Subgroups — #30772178

Order 17152.

366 conjugacy classes of subgroups.

The subgroup lattice has 170 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	2	yes	maximal
2	#22558287	C₂	1	4	yes	maximal
3	C₄²×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₆₇⋊C₁₆	C₂²	1	4	yes	—
5	C₄⋊(C₆₇⋊C₁₆)	—	2	4	no	—
6	C₂×C₄×C₈×C₆₇	C₄	1	2	yes	—
7	C₂×C₄×C₈×C₆₇	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
10	C₄³×C₆₇	C₄	1	1	yes	characteristic
11	C₂×C₆₇⋊C₁₆	—	2	4	no	—
12	C₂×C₆₇⋊C₁₆	—	2	8	no	—
13	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
15	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
16	C₂×C₄²×C₆₇	C₈	1	2	yes	—
17	C₂×C₄²×C₆₇	C₈	1	1	yes	characteristic
18	C₂×C₄²×C₆₇	C₈	1	1	yes	characteristic
19	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
20	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
21	C₄×C₈×C₆₇	—	2	4	no	—
22	C₄×C₈×C₆₇	—	2	4	no	—
23	C₄×C₈×C₆₇	—	2	2	no	—
24	C₄×C₈×C₆₇	D₄	1	2	yes	—
25	C₄×C₈×C₆₇	Q₈	1	2	yes	—
26	C₆₇⋊C₁₆	—	4	4	no	—
27	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
28	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
29	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
30	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
31	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₈×C₆₇	—	2	2	no	—
33	C₂×C₈×C₆₇	—	2	2	no	—
34	C₂×C₈×C₆₇	—	2	2	no	—
35	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
36	C₂×C₈×C₆₇	M₁₆	1	4	yes	—
37	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
38	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₈×C₆₇	#41	1	4	yes	—
40	C₄²×C₆₇	—	2	4	no	—
41	C₄²×C₆₇	—	2	2	no	—
42	C₄²×C₆₇	—	2	2	no	—
43	C₄²×C₆₇	—	2	2	no	—
44	C₄²×C₆₇	C₁₆	1	4	yes	complemented
45	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
46	C₄²×C₆₇	M₁₆	1	2	yes	—
47	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
48	C₂×C₄×C₆₇	—	2	2	no	—
49	C₂×C₄×C₆₇	—	2	2	no	—
50	C₂×C₄×C₆₇	—	2	2	no	—
51	C₂×C₄×C₆₇	—	2	2	no	—
52	C₂×C₄×C₆₇	—	2	2	no	—
53	C₂×C₄×C₆₇	C₄⋊C₈	1	2	yes	—
54	C₂×C₄×C₆₇	C₄⋊C₈	1	2	yes	—
55	C₂×C₄×C₆₇	C₂×C₁₆	1	4	yes	—
56	C₂×C₄×C₆₇	C₂×C₁₆	1	2	yes	—
57	C₂×C₄×C₆₇	M₃₂	1	4	yes	—
58	C₂×C₄×C₆₇	M₃₂	1	2	yes	—
59	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₆₇	#117	1	2	yes	—
61	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
62	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
63	C₈×C₆₇	—	2	2	no	—
64	C₈×C₆₇	—	2	4	no	—
65	C₈×C₆₇	#124	1	2	yes	—
66	C₈×C₆₇	#125	1	2	yes	—
67	C₂²×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
68	C₂²×C₆₇	#431	1	2	yes	—
69	C₂²×C₆₇	C₂²×C₁₆	1	1	yes	characteristic
70	C₂²×C₆₇	C₂×M₃₂	1	1	yes	characteristic
71	C₂²×C₆₇	C₂×M₃₂	1	2	yes	—
72	C₄×C₆₇	—	2	2	no	—
73	C₄×C₆₇	—	2	4	no	—
74	C₄×C₆₇	—	2	4	no	—
75	C₄×C₆₇	C₄⋊C₁₆	1	4	yes	—
76	C₄×C₆₇	#382	1	1	yes	characteristic
77	C₄×C₆₇	#423	1	1	yes	characteristic
78	C₄×C₆₇	#432	1	2	yes	—
79	#12879	—	67	1	no	maximal, Sylow
80	C₂×C₆₇	#1831	1	1	yes	characteristic
81	C₂×C₆₇	C₂×C₄⋊C₁₆	1	1	yes	characteristic
82	C₂×C₆₇	#2150	1	1	yes	characteristic
83	C₂×C₆₇	#2162	1	2	yes	—
84	C₂×C₆₇	#2163	1	2	yes	—
85	C₂×C₄⋊C₁₆	—	67	2	no	—
86	C₄²×C₈	D₆₇	1	1	yes	characteristic
87	#1374	—	67	4	no	—
88	C₆₇	#12879	1	1	yes	characteristic, complemented, Sylow
89	C₂²×C₁₆	—	67	4	no	—
90	C₂×C₄×C₈	Dic₆₇	1	2	yes	—
91	C₂×C₄×C₈	Dic₆₇	1	1	yes	characteristic
92	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
93	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
94	C₄³	Dic₆₇	1	1	yes	characteristic
95	C₄⋊C₁₆	—	134	4	no	—
96	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
97	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
98	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
99	C₂×C₄²	C₆₇⋊C₈	1	2	yes	—
100	C₂×C₄²	C₆₇⋊C₈	1	1	yes	characteristic
101	C₂×C₄²	C₆₇⋊C₈	1	1	yes	characteristic
102	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
103	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
104	C₂×C₁₆	—	134	4	no	—
105	C₂×C₁₆	—	134	8	no	—
106	C₄×C₈	—	2	4	no	—
107	C₄×C₈	—	2	4	no	—
108	C₄×C₈	—	2	2	no	—
109	C₄×C₈	Dic₁₃₄	1	2	yes	—
110	C₄×C₈	D₂₆₈	1	2	yes	—
111	C₂²×C₄	C₂×C₆₇⋊C₈	1	1	yes	characteristic
112	C₂²×C₄	C₂×C₆₇⋊C₈	1	2	yes	—
113	C₂²×C₄	C₂×C₆₇⋊C₈	1	2	yes	—
114	C₂²×C₄	C₂×C₆₇⋊C₈	1	1	yes	characteristic
115	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	C₄⋊Dic₆₇	1	2	yes	—
120	C₂×C₈	#11761612	1	4	yes	—
121	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
122	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
123	C₂×C₈	#11761628	1	4	yes	—
124	C₄²	—	2	4	no	—
125	C₄²	—	2	2	no	—
126	C₄²	—	2	2	no	—
127	C₄²	—	2	2	no	—
128	C₄²	C₆₇⋊C₁₆	1	4	yes	complemented
129	C₄²	C₄⋊Dic₆₇	1	2	yes	—
130	C₄²	#11761612	1	2	yes	—
131	C₁₆	—	268	4	no	—
132	C₂³	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	C₄⋊(C₆₇⋊C₈)	1	2	yes	—
139	C₂×C₄	C₄⋊(C₆₇⋊C₈)	1	2	yes	—
140	C₂×C₄	C₂×C₆₇⋊C₁₆	1	4	yes	—
141	C₂×C₄	C₂×C₆₇⋊C₁₆	1	2	yes	—
142	C₂×C₄	#14523983	1	4	yes	—
143	C₂×C₄	#14523983	1	2	yes	—
144	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
145	C₂×C₄	#14524002	1	2	yes	—
146	C₂×C₄	C₂×#11761612	1	2	yes	—
147	C₂×C₄	C₂×#11761612	1	1	yes	characteristic
148	C₈	—	2	2	no	—
149	C₈	—	2	4	no	—
150	C₈	#14524031	1	2	yes	—
151	C₈	#14524035	1	2	yes	—
152	C₂²	C₂×C₄⋊(C₆₇⋊C₈)	1	1	yes	characteristic
153	C₂²	#16863449	1	2	yes	—
154	C₂²	C₂²×C₆₇⋊C₁₆	1	1	yes	characteristic
155	C₂²	C₂×#14523983	1	1	yes	characteristic
156	C₂²	C₂×#14523983	1	2	yes	—
157	C₄	—	2	2	no	—
158	C₄	—	2	4	no	—
159	C₄	—	2	4	no	—
160	C₄	C₄⋊(C₆₇⋊C₁₆)	1	4	yes	—
161	C₄	#16863262	1	1	yes	characteristic
162	C₄	#16863420	1	1	yes	characteristic
163	C₄	#16863453	1	2	yes	—
164	C₂	#22560326	1	1	yes	characteristic
165	C₂	C₂×C₄⋊(C₆₇⋊C₁₆)	1	1	yes	characteristic
166	C₂	#22561965	1	1	yes	characteristic
167	C₂	#22562008	1	2	yes	—
168	C₂	#22562012	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₆₇
commutator	C₂²×C₆₇	C₂²×C₁₆
Fitting	C₄²×C₈×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂²×C₈
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₄²×C₈	D₆₇
hypocenter	C₆₇	#12879
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
44. C₄²×C₆₇	C₁₆	semidirect
88. C₆₇	#12879	semidirect
128. C₄²	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
#22558287	2
C₄²×C₈×C₆₇	2
#12879	67