Finite Group Databasebeta

Subgroups — #17978785

Order 4352.

460 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 218 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	#14545200	C₂	1	4	yes	maximal, complemented
2	#14545198	C₂	1	4	yes	maximal, complemented
3	#14545150	C₂	1	4	yes	maximal
4	#14545141	C₂	1	2	yes	maximal
5	#14537714	C₂	1	1	yes	maximal, characteristic
6	#11763730	C₂²	1	8	yes	—
7	C₄⋊#63216	C₂²	1	2	yes	—
8	C₄⋊#63216	C₂²	1	2	yes	—
9	C₂²⋊#63216	C₂²	1	4	yes	—
10	#11763698	C₂²	1	4	yes	—
11	#11763695	C₂²	1	4	yes	—
12	C₄×#63216	C₂²	1	4	yes	—
13	#11763366	C₄	1	1	yes	characteristic
14	#11762936	C₄	1	4	yes	—
15	#11762933	C₂²	1	4	yes	—
16	#11762837	C₄	1	2	yes	—
17	#11762767	C₂²	1	2	yes	—
18	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
19	C₁₇×#423	C₄	1	1	yes	characteristic
20	C₂×M₁₆×C₁₇	C₂×C₄	1	2	yes	—
21	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
22	#10587612	C₂×C₄	1	2	yes	—
23	#10587610	C₈	1	4	yes	—
24	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
25	C₂²×#3622	C₂³	1	1	yes	characteristic
26	#10587556	C₈	1	2	yes	—
27	#10587555	C₈	1	2	yes	—
28	C₄×Dic₃₄	C₂×C₄	1	4	yes	—
29	C₂×#63216	C₂³	1	4	yes	—
30	C₂×#63216	C₂³	1	4	yes	—
31	C₁₇×C₄⋊C₈	C₂×C₄	1	4	yes	—
32	#10587486	C₂×C₄	1	4	yes	—
33	#10587485	C₂×C₄	1	8	yes	—
34	C₈×Dic₁₇	C₂³	1	4	yes	—
35	C₄×#3622	C₂×C₄	1	2	yes	—
36	C₄×#3622	C₂³	1	2	yes	—
37	C₂²×C₄×C₁₇	C₂×C₈	1	2	yes	—
38	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
39	C₂×Dic₃₄	C₂×C₈	1	4	yes	—
40	M₁₆×C₁₇	—	4	2	no	—
41	C₂×C₈×C₁₇	C₂²×C₄	1	4	yes	—
42	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
43	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
44	C₄²×C₁₇	C₂²×C₄	1	2	yes	—
45	C₂²⋊Dic₁₇	C₂×C₈	1	4	yes	—
46	C₄⋊Dic₁₇	C₂×C₈	1	4	yes	—
47	C₄⋊Dic₁₇	C₂×C₈	1	4	yes	—
48	#63225	C₂×C₈	1	8	yes	—
49	C₄×Dic₁₇	C₂²×C₄	1	4	yes	—
50	C₂×#3622	—	2	2	no	—
51	C₂×#3622	—	4	1	no	—
52	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
53	C₂×#3622	C₂²×C₄	1	2	yes	—
54	C₂×#3622	C₂⁴	1	1	yes	characteristic
55	#63216	—	2	4	no	—
56	#63216	—	2	4	no	—
57	#33500	—	17	1	no	maximal, Sylow
58	C₂³×C₁₇	C₂²×C₈	1	1	yes	characteristic
59	C₂×C₄×C₁₇	—	2	1	no	—
60	C₂×C₄×C₁₇	—	2	1	no	—
61	C₂×C₄×C₁₇	—	2	4	no	—
62	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
63	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
64	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
65	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
66	C₂×Dic₁₇	C₂²×C₈	1	8	yes	—
67	Dic₃₄	—	2	8	no	—
68	C₈×C₁₇	—	2	4	no	—
69	#3622	—	2	1	no	—
70	#3622	—	2	2	no	—
71	#3622	2₋¹⁺⁴	1	2	yes	—
72	#2960	—	17	1	no	—
73	#2183	—	17	4	no	—
74	Q₈×C₁₆	—	17	4	no	—
75	#2164	—	17	4	no	—
76	#2162	—	17	2	no	—
77	C₂²×C₁₇	—	2	2	no	—
78	C₂²×C₁₇	—	2	1	no	—
79	C₂²×C₁₇	C₂³×C₈	1	1	yes	characteristic
80	C₄×C₁₇	—	2	2	no	—
81	C₄×C₁₇	—	2	1	no	—
82	C₄×C₁₇	#504	1	4	yes	—
83	C₄×C₁₇	#519	1	1	yes	characteristic
84	C₄×C₁₇	#568	1	1	yes	characteristic
85	Dic₁₇	—	2	8	no	—
86	#533	—	17	1	no	—
87	#446	—	17	4	no	—
88	C₈×Q₈	—	17	4	no	—
89	#424	—	17	2	no	—
90	#423	#595	1	1	yes	characteristic
91	#405	—	17	2	no	—
92	C₂×C₄×C₈	—	17	1	no	—
93	C₄⋊C₁₆	—	17	2	no	—
94	C₄⋊C₁₆	—	17	2	no	—
95	C₄⋊C₁₆	—	17	8	no	—
96	C₂²⋊C₁₆	—	17	4	no	—
97	#346	—	17	4	no	—
98	C₄×C₁₆	—	17	4	no	—
99	C₄×C₁₆	—	17	4	no	—
100	C₂×C₁₇	—	4	1	no	—
101	C₂×C₁₇	#2982	1	1	yes	characteristic
102	C₂×C₁₇	C₂×#504	1	2	yes	—
103	C₂×M₁₆	C₂×#595	1	2	yes	—
104	C₂²×C₈	—	17	1	no	—
105	C₂²×C₈	—	17	1	no	—
106	C₄⋊Q₈	—	17	2	no	—
107	#125	—	17	2	no	—
108	C₂²⋊Q₈	—	17	4	no	—
109	C₄×Q₈	—	17	4	no	—
110	#117	—	17	2	no	—
111	C₂×C₄²	C₂×#595	1	1	yes	characteristic
112	C₂×C₁₆	—	17	4	no	—
113	C₂×C₁₆	—	17	4	no	—
114	C₄⋊C₈	—	17	8	no	—
115	C₄⋊C₈	C₂×#595	1	4	yes	—
116	#97	—	17	4	no	—
117	C₄×C₈	—	17	4	no	—
118	C₄×C₈	—	17	2	no	—
119	C₄×C₈	—	17	2	no	—
120	C₁₇	#33500	1	1	yes	characteristic, complemented, Sylow
121	C₂×Q₈	—	17	4	no	—
122	C₂²×C₄	#63242	1	2	yes	—
123	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
124	M₁₆	—	4	2	no	—
125	C₂×C₈	—	17	1	no	—
126	C₂×C₈	—	17	1	no	—
127	C₂×C₈	—	17	2	no	—
128	C₂×C₈	—	34	2	no	—
129	C₂×C₈	—	68	1	no	—
130	C₂×C₈	C₂²×#595	1	4	yes	—
131	C₄⋊C₄	—	17	4	no	—
132	C₄⋊C₄	—	17	8	no	—
133	C₄⋊C₄	—	17	4	no	—
134	C₂²⋊C₄	—	17	4	no	—
135	C₄²	—	17	4	no	—
136	C₄²	#63242	1	1	yes	characteristic
137	C₄²	#63242	1	1	yes	characteristic
138	C₄²	C₂²×#595	1	2	yes	—
139	C₁₆	—	34	4	no	—
140	C₁₆	—	34	4	no	—
141	C₂³	C₂×#63242	1	1	yes	characteristic
142	Q₈	—	34	8	no	—
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	17	8	no	—
147	C₂×C₄	C₂×#63242	1	1	yes	characteristic
148	C₂×C₄	C₂×#63242	1	2	yes	—
149	C₂×C₄	C₂×#63242	1	2	yes	—
150	C₂×C₄	C₂³×#595	1	1	yes	characteristic
151	C₈	—	2	4	no	—
152	C₈	—	17	2	no	—
153	C₈	—	34	1	no	—
154	C₈	—	34	2	no	—
155	C₂²	—	2	1	no	—
156	C₂²	—	2	2	no	—
157	C₂²	C₂²×#63242	1	1	yes	characteristic
158	C₄	—	2	1	no	—
159	C₄	—	2	2	no	—
160	C₄	—	34	8	no	—
161	C₄	#11763868	1	4	yes	—
162	C₄	#11763877	1	1	yes	characteristic
163	C₄	#11763887	1	1	yes	characteristic
164	C₂	—	4	1	no	—
165	C₂	#14545679	1	1	yes	characteristic
166	C₂	C₂×#11763868	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#595
commutator, socle	C₂²×C₁₇	C₂³×C₈
Fitting	C₁₇×#423	C₄
prime radical	C₂³×C₁₇	C₂²×C₈
cosocle	C₂×#3622	C₂⁴
hypercenter	#423	#595
hypocenter	C₁₇	#33500
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ #423

Complements

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

N	H ≅ G/N	type
1. #14545200	C₂	semidirect
2. #14545198	C₂	semidirect
120. C₁₇	#33500	semidirect

Maximal subgroups

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

subgroup	index
#14545200	2
#14545198	2
#14545150	2
#14545141	2
#14537714	2
#33500	17