Subgroups — C₂×C₆₇×#2956 #30739431

Order 17152.

2674 conjugacy classes of subgroups.

The subgroup lattice has 434 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	1	yes	maximal, characteristic
2	C₂×#568×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₆₇×#2956	C₂	1	16	yes	maximal, direct product factor
4	C₂×#444×C₆₇	C₂	1	4	yes	maximal, complemented
5	C₂×#443×C₆₇	C₂	1	4	yes	maximal, complemented
6	C₂×#430×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#104×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	#575×C₆₇	C₄	1	8	yes	complemented
11	#575×C₆₇	C₂²	1	8	yes	—
12	C₂×#137×C₆₇	C₄	1	4	yes	complemented
13	C₂×AGL(1,ℤ/8)×C₆₇	C₄	1	4	yes	complemented
14	C₂×#135×C₆₇	C₂²	1	4	yes	—
15	#568×C₆₇	—	2	8	no	—
16	#568×C₆₇	C₂²	1	16	yes	complemented
17	C₂×#131×C₆₇	—	2	4	no	—
18	C₂×#131×C₆₇	C₂²	1	8	yes	complemented
19	C₂²×M₁₆×C₆₇	—	2	4	no	—
20	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
21	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
22	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
23	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
24	#444×C₆₇	C₂²	1	32	yes	complemented
25	#443×C₆₇	C₂²	1	32	yes	complemented
26	#430×C₆₇	C₂²	1	8	yes	complemented
27	C₂×#108×C₆₇	C₂²	1	4	yes	complemented
28	C₂×#104×C₆₇	C₂²	1	4	yes	complemented
29	C₂×#104×C₆₇	C₂²	1	4	yes	complemented
30	C₂×#104×C₆₇	C₂²	1	16	yes	complemented
31	C₄×M₁₆×C₆₇	C₂²	1	8	yes	complemented
32	C₂×#97×C₆₇	C₂²	1	2	yes	complemented
33	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	complemented
34	C₂×#41×C₆₇	—	2	2	no	—
35	C₂×#41×C₆₇	—	2	4	no	—
36	C₂×#41×C₆₇	—	2	2	no	—
37	C₂×#41×C₆₇	—	2	4	no	—
38	C₂×#41×C₆₇	C₂×C₄	1	4	yes	—
39	C₂×#41×C₆₇	C₂×C₄	1	8	yes	—
40	C₂×#41×C₆₇	C₂³	1	4	yes	—
41	C₂×#41×C₆₇	C₂³	1	8	yes	—
42	C₂²×Q₈×C₆₇	C₂×C₄	1	2	yes	—
43	C₂²×D₄×C₆₇	—	2	2	no	—
44	C₂²×D₄×C₆₇	C₂×C₄	1	2	yes	—
45	C₂³×C₄×C₆₇	—	2	2	no	—
46	C₂³×C₄×C₆₇	—	2	1	no	—
47	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
48	#137×C₆₇	C₂×C₄	1	32	yes	complemented
49	AGL(1,ℤ/8)×C₆₇	C₂×C₄	1	32	yes	complemented
50	#135×C₆₇	C₂×C₄	1	16	yes	complemented
51	#135×C₆₇	C₂³	1	16	yes	—
52	C₂×Q₁₆×C₆₇	C₂×C₄	1	4	yes	complemented
53	C₂×SD₁₆×C₆₇	C₂×C₄	1	8	yes	complemented
54	C₂×D₈×C₆₇	C₂×C₄	1	4	yes	complemented
55	#131×C₆₇	—	2	32	no	—
56	#131×C₆₇	—	2	16	no	—
57	#131×C₆₇	C₂³	1	32	yes	complemented
58	C₂×M₁₆×C₆₇	—	2	2	no	—
59	C₂×M₁₆×C₆₇	—	2	4	no	—
60	C₂×M₁₆×C₆₇	—	2	32	no	—
61	C₂×M₁₆×C₆₇	—	2	8	no	—
62	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
63	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
64	C₂×M₁₆×C₆₇	D₄	1	4	yes	—
65	C₂×M₁₆×C₆₇	D₄	1	4	yes	—
66	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
67	C₂×M₁₆×C₆₇	C₂³	1	8	yes	—
68	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
69	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
70	C₂²×C₈×C₆₇	—	2	2	no	—
71	C₂²×C₈×C₆₇	—	2	8	no	—
72	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
73	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
74	C₂×C₄²×C₆₇	—	2	1	no	—
75	C₂×C₄²×C₆₇	—	2	2	no	—
76	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
77	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
78	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
79	#108×C₆₇	C₂³	1	16	yes	complemented
80	#104×C₆₇	C₂³	1	16	yes	complemented
81	#104×C₆₇	C₂³	1	16	yes	complemented
82	#97×C₆₇	C₂³	1	8	yes	complemented
83	C₄×C₈×C₆₇	C₂³	1	8	yes	complemented
84	C₂⁴×C₆₇	—	2	2	no	—
85	C₂⁴×C₆₇	#41	1	1	yes	characteristic
86	#41×C₆₇	—	2	8	no	—
87	#41×C₆₇	—	2	8	no	—
88	#41×C₆₇	—	2	8	no	—
89	#41×C₆₇	—	2	8	no	—
90	#41×C₆₇	—	2	4	no	—
91	#41×C₆₇	—	2	4	no	—
92	#41×C₆₇	—	2	8	no	—
93	#41×C₆₇	C₂²×C₄	1	8	yes	—
94	#41×C₆₇	C₂²×C₄	1	16	yes	—
95	#41×C₆₇	C₂⁴	1	8	yes	—
96	C₂×Q₈×C₆₇	—	2	2	no	—
97	C₂×Q₈×C₆₇	—	2	4	no	—
98	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
99	C₂×Q₈×C₆₇	C₂²×C₄	1	8	yes	—
100	C₂×D₄×C₆₇	—	2	4	no	—
101	C₂×D₄×C₆₇	—	2	8	no	—
102	C₂×D₄×C₆₇	—	2	4	no	—
103	C₂×D₄×C₆₇	—	2	8	no	—
104	C₂×D₄×C₆₇	—	2	2	no	—
105	C₂×D₄×C₆₇	—	2	4	no	—
106	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
107	C₂×D₄×C₆₇	C₂²×C₄	1	8	yes	—
108	C₂²×C₄×C₆₇	—	2	2	no	—
109	C₂²×C₄×C₆₇	—	2	1	no	—
110	C₂²×C₄×C₆₇	—	2	4	no	—
111	C₂²×C₄×C₆₇	—	2	8	no	—
112	C₂²×C₄×C₆₇	—	2	8	no	—
113	C₂²×C₄×C₆₇	—	2	4	no	—
114	C₂²×C₄×C₆₇	—	2	4	no	—
115	C₂²×C₄×C₆₇	—	2	1	no	—
116	C₂²×C₄×C₆₇	—	4	4	no	—
117	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
118	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
119	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
120	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
121	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
122	C₂²×C₄×C₆₇	#41	1	2	yes	—
123	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
124	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
125	Q₁₆×C₆₇	C₂²×C₄	1	16	yes	complemented
126	SD₁₆×C₆₇	C₂²×C₄	1	32	yes	complemented
127	D₈×C₆₇	C₂²×C₄	1	16	yes	complemented
128	M₁₆×C₆₇	—	2	8	no	—
129	M₁₆×C₆₇	—	2	4	no	—
130	M₁₆×C₆₇	—	2	32	no	—
131	M₁₆×C₆₇	C₂²×C₄	1	16	yes	—
132	M₁₆×C₆₇	C₂×D₄	1	16	yes	—
133	M₁₆×C₆₇	C₂⁴	1	8	yes	—
134	C₂×C₈×C₆₇	—	2	8	no	—
135	C₂×C₈×C₆₇	—	2	4	no	—
136	C₂×C₈×C₆₇	—	2	32	no	—
137	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
138	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
139	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
140	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
141	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
142	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
143	C₄²×C₆₇	—	2	1	no	—
144	C₄²×C₆₇	—	2	4	no	—
145	C₄²×C₆₇	—	2	1	no	—
146	C₄²×C₆₇	C₂⁴	1	2	yes	—
147	C₄²×C₆₇	C₂⁴	1	2	yes	—
148	C₂³×C₆₇	—	2	4	no	—
149	C₂³×C₆₇	—	2	8	no	—
150	C₂³×C₆₇	—	4	2	no	—
151	C₂³×C₆₇	—	8	4	no	—
152	C₂³×C₆₇	C₄×D₄	1	2	yes	—
153	C₂³×C₆₇	C₄×D₄	1	2	yes	—
154	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
155	C₂³×C₆₇	C₂×#41	1	2	yes	—
156	Q₈×C₆₇	—	2	8	no	—
157	Q₈×C₆₇	—	2	4	no	—
158	Q₈×C₆₇	C₂³×C₄	1	8	yes	—
159	D₄×C₆₇	—	2	16	no	—
160	D₄×C₆₇	—	2	8	no	—
161	D₄×C₆₇	—	2	8	no	—
162	D₄×C₆₇	—	2	8	no	—
163	D₄×C₆₇	—	2	4	no	—
164	D₄×C₆₇	C₂³×C₄	1	8	yes	—
165	C₂×C₄×C₆₇	—	2	4	no	—
166	C₂×C₄×C₆₇	—	2	8	no	—
167	C₂×C₄×C₆₇	—	2	4	no	—
168	C₂×C₄×C₆₇	—	2	2	no	—
169	C₂×C₄×C₆₇	—	2	8	no	—
170	C₂×C₄×C₆₇	—	2	16	no	—
171	C₂×C₄×C₆₇	—	2	8	no	—
172	C₂×C₄×C₆₇	—	2	2	no	—
173	C₂×C₄×C₆₇	—	4	8	no	—
174	C₂×C₄×C₆₇	—	4	2	no	—
175	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
176	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
177	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
178	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
179	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
180	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
181	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
182	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
183	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
184	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
185	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
186	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
187	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
188	C₈×C₆₇	—	2	8	no	—
189	C₈×C₆₇	C₂³×C₄	1	8	yes	—
190	C₈×C₆₇	C₂²×D₄	1	8	yes	—
191	C₂²×C₆₇	—	2	2	no	—
192	C₂²×C₆₇	—	2	4	no	—
193	C₂²×C₆₇	—	2	8	no	—
194	C₂²×C₆₇	—	4	2	no	—
195	C₂²×C₆₇	—	4	2	no	—
196	C₂²×C₆₇	—	4	8	no	—
197	C₂²×C₆₇	—	8	8	no	—
198	C₂²×C₆₇	C₂×C₄×D₄	1	4	yes	—
199	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
200	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
201	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
202	C₄×C₆₇	—	2	4	no	—
203	C₄×C₆₇	—	2	8	no	—
204	C₄×C₆₇	—	4	2	no	—
205	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
206	C₄×C₆₇	C₂×C₄×D₄	1	4	yes	—
207	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
208	C₄×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
209	C₄×C₆₇	C₂³×D₄	1	1	yes	characteristic
210	C₂×#2956	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
211	C₂×C₆₇	—	2	1	no	—
212	C₂×C₆₇	—	2	4	no	—
213	C₂×C₆₇	—	2	1	no	—
214	C₂×C₆₇	—	4	8	no	—
215	C₂×C₆₇	#2956	1	2	yes	direct product factor
216	C₂×C₆₇	C₂²×C₄×D₄	1	1	yes	characteristic
217	C₂×#575	C₂×C₆₇	1	1	yes	characteristic
218	C₂×#568	C₂×C₆₇	1	2	yes	complemented
219	#2956	C₂×C₆₇	1	16	yes	direct product factor
220	C₂×#444	C₂×C₆₇	1	4	yes	complemented
221	C₂×#443	C₂×C₆₇	1	4	yes	complemented
222	C₂×#430	C₂×C₆₇	1	1	yes	characteristic, complemented
223	C₂²×#104	C₂×C₆₇	1	2	yes	complemented
224	C₂×C₄×M₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
225	C₆₇	C₂×#2956	1	1	yes	characteristic, direct product factor, Sylow
226	C₂²×#41	C₂²×C₆₇	1	2	yes	—
227	#575	C₄×C₆₇	1	8	yes	complemented
228	#575	C₂²×C₆₇	1	8	yes	—
229	C₂×#137	C₄×C₆₇	1	4	yes	complemented
230	C₂×AGL(1,ℤ/8)	C₄×C₆₇	1	4	yes	complemented
231	C₂×#135	C₂²×C₆₇	1	4	yes	—
232	#568	—	2	8	no	—
233	#568	C₂²×C₆₇	1	16	yes	complemented
234	C₂×#131	—	2	4	no	—
235	C₂×#131	C₂²×C₆₇	1	8	yes	complemented
236	C₂²×M₁₆	—	2	4	no	—
237	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
238	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
239	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
240	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
241	#444	C₂²×C₆₇	1	32	yes	complemented
242	#443	C₂²×C₆₇	1	32	yes	complemented
243	#430	C₂²×C₆₇	1	8	yes	complemented
244	C₂×#108	C₂²×C₆₇	1	4	yes	complemented
245	C₂×#104	C₂²×C₆₇	1	4	yes	complemented
246	C₂×#104	C₂²×C₆₇	1	4	yes	complemented
247	C₂×#104	C₂²×C₆₇	1	16	yes	complemented
248	C₄×M₁₆	C₂²×C₆₇	1	8	yes	complemented
249	C₂×#97	C₂²×C₆₇	1	2	yes	complemented
250	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	complemented
251	C₂×#41	—	2	2	no	—
252	C₂×#41	—	2	4	no	—
253	C₂×#41	—	2	2	no	—
254	C₂×#41	—	2	4	no	—
255	C₂×#41	C₂×C₄×C₆₇	1	4	yes	—
256	C₂×#41	C₂×C₄×C₆₇	1	8	yes	—
257	C₂×#41	C₂³×C₆₇	1	4	yes	—
258	C₂×#41	C₂³×C₆₇	1	8	yes	—
259	C₂²×Q₈	C₂×C₄×C₆₇	1	2	yes	—
260	C₂²×D₄	—	2	2	no	—
261	C₂²×D₄	C₂×C₄×C₆₇	1	2	yes	—
262	C₂³×C₄	—	2	2	no	—
263	C₂³×C₄	—	2	1	no	—
264	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
265	#137	C₂×C₄×C₆₇	1	32	yes	complemented
266	AGL(1,ℤ/8)	C₂×C₄×C₆₇	1	32	yes	complemented
267	#135	C₂×C₄×C₆₇	1	16	yes	complemented
268	#135	C₂³×C₆₇	1	16	yes	—
269	C₂×Q₁₆	C₂×C₄×C₆₇	1	4	yes	complemented
270	C₂×SD₁₆	C₂×C₄×C₆₇	1	8	yes	complemented
271	C₂×D₈	C₂×C₄×C₆₇	1	4	yes	complemented
272	#131	—	2	32	no	—
273	#131	—	2	16	no	—
274	#131	C₂³×C₆₇	1	32	yes	complemented
275	C₂×M₁₆	—	2	2	no	—
276	C₂×M₁₆	—	2	4	no	—
277	C₂×M₁₆	—	2	32	no	—
278	C₂×M₁₆	—	2	8	no	—
279	C₂×M₁₆	C₂×C₄×C₆₇	1	4	yes	—
280	C₂×M₁₆	C₂×C₄×C₆₇	1	4	yes	—
281	C₂×M₁₆	D₄×C₆₇	1	4	yes	—
282	C₂×M₁₆	D₄×C₆₇	1	4	yes	—
283	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
284	C₂×M₁₆	C₂³×C₆₇	1	8	yes	—
285	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
286	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
287	C₂²×C₈	—	2	2	no	—
288	C₂²×C₈	—	2	8	no	—
289	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
290	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
291	C₂×C₄²	—	2	1	no	—
292	C₂×C₄²	—	2	2	no	—
293	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
294	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
295	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
296	#108	C₂³×C₆₇	1	16	yes	complemented
297	#104	C₂³×C₆₇	1	16	yes	complemented
298	#104	C₂³×C₆₇	1	16	yes	complemented
299	#97	C₂³×C₆₇	1	8	yes	complemented
300	C₄×C₈	C₂³×C₆₇	1	8	yes	complemented
301	C₂⁴	—	2	2	no	—
302	C₂⁴	#41×C₆₇	1	1	yes	characteristic
303	#41	—	2	8	no	—
304	#41	—	2	8	no	—
305	#41	—	2	8	no	—
306	#41	—	2	8	no	—
307	#41	—	2	4	no	—
308	#41	—	2	4	no	—
309	#41	—	2	8	no	—
310	#41	C₂²×C₄×C₆₇	1	8	yes	—
311	#41	C₂²×C₄×C₆₇	1	16	yes	—
312	#41	C₂⁴×C₆₇	1	8	yes	—
313	C₂×Q₈	—	2	2	no	—
314	C₂×Q₈	—	2	4	no	—
315	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
316	C₂×Q₈	C₂²×C₄×C₆₇	1	8	yes	—
317	C₂×D₄	—	2	4	no	—
318	C₂×D₄	—	2	8	no	—
319	C₂×D₄	—	2	4	no	—
320	C₂×D₄	—	2	8	no	—
321	C₂×D₄	—	2	2	no	—
322	C₂×D₄	—	2	4	no	—
323	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
324	C₂×D₄	C₂²×C₄×C₆₇	1	8	yes	—
325	C₂²×C₄	—	2	2	no	—
326	C₂²×C₄	—	2	1	no	—
327	C₂²×C₄	—	2	4	no	—
328	C₂²×C₄	—	2	8	no	—
329	C₂²×C₄	—	2	4	no	—
330	C₂²×C₄	—	2	8	no	—
331	C₂²×C₄	—	2	4	no	—
332	C₂²×C₄	—	2	1	no	—
333	C₂²×C₄	—	4	4	no	—
334	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
335	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
336	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
337	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
338	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
339	C₂²×C₄	#41×C₆₇	1	2	yes	—
340	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
341	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
342	Q₁₆	C₂²×C₄×C₆₇	1	16	yes	complemented
343	SD₁₆	C₂²×C₄×C₆₇	1	32	yes	complemented
344	D₈	C₂²×C₄×C₆₇	1	16	yes	complemented
345	M₁₆	—	2	8	no	—
346	M₁₆	—	2	4	no	—
347	M₁₆	—	2	32	no	—
348	M₁₆	C₂²×C₄×C₆₇	1	16	yes	—
349	M₁₆	C₂×D₄×C₆₇	1	16	yes	—
350	M₁₆	C₂⁴×C₆₇	1	8	yes	—
351	C₂×C₈	—	2	4	no	—
352	C₂×C₈	—	2	8	no	—
353	C₂×C₈	—	2	32	no	—
354	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
355	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
356	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
357	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
358	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
359	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
360	C₄²	—	2	1	no	—
361	C₄²	—	2	4	no	—
362	C₄²	—	2	1	no	—
363	C₄²	C₂⁴×C₆₇	1	2	yes	—
364	C₄²	C₂⁴×C₆₇	1	2	yes	—
365	C₂³	—	2	4	no	—
366	C₂³	—	2	8	no	—
367	C₂³	—	4	2	no	—
368	C₂³	—	8	4	no	—
369	C₂³	C₄×D₄×C₆₇	1	2	yes	—
370	C₂³	C₄×D₄×C₆₇	1	2	yes	—
371	C₂³	C₂×#41×C₆₇	1	1	yes	characteristic
372	C₂³	C₂×#41×C₆₇	1	2	yes	—
373	Q₈	—	2	8	no	—
374	Q₈	—	2	4	no	—
375	Q₈	C₂³×C₄×C₆₇	1	8	yes	—
376	D₄	—	2	16	no	—
377	D₄	—	2	8	no	—
378	D₄	—	2	8	no	—
379	D₄	—	2	8	no	—
380	D₄	—	2	4	no	—
381	D₄	C₂³×C₄×C₆₇	1	8	yes	—
382	C₂×C₄	—	2	4	no	—
383	C₂×C₄	—	2	8	no	—
384	C₂×C₄	—	2	4	no	—
385	C₂×C₄	—	2	2	no	—
386	C₂×C₄	—	2	8	no	—
387	C₂×C₄	—	2	16	no	—
388	C₂×C₄	—	2	8	no	—
389	C₂×C₄	—	2	2	no	—
390	C₂×C₄	—	4	8	no	—
391	C₂×C₄	—	4	2	no	—
392	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
393	C₂×C₄	C₄×D₄×C₆₇	1	4	yes	—
394	C₂×C₄	C₄×D₄×C₆₇	1	4	yes	—
395	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
396	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
397	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
398	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
399	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
400	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
401	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
402	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
403	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
404	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
405	C₈	—	2	8	no	—
406	C₈	C₂³×C₄×C₆₇	1	8	yes	—
407	C₈	C₂²×D₄×C₆₇	1	8	yes	—
408	C₂²	—	2	2	no	—
409	C₂²	—	2	4	no	—
410	C₂²	—	2	8	no	—
411	C₂²	—	4	2	no	—
412	C₂²	—	4	2	no	—
413	C₂²	—	4	8	no	—
414	C₂²	—	8	8	no	—
415	C₂²	C₂×C₄×D₄×C₆₇	1	4	yes	—
416	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
417	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
418	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
419	C₄	—	2	4	no	—
420	C₄	—	2	8	no	—
421	C₄	—	4	2	no	—
422	C₄	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
423	C₄	C₂×C₄×D₄×C₆₇	1	4	yes	—
424	C₄	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
425	C₄	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
426	C₄	C₂³×D₄×C₆₇	1	1	yes	characteristic
427	C₂	—	2	1	no	—
428	C₂	—	2	4	no	—
429	C₂	—	2	1	no	—
430	C₂	—	4	8	no	—
431	C₂	C₆₇×#2956	1	2	yes	direct product factor
432	C₂	C₂²×C₄×D₄×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂²×D₄
commutator	C₄	C₂⁴×C₄×C₆₇
Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂×#575×C₆₇	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂²×M₁₆×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#568×C₆₇	C₂	semidirect
3. C₆₇×#2956	C₂	direct (N × H)
4. C₂×#444×C₆₇	C₂	semidirect
5. C₂×#443×C₆₇	C₂	semidirect
6. C₂×#430×C₆₇	C₂	semidirect
7. C₂²×#104×C₆₇	C₂	semidirect
8. C₂×C₄×M₁₆×C₆₇	C₂	semidirect
10. #575×C₆₇	C₄	semidirect
12. C₂×#137×C₆₇	C₄	semidirect
13. C₂×AGL(1,ℤ/8)×C₆₇	C₄	semidirect
16. #568×C₆₇	C₂²	semidirect
18. C₂×#131×C₆₇	C₂²	semidirect
24. #444×C₆₇	C₂²	semidirect
25. #443×C₆₇	C₂²	semidirect
26. #430×C₆₇	C₂²	semidirect
27. C₂×#108×C₆₇	C₂²	semidirect
28. C₂×#104×C₆₇	C₂²	semidirect
29. C₂×#104×C₆₇	C₂²	semidirect
30. C₂×#104×C₆₇	C₂²	semidirect
31. C₄×M₁₆×C₆₇	C₂²	semidirect
32. C₂×#97×C₆₇	C₂²	semidirect
33. C₂×C₄×C₈×C₆₇	C₂²	semidirect
48. #137×C₆₇	C₂×C₄	semidirect
49. AGL(1,ℤ/8)×C₆₇	C₂×C₄	semidirect
50. #135×C₆₇	C₂×C₄	semidirect
52. C₂×Q₁₆×C₆₇	C₂×C₄	semidirect
53. C₂×SD₁₆×C₆₇	C₂×C₄	semidirect
54. C₂×D₈×C₆₇	C₂×C₄	semidirect
57. #131×C₆₇	C₂³	semidirect
79. #108×C₆₇	C₂³	semidirect
80. #104×C₆₇	C₂³	semidirect
81. #104×C₆₇	C₂³	semidirect
82. #97×C₆₇	C₂³	semidirect
83. C₄×C₈×C₆₇	C₂³	semidirect
125. Q₁₆×C₆₇	C₂²×C₄	semidirect
126. SD₁₆×C₆₇	C₂²×C₄	semidirect
127. D₈×C₆₇	C₂²×C₄	semidirect
210. C₂×#2956	C₆₇	direct (N × H)
215. C₂×C₆₇	#2956	direct (N × H)
218. C₂×#568	C₂×C₆₇	semidirect
219. #2956	C₂×C₆₇	direct (N × H)
220. C₂×#444	C₂×C₆₇	semidirect
221. C₂×#443	C₂×C₆₇	semidirect
222. C₂×#430	C₂×C₆₇	semidirect
223. C₂²×#104	C₂×C₆₇	semidirect
224. C₂×C₄×M₁₆	C₂×C₆₇	semidirect
225. C₆₇	C₂×#2956	direct (N × H)
227. #575	C₄×C₆₇	semidirect
229. C₂×#137	C₄×C₆₇	semidirect
230. C₂×AGL(1,ℤ/8)	C₄×C₆₇	semidirect
233. #568	C₂²×C₆₇	semidirect
235. C₂×#131	C₂²×C₆₇	semidirect
241. #444	C₂²×C₆₇	semidirect
242. #443	C₂²×C₆₇	semidirect
243. #430	C₂²×C₆₇	semidirect
244. C₂×#108	C₂²×C₆₇	semidirect
245. C₂×#104	C₂²×C₆₇	semidirect
246. C₂×#104	C₂²×C₆₇	semidirect
247. C₂×#104	C₂²×C₆₇	semidirect
248. C₄×M₁₆	C₂²×C₆₇	semidirect
249. C₂×#97	C₂²×C₆₇	semidirect
250. C₂×C₄×C₈	C₂²×C₆₇	semidirect
265. #137	C₂×C₄×C₆₇	semidirect
266. AGL(1,ℤ/8)	C₂×C₄×C₆₇	semidirect
267. #135	C₂×C₄×C₆₇	semidirect
269. C₂×Q₁₆	C₂×C₄×C₆₇	semidirect
270. C₂×SD₁₆	C₂×C₄×C₆₇	semidirect
271. C₂×D₈	C₂×C₄×C₆₇	semidirect
274. #131	C₂³×C₆₇	semidirect
296. #108	C₂³×C₆₇	semidirect
297. #104	C₂³×C₆₇	semidirect
298. #104	C₂³×C₆₇	semidirect
299. #97	C₂³×C₆₇	semidirect
300. C₄×C₈	C₂³×C₆₇	semidirect
342. Q₁₆	C₂²×C₄×C₆₇	semidirect
343. SD₁₆	C₂²×C₄×C₆₇	semidirect
344. D₈	C₂²×C₄×C₆₇	semidirect
431. C₂	C₆₇×#2956	direct (N × H)

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×#568×C₆₇	2
C₆₇×#2956	2
C₂×#444×C₆₇	2
C₂×#443×C₆₇	2
C₂×#430×C₆₇	2
C₂²×#104×C₆₇	2
C₂×C₄×M₁₆×C₆₇	2
C₂×#2956	67

Subgroups — C2×C67×#2956 #30739431

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2956 #30739431