Finite Group Databasebeta

Subgroups — #17982811

Order 4352.

184 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 71 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	#14546018	C₂	1	1	yes	maximal, characteristic, complemented
2	#14546017	C₂	1	1	yes	maximal, characteristic, complemented
3	#14545885	C₂	1	1	yes	maximal, characteristic, complemented
4	#14545881	C₂	1	1	yes	maximal, characteristic, complemented
5	#14545880	C₂	1	1	yes	maximal, characteristic, complemented
6	#14545869	C₂	1	1	yes	maximal, characteristic, complemented
7	#14545172	C₂	1	1	yes	maximal, characteristic
8	#11763918	—	2	1	no	—
9	#11763917	—	2	1	no	—
10	#11763917	C₂²	1	1	yes	characteristic
11	#11763917	C₂²	1	1	yes	characteristic
12	#11763916	—	2	1	no	—
13	C₂×#10587528	—	2	1	no	—
14	C₂×#10587528	C₂²	1	1	yes	characteristic
15	C₂×#10587528	C₂²	1	1	yes	characteristic
16	#11763868	C₂²	1	1	yes	characteristic
17	#11763867	C₂²	1	1	yes	characteristic
18	#11763732	C₄	1	1	yes	characteristic
19	#11763708	C₄	1	1	yes	characteristic
20	#11763707	C₄	1	1	yes	characteristic
21	C₄×#63216	C₂²	1	1	yes	characteristic
22	#11762923	C₄	1	1	yes	characteristic
23	#10587636	C₈	1	1	yes	characteristic
24	#10587634	C₂×C₄	1	1	yes	characteristic
25	#10587593	C₂×C₄	1	1	yes	characteristic
26	#10587538	—	2	1	no	—
27	#10587538	C₂×C₄	1	1	yes	characteristic
28	#10587538	C₂×C₄	1	1	yes	characteristic
29	C₂×#63216	—	2	1	no	—
30	C₂×#63216	C₂×C₄	1	1	yes	characteristic
31	C₂×#63216	C₂³	1	1	yes	characteristic
32	#10587530	—	2	1	no	—
33	#10587529	—	2	1	no	—
34	#10587528	—	2	1	no	—
35	#10587528	—	2	1	no	—
36	#10587528	—	2	1	no	—
37	#10587528	D₄	1	1	yes	characteristic
38	#10587528	D₄	1	1	yes	characteristic
39	C₁₇×#104	C₈	1	1	yes	characteristic
40	#10587493	C₈	1	1	yes	characteristic
41	#10587476	C₈	1	1	yes	characteristic
42	C₄×#3622	C₂×C₄	1	1	yes	characteristic
43	#41×C₁₇	C₂×C₈	1	1	yes	characteristic
44	#63252	C₂×C₈	1	1	yes	characteristic
45	M₁₆×C₁₇	C₂×C₈	1	1	yes	characteristic
46	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
47	#63231	C₁₆	1	1	yes	characteristic
48	#63230	C₁₆	1	1	yes	characteristic
49	#63229	C₁₆	1	1	yes	characteristic
50	#63228	C₁₆	1	1	yes	characteristic
51	#63223	—	2	1	no	—
52	#63223	C₂×C₈	1	1	yes	characteristic
53	C₂×#3622	—	2	1	no	—
54	C₂×#3622	C₂×C₈	1	1	yes	characteristic
55	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
56	#63218	—	2	1	no	—
57	C₈×D₁₇	—	2	1	no	—
58	#63216	—	2	1	no	—
59	#63216	—	2	1	no	—
60	#63216	—	2	1	no	—
61	#63216	C₂×D₄	1	1	yes	characteristic
62	#63216	#41	1	1	yes	characteristic
63	#13648	—	17	1	no	maximal, Sylow
64	Q₈×C₁₇	C₂×C₁₆	1	1	yes	characteristic
65	D₄×C₁₇	—	2	1	no	—
66	D₄×C₁₇	C₂×C₁₆	1	1	yes	characteristic
67	C₂×C₄×C₁₇	—	2	1	no	—
68	C₂×C₄×C₁₇	—	2	1	no	—
69	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
70	C₂²⋊D₁₇	—	2	1	no	—
71	D₆₈	C₂×C₁₆	1	1	yes	characteristic
72	C₄×D₁₇	—	2	1	no	—
73	Dic₃₄	C₂×C₁₆	1	1	yes	characteristic
74	C₈×C₁₇	—	2	1	no	—
75	#3622	—	2	1	no	—
76	#3622	C₂×C₁₆	1	1	yes	characteristic
77	#3622	C₂×C₁₆	1	1	yes	characteristic
78	#3622	C₄×D₄	1	1	yes	characteristic
79	#3622	#131	1	1	yes	characteristic
80	#2258	—	17	1	no	—
81	#2258	—	17	1	no	—
82	#2170	—	17	1	no	—
83	#1422	—	17	1	no	—
84	#1401	—	17	1	no	—
85	#1401	—	17	1	no	—
86	#1397	—	17	1	no	—
87	C₂²×C₁₇	—	2	1	no	—
88	C₂²×C₁₇	#504	1	1	yes	characteristic
89	C₂×D₁₇	—	2	1	no	—
90	C₄×C₁₇	—	2	1	no	—
91	C₄×C₁₇	—	4	1	no	—
92	C₄×C₁₇	C₈×D₄	1	1	yes	characteristic
93	C₄×C₁₇	C₂²×C₁₆	1	1	yes	characteristic
94	Dic₁₇	—	2	1	no	—
95	#504	—	17	1	no	—
96	#504	—	17	1	no	—
97	#443	—	17	1	no	—
98	M₆₄	—	17	1	no	—
99	M₆₄	—	17	1	no	—
100	M₆₄	—	34	1	no	—
101	M₆₄	—	34	1	no	—
102	C₂×C₃₂	—	17	1	no	—
103	C₂×C₃₂	—	17	1	no	—
104	C₂×C₃₂	—	34	1	no	—
105	C₂×C₃₂	—	34	1	no	—
106	#364	—	17	1	no	—
107	#350	—	17	1	no	—
108	#350	—	17	1	no	—
109	C₄×C₁₆	—	17	1	no	—
110	C₂×C₁₇	—	2	1	no	—
111	C₂×C₁₇	—	4	1	no	—
112	C₂×C₁₇	D₄×C₁₆	1	1	yes	characteristic
113	D₁₇	—	4	1	no	—
114	#135	—	17	1	no	—
115	#131	—	17	1	no	—
116	#131	—	17	1	no	—
117	M₃₂	—	17	1	no	—
118	M₃₂	—	17	1	no	—
119	M₃₂	—	34	1	no	—
120	M₃₂	—	34	1	no	—
121	C₂×C₁₆	—	17	1	no	—
122	C₂×C₁₆	—	17	1	no	—
123	C₂×C₁₆	—	34	1	no	—
124	C₂×C₁₆	—	34	1	no	—
125	#108	—	17	1	no	—
126	#104	—	17	1	no	—
127	#104	#3633	1	1	yes	characteristic
128	C₄×C₈	—	17	1	no	—
129	C₃₂	—	17	1	no	—
130	C₃₂	—	17	1	no	—
131	C₃₂	—	34	1	no	—
132	C₃₂	—	34	1	no	—
133	C₃₂	—	34	1	no	—
134	C₁₇	#13648	1	1	yes	characteristic, complemented, Sylow
135	#41	—	17	1	no	—
136	#41	C₂×#3633	1	1	yes	characteristic
137	Q₁₆	—	17	1	no	—
138	SD₁₆	—	17	1	no	—
139	SD₁₆	—	17	1	no	—
140	D₈	—	17	1	no	—
141	M₁₆	—	17	1	no	—
142	M₁₆	—	34	1	no	—
143	M₁₆	—	34	1	no	—
144	M₁₆	C₂×#3633	1	1	yes	characteristic
145	C₂×C₈	—	17	1	no	—
146	C₂×C₈	—	17	1	no	—
147	C₂×C₈	—	34	1	no	—
148	C₂×C₈	—	34	1	no	—
149	C₄²	C₂×#3633	1	1	yes	characteristic
150	C₁₆	—	17	1	no	—
151	C₁₆	—	17	1	no	—
152	C₁₆	—	34	1	no	—
153	C₁₆	—	34	1	no	—
154	C₁₆	—	34	1	no	—
155	Q₈	—	17	1	no	—
156	Q₈	#10587665	1	1	yes	characteristic
157	D₄	—	2	1	no	—
158	D₄	—	17	1	no	—
159	D₄	—	34	1	no	—
160	D₄	#10587665	1	1	yes	characteristic
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	34	1	no	—
164	C₂×C₄	C₂²×#3633	1	1	yes	characteristic
165	C₈	—	2	1	no	—
166	C₈	—	17	1	no	—
167	C₈	—	17	1	no	—
168	C₈	—	17	1	no	—
169	C₈	—	17	1	no	—
170	C₈	—	34	1	no	—
171	C₂²	—	2	1	no	—
172	C₂²	—	34	1	no	—
173	C₂²	#11763930	1	1	yes	characteristic
174	C₄	—	2	1	no	—
175	C₄	—	4	1	no	—
176	C₄	—	34	1	no	—
177	C₄	D₄×#3633	1	1	yes	characteristic
178	C₄	C₂×#10587665	1	1	yes	characteristic
179	C₂	—	2	1	no	—
180	C₂	—	4	1	no	—
181	C₂	—	68	1	no	—
182	C₂	#14545992	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄×#3633
commutator	C₄×C₁₇	C₂²×C₁₆
Frattini	C₂×C₄	C₂²×#3633
Fitting	C₁₇×#104	C₈
socle	C₂×C₁₇	D₄×C₁₆
prime radical	#10587636	C₈
cosocle	C₂×#63216	C₂³
hypercenter	#104	#3633
hypocenter	C₁₇	#13648
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #104

Complements

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

N	H ≅ G/N	type
1. #14546018	C₂	semidirect
2. #14546017	C₂	semidirect
3. #14545885	C₂	semidirect
4. #14545881	C₂	semidirect
5. #14545880	C₂	semidirect
6. #14545869	C₂	semidirect
134. C₁₇	#13648	semidirect

Maximal subgroups

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

subgroup	index
#14546018	2
#14546017	2
#14545885	2
#14545881	2
#14545880	2
#14545869	2
#14545172	2
#13648	17