Finite Group Databasebeta

Subgroups — #17982728

Order 4352.

238 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 129 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	C₂×#11763918	C₂	1	1	yes	maximal, characteristic
2	C₂×#11763916	C₂	1	1	yes	maximal, characteristic
3	#14545870	C₂	1	1	yes	maximal, characteristic, complemented
4	#14545869	C₂	1	1	yes	maximal, characteristic, complemented
5	#14545867	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#10587528	C₂	1	1	yes	maximal, characteristic, complemented
7	#14544994	C₂	1	1	yes	maximal, characteristic
8	#11763918	C₄	1	4	yes	complemented
9	#11763916	C₄	1	4	yes	complemented
10	C₂×#10587528	C₂²	1	1	yes	characteristic
11	C₂×#10587528	C₂²	1	1	yes	characteristic
12	C₂×#10587528	C₂²	1	1	yes	characteristic
13	C₂×#10587528	C₂²	1	1	yes	characteristic
14	C₂×#10587530	C₄	1	1	yes	characteristic
15	C₂×#10587529	C₂²	1	1	yes	characteristic
16	#11763698	C₄	1	1	yes	characteristic
17	#11763697	C₄	1	1	yes	characteristic
18	#11763695	C₂²	1	1	yes	characteristic
19	C₄×#63216	C₂²	1	1	yes	characteristic
20	#11762766	C₄	1	1	yes	characteristic
21	C₂×#63218	C₈	1	1	yes	characteristic
22	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
23	#10587558	C₈	1	1	yes	characteristic
24	C₂×#63216	C₂×C₄	1	1	yes	characteristic
25	C₂×#63216	C₂×C₄	1	1	yes	characteristic
26	C₂×#63216	C₂×C₄	1	1	yes	characteristic
27	C₂×#63216	C₂³	1	1	yes	characteristic
28	#10587530	C₈	1	4	yes	—
29	#10587529	C₂×C₄	1	2	yes	—
30	#10587529	C₂×C₄	1	2	yes	—
31	#10587528	C₂×C₄	1	2	yes	—
32	#10587528	C₂×C₄	1	2	yes	—
33	#10587528	C₂×C₄	1	2	yes	—
34	#10587528	C₂×C₄	1	2	yes	—
35	C₁₇×#97	C₈	1	1	yes	characteristic
36	#10587486	C₈	1	1	yes	characteristic
37	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
38	C₄×#3622	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×D₁₇	C₂×C₈	1	1	yes	characteristic
40	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
41	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
42	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
43	#63227	C₁₆	1	2	yes	—
44	#63225	C₁₆	1	2	yes	—
45	C₄×Dic₁₇	C₂×C₈	1	1	yes	characteristic
46	C₂×#3622	C₂×C₈	1	1	yes	characteristic
47	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
48	#63218	C₁₆	1	4	yes	—
49	C₈×D₁₇	C₄²	1	2	yes	—
50	C₈×D₁₇	C₂×C₈	1	2	yes	—
51	#63216	C₄²	1	2	yes	—
52	#63216	C₂×C₈	1	2	yes	—
53	#63216	C₂×C₈	1	2	yes	—
54	#63216	C₂²×C₄	1	1	yes	characteristic
55	#63216	C₂²×C₄	1	1	yes	characteristic
56	#13549	—	17	1	no	maximal, Sylow
57	C₂²×D₁₇	C₂×C₁₆	1	1	yes	characteristic
58	C₂×C₄×C₁₇	C₂×C₁₆	1	2	yes	—
59	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
60	C₂×Dic₁₇	C₂×C₁₆	1	2	yes	—
61	C₂×Dic₁₇	C₂×C₁₆	1	1	yes	characteristic
62	C₄×D₁₇	C₄×C₈	1	2	yes	—
63	C₄×D₁₇	C₂×C₁₆	1	2	yes	—
64	C₈×C₁₇	C₄×C₈	1	2	yes	—
65	C₈×C₁₇	C₂×C₁₆	1	2	yes	—
66	#3622	C₂×C₁₆	1	2	yes	—
67	#3622	C₂×C₄²	1	1	yes	characteristic
68	#3622	C₂²×C₈	1	1	yes	characteristic
69	C₂×M₆₄	—	17	1	no	—
70	C₂²×C₃₂	—	17	1	no	—
71	#2108	—	17	1	no	—
72	#1397	—	17	1	no	—
73	#1397	—	17	1	no	—
74	C₄×C₃₂	—	17	1	no	—
75	C₄×C₃₂	—	17	1	no	—
76	C₂²×C₁₇	C₂²×C₁₆	1	1	yes	characteristic
77	C₂×D₁₇	—	2	2	no	—
78	C₂×D₁₇	C₄×C₁₆	1	2	yes	—
79	C₄×C₁₇	—	2	2	no	—
80	C₄×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
81	C₄×C₁₇	C₂²×C₁₆	1	1	yes	characteristic
82	Dic₁₇	—	2	2	no	—
83	Dic₁₇	C₄×C₁₆	1	2	yes	—
84	C₂×M₃₂	—	17	1	no	—
85	C₂²×C₁₆	—	17	1	no	—
86	#405	—	17	1	no	—
87	M₆₄	—	17	4	no	—
88	C₂×C₃₂	—	17	1	no	—
89	C₂×C₃₂	—	17	1	no	—
90	C₂×C₃₂	—	17	1	no	—
91	C₂×C₃₂	—	17	1	no	—
92	C₂×C₃₂	—	17	4	no	—
93	#346	—	17	1	no	—
94	#346	—	17	1	no	—
95	C₄×C₁₆	—	17	1	no	—
96	C₄×C₁₆	—	17	1	no	—
97	C₂×C₁₇	C₂×C₄×C₁₆	1	1	yes	characteristic
98	C₂×C₁₇	#2258	1	2	yes	—
99	D₁₇	—	2	2	no	—
100	C₂×M₁₆	—	17	1	no	—
101	C₂²×C₈	—	17	1	no	—
102	#117	—	17	1	no	—
103	M₃₂	—	17	4	no	—
104	C₂×C₁₆	—	17	1	no	—
105	C₂×C₁₆	—	17	1	no	—
106	C₂×C₁₆	—	17	1	no	—
107	C₂×C₁₆	—	17	1	no	—
108	C₂×C₁₆	—	17	2	no	—
109	C₂×C₁₆	—	17	2	no	—
110	#97	—	17	1	no	—
111	#97	#3633	1	1	yes	characteristic
112	C₄×C₈	—	17	1	no	—
113	C₄×C₈	—	17	1	no	—
114	C₃₂	—	17	2	no	—
115	C₃₂	—	17	2	no	—
116	C₃₂	—	17	2	no	—
117	C₃₂	—	17	2	no	—
118	C₁₇	#13549	1	1	yes	characteristic, complemented, Sylow
119	C₂²×C₄	—	17	1	no	—
120	M₁₆	—	17	4	no	—
121	C₂×C₈	—	17	1	no	—
122	C₂×C₈	—	17	1	no	—
123	C₂×C₈	—	17	2	no	—
124	C₂×C₈	—	17	2	no	—
125	C₂×C₈	C₂×#3633	1	1	yes	characteristic
126	C₂×C₈	C₂×#3633	1	1	yes	characteristic
127	C₄⋊C₄	—	17	2	no	—
128	C₂²⋊C₄	—	17	2	no	—
129	C₄²	—	17	1	no	—
130	C₄²	C₂×#3633	1	1	yes	characteristic
131	C₁₆	—	17	1	no	—
132	C₁₆	—	17	1	no	—
133	C₁₆	—	17	2	no	—
134	C₁₆	—	17	2	no	—
135	C₁₆	—	17	2	no	—
136	C₂³	—	17	1	no	—
137	C₂×C₄	—	17	1	no	—
138	C₂×C₄	—	17	2	no	—
139	C₂×C₄	—	17	2	no	—
140	C₂×C₄	—	17	2	no	—
141	C₂×C₄	#10587665	1	2	yes	—
142	C₂×C₄	C₂²×#3633	1	1	yes	characteristic
143	C₈	—	17	1	no	—
144	C₈	—	17	1	no	—
145	C₈	—	17	2	no	—
146	C₈	#10587665	1	2	yes	—
147	C₈	C₄×#3633	1	2	yes	—
148	C₂²	—	17	2	no	—
149	C₂²	—	34	2	no	—
150	C₂²	C₂×#10587665	1	1	yes	characteristic
151	C₄	—	2	2	no	—
152	C₄	—	17	2	no	—
153	C₄	—	34	2	no	—
154	C₄	C₂×C₄×#3633	1	1	yes	characteristic
155	C₄	C₂×#10587665	1	1	yes	characteristic
156	C₂	—	34	2	no	—
157	C₂	C₄×#10587665	1	1	yes	characteristic
158	C₂	#14546018	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#3633
commutator	C₂×C₁₇	C₂×C₄×C₁₆
Fitting	C₁₇×#97	C₈
socle	C₂²×C₁₇	C₂²×C₁₆
prime radical	C₂²×D₁₇	C₂×C₁₆
cosocle	C₂×#63216	C₂³
hypercenter	#97	#3633
hypocenter	C₁₇	#13549
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ #97

Complements

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

N	H ≅ G/N	type
3. #14545870	C₂	semidirect
4. #14545869	C₂	semidirect
5. #14545867	C₂	semidirect
6. C₄×#10587528	C₂	semidirect
8. #11763918	C₄	semidirect
9. #11763916	C₄	semidirect
118. C₁₇	#13549	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11763918	2
C₂×#11763916	2
#14545870	2
#14545869	2
#14545867	2
C₄×#10587528	2
#14544994	2
#13549	17