Subgroups — #30771180

Order 17152.

374 conjugacy classes of subgroups.

The subgroup lattice has 200 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×#16863172	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	1	yes	characteristic
7	C₂²×C₆₇⋊C₁₆	C₂²	1	2	yes	—
8	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
9	C₄⋊(C₆₇⋊C₁₆)	C₄	1	4	yes	complemented
10	#16863172	—	2	4	no	—
11	C₄×C₆₇⋊C₁₆	C₄	1	4	yes	complemented
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	2	no	—
18	C₂×C₆₇⋊C₁₆	—	2	1	no	—
19	C₂×C₆₇⋊C₁₆	—	2	1	no	—
20	C₂×C₆₇⋊C₁₆	—	2	2	no	—
21	C₂×C₆₇⋊C₁₆	C₂×C₄	1	2	yes	—
22	C₂×C₆₇⋊C₁₆	C₂×C₄	1	4	yes	—
23	C₂×C₆₇⋊C₁₆	C₂×C₄	1	2	yes	—
24	C₂×C₆₇⋊C₁₆	D₄	1	2	yes	—
25	C₂×C₆₇⋊C₁₆	Q₈	1	2	yes	—
26	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
28	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₆₇	C₈	1	2	yes	—
30	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
32	C₄⋊C₈×C₆₇	C₈	1	4	yes	—
33	C₄×C₈×C₆₇	—	2	2	no	—
34	C₄×C₈×C₆₇	—	2	2	no	—
35	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
36	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
37	#95×C₆₇	C₈	1	2	yes	—
38	C₆₇⋊C₁₆	—	2	2	no	—
39	C₆₇⋊C₁₆	—	2	4	no	—
40	C₆₇⋊C₁₆	C₄⋊C₄	1	4	yes	complemented
41	C₂²×C₄×C₆₇	C₂×C₈	1	4	yes	—
42	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
43	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
44	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₆₇	—	2	2	no	—
46	C₂×C₈×C₆₇	—	2	1	no	—
47	C₂×C₈×C₆₇	—	2	1	no	—
48	C₂×C₈×C₆₇	—	2	2	no	—
49	C₂×C₈×C₆₇	C₄²	1	2	yes	—
50	C₂×C₈×C₆₇	C₂×C₈	1	4	yes	—
51	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
54	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
56	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
57	C₄⋊C₄×C₆₇	—	2	4	no	—
58	C₄²×C₆₇	—	2	2	no	—
59	C₄²×C₆₇	—	2	2	no	—
60	C₄²×C₆₇	C₄²	1	2	yes	—
61	C₄²×C₆₇	C₂×C₈	1	2	yes	—
62	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
63	C₂×C₄×C₆₇	—	2	2	no	—
64	C₂×C₄×C₆₇	—	2	4	no	—
65	C₂×C₄×C₆₇	—	2	4	no	—
66	C₂×C₄×C₆₇	—	2	4	no	—
67	C₂×C₄×C₆₇	C₄×C₈	1	2	yes	—
68	C₂×C₄×C₆₇	C₄×C₈	1	2	yes	—
69	C₂×C₄×C₆₇	M₃₂	1	4	yes	—
70	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
71	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
72	C₂×C₄×C₆₇	C₄×Q₈	1	1	yes	characteristic
73	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
74	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
75	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
76	C₈×C₆₇	—	2	2	no	—
77	C₈×C₆₇	—	2	4	no	—
78	C₈×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₈×C₆₇	#117	1	1	yes	characteristic
80	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
81	C₈×C₆₇	C₄×Q₈	1	1	yes	characteristic
82	C₂²×C₆₇	C₂×C₄×C₈	1	1	yes	characteristic
83	C₂²×C₆₇	C₈×D₄	1	1	yes	characteristic
84	C₂²×C₆₇	C₈×Q₈	1	1	yes	characteristic
85	C₂²×C₆₇	C₂×M₃₂	1	2	yes	—
86	C₂²×C₆₇	#504	1	2	yes	—
87	C₄×C₆₇	—	2	2	no	—
88	C₄×C₆₇	—	4	4	no	—
89	C₄×C₆₇	#346	1	4	yes	complemented
90	C₄×C₆₇	C₄×C₄⋊C₄	1	1	yes	characteristic
91	C₄×C₆₇	#405	1	1	yes	characteristic
92	C₄×C₆₇	C₈×D₄	1	1	yes	characteristic
93	C₄×C₆₇	C₈×Q₈	1	1	yes	characteristic
94	#12662	—	67	1	no	maximal, Sylow
95	C₂×C₆₇	C₈×C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₆₇	C₂×#346	1	1	yes	characteristic
97	C₂×C₆₇	#2108	1	1	yes	characteristic
98	C₂×C₆₇	#2169	1	2	yes	—
99	C₂×C₆₇	#2183	1	2	yes	—
100	C₂×C₄⋊C₁₆	—	67	1	no	—
101	C₂×#346	—	67	2	no	—
102	C₂×C₄×C₁₆	—	67	1	no	—
103	C₈×C₄⋊C₄	D₆₇	1	1	yes	characteristic
104	#1374	—	67	2	no	—
105	C₆₇	#12662	1	1	yes	characteristic, complemented, Sylow
106	C₂²×C₁₆	—	67	1	no	—
107	C₂²×C₁₆	—	67	1	no	—
108	C₂²×C₁₆	—	67	2	no	—
109	C₂×C₄⋊C₈	Dic₆₇	1	1	yes	characteristic
110	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
111	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
112	C₄×C₄⋊C₄	Dic₆₇	1	1	yes	characteristic
113	C₄⋊C₁₆	—	67	4	no	—
114	#346	—	134	4	no	—
115	C₄×C₁₆	—	67	4	no	—
116	#336	Dic₆₇	1	2	yes	—
117	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
118	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
119	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
120	C₂×C₄⋊C₄	C₆₇⋊C₈	1	2	yes	—
121	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
122	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
123	C₂×C₁₆	—	67	2	no	—
124	C₂×C₁₆	—	67	2	no	—
125	C₂×C₁₆	—	67	2	no	—
126	C₂×C₁₆	—	67	2	no	—
127	C₂×C₁₆	—	67	4	no	—
128	C₂×C₁₆	—	134	2	no	—
129	C₂×C₁₆	—	134	1	no	—
130	C₂×C₁₆	—	134	1	no	—
131	C₂×C₁₆	—	134	2	no	—
132	C₄⋊C₈	C₆₇⋊C₈	1	4	yes	—
133	C₄×C₈	—	2	2	no	—
134	C₄×C₈	—	2	2	no	—
135	C₄×C₈	C₄×D₆₇	1	2	yes	—
136	C₄×C₈	C₄×D₆₇	1	2	yes	—
137	#95	C₆₇⋊C₈	1	2	yes	—
138	C₂²×C₄	C₂×C₆₇⋊C₈	1	4	yes	—
139	C₂²×C₄	C₂×C₆₇⋊C₈	1	1	yes	characteristic
140	C₂²×C₄	C₂×C₆₇⋊C₈	1	1	yes	characteristic
141	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	2	1	no	—
144	C₂×C₈	—	2	1	no	—
145	C₂×C₈	—	2	2	no	—
146	C₂×C₈	C₄×Dic₆₇	1	2	yes	—
147	C₂×C₈	C₂×C₆₇⋊C₈	1	4	yes	—
148	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
149	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
150	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
151	C₂×C₈	Q₈×D₆₇	1	1	yes	characteristic
152	C₂×C₈	#11761627	1	1	yes	characteristic
153	C₂×C₈	#11761629	1	1	yes	characteristic
154	C₄⋊C₄	—	2	4	no	—
155	C₄²	—	2	2	no	—
156	C₄²	—	2	2	no	—
157	C₄²	C₄×Dic₆₇	1	2	yes	—
158	C₄²	C₂×C₆₇⋊C₈	1	2	yes	—
159	C₁₆	—	67	4	no	—
160	C₁₆	—	134	2	no	—
161	C₁₆	—	134	4	no	—
162	C₂³	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	C₄×C₆₇⋊C₈	1	2	yes	—
168	C₂×C₄	C₄×C₆₇⋊C₈	1	2	yes	—
169	C₂×C₄	#14523983	1	4	yes	—
170	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
171	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
172	C₂×C₄	Q₈×Dic₆₇	1	1	yes	characteristic
173	C₂×C₄	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
174	C₂×C₄	#14524050	1	1	yes	characteristic
175	C₂×C₄	#14524050	1	1	yes	characteristic
176	C₈	—	2	2	no	—
177	C₈	—	2	4	no	—
178	C₈	C₄⋊C₄×D₆₇	1	1	yes	characteristic
179	C₈	#14524001	1	1	yes	characteristic
180	C₈	#14524005	1	1	yes	characteristic
181	C₈	#14524008	1	1	yes	characteristic
182	C₂²	C₂×C₄×C₆₇⋊C₈	1	1	yes	characteristic
183	C₂²	D₄×C₆₇⋊C₈	1	1	yes	characteristic
184	C₂²	Q₈×C₆₇⋊C₈	1	1	yes	characteristic
185	C₂²	C₂×#14523983	1	2	yes	—
186	C₂²	#16863813	1	2	yes	—
187	C₄	—	2	2	no	—
188	C₄	—	4	4	no	—
189	C₄	#16863172	1	4	yes	complemented
190	C₄	C₄⋊C₄×Dic₆₇	1	1	yes	characteristic
191	C₄	#16863352	1	1	yes	characteristic
192	C₄	D₄×C₆₇⋊C₈	1	1	yes	characteristic
193	C₄	Q₈×C₆₇⋊C₈	1	1	yes	characteristic
194	C₂	C₄⋊C₄×C₆₇⋊C₈	1	1	yes	characteristic
195	C₂	C₂×#16863172	1	1	yes	characteristic
196	C₂	#22561806	1	1	yes	characteristic
197	C₂	#22562041	1	2	yes	—
198	C₂	#22562118	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₆₇
commutator	C₂²×C₆₇	C₂×C₄×C₈
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	#12662
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
9. C₄⋊(C₆₇⋊C₁₆)	C₄	semidirect
11. C₄×C₆₇⋊C₁₆	C₄	semidirect
40. C₆₇⋊C₁₆	C₄⋊C₄	semidirect
89. C₄×C₆₇	#346	semidirect
105. C₆₇	#12662	semidirect
189. C₄	#16863172	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×#16863172	2
C₂×C₄×C₆₇⋊C₁₆	2
#22558287	2
C₈×C₄⋊C₄×C₆₇	2
#12662	67