Subgroups — #30742671

Order 17152.

254 conjugacy classes of subgroups.

The subgroup lattice has 152 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	#22558492	C₂	1	1	yes	maximal, characteristic
3	C₄×M₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₆₇⋊C₁₆	C₄	1	2	yes	—
5	#16863449	C₄	1	2	yes	—
6	C₄×C₆₇⋊C₁₆	—	2	1	no	—
7	C₄×C₆₇⋊C₁₆	—	2	1	no	—
8	#16863117	—	2	1	no	—
9	C₈×C₆₇⋊C₈	—	2	1	no	—
10	C₂×M₃₂×C₆₇	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
12	#346×C₆₇	—	2	1	no	—
13	C₄×C₁₆×C₆₇	—	2	1	no	—
14	C₂×C₆₇⋊C₁₆	—	2	2	no	—
15	C₂×C₆₇⋊C₁₆	—	2	2	no	—
16	C₂×C₆₇⋊C₁₆	C₈	1	4	yes	complemented
17	C₄⋊(C₆₇⋊C₈)	C₈	1	4	yes	—
18	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
19	C₄×C₆₇⋊C₈	—	2	2	no	—
20	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
23	M₃₂×C₆₇	—	2	2	no	—
24	M₃₂×C₆₇	C₈	1	4	yes	complemented
25	C₂×C₁₆×C₆₇	—	2	2	no	—
26	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
27	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
28	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
29	C₄×C₈×C₆₇	Q₈	1	1	yes	characteristic
30	C₂×C₆₇⋊C₈	—	2	4	no	—
31	C₆₇⋊C₁₆	—	2	4	no	—
32	C₂²×C₄×C₆₇	C₄²	1	1	yes	characteristic
33	C₂²×C₄×C₆₇	M₁₆	1	2	yes	—
34	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
36	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₆₇	C₂×C₈	1	2	yes	—
39	C₂×C₈×C₆₇	C₂×C₈	1	2	yes	—
40	C₂×C₈×C₆₇	M₁₆	1	2	yes	—
41	C₂×C₈×C₆₇	M₁₆	1	2	yes	—
42	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
43	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
44	C₄²×C₆₇	C₂×C₈	1	2	yes	—
45	C₁₆×C₆₇	—	2	4	no	—
46	C₂³×C₆₇	#97	1	1	yes	characteristic
47	C₂×C₄×C₆₇	—	2	4	no	—
48	C₂×C₄×C₆₇	#95	1	1	yes	characteristic
49	C₂×C₄×C₆₇	C₄×C₈	1	1	yes	characteristic
50	C₂×C₄×C₆₇	C₄×C₈	1	1	yes	characteristic
51	C₂×C₄×C₆₇	C₄×C₈	1	1	yes	characteristic
52	C₂×C₄×C₆₇	#97	1	1	yes	characteristic
53	C₂×C₄×C₆₇	#97	1	1	yes	characteristic
54	C₂×C₄×C₆₇	C₂²⋊C₈	1	2	yes	—
55	C₂×C₄×C₆₇	C₄⋊C₈	1	2	yes	—
56	C₈×C₆₇	C₂²⋊C₈	1	2	yes	—
57	C₈×C₆₇	C₂²⋊C₈	1	2	yes	—
58	C₈×C₆₇	C₄⋊C₈	1	2	yes	—
59	C₈×C₆₇	C₄⋊C₈	1	2	yes	—
60	C₆₇⋊C₈	—	4	4	no	—
61	C₂²×C₆₇	—	2	2	no	—
62	C₂²×C₆₇	#322	1	1	yes	characteristic
63	C₂²×C₆₇	#322	1	1	yes	characteristic
64	C₂²×C₆₇	#336	1	1	yes	characteristic
65	C₄×C₆₇	—	2	2	no	—
66	C₄×C₆₇	C₈²	1	1	yes	characteristic
67	C₄×C₆₇	#322	1	1	yes	characteristic
68	C₄×C₆₇	#336	1	1	yes	characteristic
69	C₄×C₆₇	#336	1	1	yes	characteristic
70	C₄×C₆₇	#350	1	4	yes	—
71	#7141	—	67	1	no	maximal, Sylow
72	C₂×C₆₇	—	2	1	no	—
73	C₂×C₆₇	—	2	1	no	—
74	C₂×C₆₇	#1273	1	1	yes	characteristic
75	C₂×C₆₇	#1379	1	2	yes	—
76	C₄×M₃₂	D₆₇	1	1	yes	characteristic
77	C₂×C₄×C₁₆	—	67	1	no	—
78	#1450	—	67	1	no	—
79	C₆₇	#7141	1	1	yes	characteristic, complemented, Sylow
80	C₂×M₃₂	Dic₆₇	1	2	yes	—
81	C₂²×C₁₆	—	67	2	no	—
82	#431	—	67	2	no	—
83	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
84	#346	—	2	1	no	—
85	C₄×C₁₆	—	2	1	no	—
86	C₄×C₁₆	—	134	1	no	—
87	C₄×C₁₆	—	134	1	no	—
88	#322	—	134	1	no	—
89	C₈²	—	134	1	no	—
90	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
91	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
92	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
93	M₃₂	—	2	2	no	—
94	M₃₂	C₆₇⋊C₈	1	4	yes	complemented
95	C₂×C₁₆	—	2	2	no	—
96	C₂×C₁₆	—	67	4	no	—
97	C₂×C₁₆	—	134	2	no	—
98	C₂×C₁₆	—	134	2	no	—
99	C₄⋊C₈	—	67	4	no	—
100	C₂²⋊C₈	—	134	2	no	—
101	C₄×C₈	—	134	2	no	—
102	C₄×C₈	Dic₁₃₄	1	1	yes	characteristic
103	C₄×C₈	D₂₆₈	1	1	yes	characteristic
104	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
105	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
106	C₂²×C₄	C₄×Dic₆₇	1	1	yes	characteristic
107	C₂²×C₄	#11761611	1	2	yes	—
108	C₂×C₈	—	134	4	no	—
109	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
110	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
111	C₂×C₈	Dic₆₇⋊C₄	1	1	yes	characteristic
112	C₂×C₈	C₄⋊Dic₆₇	1	1	yes	characteristic
113	C₂×C₈	C₈×D₆₇	1	2	yes	—
114	C₂×C₈	C₂×C₆₇⋊C₈	1	2	yes	—
115	C₂×C₈	#11761611	1	2	yes	—
116	C₂×C₈	#11761612	1	2	yes	—
117	C₄²	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
118	C₄²	Dic₆₇⋊C₄	1	1	yes	characteristic
119	C₄²	C₈×D₆₇	1	2	yes	—
120	C₁₆	—	2	4	no	—
121	C₁₆	—	134	4	no	—
122	C₂³	#14523953	1	1	yes	characteristic
123	C₂×C₄	—	2	4	no	—
124	C₂×C₄	C₆₇⋊#95	1	1	yes	characteristic
125	C₂×C₄	C₈×Dic₆₇	1	1	yes	characteristic
126	C₂×C₄	C₈×Dic₆₇	1	1	yes	characteristic
127	C₂×C₄	C₄×C₆₇⋊C₈	1	1	yes	characteristic
128	C₂×C₄	#14523953	1	1	yes	characteristic
129	C₂×C₄	#14523954	1	1	yes	characteristic
130	C₂×C₄	(C₂×D₆₇)⋊C₈	1	2	yes	—
131	C₂×C₄	Dic₆₇⋊C₈	1	2	yes	—
132	C₈	—	268	4	no	—
133	C₈	(C₂×D₆₇)⋊C₈	1	2	yes	—
134	C₈	C₂²⋊(C₆₇⋊C₈)	1	2	yes	—
135	C₈	Dic₆₇⋊C₈	1	2	yes	—
136	C₈	C₄⋊(C₆₇⋊C₈)	1	2	yes	—
137	C₂²	—	2	2	no	—
138	C₂²	#16863117	1	1	yes	characteristic
139	C₂²	#16863118	1	1	yes	characteristic
140	C₂²	#16863150	1	1	yes	characteristic
141	C₄	—	2	2	no	—
142	C₄	C₈×C₆₇⋊C₈	1	1	yes	characteristic
143	C₄	#16863118	1	1	yes	characteristic
144	C₄	#16863150	1	1	yes	characteristic
145	C₄	#16863151	1	1	yes	characteristic
146	C₄	#16863181	1	4	yes	—
147	C₂	—	2	1	no	—
148	C₂	—	2	1	no	—
149	C₂	C₆₇⋊#1273	1	1	yes	characteristic
150	C₂	#22558299	1	2	yes	—

	subgroup	G/H
center	C₄×C₈	C₂²⋊D₆₇
commutator	C₄×C₆₇	C₈²
Frattini	C₂×C₄×C₈	C₂×D₆₇
Fitting	C₄×M₃₂×C₆₇	C₂
socle	C₂²×C₆₇	#336
prime radical	C₂³×C₆₇	#97
cosocle	C₂×C₄×C₈×C₆₇	C₂²
hypercenter	C₄×M₃₂	D₆₇
hypocenter	C₆₇	#7141
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
16. C₂×C₆₇⋊C₁₆	C₈	semidirect
24. M₃₂×C₆₇	C₈	semidirect
79. C₆₇	#7141	semidirect
94. M₃₂	C₆₇⋊C₈	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
#22558492	2
C₄×M₃₂×C₆₇	2
#7141	67