Finite Group Databasebeta

Subgroups — #17978786

Order 4352.

444 conjugacy classes of subgroups; only the 250 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 250 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	#14545199	C₂	1	8	yes	maximal, complemented
2	#14545149	C₂	1	4	yes	maximal
3	#14545145	C₂	1	2	yes	maximal
4	#14537719	C₂	1	1	yes	maximal, characteristic
5	C₄⋊#63216	C₂²	1	4	yes	—
6	C₄⋊#63216	C₂²	1	8	yes	—
7	C₂²⋊#63216	C₂²	1	4	yes	—
8	#11763697	C₂²	1	4	yes	—
9	#11763697	C₂²	1	4	yes	—
10	C₄×#63216	C₂²	1	4	yes	—
11	#11762940	C₄	1	4	yes	—
12	Q₈×#3622	C₂²	1	4	yes	—
13	#11762836	C₄	1	2	yes	—
14	#11762833	C₄	1	1	yes	characteristic
15	#11762765	C₂²	1	2	yes	—
16	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
17	C₁₇×#533	C₄	1	1	yes	characteristic
18	C₁₇×C₄⋊Q₈	C₈	1	2	yes	—
19	C₁₇×#125	C₈	1	2	yes	—
20	C₁₇×C₂²⋊Q₈	C₈	1	4	yes	—
21	C₄×Q₈×C₁₇	C₂×C₄	1	4	yes	—
22	C₁₇×#117	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
24	C₂×#63223	C₂×C₄	1	2	yes	—
25	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
26	C₂²×#3622	C₂³	1	1	yes	characteristic
27	C₂×#63216	C₂³	1	8	yes	—
28	C₄⋊#3622	C₂×C₄	1	4	yes	—
29	C₄⋊#3622	C₂×C₄	1	8	yes	—
30	#10587474	C₂×C₄	1	4	yes	—
31	C₄×#3622	C₂×C₄	1	2	yes	—
32	C₄×#3622	C₂³	1	2	yes	—
33	C₄×#3622	C₂³	1	4	yes	—
34	C₂×Q₈×C₁₇	C₂×C₈	1	4	yes	—
35	C₂²×C₄×C₁₇	C₂×C₈	1	2	yes	—
36	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
37	C₄⋊C₄×C₁₇	C₂×C₈	1	8	yes	—
38	C₄⋊C₄×C₁₇	C₂×C₈	1	4	yes	—
39	C₄⋊C₄×C₁₇	C₂×C₈	1	4	yes	—
40	C₂²⋊C₄×C₁₇	C₂×C₈	1	4	yes	—
41	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
42	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
43	C₄²×C₁₇	C₂²×C₄	1	2	yes	—
44	C₄²×C₁₇	C₂²×C₄	1	4	yes	—
45	#63223	—	4	2	no	—
46	C₂×#3622	—	2	2	no	—
47	C₂×#3622	—	4	1	no	—
48	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
49	C₂×#3622	C₂²×C₄	1	2	yes	—
50	C₂×#3622	C₂²×C₄	1	4	yes	—
51	C₂×#3622	C₂⁴	1	1	yes	characteristic
52	#63216	—	2	8	no	—
53	#33501	—	17	1	no	maximal, Sylow
54	C₂³×C₁₇	C₂²×C₈	1	1	yes	characteristic
55	Q₈×C₁₇	—	4	4	no	—
56	C₂×C₄×C₁₇	—	2	1	no	—
57	C₂×C₄×C₁₇	—	2	1	no	—
58	C₂×C₄×C₁₇	—	4	2	no	—
59	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
60	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
61	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
62	C₂×C₄×C₁₇	C₂²×C₈	1	8	yes	—
63	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
64	#3622	—	2	1	no	—
65	#3622	—	2	2	no	—
66	#3622	—	2	4	no	—
67	#3622	2₋¹⁺⁴	1	2	yes	—
68	#2960	—	17	1	no	—
69	#2183	—	17	8	no	—
70	#2164	—	17	4	no	—
71	#2163	—	17	2	no	—
72	C₂²×C₁₇	—	2	2	no	—
73	C₂²×C₁₇	—	2	1	no	—
74	C₂²×C₁₇	C₂³×C₈	1	1	yes	characteristic
75	C₄×C₁₇	—	2	2	no	—
76	C₄×C₁₇	—	2	2	no	—
77	C₄×C₁₇	—	2	8	no	—
78	C₄×C₁₇	—	2	1	no	—
79	C₄×C₁₇	#519	1	1	yes	characteristic
80	C₄×C₁₇	#568	1	1	yes	characteristic
81	#533	#595	1	1	yes	characteristic
82	#446	—	17	4	no	—
83	C₈×Q₈	—	17	4	no	—
84	#424	—	17	2	no	—
85	#423	—	17	1	no	—
86	#405	—	17	2	no	—
87	C₂×C₄×C₈	—	17	1	no	—
88	C₄⋊C₁₆	—	17	4	no	—
89	C₄⋊C₁₆	—	17	8	no	—
90	C₂²⋊C₁₆	—	17	4	no	—
91	#346	—	17	4	no	—
92	#346	—	17	4	no	—
93	C₄×C₁₆	—	17	4	no	—
94	C₂×C₁₇	—	4	1	no	—
95	C₂×C₁₇	#2982	1	1	yes	characteristic
96	C₂×C₁₇	#3407	1	2	yes	—
97	C₂×M₁₆	—	17	2	no	—
98	C₂²×C₈	—	17	1	no	—
99	C₂²×C₈	—	17	1	no	—
100	C₄⋊Q₈	#3624	1	2	yes	—
101	#125	#3624	1	2	yes	—
102	C₂²⋊Q₈	#3624	1	4	yes	—
103	C₄×Q₈	C₂×#595	1	4	yes	—
104	#117	C₂×#595	1	2	yes	—
105	C₂×C₄²	C₂×#595	1	1	yes	characteristic
106	C₂×C₁₆	—	17	8	no	—
107	C₄⋊C₈	—	17	4	no	—
108	C₄⋊C₈	—	17	8	no	—
109	#97	—	17	4	no	—
110	C₄×C₈	—	17	2	no	—
111	C₄×C₈	—	17	4	no	—
112	C₄×C₈	—	17	2	no	—
113	C₁₇	#33501	1	1	yes	characteristic, complemented, Sylow
114	C₂×Q₈	C₂×#3624	1	4	yes	—
115	C₂²×C₄	C₂×#3624	1	2	yes	—
116	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
117	M₁₆	—	68	2	no	—
118	C₂×C₈	—	17	1	no	—
119	C₂×C₈	—	17	1	no	—
120	C₂×C₈	—	17	2	no	—
121	C₂×C₈	—	17	4	no	—
122	C₂×C₈	—	34	2	no	—
123	C₂×C₈	—	68	1	no	—
124	C₄⋊C₄	C₂×#3624	1	4	yes	—
125	C₄⋊C₄	C₂×#3624	1	8	yes	—
126	C₄⋊C₄	C₂×#3624	1	4	yes	—
127	C₂²⋊C₄	C₂×#3624	1	4	yes	—
128	C₄²	C₂×#3624	1	1	yes	characteristic
129	C₄²	C₂×#3624	1	1	yes	characteristic
130	C₄²	C₂²×#595	1	2	yes	—
131	C₄²	C₂²×#595	1	4	yes	—
132	C₁₆	—	34	8	no	—
133	C₂³	C₂²×#3624	1	1	yes	characteristic
134	Q₈	—	4	4	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	4	2	no	—
138	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
139	C₂×C₄	C₂²×#3624	1	2	yes	—
140	C₂×C₄	C₂²×#3624	1	2	yes	—
141	C₂×C₄	C₂²×#3624	1	8	yes	—
142	C₂×C₄	C₂³×#595	1	1	yes	characteristic
143	C₈	—	17	2	no	—
144	C₈	—	34	1	no	—
145	C₈	—	34	2	no	—
146	C₈	—	34	4	no	—
147	C₂²	—	2	1	no	—
148	C₂²	—	2	2	no	—
149	C₂²	C₂³×#3624	1	1	yes	characteristic
150	C₄	—	2	1	no	—
151	C₄	—	2	2	no	—
152	C₄	—	2	2	no	—
153	C₄	—	2	8	no	—
154	C₄	#11763877	1	1	yes	characteristic
155	C₄	#11763889	1	1	yes	characteristic
156	C₂	—	4	1	no	—
157	C₂	#14545677	1	1	yes	characteristic
158	C₂	#14545751	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#595
commutator, socle	C₂²×C₁₇	C₂³×C₈
Fitting	C₁₇×#533	C₄
prime radical	C₂³×C₁₇	C₂²×C₈
cosocle	C₂×#3622	C₂⁴
hypercenter	#533	#595
hypocenter	C₁₇	#33501
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ #533

Complements

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

N	H ≅ G/N	type
1. #14545199	C₂	semidirect
113. C₁₇	#33501	semidirect

Maximal subgroups

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

subgroup	index
#14545199	2
#14545149	2
#14545145	2
#14537719	2
#33501	17