Subgroups — C₂×C₆₇×#2951 #30739416

Order 17152.

2806 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#135×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#520×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₆₇×#2951	C₂	1	16	yes	maximal, direct product factor
4	C₂×#440×C₆₇	C₂	1	4	yes	maximal, complemented
5	C₂×C₄×Q₁₆×C₆₇	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×D₈×C₆₇	C₂	1	2	yes	maximal, complemented
7	C₂²×#107×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#417×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	C₂×#135×C₆₇	C₄	1	16	yes	complemented
12	C₂×#135×C₆₇	C₂²	1	8	yes	—
13	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
14	C₂²×SD₁₆×C₆₇	C₂²	1	2	yes	—
15	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
17	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
18	#520×C₆₇	—	2	8	no	—
19	#520×C₆₇	C₂²	1	16	yes	complemented
20	C₂×C₄×Q₈×C₆₇	C₂²	1	4	yes	—
21	C₂×C₄×D₄×C₆₇	—	2	4	no	—
22	C₂×C₄×D₄×C₆₇	C₂²	1	4	yes	—
23	C₂×#117×C₆₇	—	2	2	no	—
24	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
25	C₂²×C₄⋊C₄×C₆₇	—	2	2	no	—
26	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
27	#440×C₆₇	C₂²	1	32	yes	complemented
28	C₄×Q₁₆×C₆₇	C₂²	1	16	yes	complemented
29	C₄×D₈×C₆₇	C₂²	1	16	yes	complemented
30	C₂×#107×C₆₇	C₂²	1	2	yes	—
31	C₂×#107×C₆₇	C₂²	1	2	yes	—
32	C₂×#107×C₆₇	C₂²	1	8	yes	complemented
33	#417×C₆₇	C₂²	1	16	yes	complemented
34	C₂×#103×C₆₇	C₂²	1	4	yes	—
35	C₂×#102×C₆₇	C₂²	1	4	yes	—
36	#405×C₆₇	C₂²	1	8	yes	complemented
37	C₂×#97×C₆₇	C₂²	1	2	yes	—
38	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
39	C₂×#41×C₆₇	—	2	4	no	—
40	C₂×#41×C₆₇	—	2	8	no	—
41	C₂×#41×C₆₇	C₂×C₄	1	16	yes	complemented
42	C₂×#41×C₆₇	C₂³	1	8	yes	—
43	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
44	C₂²×D₄×C₆₇	—	2	2	no	—
45	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
46	C₂³×C₄×C₆₇	—	2	2	no	—
47	C₂³×C₄×C₆₇	—	2	2	no	—
48	C₂³×C₄×C₆₇	—	2	4	no	—
49	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
50	#135×C₆₇	C₂×C₄	1	64	yes	complemented
51	C₂×Q₁₆×C₆₇	C₂×C₄	1	8	yes	complemented
52	C₂×Q₁₆×C₆₇	C₂³	1	4	yes	—
53	C₂×SD₁₆×C₆₇	C₂×C₄	1	16	yes	complemented
54	C₂×SD₁₆×C₆₇	C₂³	1	8	yes	—
55	C₂×D₈×C₆₇	C₂×C₄	1	8	yes	complemented
56	C₂×D₈×C₆₇	C₂³	1	4	yes	—
57	C₂×M₁₆×C₆₇	—	2	4	no	—
58	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
59	C₂²×C₈×C₆₇	C₂×C₄	1	8	yes	complemented
60	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
61	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
62	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
63	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
64	C₄×Q₈×C₆₇	—	2	8	no	—
65	C₄×Q₈×C₆₇	C₂³	1	16	yes	—
66	C₄×D₄×C₆₇	—	2	32	no	—
67	C₄×D₄×C₆₇	—	2	8	no	—
68	C₄×D₄×C₆₇	C₂³	1	16	yes	—
69	#117×C₆₇	—	2	16	no	—
70	#117×C₆₇	D₄	1	4	yes	—
71	#117×C₆₇	C₂³	1	4	yes	—
72	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
73	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
74	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
75	C₂×C₄⋊C₄×C₆₇	—	2	16	no	—
76	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
77	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
78	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
79	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
80	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
81	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
82	C₂×C₄²×C₆₇	—	2	4	no	—
83	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
84	#107×C₆₇	C₂³	1	8	yes	—
85	#107×C₆₇	C₂³	1	8	yes	—
86	#103×C₆₇	C₂³	1	16	yes	—
87	#102×C₆₇	C₂³	1	16	yes	—
88	#97×C₆₇	C₂³	1	8	yes	—
89	C₄×C₈×C₆₇	C₂³	1	8	yes	—
90	C₂⁴×C₆₇	—	2	2	no	—
91	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
92	#41×C₆₇	—	2	32	no	—
93	#41×C₆₇	—	2	16	no	—
94	#41×C₆₇	C₂²×C₄	1	32	yes	complemented
95	C₂×Q₈×C₆₇	—	2	2	no	—
96	C₂×Q₈×C₆₇	—	2	4	no	—
97	C₂×Q₈×C₆₇	C₂²×C₄	1	8	yes	—
98	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
99	C₂×D₄×C₆₇	—	2	16	no	—
100	C₂×D₄×C₆₇	—	2	8	no	—
101	C₂×D₄×C₆₇	—	2	2	no	—
102	C₂×D₄×C₆₇	—	2	4	no	—
103	C₂×D₄×C₆₇	C₂²×C₄	1	8	yes	—
104	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
105	C₂²×C₄×C₆₇	—	2	2	no	—
106	C₂²×C₄×C₆₇	—	2	16	no	—
107	C₂²×C₄×C₆₇	—	2	8	no	—
108	C₂²×C₄×C₆₇	—	2	4	no	—
109	C₂²×C₄×C₆₇	—	2	8	no	—
110	C₂²×C₄×C₆₇	—	2	16	no	—
111	C₂²×C₄×C₆₇	—	8	8	no	—
112	C₂²×C₄×C₆₇	—	8	4	no	—
113	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
114	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
115	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
116	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
117	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
118	C₂²×C₄×C₆₇	#41	1	4	yes	—
119	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
120	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
121	Q₁₆×C₆₇	C₂²×C₄	1	16	yes	complemented
122	SD₁₆×C₆₇	C₂²×C₄	1	32	yes	complemented
123	D₈×C₆₇	C₂²×C₄	1	16	yes	complemented
124	M₁₆×C₆₇	—	2	8	no	—
125	C₂×C₈×C₆₇	—	2	4	no	—
126	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
127	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
128	C₂×C₈×C₆₇	C₂²×C₄	1	16	yes	complemented
129	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
130	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
131	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
132	C₄⋊C₄×C₆₇	—	2	8	no	—
133	C₄⋊C₄×C₆₇	—	2	16	no	—
134	C₄⋊C₄×C₆₇	—	2	4	no	—
135	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
136	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
137	C₄⋊C₄×C₆₇	C₂⁴	1	8	yes	—
138	C₂²⋊C₄×C₆₇	—	2	16	no	—
139	C₂²⋊C₄×C₆₇	C₂×D₄	1	8	yes	—
140	C₄²×C₆₇	—	2	16	no	—
141	C₄²×C₆₇	C₂×D₄	1	4	yes	—
142	C₄²×C₆₇	C₂⁴	1	4	yes	—
143	C₂³×C₆₇	—	2	2	no	—
144	C₂³×C₆₇	—	2	2	no	—
145	C₂³×C₆₇	—	2	8	no	—
146	C₂³×C₆₇	—	2	4	no	—
147	C₂³×C₆₇	—	4	4	no	—
148	C₂³×C₆₇	—	4	4	no	—
149	C₂³×C₆₇	C₄×D₄	1	4	yes	—
150	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
151	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
152	Q₈×C₆₇	—	2	8	no	—
153	Q₈×C₆₇	—	2	4	no	—
154	Q₈×C₆₇	C₂³×C₄	1	8	yes	—
155	D₄×C₆₇	—	2	32	no	—
156	D₄×C₆₇	—	2	8	no	—
157	D₄×C₆₇	—	2	4	no	—
158	D₄×C₆₇	C₂³×C₄	1	8	yes	—
159	C₂×C₄×C₆₇	—	2	8	no	—
160	C₂×C₄×C₆₇	—	2	4	no	—
161	C₂×C₄×C₆₇	—	2	4	no	—
162	C₂×C₄×C₆₇	—	2	4	no	—
163	C₂×C₄×C₆₇	—	2	32	no	—
164	C₂×C₄×C₆₇	—	2	8	no	—
165	C₂×C₄×C₆₇	—	4	8	no	—
166	C₂×C₄×C₆₇	—	8	16	no	—
167	C₂×C₄×C₆₇	—	8	8	no	—
168	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
169	C₂×C₄×C₆₇	C₄×D₄	1	8	yes	—
170	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
171	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
172	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
173	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
174	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
175	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
176	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
177	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
178	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
179	C₈×C₆₇	—	2	4	no	—
180	C₈×C₆₇	C₂³×C₄	1	4	yes	—
181	C₈×C₆₇	C₂³×C₄	1	4	yes	—
182	C₂²×C₆₇	—	2	8	no	—
183	C₂²×C₆₇	—	2	2	no	—
184	C₂²×C₆₇	—	2	2	no	—
185	C₂²×C₆₇	—	2	8	no	—
186	C₂²×C₆₇	—	4	16	no	—
187	C₂²×C₆₇	—	4	4	no	—
188	C₂²×C₆₇	—	4	4	no	—
189	C₂²×C₆₇	C₂×C₄×D₄	1	4	yes	—
190	C₂²×C₆₇	C₂×C₄×D₄	1	2	yes	—
191	C₂²×C₆₇	#576	1	2	yes	—
192	C₂²×C₆₇	#578	1	2	yes	—
193	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
194	C₄×C₆₇	—	2	8	no	—
195	C₄×C₆₇	—	2	4	no	—
196	C₄×C₆₇	—	2	4	no	—
197	C₄×C₆₇	—	4	8	no	—
198	C₄×C₆₇	C₂×C₄×D₄	1	4	yes	—
199	C₄×C₆₇	C₂×C₄×D₄	1	2	yes	—
200	C₄×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
201	C₄×C₆₇	C₂²×#41	1	1	yes	characteristic
202	C₂×#2951	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
203	C₂×C₆₇	—	2	4	no	—
204	C₂×C₆₇	—	4	8	no	—
205	C₂×C₆₇	#2951	1	4	yes	direct product factor
206	C₂×C₆₇	C₂²×C₄×D₄	1	1	yes	characteristic
207	C₂×C₆₇	C₂×#576	1	1	yes	characteristic
208	C₂×C₆₇	C₂×#578	1	1	yes	characteristic
209	C₂²×#135	C₂×C₆₇	1	1	yes	characteristic
210	C₂×#520	C₂×C₆₇	1	2	yes	complemented
211	#2951	C₂×C₆₇	1	16	yes	direct product factor
212	C₂×#440	C₂×C₆₇	1	4	yes	complemented
213	C₂×C₄×Q₁₆	C₂×C₆₇	1	2	yes	complemented
214	C₂×C₄×D₈	C₂×C₆₇	1	2	yes	complemented
215	C₂²×#107	C₂×C₆₇	1	1	yes	characteristic, complemented
216	C₂×#417	C₂×C₆₇	1	2	yes	complemented
217	C₂×#405	C₂×C₆₇	1	1	yes	characteristic, complemented
218	C₆₇	C₂×#2951	1	1	yes	characteristic, direct product factor, Sylow
219	C₂²×#41	C₂²×C₆₇	1	2	yes	—
220	C₂×#135	C₄×C₆₇	1	16	yes	complemented
221	C₂×#135	C₂²×C₆₇	1	8	yes	—
222	C₂²×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
223	C₂²×SD₁₆	C₂²×C₆₇	1	2	yes	—
224	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic
225	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
226	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
227	#520	—	2	8	no	—
228	#520	C₂²×C₆₇	1	16	yes	complemented
229	C₂×C₄×Q₈	C₂²×C₆₇	1	4	yes	—
230	C₂×C₄×D₄	—	2	4	no	—
231	C₂×C₄×D₄	C₂²×C₆₇	1	4	yes	—
232	C₂×#117	—	2	2	no	—
233	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
234	C₂²×C₄⋊C₄	—	2	2	no	—
235	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
236	#440	C₂²×C₆₇	1	32	yes	complemented
237	C₄×Q₁₆	C₂²×C₆₇	1	16	yes	complemented
238	C₄×D₈	C₂²×C₆₇	1	16	yes	complemented
239	C₂×#107	C₂²×C₆₇	1	2	yes	—
240	C₂×#107	C₂²×C₆₇	1	2	yes	—
241	C₂×#107	C₂²×C₆₇	1	8	yes	complemented
242	#417	C₂²×C₆₇	1	16	yes	complemented
243	C₂×#103	C₂²×C₆₇	1	4	yes	—
244	C₂×#102	C₂²×C₆₇	1	4	yes	—
245	#405	C₂²×C₆₇	1	8	yes	complemented
246	C₂×#97	C₂²×C₆₇	1	2	yes	—
247	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
248	C₂×#41	—	2	4	no	—
249	C₂×#41	—	2	8	no	—
250	C₂×#41	C₂×C₄×C₆₇	1	16	yes	complemented
251	C₂×#41	C₂³×C₆₇	1	8	yes	—
252	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
253	C₂²×D₄	—	2	2	no	—
254	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
255	C₂³×C₄	—	2	2	no	—
256	C₂³×C₄	—	2	2	no	—
257	C₂³×C₄	—	2	4	no	—
258	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
259	#135	C₂×C₄×C₆₇	1	64	yes	complemented
260	C₂×Q₁₆	C₂×C₄×C₆₇	1	8	yes	complemented
261	C₂×Q₁₆	C₂³×C₆₇	1	4	yes	—
262	C₂×SD₁₆	C₂×C₄×C₆₇	1	16	yes	complemented
263	C₂×SD₁₆	C₂³×C₆₇	1	8	yes	—
264	C₂×D₈	C₂×C₄×C₆₇	1	8	yes	complemented
265	C₂×D₈	C₂³×C₆₇	1	4	yes	—
266	C₂×M₁₆	—	2	4	no	—
267	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
268	C₂²×C₈	C₂×C₄×C₆₇	1	8	yes	complemented
269	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
270	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
271	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
272	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
273	C₄×Q₈	—	2	8	no	—
274	C₄×Q₈	C₂³×C₆₇	1	16	yes	—
275	C₄×D₄	—	2	32	no	—
276	C₄×D₄	—	2	8	no	—
277	C₄×D₄	C₂³×C₆₇	1	16	yes	—
278	#117	—	2	16	no	—
279	#117	D₄×C₆₇	1	4	yes	—
280	#117	C₂³×C₆₇	1	4	yes	—
281	C₂×C₄⋊C₄	—	2	2	no	—
282	C₂×C₄⋊C₄	—	2	4	no	—
283	C₂×C₄⋊C₄	—	2	4	no	—
284	C₂×C₄⋊C₄	—	2	16	no	—
285	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
286	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
287	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
288	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
289	C₂×C₂²⋊C₄	—	2	4	no	—
290	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
291	C₂×C₄²	—	2	4	no	—
292	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
293	#107	C₂³×C₆₇	1	8	yes	—
294	#107	C₂³×C₆₇	1	8	yes	—
295	#103	C₂³×C₆₇	1	16	yes	—
296	#102	C₂³×C₆₇	1	16	yes	—
297	#97	C₂³×C₆₇	1	8	yes	—
298	C₄×C₈	C₂³×C₆₇	1	8	yes	—
299	C₂⁴	—	2	2	no	—
300	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
301	#41	—	2	32	no	—
302	#41	—	2	16	no	—
303	#41	C₂²×C₄×C₆₇	1	32	yes	complemented
304	C₂×Q₈	—	2	2	no	—
305	C₂×Q₈	—	2	4	no	—
306	C₂×Q₈	C₂²×C₄×C₆₇	1	8	yes	—
307	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
308	C₂×D₄	—	2	16	no	—
309	C₂×D₄	—	2	8	no	—
310	C₂×D₄	—	2	2	no	—
311	C₂×D₄	—	2	4	no	—
312	C₂×D₄	C₂²×C₄×C₆₇	1	8	yes	—
313	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
314	C₂²×C₄	—	2	2	no	—
315	C₂²×C₄	—	2	16	no	—
316	C₂²×C₄	—	2	8	no	—
317	C₂²×C₄	—	2	4	no	—
318	C₂²×C₄	—	2	8	no	—
319	C₂²×C₄	—	2	16	no	—
320	C₂²×C₄	—	8	8	no	—
321	C₂²×C₄	—	8	4	no	—
322	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
323	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
324	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
325	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
326	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
327	C₂²×C₄	#41×C₆₇	1	4	yes	—
328	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
329	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
330	Q₁₆	C₂²×C₄×C₆₇	1	16	yes	complemented
331	SD₁₆	C₂²×C₄×C₆₇	1	32	yes	complemented
332	D₈	C₂²×C₄×C₆₇	1	16	yes	complemented
333	M₁₆	—	2	8	no	—
334	C₂×C₈	—	2	4	no	—
335	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
336	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
337	C₂×C₈	C₂²×C₄×C₆₇	1	16	yes	complemented
338	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
339	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
340	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
341	C₄⋊C₄	—	2	8	no	—
342	C₄⋊C₄	—	2	16	no	—
343	C₄⋊C₄	—	2	4	no	—
344	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
345	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
346	C₄⋊C₄	C₂⁴×C₆₇	1	8	yes	—
347	C₂²⋊C₄	—	2	16	no	—
348	C₂²⋊C₄	C₂×D₄×C₆₇	1	8	yes	—
349	C₄²	—	2	16	no	—
350	C₄²	C₂×D₄×C₆₇	1	4	yes	—
351	C₄²	C₂⁴×C₆₇	1	4	yes	—
352	C₂³	—	2	2	no	—
353	C₂³	—	2	2	no	—
354	C₂³	—	2	8	no	—
355	C₂³	—	2	4	no	—
356	C₂³	—	4	4	no	—
357	C₂³	—	4	4	no	—
358	C₂³	C₄×D₄×C₆₇	1	4	yes	—
359	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
360	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
361	Q₈	—	2	8	no	—
362	Q₈	—	2	4	no	—
363	Q₈	C₂³×C₄×C₆₇	1	8	yes	—
364	D₄	—	2	32	no	—
365	D₄	—	2	8	no	—
366	D₄	—	2	4	no	—
367	D₄	C₂³×C₄×C₆₇	1	8	yes	—
368	C₂×C₄	—	2	8	no	—
369	C₂×C₄	—	2	4	no	—
370	C₂×C₄	—	2	4	no	—
371	C₂×C₄	—	2	4	no	—
372	C₂×C₄	—	2	32	no	—
373	C₂×C₄	—	2	8	no	—
374	C₂×C₄	—	4	8	no	—
375	C₂×C₄	—	8	16	no	—
376	C₂×C₄	—	8	8	no	—
377	C₂×C₄	C₄×D₄×C₆₇	1	4	yes	—
378	C₂×C₄	C₄×D₄×C₆₇	1	8	yes	—
379	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
380	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
381	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
382	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
383	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
384	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
385	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
386	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
387	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
388	C₈	—	2	4	no	—
389	C₈	C₂³×C₄×C₆₇	1	4	yes	—
390	C₈	C₂³×C₄×C₆₇	1	4	yes	—
391	C₂²	—	2	2	no	—
392	C₂²	—	2	2	no	—
393	C₂²	—	2	8	no	—
394	C₂²	—	2	8	no	—
395	C₂²	—	4	4	no	—
396	C₂²	—	4	4	no	—
397	C₂²	—	4	16	no	—
398	C₂²	C₂×C₄×D₄×C₆₇	1	4	yes	—
399	C₂²	C₂×C₄×D₄×C₆₇	1	2	yes	—
400	C₂²	#576×C₆₇	1	2	yes	—
401	C₂²	#578×C₆₇	1	2	yes	—
402	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
403	C₄	—	2	4	no	—
404	C₄	—	2	4	no	—
405	C₄	—	2	8	no	—
406	C₄	—	4	8	no	—
407	C₄	C₂×C₄×D₄×C₆₇	1	4	yes	—
408	C₄	C₂×C₄×D₄×C₆₇	1	2	yes	—
409	C₄	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
410	C₄	C₂²×#41×C₆₇	1	1	yes	characteristic
411	C₂	—	2	4	no	—
412	C₂	—	4	8	no	—
413	C₂	C₆₇×#2951	1	4	yes	direct product factor
414	C₂	C₂²×C₄×D₄×C₆₇	1	1	yes	characteristic
415	C₂	C₂×#576×C₆₇	1	1	yes	characteristic
416	C₂	C₂×#578×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator	C₄	C₂⁴×C₄×C₆₇
Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
prime radical	C₂²×#135×C₆₇	C₂
LCS	G ▷ 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
2. C₂×#520×C₆₇	C₂	semidirect
3. C₆₇×#2951	C₂	direct (N × H)
4. C₂×#440×C₆₇	C₂	semidirect
5. C₂×C₄×Q₁₆×C₆₇	C₂	semidirect
6. C₂×C₄×D₈×C₆₇	C₂	semidirect
7. C₂²×#107×C₆₇	C₂	semidirect
8. C₂×#417×C₆₇	C₂	semidirect
9. C₂×#405×C₆₇	C₂	semidirect
11. C₂×#135×C₆₇	C₄	semidirect
19. #520×C₆₇	C₂²	semidirect
27. #440×C₆₇	C₂²	semidirect
28. C₄×Q₁₆×C₆₇	C₂²	semidirect
29. C₄×D₈×C₆₇	C₂²	semidirect
32. C₂×#107×C₆₇	C₂²	semidirect
33. #417×C₆₇	C₂²	semidirect
36. #405×C₆₇	C₂²	semidirect
41. C₂×#41×C₆₇	C₂×C₄	semidirect
50. #135×C₆₇	C₂×C₄	semidirect
51. C₂×Q₁₆×C₆₇	C₂×C₄	semidirect
53. C₂×SD₁₆×C₆₇	C₂×C₄	semidirect
55. C₂×D₈×C₆₇	C₂×C₄	semidirect
59. C₂²×C₈×C₆₇	C₂×C₄	semidirect
94. #41×C₆₇	C₂²×C₄	semidirect
121. Q₁₆×C₆₇	C₂²×C₄	semidirect
122. SD₁₆×C₆₇	C₂²×C₄	semidirect
123. D₈×C₆₇	C₂²×C₄	semidirect
128. C₂×C₈×C₆₇	C₂²×C₄	semidirect
202. C₂×#2951	C₆₇	direct (N × H)
205. C₂×C₆₇	#2951	direct (N × H)
210. C₂×#520	C₂×C₆₇	semidirect
211. #2951	C₂×C₆₇	direct (N × H)
212. C₂×#440	C₂×C₆₇	semidirect
213. C₂×C₄×Q₁₆	C₂×C₆₇	semidirect
214. C₂×C₄×D₈	C₂×C₆₇	semidirect
215. C₂²×#107	C₂×C₆₇	semidirect
216. C₂×#417	C₂×C₆₇	semidirect
217. C₂×#405	C₂×C₆₇	semidirect
218. C₆₇	C₂×#2951	direct (N × H)
220. C₂×#135	C₄×C₆₇	semidirect
228. #520	C₂²×C₆₇	semidirect
236. #440	C₂²×C₆₇	semidirect
237. C₄×Q₁₆	C₂²×C₆₇	semidirect
238. C₄×D₈	C₂²×C₆₇	semidirect
241. C₂×#107	C₂²×C₆₇	semidirect
242. #417	C₂²×C₆₇	semidirect
245. #405	C₂²×C₆₇	semidirect
250. C₂×#41	C₂×C₄×C₆₇	semidirect
259. #135	C₂×C₄×C₆₇	semidirect
260. C₂×Q₁₆	C₂×C₄×C₆₇	semidirect
262. C₂×SD₁₆	C₂×C₄×C₆₇	semidirect
264. C₂×D₈	C₂×C₄×C₆₇	semidirect
268. C₂²×C₈	C₂×C₄×C₆₇	semidirect
303. #41	C₂²×C₄×C₆₇	semidirect
330. Q₁₆	C₂²×C₄×C₆₇	semidirect
331. SD₁₆	C₂²×C₄×C₆₇	semidirect
332. D₈	C₂²×C₄×C₆₇	semidirect
337. C₂×C₈	C₂²×C₄×C₆₇	semidirect
413. C₂	C₆₇×#2951	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×#135×C₆₇	2
C₂×#520×C₆₇	2
C₆₇×#2951	2
C₂×#440×C₆₇	2
C₂×C₄×Q₁₆×C₆₇	2
C₂×C₄×D₈×C₆₇	2
C₂²×#107×C₆₇	2
C₂×#417×C₆₇	2
C₂×#405×C₆₇	2
C₂×#2951	67

Subgroups — C2×C67×#2951 #30739416

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2951 #30739416