Finite Group Databasebeta

Subgroups — #17960057

Order 4352.

270 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 124 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	#14545116	C₂	1	2	yes	maximal
2	#14543807	C₂	1	4	yes	maximal, complemented
3	C₂×C₈×#3622	C₂	1	1	yes	maximal, characteristic
4	C₂×#10587538	—	2	2	no	—
5	C₄⋊#63216	—	2	2	no	—
6	C₄⋊#63216	C₂²	1	4	yes	—
7	C₂×C₄×#3622	C₄	1	1	yes	characteristic
8	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
9	C₈×#3622	C₄	1	2	yes	—
10	C₈×#3622	C₂²	1	2	yes	—
11	C₂×C₄×C₈×C₁₇	C₄	1	1	yes	characteristic
12	C₂²×C₈×C₁₇	—	2	1	no	—
13	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
14	C₂²×#3622	C₈	1	2	yes	—
15	C₂²×#3622	D₄	1	1	yes	characteristic
16	C₂²×#3622	Q₈	1	1	yes	characteristic
17	#10587538	—	4	4	no	—
18	C₂×#63216	—	2	4	no	—
19	C₄×C₈×C₁₇	—	2	1	no	—
20	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
21	C₄×#3622	C₈	1	1	yes	characteristic
22	C₄×#3622	C₈	1	1	yes	characteristic
23	C₄×#3622	C₂×C₄	1	1	yes	characteristic
24	C₄×#3622	C₂×C₄	1	2	yes	—
25	C₄×#3622	D₄	1	1	yes	characteristic
26	C₄×#3622	Q₈	1	1	yes	characteristic
27	C₄×#3622	C₂³	1	1	yes	characteristic
28	C₂²×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
29	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
30	C₂²×C₄×C₁₇	M₁₆	1	1	yes	characteristic
31	C₂×C₈×C₁₇	—	2	4	no	—
32	C₂×C₈×C₁₇	—	2	2	no	—
33	C₄²×C₁₇	C₄⋊C₄	1	1	yes	characteristic
34	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
35	C₄²×C₁₇	M₁₆	1	1	yes	characteristic
36	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
37	C₂×#3622	—	2	2	no	—
38	C₂×#3622	—	2	4	no	—
39	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
40	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
41	C₂×#3622	C₂×C₈	1	2	yes	—
42	C₂×#3622	C₂×C₈	1	2	yes	—
43	C₂×#3622	SD₁₆	1	4	yes	—
44	C₂×#3622	C₂×D₄	1	1	yes	characteristic
45	C₂×#3622	C₂×Q₈	1	1	yes	characteristic
46	#63216	—	4	4	no	—
47	#11996	—	17	1	no	maximal, Sylow
48	C₂³×C₁₇	C₄⋊C₈	1	1	yes	characteristic
49	C₂×C₄×C₁₇	—	2	2	no	—
50	C₂×C₄×C₁₇	—	2	2	no	—
51	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
52	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
53	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
54	C₂×C₄×C₁₇	#106	1	2	yes	—
55	C₂×C₄×C₁₇	#108	1	2	yes	—
56	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
58	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
59	C₈×C₁₇	—	2	4	no	—
60	#3622	—	2	1	no	—
61	#3622	—	2	1	no	—
62	#3622	—	2	2	no	—
63	#3622	#106	1	2	yes	—
64	#3622	M₃₂	1	4	yes	complemented
65	#3622	C₂×SD₁₆	1	2	yes	—
66	#2150	—	17	2	no	—
67	C₂×C₈²	—	17	1	no	—
68	#1367	—	17	4	no	—
69	C₂²×C₁₇	—	2	1	no	—
70	C₂²×C₁₇	—	2	1	no	—
71	C₂²×C₁₇	#334	1	2	yes	—
72	C₂²×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
73	C₄×C₁₇	—	2	1	no	—
74	C₄×C₁₇	—	2	1	no	—
75	C₄×C₁₇	#334	1	2	yes	—
76	C₄×C₁₇	#364	1	2	yes	—
77	C₄×C₁₇	C₂×#106	1	1	yes	characteristic
78	C₄×C₁₇	C₂×#108	1	1	yes	characteristic
79	C₄×C₁₇	C₂×M₃₂	1	2	yes	—
80	C₂×M₃₂	—	34	2	no	—
81	C₂×C₄×C₈	—	17	1	no	—
82	C₂×C₄×C₈	—	17	1	no	—
83	C₂×C₄×C₈	#595	1	1	yes	characteristic
84	C₄⋊C₁₆	—	17	4	no	—
85	C₄⋊C₁₆	—	34	2	no	—
86	C₈²	—	17	2	no	—
87	C₈²	—	17	2	no	—
88	C₂×C₁₇	—	2	2	no	—
89	C₂×C₁₇	C₂×#334	1	1	yes	characteristic
90	C₂×C₁₇	#2150	1	1	yes	characteristic
91	C₂×C₁₇	C₂×#364	1	1	yes	characteristic
92	C₂²×C₈	—	2	1	no	—
93	C₂²×C₈	—	17	1	no	—
94	C₂²×C₈	—	17	2	no	—
95	C₂²×C₈	—	17	1	no	—
96	C₂×C₄²	C₂×#595	1	1	yes	characteristic
97	M₃₂	—	68	4	no	—
98	C₂×C₁₆	—	34	4	no	—
99	C₄×C₈	—	2	1	no	—
100	C₄×C₈	—	17	1	no	—
101	C₄×C₈	—	17	1	no	—
102	C₄×C₈	—	17	1	no	—
103	C₄×C₈	—	17	1	no	—
104	C₄×C₈	—	17	2	no	—
105	C₄×C₈	—	17	1	no	—
106	C₄×C₈	—	17	1	no	—
107	C₄×C₈	C₂×#595	1	2	yes	—
108	C₁₇	#11996	1	1	yes	characteristic, complemented, Sylow
109	C₂²×C₄	#63242	1	1	yes	characteristic
110	C₂²×C₄	C₄⋊#595	1	1	yes	characteristic
111	C₂²×C₄	#63247	1	1	yes	characteristic
112	C₂×C₈	—	2	4	no	—
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	17	2	no	—
115	C₂×C₈	—	17	1	no	—
116	C₂×C₈	—	17	1	no	—
117	C₂×C₈	—	17	4	no	—
118	C₂×C₈	—	17	2	no	—
119	C₂×C₈	—	17	1	no	—
120	C₂×C₈	—	17	1	no	—
121	C₂×C₈	—	34	4	no	—
122	C₂×C₈	—	34	2	no	—
123	C₄²	#63242	1	1	yes	characteristic
124	C₄²	C₄⋊#595	1	1	yes	characteristic
125	C₄²	#63247	1	1	yes	characteristic
126	C₄²	C₂²×#595	1	1	yes	characteristic
127	C₁₆	—	68	4	no	—
128	C₂³	#10587544	1	1	yes	characteristic
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	#10587533	1	2	yes	—
132	C₂×C₄	#10587536	1	2	yes	—
133	C₂×C₄	#10587544	1	1	yes	characteristic
134	C₂×C₄	#10587544	1	1	yes	characteristic
135	C₂×C₄	#10587544	1	1	yes	characteristic
136	C₂×C₄	C₂×#63242	1	1	yes	characteristic
137	C₂×C₄	C₂×C₄⋊#595	1	1	yes	characteristic
138	C₂×C₄	C₂×#63247	1	1	yes	characteristic
139	C₈	—	2	4	no	—
140	C₈	—	17	2	no	—
141	C₈	—	17	2	no	—
142	C₈	—	17	4	no	—
143	C₈	—	34	1	no	—
144	C₈	—	34	1	no	—
145	C₈	—	34	2	no	—
146	C₂²	—	2	1	no	—
147	C₂²	—	2	1	no	—
148	C₂²	#11763671	1	2	yes	—
149	C₂²	C₂×#10587544	1	1	yes	characteristic
150	C₄	—	2	1	no	—
151	C₄	—	2	1	no	—
152	C₄	#11763671	1	2	yes	—
153	C₄	#11763732	1	2	yes	—
154	C₄	C₂×#10587533	1	1	yes	characteristic
155	C₄	C₂×#10587536	1	1	yes	characteristic
156	C₄	#11763865	1	2	yes	—
157	C₂	—	2	2	no	—
158	C₂	C₂×#11763671	1	1	yes	characteristic
159	C₂	#14545118	1	1	yes	characteristic
160	C₂	C₂×#11763732	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄⋊#595
commutator	C₂×C₄×C₁₇	C₂²×C₈
Frattini	C₄²	C₂²×#595
Fitting	C₂×C₄×C₈×C₁₇	C₄
socle	C₂²×C₁₇	C₂×C₄⋊C₈
prime radical	C₂³×C₁₇	C₄⋊C₈
cosocle	C₄×#3622	C₂³
hypercenter	C₂×C₄×C₈	#595
hypocenter	C₁₇	#11996
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #14543807	C₂	semidirect
64. #3622	M₃₂	semidirect
108. C₁₇	#11996	semidirect

Maximal subgroups

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

subgroup	index
#14545116	2
#14543807	2
C₂×C₈×#3622	2
#11996	17