Finite Group Databasebeta

Subgroups — #17981237

Order 4352.

222 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 75 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	#14545908	C₂	1	1	yes	maximal, characteristic
2	C₂×#11763898	C₂	1	1	yes	maximal, characteristic
3	M₃₂×#595	C₂	1	1	yes	maximal, characteristic
4	#11763922	—	2	1	no	—
5	C₂×C₄⋊#3633	—	2	1	no	—
6	#11763899	C₄	1	2	yes	—
7	#11763898	—	2	1	no	—
8	#11763898	—	2	1	no	—
9	#11763898	C₄	1	2	yes	—
10	#11763865	C₄	1	1	yes	characteristic
11	C₂×C₈×#595	C₂²	1	1	yes	characteristic
12	#11763696	—	2	1	no	—
13	C₁₆×#595	—	2	1	no	—
14	M₃₂×D₁₇	C₄	1	1	yes	characteristic
15	C₂²×#3633	—	4	1	no	—
16	C₂×C₄×#595	D₄	1	1	yes	characteristic
17	C₂×#63242	Q₈	1	1	yes	characteristic
18	C₂²⋊#3633	—	4	1	no	—
19	C₄⋊#3633	—	2	2	no	—
20	C₄⋊#3633	—	2	2	no	—
21	C₄⋊#3633	—	4	1	no	—
22	C₄×#3633	—	4	1	no	—
23	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
24	#10587538	—	2	1	no	—
25	C₈×#595	C₂×C₄	1	1	yes	characteristic
26	C₈×#595	C₂×C₄	1	1	yes	characteristic
27	C₈×#595	D₄	1	1	yes	characteristic
28	C₈×#595	D₄	1	1	yes	characteristic
29	#10587530	C₈	1	2	yes	complemented
30	#10587529	—	2	1	no	—
31	C₁₇×M₃₂	C₈	1	1	yes	characteristic, complemented
32	#10587483	C₈	1	1	yes	characteristic, complemented
33	#10587469	—	2	1	no	—
34	C₁₆×D₁₇	—	2	1	no	—
35	C₂²×#595	D₈	1	1	yes	characteristic
36	C₂²×#595	Q₁₆	1	1	yes	characteristic
37	C₂×#3633	—	4	2	no	—
38	C₂×#3633	—	4	2	no	—
39	C₂×#3633	—	8	2	no	—
40	C₂×C₄×D₁₇	C₄⋊C₄	1	1	yes	characteristic
41	C₂×#3624	—	2	1	no	—
42	C₄×#595	C₂²⋊C₄	1	1	yes	characteristic
43	C₄×#595	C₂²⋊C₄	1	1	yes	characteristic
44	C₄×#595	SD₁₆	1	1	yes	characteristic
45	C₄×#595	SD₁₆	1	1	yes	characteristic
46	#63242	—	2	1	no	—
47	#63242	C₄⋊C₄	1	1	yes	characteristic
48	#63242	C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
50	C₂×#3622	M₁₆	1	1	yes	characteristic
51	C₈×D₁₇	C₄²	1	1	yes	characteristic
52	C₈×D₁₇	C₂²⋊C₄	1	1	yes	characteristic
53	C₈×D₁₇	C₂×C₈	1	1	yes	characteristic
54	C₈×D₁₇	M₁₆	1	1	yes	characteristic
55	#63216	—	2	2	no	—
56	C₁₆×C₁₇	—	2	1	no	—
57	#63214	—	2	1	no	—
58	#7158	—	17	1	no	maximal, Sylow
59	C₂²×D₁₇	#107	1	1	yes	characteristic
60	C₂×#595	—	2	2	no	—
61	C₂×#595	—	4	1	no	—
62	C₂×#595	#102	1	1	yes	characteristic
63	C₂×#595	#102	1	1	yes	characteristic
64	C₂×#595	#103	1	1	yes	characteristic
65	C₂×#595	#103	1	1	yes	characteristic
66	#3633	—	8	2	no	—
67	#3633	—	8	2	no	—
68	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
69	C₂×Dic₁₇	#108	1	1	yes	characteristic
70	C₄×D₁₇	#95	1	1	yes	characteristic
71	C₄×D₁₇	C₄⋊C₈	1	1	yes	characteristic
72	C₄×D₁₇	#106	1	1	yes	characteristic
73	C₄×D₁₇	#108	1	1	yes	characteristic
74	#3624	—	2	1	no	—
75	#3624	—	2	1	no	—
76	C₈×C₁₇	C₄×C₈	1	1	yes	characteristic
77	C₈×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
78	#3622	#97	1	1	yes	characteristic
79	#3622	C₂²⋊C₈	1	1	yes	characteristic
80	C₄×M₃₂	—	17	1	no	—
81	#1564	—	17	1	no	—
82	C₂×#334	—	17	1	no	—
83	C₂²×C₁₇	#335	1	1	yes	characteristic
84	C₂×D₁₇	—	2	1	no	—
85	C₂×D₁₇	—	2	1	no	—
86	C₂×D₁₇	#335	1	1	yes	characteristic
87	C₂×D₁₇	#340	1	1	yes	characteristic
88	#595	—	2	1	no	—
89	#595	—	2	1	no	—
90	#595	—	2	1	no	—
91	#595	#360	1	2	yes	—
92	C₄×C₁₇	#334	1	1	yes	characteristic
93	C₄×C₁₇	#336	1	1	yes	characteristic
94	Dic₁₇	#334	1	1	yes	characteristic
95	Dic₁₇	#341	1	1	yes	characteristic
96	C₂×M₃₂	—	17	1	no	—
97	C₂×M₃₂	—	17	1	no	—
98	#431	—	34	1	no	—
99	C₂×C₄⋊C₈	—	34	1	no	—
100	C₂×C₄×C₈	—	17	1	no	—
101	#346	—	34	1	no	—
102	C₄×C₁₆	—	34	1	no	—
103	#335	—	17	2	no	—
104	#334	—	17	2	no	—
105	#334	—	34	1	no	—
106	#334	—	34	1	no	—
107	C₂×C₁₇	—	2	1	no	—
108	C₂×C₁₇	#1277	1	1	yes	characteristic
109	D₁₇	—	2	1	no	—
110	D₁₇	#1386	1	1	yes	characteristic
111	D₁₇	#1388	1	1	yes	characteristic
112	C₂²×C₈	—	17	1	no	—
113	C₂²×C₈	—	17	1	no	—
114	C₂²×C₈	—	68	1	no	—
115	C₂×C₄²	—	17	1	no	—
116	M₃₂	—	17	2	no	—
117	M₃₂	—	17	1	no	—
118	M₃₂	—	34	1	no	—
119	M₃₂	—	34	1	no	—
120	M₃₂	#3633	1	1	yes	characteristic, complemented
121	C₂×C₁₆	—	34	1	no	—
122	C₂×C₁₆	—	34	1	no	—
123	C₄⋊C₈	—	34	2	no	—
124	C₄⋊C₈	—	34	2	no	—
125	C₄⋊C₈	—	68	1	no	—
126	C₂²⋊C₈	—	68	1	no	—
127	C₄×C₈	—	17	1	no	—
128	C₄×C₈	—	17	1	no	—
129	C₄×C₈	—	17	1	no	—
130	C₄×C₈	—	17	1	no	—
131	C₄×C₈	—	68	1	no	—
132	C₁₇	#7158	1	1	yes	characteristic, complemented, Sylow
133	C₂²×C₄	—	17	1	no	—
134	C₂²×C₄	—	17	1	no	—
135	C₂²×C₄	—	17	1	no	—
136	C₂×C₈	—	17	1	no	—
137	C₂×C₈	—	17	1	no	—
138	C₂×C₈	—	17	1	no	—
139	C₂×C₈	—	17	1	no	—
140	C₂×C₈	—	17	1	no	—
141	C₂×C₈	—	17	1	no	—
142	C₂×C₈	—	17	1	no	—
143	C₂×C₈	—	34	1	no	—
144	C₂×C₈	—	34	1	no	—
145	C₂×C₈	—	68	2	no	—
146	C₂×C₈	—	68	2	no	—
147	C₂×C₈	—	136	2	no	—
148	C₂×C₈	C₂×#3633	1	1	yes	characteristic
149	C₄²	—	17	1	no	—
150	C₄²	—	17	1	no	—
151	C₄²	—	17	1	no	—
152	C₄²	—	17	1	no	—
153	C₁₆	—	2	1	no	—
154	C₁₆	—	34	2	no	—
155	C₁₆	—	34	1	no	—
156	C₂³	—	17	1	no	—
157	C₂×C₄	—	17	1	no	—
158	C₂×C₄	—	17	1	no	—
159	C₂×C₄	—	17	1	no	—
160	C₂×C₄	—	17	1	no	—
161	C₂×C₄	—	17	1	no	—
162	C₂×C₄	—	17	1	no	—
163	C₂×C₄	—	17	1	no	—
164	C₂×C₄	—	17	1	no	—
165	C₂×C₄	—	17	1	no	—
166	C₂×C₄	—	34	2	no	—
167	C₂×C₄	—	68	1	no	—
168	C₂×C₄	C₄⋊#3633	1	1	yes	characteristic
169	C₈	—	17	1	no	—
170	C₈	—	17	1	no	—
171	C₈	—	34	1	no	—
172	C₈	—	34	1	no	—
173	C₈	—	136	2	no	—
174	C₈	—	136	2	no	—
175	C₈	C₄×#3633	1	1	yes	characteristic
176	C₈	C₂²⋊#3633	1	1	yes	characteristic
177	C₂²	—	17	1	no	—
178	C₂²	—	17	1	no	—
179	C₂²	—	34	1	no	—
180	C₂²	—	34	1	no	—
181	C₂²	#11763899	1	1	yes	characteristic
182	C₄	—	17	2	no	—
183	C₄	—	17	1	no	—
184	C₄	—	17	1	no	—
185	C₄	—	34	1	no	—
186	C₄	—	34	1	no	—
187	C₄	—	34	1	no	—
188	C₄	#11763898	1	1	yes	characteristic
189	C₄	(C₂×#3633)⋊C₄	1	1	yes	characteristic
190	C₂	—	2	1	no	—
191	C₂	—	17	1	no	—
192	C₂	—	17	1	no	—
193	C₂	—	34	1	no	—
194	C₂	#14545790	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#14545790
commutator	C₈×C₁₇	C₄×C₈
Frattini	C₂×C₈	C₂×#3633
Fitting	C₁₇×M₃₂	C₈
socle	C₂×C₁₇	#1277
prime radical	C₂²×D₁₇	#107
cosocle	C₂×C₈×#595	C₂²
hypercenter	M₃₂	#3633
hypocenter	C₁₇	#7158
LCS	G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
29. #10587530	C₈	semidirect
31. C₁₇×M₃₂	C₈	semidirect
32. #10587483	C₈	semidirect
120. M₃₂	#3633	semidirect
132. C₁₇	#7158	semidirect

Maximal subgroups

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

subgroup	index
#14545908	2
C₂×#11763898	2
M₃₂×#595	2
#7158	17