Finite Group Databasebeta

Subgroups — #30776726

Order 17152.

164 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 164 conjugacy classes of subgroups — too many to draw. Only its 59 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22562483	C₂	1	1	yes	maximal, characteristic
2	Q₃₂×Dic₆₇	C₂	1	1	yes	maximal, characteristic
3	#22558409	C₂	1	1	yes	maximal, characteristic
4	#22558408	C₂	1	1	yes	maximal, characteristic
5	Dic₆₇⋊C₃₂	C₂	1	1	yes	maximal, characteristic
6	#22558390	C₂	1	1	yes	maximal, characteristic
7	C₆₇×#1416	C₂	1	1	yes	maximal, characteristic
8	#16863800	—	2	1	no	—
9	Q₁₆×Dic₆₇	—	2	1	no	—
10	C₂×C₆₇⋊C₃₂	C₂²	1	1	yes	characteristic
11	#16863222	C₂²	1	1	yes	characteristic
12	#16863222	C₂²	1	1	yes	characteristic
13	#16863221	—	2	1	no	—
14	#16863200	—	2	1	no	—
15	C₁₆×Dic₆₇	C₂²	1	1	yes	characteristic
16	C₂×Q₃₂×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×C₃₂×C₆₇	C₂²	1	1	yes	characteristic
18	#366×C₆₇	C₂²	1	1	yes	characteristic
19	#14524042	—	4	1	no	—
20	Q₈×Dic₆₇	—	4	1	no	—
21	C₂×C₆₇⋊C₁₆	D₄	1	1	yes	characteristic
22	#14523977	—	2	1	no	—
23	#14523977	—	2	1	no	—
24	#14523976	—	4	1	no	—
25	#14523967	—	4	1	no	—
26	C₈×Dic₆₇	D₄	1	1	yes	characteristic
27	C₆₇⋊C₃₂	—	2	1	no	—
28	C₂×Q₁₆×C₆₇	—	2	1	no	—
29	Q₃₂×C₆₇	—	2	1	no	—
30	Q₃₂×C₆₇	Q₈	1	2	yes	complemented
31	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
32	#107×C₆₇	—	2	1	no	—
33	C₃₂×C₆₇	—	2	1	no	—
34	C₂×Dic₁₃₄	—	8	1	no	—
35	C₂×C₆₇⋊C₈	D₈	1	1	yes	characteristic
36	C₄⋊Dic₆₇	—	4	1	no	—
37	C₄⋊Dic₆₇	—	4	1	no	—
38	C₄⋊Dic₆₇	—	8	1	no	—
39	Dic₆₇⋊C₄	—	8	1	no	—
40	C₄×Dic₆₇	—	8	1	no	—
41	C₄×Dic₆₇	D₈	1	1	yes	characteristic
42	C₆₇⋊C₁₆	—	2	1	no	—
43	C₂×Q₈×C₆₇	—	4	1	no	—
44	Q₁₆×C₆₇	—	2	2	no	—
45	Q₁₆×C₆₇	—	4	1	no	—
46	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
47	C₄⋊C₄×C₆₇	—	4	1	no	—
48	C₁₆×C₆₇	C₂×Q₈	1	1	yes	characteristic
49	C₁₆×C₆₇	#41	1	1	yes	characteristic
50	Q₈×C₆₇	—	4	2	no	—
51	Q₈×C₆₇	—	8	1	no	—
52	C₂×C₄×C₆₇	—	8	1	no	—
53	C₂×C₄×C₆₇	—	8	1	no	—
54	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
55	C₂×Dic₆₇	—	8	1	no	—
56	C₂×Dic₆₇	—	8	1	no	—
57	C₂×Dic₆₇	D₁₆	1	1	yes	characteristic
58	C₂×Dic₆₇	D₁₆	1	1	yes	characteristic
59	Dic₁₃₄	—	16	2	no	—
60	C₈×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
61	C₈×C₆₇	#137	1	1	yes	characteristic
62	C₆₇⋊C₈	—	2	1	no	—
63	C₂²×C₆₇	C₂×D₁₆	1	1	yes	characteristic
64	C₄×C₆₇	—	8	2	no	—
65	C₄×C₆₇	—	16	1	no	—
66	C₄×C₆₇	#474	1	1	yes	characteristic
67	C₄×C₆₇	#510	1	1	yes	characteristic
68	Dic₆₇	—	2	1	no	—
69	Dic₆₇	—	16	1	no	—
70	Dic₆₇	—	16	1	no	—
71	Dic₆₇	Q₆₄	1	2	yes	—
72	#13706	—	67	1	no	maximal, Sylow
73	C₂×C₆₇	#2224	1	1	yes	characteristic
74	C₂×C₆₇	C₂×Q₆₄	1	1	yes	characteristic
75	C₂×C₆₇	#2263	1	1	yes	characteristic
76	#2252	—	67	1	no	—
77	C₄×Q₃₂	—	67	1	no	—
78	#1423	—	67	1	no	—
79	#1423	—	67	1	no	—
80	C₄⋊C₃₂	—	67	1	no	—
81	#1416	—	67	1	no	—
82	#1416	D₆₇	1	1	yes	characteristic
83	C₆₇	#13706	1	1	yes	characteristic, complemented, Sylow
84	C₂×Q₃₂	C₂×D₆₇	1	1	yes	characteristic
85	#500	—	134	1	no	—
86	C₄×Q₁₆	—	134	1	no	—
87	C₂×C₃₂	—	67	1	no	—
88	C₂×C₃₂	C₂×D₆₇	1	1	yes	characteristic
89	#366	—	67	1	no	—
90	#366	—	67	1	no	—
91	#366	—	134	1	no	—
92	#366	C₂×D₆₇	1	1	yes	characteristic
93	#358	—	134	1	no	—
94	C₄×C₁₆	—	67	1	no	—
95	C₂×Q₁₆	—	2	1	no	—
96	C₄⋊Q₈	—	268	1	no	—
97	C₄×Q₈	—	268	1	no	—
98	Q₃₂	—	2	1	no	—
99	Q₃₂	Dic₁₃₄	1	2	yes	complemented
100	C₂×C₁₆	—	67	1	no	—
101	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
102	#107	—	2	1	no	—
103	#107	—	268	1	no	—
104	#107	—	134	1	no	—
105	#107	—	134	1	no	—
106	#103	—	268	1	no	—
107	C₄×C₈	—	67	1	no	—
108	C₃₂	—	2	1	no	—
109	C₃₂	—	134	1	no	—
110	C₂×Q₈	—	4	1	no	—
111	C₂×Q₈	—	536	1	no	—
112	Q₁₆	—	2	2	no	—
113	Q₁₆	—	4	1	no	—
114	C₂×C₈	—	67	1	no	—
115	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
116	C₄⋊C₄	—	4	1	no	—
117	C₄⋊C₄	—	268	1	no	—
118	C₄⋊C₄	—	268	1	no	—
119	C₄⋊C₄	—	536	1	no	—
120	C₄⋊C₄	—	536	1	no	—
121	C₄²	—	536	1	no	—
122	C₄²	—	67	1	no	—
123	C₁₆	—	134	1	no	—
124	C₁₆	C₂×Dic₁₃₄	1	1	yes	characteristic
125	C₁₆	#11761627	1	1	yes	characteristic
126	Q₈	—	4	2	no	—
127	Q₈	—	8	1	no	—
128	Q₈	—	1072	2	no	—
129	C₂×C₄	—	8	1	no	—
130	C₂×C₄	—	8	1	no	—
131	C₂×C₄	—	536	1	no	—
132	C₂×C₄	—	536	1	no	—
133	C₂×C₄	—	67	1	no	—
134	C₂×C₄	—	67	1	no	—
135	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
136	C₈	—	134	1	no	—
137	C₈	#14524020	1	1	yes	characteristic
138	C₈	#14524072	1	1	yes	characteristic
139	C₂²	D₁₆×D₆₇	1	1	yes	characteristic
140	C₄	—	8	2	no	—
141	C₄	—	16	1	no	—
142	C₄	—	1072	1	no	—
143	C₄	—	1072	1	no	—
144	C₄	—	67	2	no	—
145	C₄	—	134	1	no	—
146	C₄	#16863674	1	1	yes	characteristic
147	C₄	#16863835	1	1	yes	characteristic
148	C₂	#22562344	1	1	yes	characteristic
149	C₂	Q₆₄×D₆₇	1	1	yes	characteristic
150	C₂	#22562523	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₁₆×D₆₇
commutator, cosocle	C₂×C₁₆×C₆₇	C₂³
Frattini	C₂×C₁₆	C₂²×D₆₇
Fitting	C₆₇×#1416	C₂
socle, prime radical	C₂²×C₆₇	C₂×D₁₆
hypercenter	#1416	D₆₇
hypocenter	C₆₇	#13706
LCS	G ▷ C₂×C₁₆×C₆₇ ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1416

Complements

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

N	H ≅ G/N	type
30. Q₃₂×C₆₇	Q₈	semidirect
83. C₆₇	#13706	semidirect
99. Q₃₂	Dic₁₃₄	semidirect

Maximal subgroups

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

subgroup	index
#22562483	2
Q₃₂×Dic₆₇	2
#22558409	2
#22558408	2
Dic₆₇⋊C₃₂	2
#22558390	2
C₆₇×#1416	2
#13706	67