Finite Group Databasebeta

Subgroups — #30845160

Order 17152.

736 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 179 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	#22570954	C₂	1	3	yes	maximal, complemented
2	#22570584	C₂	1	3	yes	maximal
3	#22559869	C₂	1	6	yes	maximal, complemented
4	#22559821	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1455	C₂	1	1	yes	maximal, characteristic, complemented
6	#16863953	C₂²	1	3	yes	—
7	C₂×#14524040	C₂²	1	1	yes	characteristic
8	#16863794	C₂²	1	3	yes	complemented
9	#16863776	C₂²	1	3	yes	complemented
10	#16863761	—	2	6	no	—
11	#16863747	C₂²	1	6	yes	complemented
12	#16863743	C₂²	1	6	yes	—
13	#16863743	C₂²	1	6	yes	—
14	#16863654	—	2	6	no	—
15	#16863504	—	2	6	no	—
16	#16863402	—	2	6	no	—
17	C₄×M₁₆×C₆₇	C₂²	1	3	yes	—
18	#322×C₆₇	C₂²	1	3	yes	complemented
19	C₈²×C₆₇	C₂²	1	1	yes	characteristic, complemented
20	C₂²×D₂₆₈	—	2	3	no	—
21	C₂×Dic₂₆₈	—	2	6	no	—
22	#14524043	C₂³	1	6	yes	—
23	#14524040	—	2	3	no	—
24	#14524040	C₂³	1	2	yes	—
25	#14524035	—	2	3	no	—
26	#14524022	—	2	6	no	—
27	C₄×Dic₁₃₄	—	2	6	no	—
28	#14524002	—	2	3	no	—
29	#14523975	—	2	6	no	—
30	#14523968	—	2	12	no	—
31	#14523965	—	2	12	no	—
32	#14523965	—	2	12	no	—
33	C₂×M₁₆×C₆₇	D₄	1	6	yes	—
34	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
35	#97×C₆₇	D₄	1	6	yes	—
36	C₄×C₈×C₆₇	C₂³	1	3	yes	—
37	C₄×C₈×C₆₇	C₂³	1	3	yes	—
38	C₂³×D₆₇	—	4	1	no	—
39	C₂×Dic₁₃₄	—	2	6	no	—
40	C₂×D₂₆₈	—	2	6	no	—
41	C₂×D₂₆₈	—	4	12	no	—
42	C₂×D₂₆₈	—	4	3	no	—
43	Dic₂₆₈	—	4	12	no	—
44	C₄⋊Dic₆₇	—	2	6	no	—
45	C₄⋊Dic₆₇	—	2	6	no	—
46	Dic₆₇⋊C₄	—	4	6	no	—
47	C₂²⋊Dic₆₇	—	4	3	no	—
48	C₄×Dic₆₇	—	4	3	no	—
49	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
50	M₁₆×C₆₇	—	4	6	no	—
51	C₂×C₈×C₆₇	C₂×D₄	1	6	yes	—
52	C₂×C₈×C₆₇	C₂×D₄	1	6	yes	—
53	C₄²×C₆₇	C₂×D₄	1	3	yes	—
54	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
55	C₂³×C₆₇	#127	1	1	yes	characteristic
56	C₂²×D₆₇	—	4	2	no	—
57	C₂²×D₆₇	—	8	6	no	—
58	C₂×C₄×C₆₇	—	2	6	no	—
59	C₂×C₄×C₆₇	#127	1	3	yes	—
60	C₂×C₄×C₆₇	C₂²×D₄	1	3	yes	—
61	C₂×Dic₆₇	—	4	6	no	—
62	D₂₆₈	—	4	12	no	—
63	D₂₆₈	—	8	6	no	—
64	Dic₁₃₄	—	4	12	no	—
65	C₈×C₆₇	—	2	6	no	—
66	C₈×C₆₇	—	2	6	no	—
67	C₂²×C₆₇	—	2	3	no	—
68	C₂²×C₆₇	C₂×#127	1	1	yes	characteristic
69	C₂×D₆₇	—	8	6	no	—
70	C₂×D₆₇	—	16	4	no	—
71	C₄×C₆₇	—	2	3	no	—
72	C₄×C₆₇	#575	1	6	yes	—
73	Dic₆₇	—	8	6	no	—
74	#19753	—	67	1	no	maximal, Sylow
75	C₂×C₆₇	—	4	1	no	—
76	C₂×C₆₇	#3153	1	3	yes	—
77	D₆₇	—	16	2	no	—
78	#3185	—	67	3	no	—
79	#3138	—	67	3	no	—
80	#1720	—	67	6	no	—
81	#1711	—	67	2	no	—
82	#1455	D₆₇	1	1	yes	characteristic, complemented
83	C₆₇	#19753	1	1	yes	characteristic, complemented, Sylow
84	#533	—	67	3	no	—
85	C₂×#127	—	67	1	no	—
86	#498	—	67	3	no	—
87	#494	—	67	3	no	—
88	#490	—	134	6	no	—
89	#487	—	67	6	no	—
90	#486	—	67	6	no	—
91	#486	—	67	6	no	—
92	#470	—	134	6	no	—
93	#441	—	134	6	no	—
94	#418	—	134	6	no	—
95	C₄×M₁₆	C₂×D₆₇	1	3	yes	—
96	#322	C₂×D₆₇	1	3	yes	complemented
97	C₈²	C₂×D₆₇	1	1	yes	characteristic, complemented
98	C₂²×D₄	—	134	3	no	—
99	C₂×Q₁₆	—	134	6	no	—
100	C₂×M₁₆	D₂₆₈	1	6	yes	—
101	C₄⋊Q₈	—	67	6	no	—
102	#127	—	67	2	no	—
103	#127	—	134	3	no	—
104	#125	—	134	3	no	—
105	C₂²⋊Q₈	—	134	6	no	—
106	C₄×Q₈	—	134	6	no	—
107	#117	—	134	3	no	—
108	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
109	#106	—	134	6	no	—
110	#103	—	134	12	no	—
111	#102	—	134	12	no	—
112	#102	—	134	12	no	—
113	#97	D₂₆₈	1	6	yes	—
114	C₄×C₈	C₂²×D₆₇	1	3	yes	—
115	C₄×C₈	C₂²×D₆₇	1	3	yes	—
116	C₂⁴	—	268	1	no	—
117	C₂×Q₈	—	134	6	no	—
118	C₂×D₄	—	268	12	no	—
119	C₂×D₄	—	268	3	no	—
120	C₂×D₄	—	134	6	no	—
121	C₂²×C₄	C₂×D₂₆₈	1	3	yes	—
122	Q₁₆	—	268	12	no	—
123	M₁₆	—	4	6	no	—
124	C₂×C₈	C₂×D₂₆₈	1	6	yes	—
125	C₂×C₈	C₂×D₂₆₈	1	6	yes	—
126	C₄⋊C₄	—	268	6	no	—
127	C₄⋊C₄	—	134	6	no	—
128	C₄⋊C₄	—	134	6	no	—
129	C₂²⋊C₄	—	268	3	no	—
130	C₄²	—	268	3	no	—
131	C₄²	C₂×D₂₆₈	1	3	yes	—
132	C₄²	C₂³×D₆₇	1	1	yes	characteristic
133	C₂³	—	268	2	no	—
134	C₂³	—	536	6	no	—
135	C₂³	#14524040	1	1	yes	characteristic
136	Q₈	—	268	12	no	—
137	D₄	—	268	12	no	—
138	D₄	—	536	6	no	—
139	C₂×C₄	—	2	6	no	—
140	C₂×C₄	—	268	6	no	—
141	C₂×C₄	#14524040	1	3	yes	—
142	C₂×C₄	C₂²×D₂₆₈	1	3	yes	—
143	C₈	—	2	6	no	—
144	C₈	—	2	6	no	—
145	C₂²	—	2	3	no	—
146	C₂²	—	536	6	no	—
147	C₂²	—	1072	4	no	—
148	C₂²	C₂×#14524040	1	1	yes	characteristic
149	C₄	—	2	3	no	—
150	C₄	—	536	6	no	—
151	C₄	#16864196	1	6	yes	—
152	C₂	—	4	1	no	—
153	C₂	—	1072	2	no	—
154	C₂	#22570675	1	3	yes	—

	subgroup	G/H
center	C₂²	C₂×#14524040
commutator, cosocle	C₄²×C₆₇	C₂⁴
Frattini	C₄²	C₂³×D₆₇
Fitting	C₆₇×#1455	C₂
socle	C₂²×C₆₇	C₂×#127
prime radical	C₂×#14524040	C₂²
hypercenter	#1455	D₆₇
hypocenter	C₆₇	#19753
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1455

Complements

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

N	H ≅ G/N	type
1. #22570954	C₂	semidirect
3. #22559869	C₂	semidirect
4. #22559821	C₂	semidirect
5. C₆₇×#1455	C₂	semidirect
8. #16863794	C₂²	semidirect
9. #16863776	C₂²	semidirect
11. #16863747	C₂²	semidirect
18. #322×C₆₇	C₂²	semidirect
19. C₈²×C₆₇	C₂²	semidirect
82. #1455	D₆₇	semidirect
83. C₆₇	#19753	semidirect
96. #322	C₂×D₆₇	semidirect
97. C₈²	C₂×D₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22570954	2
#22570584	2
#22559869	2
#22559821	2
C₆₇×#1455	2
#19753	67