Subgroups — C₂×C₆₇×#3155 #30739892

Order 17152.

3534 conjugacy classes of subgroups.

The subgroup lattice has 446 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	2	yes	maximal, complemented
2	C₂²×#135×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂³×D₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#535×C₆₇	C₂	1	2	yes	maximal, complemented
5	C₆₇×#3155	C₂	1	16	yes	maximal, direct product factor
6	C₂×#496×C₆₇	C₂	1	4	yes	maximal, complemented
7	C₂×#495×C₆₇	C₂	1	2	yes	maximal, complemented
8	C₂×#493×C₆₇	C₂	1	2	yes	maximal, complemented
9	C₂×#405×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#41×C₆₇	C₂²	1	2	yes	—
11	C₂³×D₄×C₆₇	C₂²	1	2	yes	—
12	C₂×AGL(1,ℤ/8)×C₆₇	—	2	16	no	—
13	C₂×AGL(1,ℤ/8)×C₆₇	C₂²	1	16	yes	complemented
14	C₂×#135×C₆₇	—	2	8	no	—
15	C₂×#135×C₆₇	C₂²	1	8	yes	complemented
16	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic, complemented
17	C₂²×SD₁₆×C₆₇	C₂²	1	2	yes	complemented
18	C₂²×SD₁₆×C₆₇	C₂²	1	4	yes	complemented
19	C₂²×D₈×C₆₇	—	2	8	no	—
20	C₂²×D₈×C₆₇	C₂²	1	2	yes	—
21	C₂²×D₈×C₆₇	C₂²	1	2	yes	—
22	C₂²×D₈×C₆₇	C₂²	1	8	yes	complemented
23	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
24	C₂²×D₈×C₆₇	C₂²	1	4	yes	complemented
25	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
26	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
27	#535×C₆₇	—	2	8	no	—
28	#535×C₆₇	C₂²	1	16	yes	complemented
29	C₂×#127×C₆₇	C₂²	1	4	yes	complemented
30	C₂×#124×C₆₇	C₂²	1	4	yes	complemented
31	C₂×#121×C₆₇	—	2	4	no	—
32	C₂×#120×C₆₇	—	2	4	no	—
33	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
34	#496×C₆₇	C₂²	1	32	yes	complemented
35	#495×C₆₇	C₂²	1	16	yes	complemented
36	#493×C₆₇	C₂²	1	16	yes	complemented
37	#405×C₆₇	C₂²	1	8	yes	complemented
38	C₂×#97×C₆₇	C₂²	1	2	yes	complemented
39	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	complemented
40	C₂⁵×C₆₇	—	2	2	no	—
41	C₂×#41×C₆₇	—	2	4	no	—
42	C₂×#41×C₆₇	—	2	8	no	—
43	C₂×#41×C₆₇	—	2	8	no	—
44	C₂×#41×C₆₇	C₂³	1	8	yes	—
45	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
46	C₂²×D₄×C₆₇	—	2	2	no	—
47	C₂²×D₄×C₆₇	—	2	8	no	—
48	C₂²×D₄×C₆₇	—	2	8	no	—
49	C₂²×D₄×C₆₇	—	2	2	no	—
50	C₂²×D₄×C₆₇	—	2	4	no	—
51	C₂²×D₄×C₆₇	—	2	8	no	—
52	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
53	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
54	C₂²×D₄×C₆₇	C₂³	1	8	yes	—
55	C₂³×C₄×C₆₇	—	2	2	no	—
56	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
57	AGL(1,ℤ/8)×C₆₇	—	2	64	no	—
58	#135×C₆₇	—	2	32	no	—
59	C₂×Q₁₆×C₆₇	—	2	4	no	—
60	C₂×Q₁₆×C₆₇	C₂³	1	4	yes	complemented
61	C₂×SD₁₆×C₆₇	—	2	8	no	—
62	C₂×SD₁₆×C₆₇	—	2	16	no	—
63	C₂×SD₁₆×C₆₇	C₂³	1	16	yes	complemented
64	C₂×SD₁₆×C₆₇	C₂³	1	8	yes	complemented
65	C₂×D₈×C₆₇	—	2	8	no	—
66	C₂×D₈×C₆₇	—	2	8	no	—
67	C₂×D₈×C₆₇	—	2	4	no	—
68	C₂×D₈×C₆₇	—	2	16	no	—
69	C₂×D₈×C₆₇	—	2	32	no	—
70	C₂×D₈×C₆₇	C₂³	1	8	yes	—
71	C₂×D₈×C₆₇	C₂³	1	8	yes	—
72	C₂×D₈×C₆₇	C₂³	1	4	yes	complemented
73	C₂×D₈×C₆₇	C₂³	1	16	yes	complemented
74	C₂×M₁₆×C₆₇	D₄	1	8	yes	complemented
75	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
76	C₂²×C₈×C₆₇	D₄	1	8	yes	complemented
77	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
78	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
79	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
80	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
81	#127×C₆₇	—	2	8	no	—
82	#127×C₆₇	C₂³	1	16	yes	complemented
83	#124×C₆₇	—	2	8	no	—
84	#124×C₆₇	C₂³	1	16	yes	complemented
85	#121×C₆₇	—	2	32	no	—
86	#120×C₆₇	—	2	32	no	—
87	#117×C₆₇	D₄	1	4	yes	—
88	#117×C₆₇	C₂³	1	4	yes	—
89	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
90	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
91	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
92	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
93	#97×C₆₇	C₂³	1	8	yes	complemented
94	C₄×C₈×C₆₇	C₂³	1	8	yes	complemented
95	C₂⁴×C₆₇	—	2	4	no	—
96	C₂⁴×C₆₇	—	2	2	no	—
97	C₂⁴×C₆₇	—	2	8	no	—
98	C₂⁴×C₆₇	—	4	8	no	—
99	C₂⁴×C₆₇	—	8	8	no	—
100	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
101	#41×C₆₇	—	2	16	no	—
102	#41×C₆₇	—	2	16	no	—
103	#41×C₆₇	—	4	16	no	—
104	C₂×Q₈×C₆₇	—	2	2	no	—
105	C₂×Q₈×C₆₇	—	2	4	no	—
106	C₂×Q₈×C₆₇	—	2	4	no	—
107	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
108	C₂×D₄×C₆₇	—	2	8	no	—
109	C₂×D₄×C₆₇	—	2	8	no	—
110	C₂×D₄×C₆₇	—	2	16	no	—
111	C₂×D₄×C₆₇	—	2	4	no	—
112	C₂×D₄×C₆₇	—	2	8	no	—
113	C₂×D₄×C₆₇	—	2	4	no	—
114	C₂×D₄×C₆₇	—	2	16	no	—
115	C₂×D₄×C₆₇	—	2	8	no	—
116	C₂×D₄×C₆₇	—	2	4	no	—
117	C₂×D₄×C₆₇	—	2	2	no	—
118	C₂×D₄×C₆₇	—	2	32	no	—
119	C₂×D₄×C₆₇	—	4	8	no	—
120	C₂×D₄×C₆₇	—	4	8	no	—
121	C₂×D₄×C₆₇	—	4	16	no	—
122	C₂×D₄×C₆₇	—	8	16	no	—
123	C₂×D₄×C₆₇	C₂⁴	1	8	yes	—
124	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
125	C₂²×C₄×C₆₇	—	2	2	no	—
126	C₂²×C₄×C₆₇	—	2	8	no	—
127	C₂²×C₄×C₆₇	—	4	8	no	—
128	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
129	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
130	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
131	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
132	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
133	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
134	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
135	Q₁₆×C₆₇	—	2	8	no	—
136	SD₁₆×C₆₇	—	2	16	no	—
137	SD₁₆×C₆₇	—	2	32	no	—
138	D₈×C₆₇	—	2	16	no	—
139	D₈×C₆₇	—	2	16	no	—
140	D₈×C₆₇	—	2	8	no	—
141	D₈×C₆₇	—	2	32	no	—
142	M₁₆×C₆₇	C₂×D₄	1	16	yes	complemented
143	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
144	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
145	C₂×C₈×C₆₇	C₂×D₄	1	8	yes	—
146	C₂×C₈×C₆₇	C₂×D₄	1	16	yes	complemented
147	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
148	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
149	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
150	C₄⋊C₄×C₆₇	C₂×D₄	1	8	yes	—
151	C₂²⋊C₄×C₆₇	—	2	32	no	—
152	C₂²⋊C₄×C₆₇	C₂×D₄	1	8	yes	—
153	C₄²×C₆₇	C₂×D₄	1	4	yes	—
154	C₄²×C₆₇	C₂⁴	1	4	yes	—
155	C₂³×C₆₇	—	2	4	no	—
156	C₂³×C₆₇	—	2	8	no	—
157	C₂³×C₆₇	—	2	4	no	—
158	C₂³×C₆₇	—	4	4	no	—
159	C₂³×C₆₇	—	4	16	no	—
160	C₂³×C₆₇	—	4	8	no	—
161	C₂³×C₆₇	—	4	8	no	—
162	C₂³×C₆₇	—	4	16	no	—
163	C₂³×C₆₇	—	8	16	no	—
164	C₂³×C₆₇	—	8	32	no	—
165	C₂³×C₆₇	#127	1	4	yes	—
166	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
167	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
168	Q₈×C₆₇	—	2	4	no	—
169	Q₈×C₆₇	—	2	4	no	—
170	Q₈×C₆₇	—	4	4	no	—
171	D₄×C₆₇	—	2	8	no	—
172	D₄×C₆₇	—	2	4	no	—
173	D₄×C₆₇	—	2	8	no	—
174	D₄×C₆₇	—	2	4	no	—
175	D₄×C₆₇	—	4	16	no	—
176	D₄×C₆₇	—	4	16	no	—
177	D₄×C₆₇	—	4	8	no	—
178	D₄×C₆₇	—	4	4	no	—
179	D₄×C₆₇	—	8	32	no	—
180	C₂×C₄×C₆₇	—	2	8	no	—
181	C₂×C₄×C₆₇	—	2	4	no	—
182	C₂×C₄×C₆₇	—	4	4	no	—
183	C₂×C₄×C₆₇	—	4	16	no	—
184	C₂×C₄×C₆₇	#127	1	4	yes	—
185	C₂×C₄×C₆₇	#127	1	8	yes	—
186	C₂×C₄×C₆₇	C₂²×D₄	1	8	yes	—
187	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
188	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
189	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
190	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
191	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
192	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
193	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
194	C₈×C₆₇	C₂²×D₄	1	4	yes	—
195	C₈×C₆₇	C₂²×D₄	1	4	yes	—
196	C₈×C₆₇	C₂²×D₄	1	8	yes	—
197	C₂²×C₆₇	—	2	8	no	—
198	C₂²×C₆₇	—	2	4	no	—
199	C₂²×C₆₇	—	4	4	no	—
200	C₂²×C₆₇	—	4	8	no	—
201	C₂²×C₆₇	—	4	8	no	—
202	C₂²×C₆₇	—	4	16	no	—
203	C₂²×C₆₇	—	8	32	no	—
204	C₂²×C₆₇	—	8	8	no	—
205	C₂²×C₆₇	—	8	16	no	—
206	C₂²×C₆₇	C₂×#127	1	4	yes	—
207	C₂²×C₆₇	C₂×#127	1	2	yes	—
208	C₂²×C₆₇	#576	1	4	yes	—
209	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
210	C₄×C₆₇	—	2	8	no	—
211	C₄×C₆₇	—	4	4	no	—
212	C₄×C₆₇	C₂×#127	1	4	yes	—
213	C₄×C₆₇	C₂×#127	1	2	yes	—
214	C₄×C₆₇	C₂³×D₄	1	1	yes	characteristic
215	C₄×C₆₇	C₂³×D₄	1	1	yes	characteristic
216	C₂×#3155	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
217	C₂×C₆₇	—	2	4	no	—
218	C₂×C₆₇	—	8	4	no	—
219	C₂×C₆₇	—	8	8	no	—
220	C₂×C₆₇	#3155	1	4	yes	direct product factor
221	C₂×C₆₇	C₂²×#127	1	1	yes	characteristic
222	C₂×C₆₇	C₂×#576	1	2	yes	—
223	C₂²×AGL(1,ℤ/8)	C₂×C₆₇	1	2	yes	complemented
224	C₂²×#135	C₂×C₆₇	1	1	yes	characteristic, complemented
225	C₂³×D₈	C₂×C₆₇	1	1	yes	characteristic, complemented
226	C₂×#535	C₂×C₆₇	1	2	yes	complemented
227	#3155	C₂×C₆₇	1	16	yes	direct product factor
228	C₂×#496	C₂×C₆₇	1	4	yes	complemented
229	C₂×#495	C₂×C₆₇	1	2	yes	complemented
230	C₂×#493	C₂×C₆₇	1	2	yes	complemented
231	C₂×#405	C₂×C₆₇	1	1	yes	characteristic, complemented
232	C₆₇	C₂×#3155	1	1	yes	characteristic, direct product factor, Sylow
233	C₂²×#41	C₂²×C₆₇	1	2	yes	—
234	C₂³×D₄	C₂²×C₆₇	1	2	yes	—
235	C₂×AGL(1,ℤ/8)	—	2	16	no	—
236	C₂×AGL(1,ℤ/8)	C₂²×C₆₇	1	16	yes	complemented
237	C₂×#135	—	2	8	no	—
238	C₂×#135	C₂²×C₆₇	1	8	yes	complemented
239	C₂²×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic, complemented
240	C₂²×SD₁₆	C₂²×C₆₇	1	4	yes	complemented
241	C₂²×SD₁₆	C₂²×C₆₇	1	2	yes	complemented
242	C₂²×D₈	—	2	8	no	—
243	C₂²×D₈	C₂²×C₆₇	1	2	yes	—
244	C₂²×D₈	C₂²×C₆₇	1	2	yes	—
245	C₂²×D₈	C₂²×C₆₇	1	8	yes	complemented
246	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
247	C₂²×D₈	C₂²×C₆₇	1	4	yes	complemented
248	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
249	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
250	#535	—	2	8	no	—
251	#535	C₂²×C₆₇	1	16	yes	complemented
252	C₂×#127	C₂²×C₆₇	1	4	yes	complemented
253	C₂×#124	C₂²×C₆₇	1	4	yes	complemented
254	C₂×#121	—	2	4	no	—
255	C₂×#120	—	2	4	no	—
256	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
257	#496	C₂²×C₆₇	1	32	yes	complemented
258	#495	C₂²×C₆₇	1	16	yes	complemented
259	#493	C₂²×C₆₇	1	16	yes	complemented
260	#405	C₂²×C₆₇	1	8	yes	complemented
261	C₂×#97	C₂²×C₆₇	1	2	yes	complemented
262	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	complemented
263	C₂⁵	—	2	2	no	—
264	C₂×#41	—	2	4	no	—
265	C₂×#41	—	2	8	no	—
266	C₂×#41	—	2	8	no	—
267	C₂×#41	C₂³×C₆₇	1	8	yes	—
268	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
269	C₂²×D₄	—	2	2	no	—
270	C₂²×D₄	—	2	8	no	—
271	C₂²×D₄	—	2	8	no	—
272	C₂²×D₄	—	2	2	no	—
273	C₂²×D₄	—	2	4	no	—
274	C₂²×D₄	—	2	8	no	—
275	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
276	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
277	C₂²×D₄	C₂³×C₆₇	1	8	yes	—
278	C₂³×C₄	—	2	2	no	—
279	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
280	AGL(1,ℤ/8)	—	2	64	no	—
281	#135	—	2	32	no	—
282	C₂×Q₁₆	—	2	4	no	—
283	C₂×Q₁₆	C₂³×C₆₇	1	4	yes	complemented
284	C₂×SD₁₆	—	2	8	no	—
285	C₂×SD₁₆	—	2	16	no	—
286	C₂×SD₁₆	C₂³×C₆₇	1	8	yes	complemented
287	C₂×SD₁₆	C₂³×C₆₇	1	16	yes	complemented
288	C₂×D₈	—	2	8	no	—
289	C₂×D₈	—	2	8	no	—
290	C₂×D₈	—	2	4	no	—
291	C₂×D₈	—	2	16	no	—
292	C₂×D₈	—	2	32	no	—
293	C₂×D₈	C₂³×C₆₇	1	8	yes	—
294	C₂×D₈	C₂³×C₆₇	1	8	yes	—
295	C₂×D₈	C₂³×C₆₇	1	4	yes	complemented
296	C₂×D₈	C₂³×C₆₇	1	16	yes	complemented
297	C₂×M₁₆	D₄×C₆₇	1	8	yes	complemented
298	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
299	C₂²×C₈	D₄×C₆₇	1	8	yes	complemented
300	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
301	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
302	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
303	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
304	#127	—	2	8	no	—
305	#127	C₂³×C₆₇	1	16	yes	complemented
306	#124	—	2	8	no	—
307	#124	C₂³×C₆₇	1	16	yes	complemented
308	#121	—	2	32	no	—
309	#120	—	2	32	no	—
310	#117	D₄×C₆₇	1	4	yes	—
311	#117	C₂³×C₆₇	1	4	yes	—
312	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
313	C₂×C₂²⋊C₄	—	2	8	no	—
314	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
315	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
316	#97	C₂³×C₆₇	1	8	yes	complemented
317	C₄×C₈	C₂³×C₆₇	1	8	yes	complemented
318	C₂⁴	—	2	4	no	—
319	C₂⁴	—	2	2	no	—
320	C₂⁴	—	2	8	no	—
321	C₂⁴	—	4	8	no	—
322	C₂⁴	—	8	8	no	—
323	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
324	#41	—	2	16	no	—
325	#41	—	2	16	no	—
326	#41	—	4	16	no	—
327	C₂×Q₈	—	2	2	no	—
328	C₂×Q₈	—	2	4	no	—
329	C₂×Q₈	—	2	4	no	—
330	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
331	C₂×D₄	—	2	8	no	—
332	C₂×D₄	—	2	8	no	—
333	C₂×D₄	—	2	16	no	—
334	C₂×D₄	—	2	4	no	—
335	C₂×D₄	—	2	8	no	—
336	C₂×D₄	—	2	4	no	—
337	C₂×D₄	—	2	16	no	—
338	C₂×D₄	—	2	8	no	—
339	C₂×D₄	—	2	4	no	—
340	C₂×D₄	—	2	2	no	—
341	C₂×D₄	—	2	32	no	—
342	C₂×D₄	—	4	8	no	—
343	C₂×D₄	—	4	8	no	—
344	C₂×D₄	—	4	16	no	—
345	C₂×D₄	—	8	16	no	—
346	C₂×D₄	C₂⁴×C₆₇	1	8	yes	—
347	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
348	C₂²×C₄	—	2	2	no	—
349	C₂²×C₄	—	2	8	no	—
350	C₂²×C₄	—	4	8	no	—
351	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
352	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
353	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
354	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
355	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
356	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
357	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
358	Q₁₆	—	2	8	no	—
359	SD₁₆	—	2	16	no	—
360	SD₁₆	—	2	32	no	—
361	D₈	—	2	16	no	—
362	D₈	—	2	16	no	—
363	D₈	—	2	8	no	—
364	D₈	—	2	32	no	—
365	M₁₆	C₂×D₄×C₆₇	1	16	yes	complemented
366	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
367	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
368	C₂×C₈	C₂×D₄×C₆₇	1	8	yes	—
369	C₂×C₈	C₂×D₄×C₆₇	1	16	yes	complemented
370	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
371	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
372	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
373	C₄⋊C₄	C₂×D₄×C₆₇	1	8	yes	—
374	C₂²⋊C₄	—	2	32	no	—
375	C₂²⋊C₄	C₂×D₄×C₆₇	1	8	yes	—
376	C₄²	C₂×D₄×C₆₇	1	4	yes	—
377	C₄²	C₂⁴×C₆₇	1	4	yes	—
378	C₂³	—	2	4	no	—
379	C₂³	—	2	8	no	—
380	C₂³	—	2	4	no	—
381	C₂³	—	4	4	no	—
382	C₂³	—	4	16	no	—
383	C₂³	—	4	8	no	—
384	C₂³	—	4	8	no	—
385	C₂³	—	4	16	no	—
386	C₂³	—	8	16	no	—
387	C₂³	—	8	32	no	—
388	C₂³	#127×C₆₇	1	4	yes	—
389	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
390	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
391	Q₈	—	2	4	no	—
392	Q₈	—	2	4	no	—
393	Q₈	—	4	4	no	—
394	D₄	—	2	8	no	—
395	D₄	—	2	4	no	—
396	D₄	—	2	8	no	—
397	D₄	—	2	4	no	—
398	D₄	—	4	16	no	—
399	D₄	—	4	16	no	—
400	D₄	—	4	8	no	—
401	D₄	—	4	4	no	—
402	D₄	—	8	32	no	—
403	C₂×C₄	—	2	8	no	—
404	C₂×C₄	—	2	4	no	—
405	C₂×C₄	—	4	4	no	—
406	C₂×C₄	—	4	16	no	—
407	C₂×C₄	#127×C₆₇	1	4	yes	—
408	C₂×C₄	#127×C₆₇	1	8	yes	—
409	C₂×C₄	C₂²×D₄×C₆₇	1	8	yes	—
410	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
411	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
412	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
413	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
414	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
415	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
416	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
417	C₈	C₂²×D₄×C₆₇	1	4	yes	—
418	C₈	C₂²×D₄×C₆₇	1	4	yes	—
419	C₈	C₂²×D₄×C₆₇	1	8	yes	—
420	C₂²	—	2	4	no	—
421	C₂²	—	2	8	no	—
422	C₂²	—	4	8	no	—
423	C₂²	—	4	16	no	—
424	C₂²	—	4	4	no	—
425	C₂²	—	4	8	no	—
426	C₂²	—	8	32	no	—
427	C₂²	—	8	16	no	—
428	C₂²	—	8	8	no	—
429	C₂²	C₂×#127×C₆₇	1	4	yes	—
430	C₂²	C₂×#127×C₆₇	1	2	yes	—
431	C₂²	#576×C₆₇	1	4	yes	—
432	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
433	C₄	—	2	8	no	—
434	C₄	—	4	4	no	—
435	C₄	C₂×#127×C₆₇	1	4	yes	—
436	C₄	C₂×#127×C₆₇	1	2	yes	—
437	C₄	C₂³×D₄×C₆₇	1	1	yes	characteristic
438	C₄	C₂³×D₄×C₆₇	1	1	yes	characteristic
439	C₂	—	2	4	no	—
440	C₂	—	8	8	no	—
441	C₂	—	8	4	no	—
442	C₂	C₆₇×#3155	1	4	yes	direct product factor
443	C₂	C₂²×#127×C₆₇	1	1	yes	characteristic
444	C₂	C₂×#576×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 ◁ C₂×#117×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. C₂²×#135×C₆₇	C₂	semidirect
3. C₂³×D₈×C₆₇	C₂	semidirect
4. C₂×#535×C₆₇	C₂	semidirect
5. C₆₇×#3155	C₂	direct (N × H)
6. C₂×#496×C₆₇	C₂	semidirect
7. C₂×#495×C₆₇	C₂	semidirect
8. C₂×#493×C₆₇	C₂	semidirect
9. C₂×#405×C₆₇	C₂	semidirect
13. C₂×AGL(1,ℤ/8)×C₆₇	C₂²	semidirect
15. C₂×#135×C₆₇	C₂²	semidirect
16. C₂²×Q₁₆×C₆₇	C₂²	semidirect
17. C₂²×SD₁₆×C₆₇	C₂²	semidirect
18. C₂²×SD₁₆×C₆₇	C₂²	semidirect
22. C₂²×D₈×C₆₇	C₂²	semidirect
23. C₂²×D₈×C₆₇	C₂²	semidirect
24. C₂²×D₈×C₆₇	C₂²	semidirect
28. #535×C₆₇	C₂²	semidirect
29. C₂×#127×C₆₇	C₂²	semidirect
30. C₂×#124×C₆₇	C₂²	semidirect
34. #496×C₆₇	C₂²	semidirect
35. #495×C₆₇	C₂²	semidirect
36. #493×C₆₇	C₂²	semidirect
37. #405×C₆₇	C₂²	semidirect
38. C₂×#97×C₆₇	C₂²	semidirect
39. C₂×C₄×C₈×C₆₇	C₂²	semidirect
60. C₂×Q₁₆×C₆₇	C₂³	semidirect
63. C₂×SD₁₆×C₆₇	C₂³	semidirect
64. C₂×SD₁₆×C₆₇	C₂³	semidirect
72. C₂×D₈×C₆₇	C₂³	semidirect
73. C₂×D₈×C₆₇	C₂³	semidirect
74. C₂×M₁₆×C₆₇	D₄	semidirect
76. C₂²×C₈×C₆₇	D₄	semidirect
82. #127×C₆₇	C₂³	semidirect
84. #124×C₆₇	C₂³	semidirect
93. #97×C₆₇	C₂³	semidirect
94. C₄×C₈×C₆₇	C₂³	semidirect
142. M₁₆×C₆₇	C₂×D₄	semidirect
146. C₂×C₈×C₆₇	C₂×D₄	semidirect
216. C₂×#3155	C₆₇	direct (N × H)
220. C₂×C₆₇	#3155	direct (N × H)
223. C₂²×AGL(1,ℤ/8)	C₂×C₆₇	semidirect
224. C₂²×#135	C₂×C₆₇	semidirect
225. C₂³×D₈	C₂×C₆₇	semidirect
226. C₂×#535	C₂×C₆₇	semidirect
227. #3155	C₂×C₆₇	direct (N × H)
228. C₂×#496	C₂×C₆₇	semidirect
229. C₂×#495	C₂×C₆₇	semidirect
230. C₂×#493	C₂×C₆₇	semidirect
231. C₂×#405	C₂×C₆₇	semidirect
232. C₆₇	C₂×#3155	direct (N × H)
236. C₂×AGL(1,ℤ/8)	C₂²×C₆₇	semidirect
238. C₂×#135	C₂²×C₆₇	semidirect
239. C₂²×Q₁₆	C₂²×C₆₇	semidirect
240. C₂²×SD₁₆	C₂²×C₆₇	semidirect
241. C₂²×SD₁₆	C₂²×C₆₇	semidirect
245. C₂²×D₈	C₂²×C₆₇	semidirect
246. C₂²×D₈	C₂²×C₆₇	semidirect
247. C₂²×D₈	C₂²×C₆₇	semidirect
251. #535	C₂²×C₆₇	semidirect
252. C₂×#127	C₂²×C₆₇	semidirect
253. C₂×#124	C₂²×C₆₇	semidirect
257. #496	C₂²×C₆₇	semidirect
258. #495	C₂²×C₆₇	semidirect
259. #493	C₂²×C₆₇	semidirect
260. #405	C₂²×C₆₇	semidirect
261. C₂×#97	C₂²×C₆₇	semidirect
262. C₂×C₄×C₈	C₂²×C₆₇	semidirect
283. C₂×Q₁₆	C₂³×C₆₇	semidirect
286. C₂×SD₁₆	C₂³×C₆₇	semidirect
287. C₂×SD₁₆	C₂³×C₆₇	semidirect
295. C₂×D₈	C₂³×C₆₇	semidirect
296. C₂×D₈	C₂³×C₆₇	semidirect
297. C₂×M₁₆	D₄×C₆₇	semidirect
299. C₂²×C₈	D₄×C₆₇	semidirect
305. #127	C₂³×C₆₇	semidirect
307. #124	C₂³×C₆₇	semidirect
316. #97	C₂³×C₆₇	semidirect
317. C₄×C₈	C₂³×C₆₇	semidirect
365. M₁₆	C₂×D₄×C₆₇	semidirect
369. C₂×C₈	C₂×D₄×C₆₇	semidirect
442. C₂	C₆₇×#3155	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
C₂²×#135×C₆₇	2
C₂³×D₈×C₆₇	2
C₂×#535×C₆₇	2
C₆₇×#3155	2
C₂×#496×C₆₇	2
C₂×#495×C₆₇	2
C₂×#493×C₆₇	2
C₂×#405×C₆₇	2
C₂×#3155	67

Subgroups — C2×C67×#3155 #30739892

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#3155 #30739892