Finite Group Databasebeta

Subgroups — #17982730

Order 4352.

238 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 123 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	#14546015	C₂	1	1	yes	maximal, characteristic
2	C₂×#11763916	C₂	1	1	yes	maximal, characteristic
3	#14545871	C₂	1	2	yes	maximal, complemented
4	#14545868	C₂	1	2	yes	maximal, complemented
5	#14544996	C₂	1	1	yes	maximal, characteristic
6	#11763918	C₄	1	2	yes	complemented
7	#11763917	C₄	1	2	yes	complemented
8	#11763916	C₄	1	2	yes	complemented
9	#11763916	C₂²	1	1	yes	characteristic
10	#11763916	C₂²	1	1	yes	characteristic
11	#11763916	C₂²	1	2	yes	—
12	C₂×#10587528	C₄	1	2	yes	complemented
13	C₂×#10587529	C₂²	1	1	yes	characteristic
14	#11763772	C₄	1	1	yes	characteristic
15	#11763696	C₄	1	2	yes	—
16	C₁₆×#595	C₂²	1	2	yes	—
17	M₃₂×D₁₇	C₄	1	1	yes	characteristic
18	#10587677	C₈	1	1	yes	characteristic
19	D₁₇⋊M₁₆	C₈	1	1	yes	characteristic
20	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
21	C₂×#63216	C₂×C₄	1	1	yes	characteristic
22	C₂×#63216	C₂×C₄	1	1	yes	characteristic
23	#10587532	C₈	1	2	yes	—
24	C₈×#595	C₂×C₄	1	2	yes	—
25	#10587529	C₂×C₄	1	1	yes	characteristic
26	#10587529	C₂×C₄	1	1	yes	characteristic
27	#10587529	C₂×C₄	1	1	yes	characteristic
28	#10587529	C₂³	1	1	yes	characteristic
29	#10587528	C₂×C₄	1	2	yes	—
30	#10587528	C₂×C₄	1	2	yes	—
31	#10587528	C₂×C₄	1	4	yes	—
32	C₁₇×M₃₂	C₈	1	1	yes	characteristic
33	#10587483	C₈	1	1	yes	characteristic
34	#10587469	C₈	1	2	yes	—
35	C₁₆×D₁₇	C₂×C₄	1	2	yes	—
36	C₂×C₄×D₁₇	C₂×C₈	1	1	yes	characteristic
37	C₂²⋊#595	C₁₆	1	2	yes	—
38	#63247	C₁₆	1	2	yes	—
39	C₄⋊#595	C₁₆	1	2	yes	—
40	C₄×#595	C₂×C₈	1	2	yes	—
41	#63243	C₁₆	1	2	yes	—
42	#63242	C₂×C₈	1	2	yes	—
43	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
44	C₂×#3622	C₄²	1	1	yes	characteristic
45	C₈×D₁₇	C₄²	1	1	yes	characteristic
46	C₈×D₁₇	C₂×C₈	1	1	yes	characteristic
47	C₈×D₁₇	C₂×C₈	1	1	yes	characteristic
48	C₈×D₁₇	C₂²×C₄	1	1	yes	characteristic
49	#63216	C₄²	1	1	yes	characteristic
50	#63216	C₄²	1	1	yes	characteristic
51	#63216	C₂²×C₄	1	1	yes	characteristic
52	#63216	C₂²×C₄	1	1	yes	characteristic
53	C₁₆×C₁₇	C₂×C₈	1	2	yes	—
54	#63214	C₂×C₈	1	2	yes	—
55	#13549	—	17	1	no	maximal, Sylow
56	C₂²×D₁₇	C₂×C₁₆	1	1	yes	characteristic
57	C₂×#595	C₂×C₁₆	1	4	yes	—
58	C₂×C₄×C₁₇	C₄×C₈	1	1	yes	characteristic
59	C₂×Dic₁₇	C₂×C₁₆	1	1	yes	characteristic
60	C₄×D₁₇	C₄×C₈	1	1	yes	characteristic
61	C₄×D₁₇	C₂×C₁₆	1	1	yes	characteristic
62	C₄×D₁₇	C₂×C₁₆	1	1	yes	characteristic
63	C₄×D₁₇	C₂²×C₈	1	1	yes	characteristic
64	#3624	C₂×C₁₆	1	4	yes	—
65	C₈×C₁₇	C₄×C₈	1	1	yes	characteristic
66	C₈×C₁₇	C₂²×C₈	1	1	yes	characteristic
67	#3622	C₄×C₈	1	1	yes	characteristic
68	#3622	C₂×C₄²	1	1	yes	characteristic
69	C₂×M₆₄	—	17	1	no	—
70	C₂²×C₃₂	—	17	1	no	—
71	#2108	—	17	1	no	—
72	#1397	—	17	2	no	—
73	C₄×C₃₂	—	17	2	no	—
74	C₂²×C₁₇	C₄×C₁₆	1	1	yes	characteristic
75	C₂×D₁₇	—	2	2	no	—
76	C₂×D₁₇	C₄×C₁₆	1	1	yes	characteristic
77	C₂×D₁₇	C₂²×C₁₆	1	1	yes	characteristic
78	#595	—	2	4	no	—
79	C₄×C₁₇	C₄×C₁₆	1	1	yes	characteristic
80	C₄×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
81	Dic₁₇	C₄×C₁₆	1	1	yes	characteristic
82	Dic₁₇	C₂²×C₁₆	1	1	yes	characteristic
83	C₂×M₃₂	—	17	1	no	—
84	C₂²×C₁₆	—	17	1	no	—
85	#405	—	17	1	no	—
86	M₆₄	—	17	2	no	—
87	M₆₄	—	17	2	no	—
88	C₂×C₃₂	—	17	1	no	—
89	C₂×C₃₂	—	17	2	no	—
90	C₂×C₃₂	—	17	1	no	—
91	C₂×C₃₂	—	17	2	no	—
92	C₂×C₃₂	—	17	2	no	—
93	#346	—	17	2	no	—
94	C₄×C₁₆	—	17	2	no	—
95	C₂×C₁₇	—	2	1	no	—
96	C₂×C₁₇	C₂×C₄×C₁₆	1	1	yes	characteristic
97	D₁₇	—	2	1	no	—
98	D₁₇	#2258	1	2	yes	—
99	C₂×M₁₆	—	17	1	no	—
100	C₂²×C₈	—	17	1	no	—
101	#117	—	17	1	no	—
102	M₃₂	—	17	1	no	—
103	M₃₂	—	17	2	no	—
104	M₃₂	#3633	1	1	yes	characteristic
105	C₂×C₁₆	—	17	1	no	—
106	C₂×C₁₆	—	17	1	no	—
107	C₂×C₁₆	—	17	1	no	—
108	C₂×C₁₆	—	17	1	no	—
109	C₂×C₁₆	—	17	1	no	—
110	C₂×C₁₆	—	17	1	no	—
111	C₂×C₁₆	—	17	2	no	—
112	#97	—	17	2	no	—
113	C₄×C₈	—	17	2	no	—
114	C₃₂	—	17	2	no	—
115	C₃₂	—	17	2	no	—
116	C₃₂	—	17	4	no	—
117	C₁₇	#13549	1	1	yes	characteristic, complemented, Sylow
118	C₂²×C₄	—	17	1	no	—
119	M₁₆	—	17	2	no	—
120	M₁₆	—	17	2	no	—
121	C₂×C₈	—	17	2	no	—
122	C₂×C₈	—	17	1	no	—
123	C₂×C₈	—	17	1	no	—
124	C₂×C₈	—	17	1	no	—
125	C₂×C₈	—	17	1	no	—
126	C₂×C₈	—	17	1	no	—
127	C₂×C₈	C₂×#3633	1	1	yes	characteristic
128	C₄⋊C₄	—	17	2	no	—
129	C₂²⋊C₄	—	17	2	no	—
130	C₄²	—	17	2	no	—
131	C₁₆	—	17	1	no	—
132	C₁₆	—	17	1	no	—
133	C₁₆	—	17	1	no	—
134	C₁₆	—	17	1	no	—
135	C₁₆	—	17	2	no	—
136	C₁₆	C₂×#3633	1	2	yes	—
137	C₂³	—	17	1	no	—
138	C₂×C₄	—	17	1	no	—
139	C₂×C₄	—	17	1	no	—
140	C₂×C₄	—	17	1	no	—
141	C₂×C₄	—	17	1	no	—
142	C₂×C₄	—	17	1	no	—
143	C₂×C₄	—	17	4	no	—
144	C₂×C₄	C₄×#3633	1	1	yes	characteristic
145	C₈	—	17	4	no	—
146	C₈	—	17	1	no	—
147	C₈	—	17	1	no	—
148	C₈	C₄×#3633	1	1	yes	characteristic
149	C₈	C₂²×#3633	1	1	yes	characteristic
150	C₂²	—	17	1	no	—
151	C₂²	—	17	1	no	—
152	C₂²	—	34	2	no	—
153	C₂²	#11763902	1	1	yes	characteristic
154	C₄	—	17	1	no	—
155	C₄	—	17	1	no	—
156	C₄	—	34	4	no	—
157	C₄	#11763902	1	1	yes	characteristic
158	C₄	C₂×C₄×#3633	1	1	yes	characteristic
159	C₂	—	2	1	no	—
160	C₂	—	17	2	no	—
161	C₂	—	34	1	no	—
162	C₂	C₂×#11763902	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₈	C₂²×#3633
commutator, socle	C₂×C₁₇	C₂×C₄×C₁₆
Fitting	C₁₇×M₃₂	C₈
prime radical	C₂²×D₁₇	C₂×C₁₆
cosocle	#10587529	C₂³
hypercenter	M₃₂	#3633
hypocenter	C₁₇	#13549
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₈ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
3. #14545871	C₂	semidirect
4. #14545868	C₂	semidirect
6. #11763918	C₄	semidirect
7. #11763917	C₄	semidirect
8. #11763916	C₄	semidirect
12. C₂×#10587528	C₄	semidirect
117. C₁₇	#13549	semidirect

Maximal subgroups

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

subgroup	index
#14546015	2
C₂×#11763916	2
#14545871	2
#14545868	2
#14544996	2
#13549	17