Subgroups — #30773832

Order 17152.

350 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₆₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₆₇⋊C₁₆)	C₂	1	2	yes	maximal
3	C₂×C₄×C₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	#22558287	C₂	1	2	yes	maximal
5	C₈×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₆₇⋊C₁₆	C₂²	1	2	yes	—
7	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
8	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
9	C₄⋊(C₆₇⋊C₁₆)	—	2	2	no	—
10	C₄⋊(C₆₇⋊C₁₆)	—	2	4	no	—
11	C₄×C₆₇⋊C₁₆	—	2	2	no	—
12	C₂×C₄⋊C₈×C₆₇	C₄	1	1	yes	characteristic
13	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₄×C₄⋊C₄×C₆₇	C₄	1	1	yes	characteristic
16	#336×C₆₇	C₄	1	2	yes	—
17	C₂×C₆₇⋊C₁₆	—	2	1	no	—
18	C₂×C₆₇⋊C₁₆	—	2	1	no	—
19	C₂×C₆₇⋊C₁₆	—	2	2	no	—
20	C₂×C₆₇⋊C₁₆	—	2	2	no	—
21	C₂×C₆₇⋊C₁₆	—	2	1	no	—
22	C₂×C₆₇⋊C₁₆	—	2	1	no	—
23	C₂×C₆₇⋊C₁₆	—	2	2	no	—
24	C₂×C₆₇⋊C₁₆	D₄	1	2	yes	—
25	C₂×C₆₇⋊C₁₆	Q₈	1	2	yes	—
26	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
27	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₆₇	C₈	1	1	yes	characteristic
30	C₂×C₄⋊C₄×C₆₇	C₈	1	1	yes	characteristic
31	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
33	C₄⋊C₈×C₆₇	—	2	2	no	—
34	C₄×C₈×C₆₇	—	2	2	no	—
35	C₄×C₈×C₆₇	—	2	2	no	—
36	C₄×C₈×C₆₇	D₄	1	2	yes	—
37	C₄×C₈×C₆₇	Q₈	1	2	yes	—
38	#95×C₆₇	C₈	1	2	yes	—
39	C₆₇⋊C₁₆	—	2	2	no	—
40	C₆₇⋊C₁₆	—	2	2	no	—
41	C₆₇⋊C₁₆	—	4	2	no	—
42	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
43	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
44	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
45	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
46	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₈×C₆₇	—	2	1	no	—
48	C₂×C₈×C₆₇	—	2	1	no	—
49	C₂×C₈×C₆₇	—	2	2	no	—
50	C₂×C₈×C₆₇	—	2	2	no	—
51	C₂×C₈×C₆₇	—	2	2	no	—
52	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
53	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
54	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
55	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
58	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
59	C₄⋊C₄×C₆₇	—	2	2	no	—
60	C₄⋊C₄×C₆₇	C₁₆	1	4	yes	complemented
61	C₄²×C₆₇	—	2	2	no	—
62	C₄²×C₆₇	—	2	2	no	—
63	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
64	C₄²×C₆₇	M₁₆	1	2	yes	—
65	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
66	C₂×C₄×C₆₇	—	2	2	no	—
67	C₂×C₄×C₆₇	—	2	1	no	—
68	C₂×C₄×C₆₇	—	2	1	no	—
69	C₂×C₄×C₆₇	—	2	2	no	—
70	C₂×C₄×C₆₇	—	2	2	no	—
71	C₂×C₄×C₆₇	—	2	2	no	—
72	C₂×C₄×C₆₇	—	2	2	no	—
73	C₂×C₄×C₆₇	C₄⋊C₈	1	2	yes	—
74	C₂×C₄×C₆₇	C₄⋊C₈	1	2	yes	—
75	C₂×C₄×C₆₇	C₂×C₁₆	1	4	yes	—
76	C₂×C₄×C₆₇	C₂×C₁₆	1	2	yes	—
77	C₂×C₄×C₆₇	M₃₂	1	2	yes	—
78	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
80	C₂×C₄×C₆₇	C₄×Q₈	1	1	yes	characteristic
81	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
82	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
83	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
84	C₈×C₆₇	—	2	2	no	—
85	C₈×C₆₇	—	4	2	no	—
86	C₈×C₆₇	#121	1	1	yes	characteristic
87	C₈×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
88	C₈×C₆₇	#125	1	1	yes	characteristic
89	C₈×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
90	C₂²×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
91	C₂²×C₆₇	C₈×D₄	1	1	yes	characteristic
92	C₂²×C₆₇	C₈×Q₈	1	1	yes	characteristic
93	C₂²×C₆₇	C₂²×C₁₆	1	1	yes	characteristic
94	C₂²×C₆₇	C₂×M₃₂	1	1	yes	characteristic
95	C₂²×C₆₇	#504	1	1	yes	characteristic
96	C₂²×C₆₇	#504	1	1	yes	characteristic
97	C₄×C₆₇	—	2	2	no	—
98	C₄×C₆₇	—	2	4	no	—
99	C₄×C₆₇	—	4	2	no	—
100	C₄×C₆₇	C₄⋊C₁₆	1	4	yes	complemented
101	C₄×C₆₇	#389	1	1	yes	characteristic
102	C₄×C₆₇	#424	1	1	yes	characteristic
103	C₄×C₆₇	#436	1	1	yes	characteristic
104	C₄×C₆₇	#446	1	1	yes	characteristic
105	#13159	—	67	1	no	maximal, Sylow
106	C₂×C₆₇	#1939	1	1	yes	characteristic
107	C₂×C₆₇	C₂×C₄⋊C₁₆	1	1	yes	characteristic
108	C₂×C₆₇	#2151	1	1	yes	characteristic
109	C₂×C₆₇	D₄×C₁₆	1	1	yes	characteristic
110	C₂×C₆₇	#2169	1	1	yes	characteristic
111	C₂×C₆₇	Q₈×C₁₆	1	1	yes	characteristic
112	C₂×C₆₇	#2183	1	1	yes	characteristic
113	C₂×C₄⋊C₁₆	—	67	2	no	—
114	C₂×C₄⋊C₁₆	—	67	1	no	—
115	C₂×C₄×C₁₆	—	67	1	no	—
116	C₈×C₄⋊C₄	D₆₇	1	1	yes	characteristic
117	#1374	—	67	2	no	—
118	C₆₇	#13159	1	1	yes	characteristic, complemented, Sylow
119	C₂²×C₁₆	—	67	1	no	—
120	C₂²×C₁₆	—	67	1	no	—
121	C₂²×C₁₆	—	67	2	no	—
122	C₂×C₄⋊C₈	Dic₆₇	1	1	yes	characteristic
123	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
124	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
125	C₄×C₄⋊C₄	Dic₆₇	1	1	yes	characteristic
126	C₄⋊C₁₆	—	134	4	no	—
127	C₄⋊C₁₆	—	134	2	no	—
128	C₄×C₁₆	—	134	2	no	—
129	#336	Dic₆₇	1	2	yes	—
130	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
131	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
132	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
133	C₂×C₄⋊C₄	C₆₇⋊C₈	1	1	yes	characteristic
134	C₂×C₄⋊C₄	C₆₇⋊C₈	1	1	yes	characteristic
135	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
136	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
137	C₂×C₁₆	—	67	2	no	—
138	C₂×C₁₆	—	67	2	no	—
139	C₂×C₁₆	—	134	2	no	—
140	C₂×C₁₆	—	134	2	no	—
141	C₂×C₁₆	—	134	2	no	—
142	C₂×C₁₆	—	134	1	no	—
143	C₂×C₁₆	—	134	1	no	—
144	C₂×C₁₆	—	134	1	no	—
145	C₂×C₁₆	—	134	1	no	—
146	C₄⋊C₈	—	2	2	no	—
147	C₄×C₈	—	2	2	no	—
148	C₄×C₈	—	2	2	no	—
149	C₄×C₈	Dic₁₃₄	1	2	yes	—
150	C₄×C₈	D₂₆₈	1	2	yes	—
151	#95	C₆₇⋊C₈	1	2	yes	—
152	C₂²×C₄	C₂×C₆₇⋊C₈	1	2	yes	—
153	C₂²×C₄	C₂×C₆₇⋊C₈	1	1	yes	characteristic
154	C₂²×C₄	C₂×C₆₇⋊C₈	1	2	yes	—
155	C₂²×C₄	C₂×C₆₇⋊C₈	1	1	yes	characteristic
156	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
157	C₂×C₈	—	2	1	no	—
158	C₂×C₈	—	2	1	no	—
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	C₄⋊Dic₆₇	1	2	yes	—
163	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
164	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
165	C₂×C₈	Q₈×D₆₇	1	1	yes	characteristic
166	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
167	C₂×C₈	#11761627	1	1	yes	characteristic
168	C₂×C₈	#11761629	1	1	yes	characteristic
169	C₄⋊C₄	—	2	2	no	—
170	C₄⋊C₄	C₆₇⋊C₁₆	1	4	yes	complemented
171	C₄²	—	2	2	no	—
172	C₄²	—	2	2	no	—
173	C₄²	C₄⋊Dic₆₇	1	2	yes	—
174	C₄²	#11761612	1	2	yes	—
175	C₁₆	—	268	2	no	—
176	C₁₆	—	134	2	no	—
177	C₁₆	—	134	2	no	—
178	C₂³	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	2	1	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	C₄⋊(C₆₇⋊C₈)	1	2	yes	—
187	C₂×C₄	C₄⋊(C₆₇⋊C₈)	1	2	yes	—
188	C₂×C₄	C₂×C₆₇⋊C₁₆	1	4	yes	—
189	C₂×C₄	C₂×C₆₇⋊C₁₆	1	2	yes	—
190	C₂×C₄	#14523983	1	2	yes	—
191	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
192	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
193	C₂×C₄	Q₈×Dic₆₇	1	1	yes	characteristic
194	C₂×C₄	C₂×#11761612	1	1	yes	characteristic
195	C₂×C₄	#14524050	1	1	yes	characteristic
196	C₂×C₄	#14524050	1	1	yes	characteristic
197	C₈	—	2	2	no	—
198	C₈	—	4	2	no	—
199	C₈	#14524018	1	1	yes	characteristic
200	C₈	#14524023	1	1	yes	characteristic
201	C₈	#14524034	1	1	yes	characteristic
202	C₈	#14524042	1	1	yes	characteristic
203	C₂²	C₂×C₄⋊(C₆₇⋊C₈)	1	1	yes	characteristic
204	C₂²	D₄×C₆₇⋊C₈	1	1	yes	characteristic
205	C₂²	Q₈×C₆₇⋊C₈	1	1	yes	characteristic
206	C₂²	C₂²×C₆₇⋊C₁₆	1	1	yes	characteristic
207	C₂²	C₂×#14523983	1	1	yes	characteristic
208	C₂²	#16863813	1	1	yes	characteristic
209	C₂²	#16863813	1	1	yes	characteristic
210	C₄	—	2	2	no	—
211	C₄	—	2	4	no	—
212	C₄	—	4	2	no	—
213	C₄	C₄⋊(C₆₇⋊C₁₆)	1	4	yes	complemented
214	C₄	#16863295	1	1	yes	characteristic
215	C₄	#16863423	1	1	yes	characteristic
216	C₄	#16863474	1	1	yes	characteristic
217	C₄	#16863527	1	1	yes	characteristic
218	C₂	#22560928	1	1	yes	characteristic
219	C₂	C₂×C₄⋊(C₆₇⋊C₁₆)	1	1	yes	characteristic
220	C₂	#22561968	1	1	yes	characteristic
221	C₂	D₄×C₆₇⋊C₁₆	1	1	yes	characteristic
222	C₂	#22562041	1	1	yes	characteristic
223	C₂	Q₈×C₆₇⋊C₁₆	1	1	yes	characteristic
224	C₂	#22562118	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₆₇
commutator	C₂²×C₆₇	C₂²×C₁₆
Fitting	C₈×C₄⋊C₄×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂²×C₈
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₈×C₄⋊C₄	D₆₇
hypocenter	C₆₇	#13159
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
60. C₄⋊C₄×C₆₇	C₁₆	semidirect
100. C₄×C₆₇	C₄⋊C₁₆	semidirect
118. C₆₇	#13159	semidirect
170. C₄⋊C₄	C₆₇⋊C₁₆	semidirect
213. C₄	C₄⋊(C₆₇⋊C₁₆)	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊(C₆₇⋊C₁₆)	2
C₂×C₄×C₆₇⋊C₁₆	2
#22558287	2
C₈×C₄⋊C₄×C₆₇	2
#13159	67