Subgroups — C₂×C₆₇×#2955 #30739430

Order 17152.

2666 conjugacy classes of subgroups.

The subgroup lattice has 452 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	C₂²×#131×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₆₇×#2955	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	C₂×#421×C₆₇	C₂	1	2	yes	maximal, complemented
8	C₂×#405×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	C₂×#131×C₆₇	—	2	2	no	—
16	C₂×#131×C₆₇	—	2	8	no	—
17	C₂×#131×C₆₇	—	2	2	no	—
18	C₂×#131×C₆₇	C₂²	1	4	yes	complemented
19	C₂×#131×C₆₇	C₂²	1	4	yes	complemented
20	C₂×#131×C₆₇	C₂²	1	16	yes	complemented
21	C₂²×M₁₆×C₆₇	—	2	2	no	—
22	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
23	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
24	C₂³×C₈×C₆₇	—	2	2	no	—
25	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
27	#444×C₆₇	C₂²	1	32	yes	complemented
28	#443×C₆₇	C₂²	1	32	yes	complemented
29	#430×C₆₇	C₂²	1	8	yes	complemented
30	C₂×#108×C₆₇	C₂²	1	4	yes	complemented
31	#421×C₆₇	C₂²	1	16	yes	complemented
32	C₂×#104×C₆₇	C₂²	1	8	yes	complemented
33	#405×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	C₂×#41×C₆₇	—	2	2	no	—
37	C₂×#41×C₆₇	—	2	4	no	—
38	C₂×#41×C₆₇	—	2	2	no	—
39	C₂×#41×C₆₇	—	2	4	no	—
40	C₂×#41×C₆₇	C₂×C₄	1	4	yes	—
41	C₂×#41×C₆₇	C₂×C₄	1	8	yes	—
42	C₂×#41×C₆₇	C₂³	1	4	yes	—
43	C₂×#41×C₆₇	C₂³	1	8	yes	—
44	C₂²×Q₈×C₆₇	C₂×C₄	1	2	yes	—
45	C₂²×D₄×C₆₇	—	2	2	no	—
46	C₂²×D₄×C₆₇	C₂×C₄	1	2	yes	—
47	C₂³×C₄×C₆₇	—	2	2	no	—
48	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
49	#137×C₆₇	C₂×C₄	1	32	yes	complemented
50	AGL(1,ℤ/8)×C₆₇	C₂×C₄	1	32	yes	complemented
51	#135×C₆₇	C₂×C₄	1	16	yes	—
52	#135×C₆₇	C₂³	1	16	yes	—
53	C₂×Q₁₆×C₆₇	C₂×C₄	1	4	yes	—
54	C₂×SD₁₆×C₆₇	C₂×C₄	1	8	yes	—
55	C₂×D₈×C₆₇	C₂×C₄	1	4	yes	—
56	#131×C₆₇	—	2	16	no	—
57	#131×C₆₇	—	2	16	no	—
58	#131×C₆₇	—	2	8	no	—
59	#131×C₆₇	—	2	8	no	—
60	#131×C₆₇	C₂³	1	16	yes	complemented
61	#131×C₆₇	C₂³	1	16	yes	complemented
62	C₂×M₁₆×C₆₇	—	2	2	no	—
63	C₂×M₁₆×C₆₇	—	2	4	no	—
64	C₂×M₁₆×C₆₇	—	2	4	no	—
65	C₂×M₁₆×C₆₇	—	2	4	no	—
66	C₂×M₁₆×C₆₇	—	2	16	no	—
67	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
68	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
69	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
70	C₂×M₁₆×C₆₇	C₂³	1	8	yes	—
71	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
72	C₂²×C₈×C₆₇	—	2	2	no	—
73	C₂²×C₈×C₆₇	—	2	4	no	—
74	C₂²×C₈×C₆₇	—	2	16	no	—
75	C₂²×C₈×C₆₇	—	2	4	no	—
76	C₂²×C₈×C₆₇	D₄	1	2	yes	—
77	C₂²×C₈×C₆₇	D₄	1	4	yes	—
78	C₂²×C₈×C₆₇	D₄	1	2	yes	—
79	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
80	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
81	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
82	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
83	#117×C₆₇	—	2	2	no	—
84	#117×C₆₇	C₂³	1	4	yes	—
85	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
86	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
87	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
88	#108×C₆₇	C₂³	1	16	yes	complemented
89	#104×C₆₇	C₂³	1	32	yes	complemented
90	#97×C₆₇	C₂³	1	8	yes	complemented
91	C₄×C₈×C₆₇	C₂³	1	8	yes	complemented
92	C₂⁴×C₆₇	—	2	2	no	—
93	C₂⁴×C₆₇	#41	1	1	yes	characteristic
94	#41×C₆₇	—	2	8	no	—
95	#41×C₆₇	—	2	8	no	—
96	#41×C₆₇	—	2	8	no	—
97	#41×C₆₇	—	2	8	no	—
98	#41×C₆₇	—	2	4	no	—
99	#41×C₆₇	—	2	4	no	—
100	#41×C₆₇	—	2	8	no	—
101	#41×C₆₇	C₂²×C₄	1	8	yes	—
102	#41×C₆₇	C₂²×C₄	1	16	yes	—
103	#41×C₆₇	C₂⁴	1	8	yes	—
104	C₂×Q₈×C₆₇	—	2	2	no	—
105	C₂×Q₈×C₆₇	—	2	4	no	—
106	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
107	C₂×Q₈×C₆₇	C₂²×C₄	1	8	yes	—
108	C₂×D₄×C₆₇	—	2	4	no	—
109	C₂×D₄×C₆₇	—	2	8	no	—
110	C₂×D₄×C₆₇	—	2	4	no	—
111	C₂×D₄×C₆₇	—	2	8	no	—
112	C₂×D₄×C₆₇	—	2	2	no	—
113	C₂×D₄×C₆₇	—	2	4	no	—
114	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
115	C₂×D₄×C₆₇	C₂²×C₄	1	8	yes	—
116	C₂²×C₄×C₆₇	—	2	2	no	—
117	C₂²×C₄×C₆₇	—	2	4	no	—
118	C₂²×C₄×C₆₇	—	2	8	no	—
119	C₂²×C₄×C₆₇	—	2	4	no	—
120	C₂²×C₄×C₆₇	—	2	8	no	—
121	C₂²×C₄×C₆₇	—	2	2	no	—
122	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
123	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
124	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
125	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
126	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
127	C₂²×C₄×C₆₇	#41	1	2	yes	—
128	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
129	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
130	Q₁₆×C₆₇	C₂²×C₄	1	16	yes	—
131	SD₁₆×C₆₇	C₂²×C₄	1	32	yes	—
132	D₈×C₆₇	C₂²×C₄	1	16	yes	—
133	M₁₆×C₆₇	—	2	8	no	—
134	M₁₆×C₆₇	—	2	4	no	—
135	M₁₆×C₆₇	—	2	16	no	—
136	M₁₆×C₆₇	—	2	16	no	—
137	M₁₆×C₆₇	C₂²×C₄	1	16	yes	—
138	M₁₆×C₆₇	C₂⁴	1	8	yes	—
139	C₂×C₈×C₆₇	—	2	4	no	—
140	C₂×C₈×C₆₇	—	2	8	no	—
141	C₂×C₈×C₆₇	—	2	16	no	—
142	C₂×C₈×C₆₇	—	2	16	no	—
143	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
144	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
145	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
146	C₂×C₈×C₆₇	C₂×D₄	1	8	yes	—
147	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
148	C₂×C₈×C₆₇	C₂×D₄	1	8	yes	—
149	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
150	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
151	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
152	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
153	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
154	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
155	C₄⋊C₄×C₆₇	—	2	4	no	—
156	C₂²⋊C₄×C₆₇	—	2	4	no	—
157	C₄²×C₆₇	—	2	2	no	—
158	C₄²×C₆₇	C₂⁴	1	4	yes	—
159	C₂³×C₆₇	—	2	8	no	—
160	C₂³×C₆₇	—	2	4	no	—
161	C₂³×C₆₇	—	4	2	no	—
162	C₂³×C₆₇	—	4	8	no	—
163	C₂³×C₆₇	C₄×D₄	1	2	yes	—
164	C₂³×C₆₇	C₄×D₄	1	2	yes	—
165	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
166	C₂³×C₆₇	C₂×#41	1	2	yes	—
167	Q₈×C₆₇	—	2	8	no	—
168	Q₈×C₆₇	—	2	4	no	—
169	Q₈×C₆₇	C₂³×C₄	1	8	yes	—
170	D₄×C₆₇	—	2	16	no	—
171	D₄×C₆₇	—	2	8	no	—
172	D₄×C₆₇	—	2	8	no	—
173	D₄×C₆₇	—	2	8	no	—
174	D₄×C₆₇	—	2	4	no	—
175	D₄×C₆₇	C₂³×C₄	1	8	yes	—
176	C₂×C₄×C₆₇	—	2	8	no	—
177	C₂×C₄×C₆₇	—	2	4	no	—
178	C₂×C₄×C₆₇	—	2	8	no	—
179	C₂×C₄×C₆₇	—	2	16	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	C₄×D₄	1	2	yes	—
184	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
185	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
186	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
187	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
188	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
189	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
190	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
191	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
192	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
193	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
194	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
195	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
196	C₈×C₆₇	—	2	8	no	—
197	C₈×C₆₇	C₂³×C₄	1	8	yes	—
198	C₈×C₆₇	C₂²×D₄	1	2	yes	—
199	C₈×C₆₇	C₂²×D₄	1	4	yes	—
200	C₈×C₆₇	C₂²×D₄	1	2	yes	—
201	C₂²×C₆₇	—	2	2	no	—
202	C₂²×C₆₇	—	2	4	no	—
203	C₂²×C₆₇	—	2	8	no	—
204	C₂²×C₆₇	—	4	2	no	—
205	C₂²×C₆₇	—	4	2	no	—
206	C₂²×C₆₇	—	4	16	no	—
207	C₂²×C₆₇	—	4	8	no	—
208	C₂²×C₆₇	C₂×C₄×D₄	1	4	yes	—
209	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
210	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
211	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
212	C₄×C₆₇	—	2	8	no	—
213	C₄×C₆₇	—	4	4	no	—
214	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
215	C₄×C₆₇	C₂×C₄×D₄	1	4	yes	—
216	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
217	C₄×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
218	C₄×C₆₇	C₂³×D₄	1	1	yes	characteristic
219	C₂×#2955	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
220	C₂×C₆₇	—	2	1	no	—
221	C₂×C₆₇	—	2	4	no	—
222	C₂×C₆₇	—	2	1	no	—
223	C₂×C₆₇	—	4	8	no	—
224	C₂×C₆₇	#2955	1	2	yes	direct product factor
225	C₂×C₆₇	C₂²×C₄×D₄	1	1	yes	characteristic
226	C₂×#575	C₂×C₆₇	1	1	yes	characteristic
227	C₂²×#131	C₂×C₆₇	1	2	yes	complemented
228	#2955	C₂×C₆₇	1	16	yes	direct product factor
229	C₂×#444	C₂×C₆₇	1	4	yes	complemented
230	C₂×#443	C₂×C₆₇	1	4	yes	complemented
231	C₂×#430	C₂×C₆₇	1	1	yes	characteristic, complemented
232	C₂×#421	C₂×C₆₇	1	2	yes	complemented
233	C₂×#405	C₂×C₆₇	1	1	yes	characteristic, complemented
234	C₆₇	C₂×#2955	1	1	yes	characteristic, direct product factor, Sylow
235	C₂²×#41	C₂²×C₆₇	1	2	yes	—
236	#575	C₄×C₆₇	1	8	yes	complemented
237	#575	C₂²×C₆₇	1	8	yes	—
238	C₂×#137	C₄×C₆₇	1	4	yes	complemented
239	C₂×AGL(1,ℤ/8)	C₄×C₆₇	1	4	yes	complemented
240	C₂×#135	C₂²×C₆₇	1	4	yes	—
241	C₂×#131	—	2	2	no	—
242	C₂×#131	—	2	2	no	—
243	C₂×#131	—	2	8	no	—
244	C₂×#131	C₂²×C₆₇	1	4	yes	complemented
245	C₂×#131	C₂²×C₆₇	1	4	yes	complemented
246	C₂×#131	C₂²×C₆₇	1	16	yes	complemented
247	C₂²×M₁₆	—	2	2	no	—
248	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
249	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
250	C₂³×C₈	—	2	2	no	—
251	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
252	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
253	#444	C₂²×C₆₇	1	32	yes	complemented
254	#443	C₂²×C₆₇	1	32	yes	complemented
255	#430	C₂²×C₆₇	1	8	yes	complemented
256	C₂×#108	C₂²×C₆₇	1	4	yes	complemented
257	#421	C₂²×C₆₇	1	16	yes	complemented
258	C₂×#104	C₂²×C₆₇	1	8	yes	complemented
259	#405	C₂²×C₆₇	1	8	yes	complemented
260	C₂×#97	C₂²×C₆₇	1	2	yes	complemented
261	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	complemented
262	C₂×#41	—	2	2	no	—
263	C₂×#41	—	2	4	no	—
264	C₂×#41	—	2	2	no	—
265	C₂×#41	—	2	4	no	—
266	C₂×#41	C₂×C₄×C₆₇	1	4	yes	—
267	C₂×#41	C₂×C₄×C₆₇	1	8	yes	—
268	C₂×#41	C₂³×C₆₇	1	4	yes	—
269	C₂×#41	C₂³×C₆₇	1	8	yes	—
270	C₂²×Q₈	C₂×C₄×C₆₇	1	2	yes	—
271	C₂²×D₄	—	2	2	no	—
272	C₂²×D₄	C₂×C₄×C₆₇	1	2	yes	—
273	C₂³×C₄	—	2	2	no	—
274	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
275	#137	C₂×C₄×C₆₇	1	32	yes	complemented
276	AGL(1,ℤ/8)	C₂×C₄×C₆₇	1	32	yes	complemented
277	#135	C₂×C₄×C₆₇	1	16	yes	—
278	#135	C₂³×C₆₇	1	16	yes	—
279	C₂×Q₁₆	C₂×C₄×C₆₇	1	4	yes	—
280	C₂×SD₁₆	C₂×C₄×C₆₇	1	8	yes	—
281	C₂×D₈	C₂×C₄×C₆₇	1	4	yes	—
282	#131	—	2	16	no	—
283	#131	—	2	16	no	—
284	#131	—	2	8	no	—
285	#131	—	2	8	no	—
286	#131	C₂³×C₆₇	1	16	yes	complemented
287	#131	C₂³×C₆₇	1	16	yes	complemented
288	C₂×M₁₆	—	2	2	no	—
289	C₂×M₁₆	—	2	4	no	—
290	C₂×M₁₆	—	2	4	no	—
291	C₂×M₁₆	—	2	4	no	—
292	C₂×M₁₆	—	2	16	no	—
293	C₂×M₁₆	C₂×C₄×C₆₇	1	4	yes	—
294	C₂×M₁₆	C₂×C₄×C₆₇	1	4	yes	—
295	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
296	C₂×M₁₆	C₂³×C₆₇	1	8	yes	—
297	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
298	C₂²×C₈	—	2	2	no	—
299	C₂²×C₈	—	2	4	no	—
300	C₂²×C₈	—	2	4	no	—
301	C₂²×C₈	—	2	16	no	—
302	C₂²×C₈	D₄×C₆₇	1	2	yes	—
303	C₂²×C₈	D₄×C₆₇	1	2	yes	—
304	C₂²×C₈	D₄×C₆₇	1	4	yes	—
305	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
306	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
307	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
308	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
309	#117	—	2	2	no	—
310	#117	C₂³×C₆₇	1	4	yes	—
311	C₂×C₄⋊C₄	—	2	1	no	—
312	C₂×C₂²⋊C₄	—	2	1	no	—
313	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
314	#108	C₂³×C₆₇	1	16	yes	complemented
315	#104	C₂³×C₆₇	1	32	yes	complemented
316	#97	C₂³×C₆₇	1	8	yes	complemented
317	C₄×C₈	C₂³×C₆₇	1	8	yes	complemented
318	C₂⁴	—	2	2	no	—
319	C₂⁴	#41×C₆₇	1	1	yes	characteristic
320	#41	—	2	8	no	—
321	#41	—	2	8	no	—
322	#41	—	2	8	no	—
323	#41	—	2	8	no	—
324	#41	—	2	4	no	—
325	#41	—	2	4	no	—
326	#41	—	2	8	no	—
327	#41	C₂²×C₄×C₆₇	1	8	yes	—
328	#41	C₂²×C₄×C₆₇	1	16	yes	—
329	#41	C₂⁴×C₆₇	1	8	yes	—
330	C₂×Q₈	—	2	2	no	—
331	C₂×Q₈	—	2	4	no	—
332	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
333	C₂×Q₈	C₂²×C₄×C₆₇	1	8	yes	—
334	C₂×D₄	—	2	4	no	—
335	C₂×D₄	—	2	8	no	—
336	C₂×D₄	—	2	4	no	—
337	C₂×D₄	—	2	8	no	—
338	C₂×D₄	—	2	2	no	—
339	C₂×D₄	—	2	4	no	—
340	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
341	C₂×D₄	C₂²×C₄×C₆₇	1	8	yes	—
342	C₂²×C₄	—	2	2	no	—
343	C₂²×C₄	—	2	4	no	—
344	C₂²×C₄	—	2	8	no	—
345	C₂²×C₄	—	2	4	no	—
346	C₂²×C₄	—	2	8	no	—
347	C₂²×C₄	—	2	2	no	—
348	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
349	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
350	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
351	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
352	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
353	C₂²×C₄	#41×C₆₇	1	2	yes	—
354	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
355	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
356	Q₁₆	C₂²×C₄×C₆₇	1	16	yes	—
357	SD₁₆	C₂²×C₄×C₆₇	1	32	yes	—
358	D₈	C₂²×C₄×C₆₇	1	16	yes	—
359	M₁₆	—	2	8	no	—
360	M₁₆	—	2	4	no	—
361	M₁₆	—	2	16	no	—
362	M₁₆	—	2	16	no	—
363	M₁₆	C₂²×C₄×C₆₇	1	16	yes	—
364	M₁₆	C₂⁴×C₆₇	1	8	yes	—
365	C₂×C₈	—	2	4	no	—
366	C₂×C₈	—	2	8	no	—
367	C₂×C₈	—	2	16	no	—
368	C₂×C₈	—	2	16	no	—
369	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
370	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
371	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
372	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
373	C₂×C₈	C₂×D₄×C₆₇	1	8	yes	—
374	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
375	C₂×C₈	C₂×D₄×C₆₇	1	8	yes	—
376	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
377	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
378	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
379	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
380	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
381	C₄⋊C₄	—	2	4	no	—
382	C₂²⋊C₄	—	2	4	no	—
383	C₄²	—	2	2	no	—
384	C₄²	C₂⁴×C₆₇	1	4	yes	—
385	C₂³	—	2	8	no	—
386	C₂³	—	2	4	no	—
387	C₂³	—	4	2	no	—
388	C₂³	—	4	8	no	—
389	C₂³	C₄×D₄×C₆₇	1	2	yes	—
390	C₂³	C₄×D₄×C₆₇	1	2	yes	—
391	C₂³	C₂×#41×C₆₇	1	1	yes	characteristic
392	C₂³	C₂×#41×C₆₇	1	2	yes	—
393	Q₈	—	2	8	no	—
394	Q₈	—	2	4	no	—
395	Q₈	C₂³×C₄×C₆₇	1	8	yes	—
396	D₄	—	2	16	no	—
397	D₄	—	2	8	no	—
398	D₄	—	2	8	no	—
399	D₄	—	2	8	no	—
400	D₄	—	2	4	no	—
401	D₄	C₂³×C₄×C₆₇	1	8	yes	—
402	C₂×C₄	—	2	8	no	—
403	C₂×C₄	—	2	4	no	—
404	C₂×C₄	—	2	8	no	—
405	C₂×C₄	—	2	16	no	—
406	C₂×C₄	—	2	8	no	—
407	C₂×C₄	—	2	4	no	—
408	C₂×C₄	—	4	4	no	—
409	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
410	C₂×C₄	C₄×D₄×C₆₇	1	4	yes	—
411	C₂×C₄	C₄×D₄×C₆₇	1	4	yes	—
412	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
413	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
414	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
415	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
416	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
417	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
418	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
419	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
420	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
421	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
422	C₈	—	2	8	no	—
423	C₈	C₂³×C₄×C₆₇	1	8	yes	—
424	C₈	C₂²×D₄×C₆₇	1	2	yes	—
425	C₈	C₂²×D₄×C₆₇	1	4	yes	—
426	C₈	C₂²×D₄×C₆₇	1	2	yes	—
427	C₂²	—	2	2	no	—
428	C₂²	—	2	4	no	—
429	C₂²	—	2	8	no	—
430	C₂²	—	4	2	no	—
431	C₂²	—	4	2	no	—
432	C₂²	—	4	16	no	—
433	C₂²	—	4	8	no	—
434	C₂²	C₂×C₄×D₄×C₆₇	1	4	yes	—
435	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
436	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
437	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
438	C₄	—	2	8	no	—
439	C₄	—	4	4	no	—
440	C₄	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
441	C₄	C₂×C₄×D₄×C₆₇	1	4	yes	—
442	C₄	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
443	C₄	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
444	C₄	C₂³×D₄×C₆₇	1	1	yes	characteristic
445	C₂	—	2	1	no	—
446	C₂	—	2	4	no	—
447	C₂	—	2	1	no	—
448	C₂	—	4	8	no	—
449	C₂	C₆₇×#2955	1	2	yes	direct product factor
450	C₂	C₂²×C₄×D₄×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₆₇	C₂×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×C₈×C₆₇ ◁ C₂³×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×#131×C₆₇	C₂	semidirect
3. C₆₇×#2955	C₂	direct (N × H)
4. C₂×#444×C₆₇	C₂	semidirect
5. C₂×#443×C₆₇	C₂	semidirect
6. C₂×#430×C₆₇	C₂	semidirect
7. C₂×#421×C₆₇	C₂	semidirect
8. C₂×#405×C₆₇	C₂	semidirect
10. #575×C₆₇	C₄	semidirect
12. C₂×#137×C₆₇	C₄	semidirect
13. C₂×AGL(1,ℤ/8)×C₆₇	C₄	semidirect
18. C₂×#131×C₆₇	C₂²	semidirect
19. C₂×#131×C₆₇	C₂²	semidirect
20. C₂×#131×C₆₇	C₂²	semidirect
27. #444×C₆₇	C₂²	semidirect
28. #443×C₆₇	C₂²	semidirect
29. #430×C₆₇	C₂²	semidirect
30. C₂×#108×C₆₇	C₂²	semidirect
31. #421×C₆₇	C₂²	semidirect
32. C₂×#104×C₆₇	C₂²	semidirect
33. #405×C₆₇	C₂²	semidirect
34. C₂×#97×C₆₇	C₂²	semidirect
35. C₂×C₄×C₈×C₆₇	C₂²	semidirect
49. #137×C₆₇	C₂×C₄	semidirect
50. AGL(1,ℤ/8)×C₆₇	C₂×C₄	semidirect
60. #131×C₆₇	C₂³	semidirect
61. #131×C₆₇	C₂³	semidirect
88. #108×C₆₇	C₂³	semidirect
89. #104×C₆₇	C₂³	semidirect
90. #97×C₆₇	C₂³	semidirect
91. C₄×C₈×C₆₇	C₂³	semidirect
219. C₂×#2955	C₆₇	direct (N × H)
224. C₂×C₆₇	#2955	direct (N × H)
227. C₂²×#131	C₂×C₆₇	semidirect
228. #2955	C₂×C₆₇	direct (N × H)
229. C₂×#444	C₂×C₆₇	semidirect
230. C₂×#443	C₂×C₆₇	semidirect
231. C₂×#430	C₂×C₆₇	semidirect
232. C₂×#421	C₂×C₆₇	semidirect
233. C₂×#405	C₂×C₆₇	semidirect
234. C₆₇	C₂×#2955	direct (N × H)
236. #575	C₄×C₆₇	semidirect
238. C₂×#137	C₄×C₆₇	semidirect
239. C₂×AGL(1,ℤ/8)	C₄×C₆₇	semidirect
244. C₂×#131	C₂²×C₆₇	semidirect
245. C₂×#131	C₂²×C₆₇	semidirect
246. C₂×#131	C₂²×C₆₇	semidirect
253. #444	C₂²×C₆₇	semidirect
254. #443	C₂²×C₆₇	semidirect
255. #430	C₂²×C₆₇	semidirect
256. C₂×#108	C₂²×C₆₇	semidirect
257. #421	C₂²×C₆₇	semidirect
258. C₂×#104	C₂²×C₆₇	semidirect
259. #405	C₂²×C₆₇	semidirect
260. C₂×#97	C₂²×C₆₇	semidirect
261. C₂×C₄×C₈	C₂²×C₆₇	semidirect
275. #137	C₂×C₄×C₆₇	semidirect
276. AGL(1,ℤ/8)	C₂×C₄×C₆₇	semidirect
286. #131	C₂³×C₆₇	semidirect
287. #131	C₂³×C₆₇	semidirect
314. #108	C₂³×C₆₇	semidirect
315. #104	C₂³×C₆₇	semidirect
316. #97	C₂³×C₆₇	semidirect
317. C₄×C₈	C₂³×C₆₇	semidirect
449. C₂	C₆₇×#2955	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
C₂²×#131×C₆₇	2
C₆₇×#2955	2
C₂×#444×C₆₇	2
C₂×#443×C₆₇	2
C₂×#430×C₆₇	2
C₂×#421×C₆₇	2
C₂×#405×C₆₇	2
C₂×#2955	67

Subgroups — C2×C67×#2955 #30739430

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2955 #30739430