₇	1	4	yes	—
269	C₂×C₄²	D₄×C₆₇	1	2	yes	—
270	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
271	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
272	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
273	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
274	#97	C₂³×C₆₇	1	8	yes	—
275	C₄×C₈	C₂³×C₆₇	1	4	yes	—
276	C₄×C₈	C₂³×C₆₇	1	4	yes	—
277	C₂⁴	—	2	4	no	—
278	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
279	#41	—	2	32	no	—
280	#41	—	2	32	no	—
281	#41	—	4	16	no	—
282	#41	—	4	16	no	—
283	C₂×Q₈	—	2	4	no	—
284	C₂×Q₈	—	2	16	no	—
285	C₂×Q₈	C₂⁴×C₆₇	1	8	yes	—
286	C₂×D₄	—	2	8	no	—
287	C₂×D₄	—	2	8	no	—
288	C₂×D₄	—	2	16	no	—
289	C₂×D₄	—	2	8	no	—
290	C₂×D₄	—	2	8	no	—
291	C₂×D₄	—	2	4	no	—
292	C₂×D₄	—	4	16	no	—
293	C₂×D₄	—	4	16	no	—
294	C₂×D₄	C₂⁴×C₆₇	1	8	yes	—
295	C₂²×C₄	—	2	4	no	—
296	C₂²×C₄	—	2	8	no	—
297	C₂²×C₄	—	2	8	no	—
298	C₂²×C₄	—	4	4	no	—
299	C₂²×C₄	—	4	16	no	—
300	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
301	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
302	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
303	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
304	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
305	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
306	C₂²×C₄	C₂×D₄×C₆₇	1	8	yes	—
307	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
308	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
309	Q₁₆	—	2	32	no	—
310	SD₁₆	—	2	64	no	—
311	D₈	—	2	32	no	—
312	M₁₆	C₂×D₄×C₆₇	1	32	yes	complemented
313	C₂×C₈	C₂×D₄×C₆₇	1	16	yes	complemented
314	C₂×C₈	C₂⁴×C₆₇	1	8	yes	—
315	C₄⋊C₄	—	2	8	no	—
316	C₄⋊C₄	—	2	8	no	—
317	C₂²⋊C₄	—	2	16	no	—
318	C₂²⋊C₄	—	2	16	no	—
319	C₄²	C₂×D₄×C₆₇	1	2	yes	—
320	C₄²	C₂×D₄×C₆₇	1	2	yes	—
321	C₄²	C₂×D₄×C₆₇	1	4	yes	—
322	C₄²	C₂×D₄×C₆₇	1	4	yes	—
323	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
324	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
325	C₄²	C₂⁴×C₆₇	1	2	yes	—
326	C₂³	—	2	8	no	—
327	C₂³	—	4	2	no	—
328	C₂³	—	4	16	no	—
329	C₂³	—	4	8	no	—
330	C₂³	#127×C₆₇	1	4	yes	—
331	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
332	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
333	Q₈	—	2	16	no	—
334	Q₈	—	4	8	no	—
335	D₄	—	2	16	no	—
336	D₄	—	4	16	no	—
337	D₄	—	4	16	no	—
338	D₄	—	4	16	no	—
339	D₄	—	4	8	no	—
340	D₄	—	4	16	no	—
341	C₂×C₄	—	2	8	no	—
342	C₂×C₄	—	2	4	no	—
343	C₂×C₄	—	2	8	no	—
344	C₂×C₄	—	4	2	no	—
345	C₂×C₄	—	4	4	no	—
346	C₂×C₄	—	4	8	no	—
347	C₂×C₄	—	4	16	no	—
348	C₂×C₄	—	4	16	no	—
349	C₂×C₄	#127×C₆₇	1	8	yes	—
350	C₂×C₄	#127×C₆₇	1	4	yes	—
351	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
352	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
353	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
354	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
355	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
356	C₂×C₄	C₂²×D₄×C₆₇	1	8	yes	—
357	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
358	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
359	C₈	C₂²×D₄×C₆₇	1	16	yes	—
360	C₂²	—	2	4	no	—
361	C₂²	—	2	2	no	—
362	C₂²	—	4	4	no	—
363	C₂²	—	4	16	no	—
364	C₂²	—	4	8	no	—
365	C₂²	—	8	16	no	—
366	C₂²	C₂×#127×C₆₇	1	4	yes	—
367	C₂²	C₂×#127×C₆₇	1	2	yes	—
368	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
369	C₄	—	2	4	no	—
370	C₄	—	2	2	no	—
371	C₄	—	4	8	no	—
372	C₄	C₂×#127×C₆₇	1	2	yes	—
373	C₄	C₂×#127×C₆₇	1	4	yes	—
374	C₄	#575×C₆₇	1	4	yes	complemented
375	C₄	C₂³×D₄×C₆₇	1	2	yes	—
376	C₂	—	2	4	no	—
377	C₂	—	2	2	no	—
378	C₂	—	8	8	no	—
379	C₂	C₂²×#127×C₆₇	1	1	yes	characteristic
380	C₂	C₂×#575×C₆₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
socle	C₂²×C₆₇	C₂³×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#575×C₆₇	C₂	semidirect
2. C₂×#532×C₆₇	C₂	semidirect
3. C₆₇×#3153	C₂	semidirect
4. C₆₇×#3152	C₂	semidirect
5. C₆₇×#3151	C₂	semidirect
6. C₆₇×#3150	C₂	semidirect
7. C₆₇×#3147	C₂	semidirect
8. C₂×C₄×M₁₆×C₆₇	C₂	semidirect
11. C₂×#137×C₆₇	C₂²	semidirect
12. C₂×AGL(1,ℤ/8)×C₆₇	C₂²	semidirect
13. C₂×#135×C₆₇	C₂²	semidirect
19. #532×C₆₇	C₂²	semidirect
20. #532×C₆₇	C₂²	semidirect
21. C₂×C₄⋊Q₈×C₆₇	C₂²	semidirect
22. C₂×#127×C₆₇	C₂²	semidirect
23. C₂×#124×C₆₇	C₂²	semidirect
27. #497×C₆₇	C₂²	semidirect
28. #496×C₆₇	C₂²	semidirect
29. #495×C₆₇	C₂²	semidirect
30. #494×C₆₇	C₂²	semidirect
31. #493×C₆₇	C₂²	semidirect
32. #492×C₆₇	C₂²	semidirect
33. C₄×M₁₆×C₆₇	C₂²	semidirect
34. C₄×M₁₆×C₆₇	C₂²	semidirect
35. C₂×#97×C₆₇	C₂²	semidirect
36. C₂×C₄×C₈×C₆₇	C₂²	semidirect
55. C₂×M₁₆×C₆₇	D₄	semidirect
121. M₁₆×C₆₇	C₂×D₄	semidirect
122. C₂×C₈×C₆₇	C₂×D₄	semidirect
183. C₄×C₆₇	#575	semidirect
185. #60639	C₆₇	direct (N × H)
191. C₂×#575	C₂×C₆₇	semidirect
192. C₂×#532	C₂×C₆₇	semidirect
193. #3153	C₂×C₆₇	semidirect
194. #3152	C₂×C₆₇	semidirect
195. #3151	C₂×C₆₇	semidirect
196. #3150	C₂×C₆₇	semidirect
197. #3147	C₂×C₆₇	semidirect
198. C₂×C₄×M₁₆	C₂×C₆₇	semidirect
199. C₆₇	#60639	direct (N × H)
202. C₂×#137	C₂²×C₆₇	semidirect
203. C₂×AGL(1,ℤ/8)	C₂²×C₆₇	semidirect
204. C₂×#135	C₂²×C₆₇	semidirect
210. #532	C₂²×C₆₇	semidirect
211. #532	C₂²×C₆₇	semidirect
212. C₂×C₄⋊Q₈	C₂²×C₆₇	semidirect
213. C₂×#127	C₂²×C₆₇	semidirect
214. C₂×#124	C₂²×C₆₇	semidirect
218. #497	C₂²×C₆₇	semidirect
219. #496	C₂²×C₆₇	semidirect
220. #495	C₂²×C₆₇	semidirect
221. #494	C₂²×C₆₇	semidirect
222. #493	C₂²×C₆₇	semidirect
223. #492	C₂²×C₆₇	semidirect
224. C₄×M₁₆	C₂²×C₆₇	semidirect
225. C₄×M₁₆	C₂²×C₆₇	semidirect
226. C₂×#97	C₂²×C₆₇	semidirect
227. C₂×C₄×C₈	C₂²×C₆₇	semidirect
246. C₂×M₁₆	D₄×C₆₇	semidirect
312. M₁₆	C₂×D₄×C₆₇	semidirect
313. C₂×C₈	C₂×D₄×C₆₇	semidirect
374. C₄	#575×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#575×C₆₇	2
C₂×#532×C₆₇	2
C₆₇×#3153	2
C₆₇×#3152	2
C₆₇×#3151	2
C₆₇×#3150	2
C₆₇×#3147	2
C₂×C₄×M₁₆×C₆₇	2
#60639	67

Subgroups — C67×#60639 #30739890

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#60639 #30739890