Subgroups — C₂×C₆₇×#3151 #30739887

Order 17152.

3950 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

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

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

Complements

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

N	H ≅ G/N	type
1. C₂²×AGL(1,ℤ/8)×C₆₇	C₂	semidirect
2. C₂×#532×C₆₇	C₂	semidirect
3. C₂²×#127×C₆₇	C₂	semidirect
4. C₆₇×#3151	C₂	direct (N × H)
5. C₂×#496×C₆₇	C₂	semidirect
6. C₂×#493×C₆₇	C₂	semidirect
7. C₂×#492×C₆₇	C₂	semidirect
8. C₂×C₄×M₁₆×C₆₇	C₂	semidirect
13. C₂×AGL(1,ℤ/8)×C₆₇	C₂²	semidirect
14. C₂²×SD₁₆×C₆₇	C₂²	semidirect
15. C₂²×D₈×C₆₇	C₂²	semidirect
18. #532×C₆₇	C₂²	semidirect
19. C₂×C₄⋊Q₈×C₆₇	C₂²	semidirect
24. C₂×#127×C₆₇	C₂²	semidirect
25. C₂×#127×C₆₇	C₂²	semidirect
26. C₂×#124×C₆₇	C₂²	semidirect
30. #496×C₆₇	C₂²	semidirect
31. #493×C₆₇	C₂²	semidirect
32. #492×C₆₇	C₂²	semidirect
33. C₄×M₁₆×C₆₇	C₂²	semidirect
34. C₂×#97×C₆₇	C₂²	semidirect
35. C₂×C₄×C₈×C₆₇	C₂²	semidirect
58. C₂×SD₁₆×C₆₇	C₂³	semidirect
60. C₂×D₈×C₆₇	C₂³	semidirect
61. C₂×M₁₆×C₆₇	D₄	semidirect
65. C₄⋊Q₈×C₆₇	C₂³	semidirect
72. #127×C₆₇	C₂³	semidirect
74. #124×C₆₇	C₂³	semidirect
84. #97×C₆₇	C₂³	semidirect
85. C₄×C₈×C₆₇	C₂³	semidirect
130. M₁₆×C₆₇	C₂×D₄	semidirect
131. C₂×C₈×C₆₇	C₂×D₄	semidirect
171. C₂×C₄×C₆₇	AGL(1,ℤ/8)	semidirect
179. C₈×C₆₇	C₂²×D₄	semidirect
197. C₄×C₆₇	C₂×AGL(1,ℤ/8)	semidirect
199. C₂×#3151	C₆₇	direct (N × H)
203. C₂×C₆₇	#3151	direct (N × H)
206. C₂²×AGL(1,ℤ/8)	C₂×C₆₇	semidirect
207. C₂×#532	C₂×C₆₇	semidirect
208. C₂²×#127	C₂×C₆₇	semidirect
209. #3151	C₂×C₆₇	direct (N × H)
210. C₂×#496	C₂×C₆₇	semidirect
211. C₂×#493	C₂×C₆₇	semidirect
212. C₂×#492	C₂×C₆₇	semidirect
213. C₂×C₄×M₁₆	C₂×C₆₇	semidirect
214. C₆₇	C₂×#3151	direct (N × H)
219. C₂×AGL(1,ℤ/8)	C₂²×C₆₇	semidirect
220. C₂²×SD₁₆	C₂²×C₆₇	semidirect
221. C₂²×D₈	C₂²×C₆₇	semidirect
224. #532	C₂²×C₆₇	semidirect
225. C₂×C₄⋊Q₈	C₂²×C₆₇	semidirect
230. C₂×#127	C₂²×C₆₇	semidirect
231. C₂×#127	C₂²×C₆₇	semidirect
232. C₂×#124	C₂²×C₆₇	semidirect
236. #496	C₂²×C₆₇	semidirect
237. #493	C₂²×C₆₇	semidirect
238. #492	C₂²×C₆₇	semidirect
239. C₄×M₁₆	C₂²×C₆₇	semidirect
240. C₂×#97	C₂²×C₆₇	semidirect
241. C₂×C₄×C₈	C₂²×C₆₇	semidirect
264. C₂×SD₁₆	C₂³×C₆₇	semidirect
266. C₂×D₈	C₂³×C₆₇	semidirect
267. C₂×M₁₆	D₄×C₆₇	semidirect
271. C₄⋊Q₈	C₂³×C₆₇	semidirect
278. #127	C₂³×C₆₇	semidirect
280. #124	C₂³×C₆₇	semidirect
290. #97	C₂³×C₆₇	semidirect
291. C₄×C₈	C₂³×C₆₇	semidirect
336. M₁₆	C₂×D₄×C₆₇	semidirect
337. C₂×C₈	C₂×D₄×C₆₇	semidirect
377. C₂×C₄	AGL(1,ℤ/8)×C₆₇	semidirect
385. C₈	C₂²×D₄×C₆₇	semidirect
403. C₄	C₂×AGL(1,ℤ/8)×C₆₇	semidirect
408. C₂	C₆₇×#3151	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×AGL(1,ℤ/8)×C₆₇	2
C₂×#532×C₆₇	2
C₂²×#127×C₆₇	2
C₆₇×#3151	2
C₂×#496×C₆₇	2
C₂×#493×C₆₇	2
C₂×#492×C₆₇	2
C₂×C₄×M₁₆×C₆₇	2
C₂×#3151	67

Subgroups — C2×C67×#3151 #30739887

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#3151 #30739887