Subgroups — #30772179

Order 17152.

322 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₆₇⋊C₁₆	C₂	1	2	yes	maximal
2	#22559895	C₂	1	1	yes	maximal, characteristic
3	#22558287	C₂	1	1	yes	maximal, characteristic
4	#22558286	C₂	1	2	yes	maximal
5	C₆₇×#1374	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₆₇⋊C₁₆	C₂²	1	2	yes	—
7	C₂×#14523954	C₄	1	1	yes	characteristic
8	C₂×#14523953	C₄	1	1	yes	characteristic
9	C₂×#14523953	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
12	#16863242	C₄	1	1	yes	characteristic
13	Dic₆₇⋊C₁₆	—	2	4	no	—
14	C₂²×C₁₆×C₆₇	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₆₇⋊C₁₆	—	2	2	no	—
21	C₂×C₆₇⋊C₁₆	—	2	2	no	—
22	C₂×C₆₇⋊C₁₆	—	2	2	no	—
23	#14523954	—	2	2	no	—
24	#14523953	—	2	2	no	—
25	#14523953	—	2	2	no	—
26	C₈×Dic₆₇	—	2	2	no	—
27	C₈×Dic₆₇	D₄	1	2	yes	—
28	C₈×Dic₆₇	Q₈	1	2	yes	—
29	C₆₇⋊#95	C₈	1	2	yes	—
30	C₆₇⋊#95	C₈	1	2	yes	—
31	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
33	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₁₆×C₆₇	—	2	2	no	—
35	C₂×C₁₆×C₆₇	—	2	2	no	—
36	C₂×C₁₆×C₆₇	—	2	2	no	—
37	C₄×C₈×C₆₇	—	2	2	no	—
38	C₂²×Dic₆₇	C₂×C₈	1	1	yes	characteristic
39	C₂²×Dic₆₇	C₂×C₈	1	1	yes	characteristic
40	C₂²×Dic₆₇	C₂×C₈	1	2	yes	—
41	C₂×C₆₇⋊C₈	—	2	1	no	—
42	C₂×C₆₇⋊C₈	—	2	1	no	—
43	C₂×C₆₇⋊C₈	—	2	1	no	—
44	C₂×C₆₇⋊C₈	—	2	1	no	—
45	C₂×C₆₇⋊C₈	C₄⋊C₄	1	2	yes	—
46	C₂×C₆₇⋊C₈	M₁₆	1	2	yes	—
47	C₄×Dic₆₇	—	2	1	no	—
48	C₄×Dic₆₇	—	2	1	no	—
49	C₄×Dic₆₇	C₄⋊C₄	1	2	yes	—
50	C₄×Dic₆₇	M₁₆	1	2	yes	—
51	C₆₇⋊C₁₆	—	4	2	no	—
52	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
53	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₈×C₆₇	—	2	1	no	—
55	C₂×C₈×C₆₇	—	2	1	no	—
56	C₂×C₈×C₆₇	M₁₆	1	2	yes	—
57	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
59	C₂×C₈×C₆₇	#41	1	2	yes	—
60	C₂×C₈×C₆₇	#41	1	2	yes	—
61	C₄²×C₆₇	—	2	1	no	—
62	C₄²×C₆₇	—	2	1	no	—
63	C₁₆×C₆₇	—	4	2	no	—
64	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
65	C₂×C₄×C₆₇	—	2	2	no	—
66	C₂×C₄×C₆₇	—	2	2	no	—
67	C₂×C₄×C₆₇	—	2	2	no	—
68	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
70	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
71	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
72	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
73	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
74	C₂×Dic₆₇	—	2	2	no	—
75	C₂×Dic₆₇	—	2	2	no	—
76	C₂×Dic₆₇	—	2	1	no	—
77	C₂×Dic₆₇	—	2	1	no	—
78	C₂×Dic₆₇	—	2	2	no	—
79	C₂×Dic₆₇	—	2	2	no	—
80	C₂×Dic₆₇	C₄⋊C₈	1	4	yes	—
81	C₈×C₆₇	—	2	2	no	—
82	C₈×C₆₇	#124	1	2	yes	—
83	C₈×C₆₇	#125	1	2	yes	—
84	C₆₇⋊C₈	—	2	2	no	—
85	C₆₇⋊C₈	—	2	2	no	—
86	C₂²×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
87	C₂²×C₆₇	#431	1	1	yes	characteristic
88	C₂²×C₆₇	#431	1	1	yes	characteristic
89	C₂²×C₆₇	#504	1	2	yes	—
90	C₂²×C₆₇	#504	1	2	yes	—
91	C₄×C₆₇	—	4	2	no	—
92	C₄×C₆₇	#382	1	1	yes	characteristic
93	C₄×C₆₇	#423	1	1	yes	characteristic
94	C₄×C₆₇	#432	1	1	yes	characteristic
95	C₄×C₆₇	#432	1	1	yes	characteristic
96	Dic₆₇	—	2	4	no	—
97	Dic₆₇	—	4	1	no	—
98	Dic₆₇	—	4	1	no	—
99	#12880	—	67	1	no	maximal, Sylow
100	C₂×C₆₇	#1831	1	1	yes	characteristic
101	C₂×C₆₇	#2151	1	2	yes	—
102	C₂×C₆₇	#2164	1	2	yes	—
103	C₂×C₆₇	#2164	1	1	yes	characteristic
104	C₂×C₆₇	#2164	1	1	yes	characteristic
105	C₂×C₄⋊C₁₆	—	67	2	no	—
106	#1726	—	67	1	no	—
107	#1374	—	67	2	no	—
108	#1374	—	67	1	no	—
109	#1374	D₆₇	1	1	yes	characteristic
110	C₆₇	#12880	1	1	yes	characteristic, complemented, Sylow
111	C₂²×C₁₆	—	67	2	no	—
112	C₂²×C₁₆	C₂×D₆₇	1	2	yes	—
113	C₂×#97	—	67	1	no	—
114	C₂×#97	—	67	1	no	—
115	C₂×#97	—	67	1	no	—
116	C₂×C₄×C₈	—	67	1	no	—
117	C₂×C₄×C₈	—	67	1	no	—
118	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
119	#376	—	67	1	no	—
120	C₄⋊C₁₆	—	134	4	no	—
121	C₂²×C₈	—	67	1	no	—
122	C₂²×C₈	—	67	1	no	—
123	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
124	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
125	C₂×C₄²	—	67	1	no	—
126	C₂×C₄²	—	67	1	no	—
127	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
128	C₂×C₁₆	—	2	2	no	—
129	C₂×C₁₆	—	2	2	no	—
130	C₂×C₁₆	—	2	2	no	—
131	C₂×C₁₆	—	134	2	no	—
132	C₂×C₁₆	—	134	2	no	—
133	C₂×C₁₆	—	134	2	no	—
134	#97	—	134	2	no	—
135	#97	—	134	2	no	—
136	#97	—	134	2	no	—
137	C₄×C₈	—	2	2	no	—
138	C₄×C₈	—	67	2	no	—
139	C₄×C₈	—	67	2	no	—
140	C₄×C₈	—	134	2	no	—
141	#95	—	67	2	no	—
142	#95	—	67	2	no	—
143	C₂²×C₄	—	67	2	no	—
144	C₂²×C₄	—	67	1	no	—
145	C₂²×C₄	—	67	1	no	—
146	C₂²×C₄	C₈×D₆₇	1	2	yes	—
147	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
148	C₂×C₈	—	2	1	no	—
149	C₂×C₈	—	2	1	no	—
150	C₂×C₈	—	67	2	no	—
151	C₂×C₈	—	67	2	no	—
152	C₂×C₈	—	134	1	no	—
153	C₂×C₈	—	134	1	no	—
154	C₂×C₈	—	134	1	no	—
155	C₂×C₈	—	134	1	no	—
156	C₂×C₈	#11761611	1	2	yes	—
157	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
158	C₂×C₈	Q₈×D₆₇	1	1	yes	characteristic
159	C₂×C₈	#11761627	1	2	yes	—
160	C₂×C₈	#11761628	1	2	yes	—
161	C₄²	—	2	1	no	—
162	C₄²	—	2	1	no	—
163	C₄²	—	67	2	no	—
164	C₄²	—	67	2	no	—
165	C₄²	—	134	1	no	—
166	C₄²	—	134	1	no	—
167	C₁₆	—	4	2	no	—
168	C₁₆	—	268	2	no	—
169	C₂³	C₂×C₈×D₆₇	1	1	yes	characteristic
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	67	4	no	—
174	C₂×C₄	—	134	2	no	—
175	C₂×C₄	—	134	2	no	—
176	C₂×C₄	—	134	1	no	—
177	C₂×C₄	—	134	1	no	—
178	C₂×C₄	—	134	2	no	—
179	C₂×C₄	—	134	2	no	—
180	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
181	C₂×C₄	#14523999	1	1	yes	characteristic
182	C₂×C₄	#14524000	1	1	yes	characteristic
183	C₂×C₄	M₁₆×D₆₇	1	1	yes	characteristic
184	C₂×C₄	M₁₆×D₆₇	1	1	yes	characteristic
185	C₂×C₄	C₂×#11761611	1	1	yes	characteristic
186	C₈	—	2	2	no	—
187	C₈	—	134	2	no	—
188	C₈	—	134	2	no	—
189	C₈	#14524029	1	2	yes	—
190	C₈	#14524033	1	2	yes	—
191	C₂²	C₄⋊C₈×D₆₇	1	1	yes	characteristic
192	C₂²	#16863446	1	1	yes	characteristic
193	C₂²	#16863447	1	1	yes	characteristic
194	C₂²	#16863812	1	2	yes	—
195	C₂²	#16863814	1	2	yes	—
196	C₄	—	4	2	no	—
197	C₄	—	268	1	no	—
198	C₄	—	268	1	no	—
199	C₄	—	134	4	no	—
200	C₄	#16863260	1	1	yes	characteristic
201	C₄	#16863418	1	1	yes	characteristic
202	C₄	#16863450	1	1	yes	characteristic
203	C₄	#16863451	1	1	yes	characteristic
204	C₂	#22560324	1	1	yes	characteristic
205	C₂	#22561966	1	2	yes	—
206	C₂	#22562013	1	2	yes	—
207	C₂	#22562014	1	1	yes	characteristic
208	C₂	#22562017	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₆₇
commutator, socle, prime radical	C₂³×C₆₇	C₂²×C₈
Fitting	C₆₇×#1374	C₂
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#1374	D₆₇
hypocenter	C₆₇	#12880
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₈ ◁ #1374

Complements

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

N	H ≅ G/N	type
110. C₆₇	#12880	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₆₇⋊C₁₆	2
#22559895	2
#22558287	2
#22558286	2
C₆₇×#1374	2
#12880	67