Finite Group Databasebeta

Subgroups — #30776750

Order 17152.

160 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 53 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	#22562490	C₂	1	1	yes	maximal, characteristic, complemented
2	#22562090	C₂	1	1	yes	maximal, characteristic
3	#22558415	C₂	1	1	yes	maximal, characteristic, complemented
4	#22558414	C₂	1	1	yes	maximal, characteristic, complemented
5	#22558406	C₂	1	1	yes	maximal, characteristic, complemented
6	#22558396	C₂	1	1	yes	maximal, characteristic
7	C₆₇×#1418	C₂	1	1	yes	maximal, characteristic
8	#16863800	—	2	1	no	—
9	#16863513	—	2	1	no	—
10	#16863230	C₂²	1	1	yes	characteristic
11	#16863226	C₂²	1	1	yes	characteristic
12	#16863224	C₂²	1	1	yes	characteristic
13	#16863223	—	2	1	no	—
14	#16863203	—	2	1	no	—
15	C₁₆×Dic₆₇	C₂²	1	1	yes	characteristic
16	#508×C₆₇	C₂²	1	1	yes	characteristic
17	M₆₄×C₆₇	C₂²	1	1	yes	characteristic
18	#367×C₆₇	C₂²	1	1	yes	characteristic
19	#14524050	—	4	1	no	—
20	#14524042	—	4	1	no	—
21	C₂×C₆₇⋊C₁₆	D₄	1	1	yes	characteristic
22	#14523979	—	2	1	no	—
23	#14523977	—	2	1	no	—
24	#14523976	—	4	1	no	—
25	#14523970	—	4	1	no	—
26	C₈×Dic₆₇	D₄	1	1	yes	characteristic
27	C₆₇⋊C₃₂	—	2	1	no	—
28	#135×C₆₇	—	2	1	no	—
29	Q₃₂×C₆₇	Q₈	1	1	yes	characteristic, complemented
30	SD₃₂×C₆₇	—	2	1	no	—
31	D₁₆×C₆₇	Q₈	1	1	yes	characteristic, complemented
32	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
33	#107×C₆₇	—	2	1	no	—
34	C₃₂×C₆₇	—	2	1	no	—
35	C₂×Dic₁₃₄	—	8	1	no	—
36	#11761612	—	4	1	no	—
37	#11761612	—	8	1	no	—
38	C₂×C₆₇⋊C₈	—	8	1	no	—
39	C₂×C₆₇⋊C₈	D₈	1	1	yes	characteristic
40	C₄⋊Dic₆₇	—	4	1	no	—
41	Dic₆₇⋊C₄	—	8	1	no	—
42	C₄×Dic₆₇	D₈	1	1	yes	characteristic
43	C₆₇⋊C₁₆	—	2	1	no	—
44	#41×C₆₇	—	4	1	no	—
45	Q₁₆×C₆₇	—	2	1	no	—
46	SD₁₆×C₆₇	—	4	1	no	—
47	D₈×C₆₇	—	2	1	no	—
48	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
49	C₄⋊C₄×C₆₇	—	4	1	no	—
50	C₁₆×C₆₇	C₂×Q₈	1	1	yes	characteristic
51	C₁₆×C₆₇	#41	1	1	yes	characteristic
52	Q₈×C₆₇	—	4	1	no	—
53	D₄×C₆₇	—	4	1	no	—
54	D₄×C₆₇	—	8	1	no	—
55	C₂×C₄×C₆₇	—	8	1	no	—
56	C₂×C₄×C₆₇	—	8	1	no	—
57	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
58	C₂×Dic₆₇	—	2	1	no	—
59	C₂×Dic₆₇	—	8	1	no	—
60	Dic₁₃₄	—	16	2	no	—
61	C₈×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
62	C₈×C₆₇	#137	1	1	yes	characteristic
63	C₆₇⋊C₈	—	8	1	no	—
64	C₆₇⋊C₈	SD₃₂	1	1	yes	characteristic
65	C₆₇⋊C₈	SD₃₂	1	1	yes	characteristic
66	C₂²×C₆₇	—	8	1	no	—
67	C₂²×C₆₇	#509	1	1	yes	characteristic
68	C₄×C₆₇	—	8	1	no	—
69	C₄×C₆₇	—	16	1	no	—
70	C₄×C₆₇	#474	1	1	yes	characteristic
71	C₄×C₆₇	C₂×SD₃₂	1	1	yes	characteristic
72	Dic₆₇	—	2	2	no	—
73	Dic₆₇	—	16	1	no	—
74	#13710	—	67	1	no	maximal, Sylow
75	C₂×C₆₇	—	2	1	no	—
76	C₂×C₆₇	—	16	1	no	—
77	C₂×C₆₇	#2226	1	1	yes	characteristic
78	#2254	—	67	1	no	—
79	#2178	—	67	1	no	—
80	#1426	—	67	1	no	—
81	#1426	—	67	1	no	—
82	#1422	—	67	1	no	—
83	#1418	—	67	1	no	—
84	#1418	D₆₇	1	1	yes	characteristic
85	C₆₇	#13710	1	1	yes	characteristic, complemented, Sylow
86	#508	C₂×D₆₇	1	1	yes	characteristic
87	#500	—	134	1	no	—
88	#443	—	134	1	no	—
89	M₆₄	—	67	1	no	—
90	M₆₄	C₂×D₆₇	1	1	yes	characteristic
91	#368	—	67	1	no	—
92	#367	—	67	1	no	—
93	#367	—	134	1	no	—
94	#367	C₂×D₆₇	1	1	yes	characteristic
95	#359	—	134	1	no	—
96	C₄×C₁₆	—	67	1	no	—
97	#135	—	2	1	no	—
98	#131	—	268	1	no	—
99	C₄⋊Q₈	—	268	1	no	—
100	Q₃₂	Dic₁₃₄	1	1	yes	characteristic, complemented
101	SD₃₂	—	2	1	no	—
102	D₁₆	Dic₁₃₄	1	1	yes	characteristic, complemented
103	C₂×C₁₆	—	67	1	no	—
104	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
105	#108	—	134	1	no	—
106	#107	—	2	1	no	—
107	#107	—	268	1	no	—
108	#107	—	134	1	no	—
109	#104	—	268	1	no	—
110	C₄×C₈	—	67	1	no	—
111	C₃₂	—	2	1	no	—
112	C₃₂	—	134	1	no	—
113	#41	—	4	1	no	—
114	C₂×Q₈	—	536	1	no	—
115	Q₁₆	—	2	1	no	—
116	SD₁₆	—	4	1	no	—
117	D₈	—	2	1	no	—
118	M₁₆	—	268	1	no	—
119	M₁₆	—	536	1	no	—
120	C₂×C₈	—	536	1	no	—
121	C₂×C₈	—	67	1	no	—
122	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
123	C₄⋊C₄	—	4	1	no	—
124	C₄⋊C₄	—	268	1	no	—
125	C₄⋊C₄	—	536	1	no	—
126	C₄²	—	67	1	no	—
127	C₁₆	—	134	1	no	—
128	C₁₆	C₂×Dic₁₃₄	1	1	yes	characteristic
129	C₁₆	#11761627	1	1	yes	characteristic
130	Q₈	—	4	1	no	—
131	Q₈	—	1072	2	no	—
132	D₄	—	4	1	no	—
133	D₄	—	8	1	no	—
134	C₂×C₄	—	8	1	no	—
135	C₂×C₄	—	8	1	no	—
136	C₂×C₄	—	536	1	no	—
137	C₂×C₄	—	134	1	no	—
138	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
139	C₈	—	536	1	no	—
140	C₈	—	67	1	no	—
141	C₈	—	67	1	no	—
142	C₈	#14524020	1	1	yes	characteristic
143	C₈	#14524072	1	1	yes	characteristic
144	C₂²	—	8	1	no	—
145	C₂²	#16863830	1	1	yes	characteristic
146	C₄	—	8	1	no	—
147	C₄	—	16	1	no	—
148	C₄	—	1072	1	no	—
149	C₄	—	134	2	no	—
150	C₄	#16863674	1	1	yes	characteristic
151	C₄	SD₃₂×D₆₇	1	1	yes	characteristic
152	C₂	—	2	1	no	—
153	C₂	—	16	1	no	—
154	C₂	#22562354	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22562354
commutator, cosocle	C₂×C₁₆×C₆₇	C₂³
Frattini	C₂×C₁₆	C₂²×D₆₇
Fitting	C₆₇×#1418	C₂
socle	C₂×C₆₇	#2226
prime radical	#508×C₆₇	C₂²
hypercenter	#1418	D₆₇
hypocenter	C₆₇	#13710
LCS	G ▷ C₂×C₁₆×C₆₇ ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1418

Complements

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

N	H ≅ G/N	type
1. #22562490	C₂	semidirect
3. #22558415	C₂	semidirect
4. #22558414	C₂	semidirect
5. #22558406	C₂	semidirect
29. Q₃₂×C₆₇	Q₈	semidirect
31. D₁₆×C₆₇	Q₈	semidirect
85. C₆₇	#13710	semidirect
100. Q₃₂	Dic₁₃₄	semidirect
102. D₁₆	Dic₁₃₄	semidirect

Maximal subgroups

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

subgroup	index
#22562490	2
#22562090	2
#22558415	2
#22558414	2
#22558406	2
#22558396	2
C₆₇×#1418	2
#13710	67