Finite Group Databasebeta

Subgroups — #17948920

Order 4352.

238 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 74 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	#14545123	C₂	1	1	yes	maximal, characteristic
2	#14544719	C₂	1	1	yes	maximal, characteristic
3	D₁₇×#364	C₂	1	1	yes	maximal, characteristic
4	#11763865	—	2	1	no	—
5	C₂×#10587533	—	2	1	no	—
6	C₂×#10587533	—	2	1	no	—
7	#11763755	C₄	1	2	yes	—
8	#11763732	C₄	1	2	yes	complemented
9	#11763731	—	2	1	no	—
10	M₃₂×D₁₇	—	2	1	no	—
11	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
12	#11762630	C₄	1	1	yes	characteristic, complemented
13	#11762629	—	2	1	no	—
14	C₁₇×#364	C₄	1	1	yes	characteristic, complemented
15	C₂×C₄⋊#595	—	2	2	no	—
16	C₂×C₄⋊#595	—	4	2	no	—
17	C₂×C₈×D₁₇	D₄	1	1	yes	characteristic
18	C₂×C₈×D₁₇	D₄	1	1	yes	characteristic
19	C₄²×D₁₇	C₂×C₄	1	1	yes	characteristic
20	#10587538	—	2	2	no	—
21	#10587533	—	4	1	no	—
22	#10587533	—	4	1	no	—
23	#10587533	—	4	1	no	—
24	#10587533	—	4	1	no	—
25	#10587530	—	4	1	no	—
26	#10587529	—	4	1	no	—
27	C₁₇×M₃₂	—	2	1	no	—
28	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
29	C₈×Dic₁₇	D₄	1	1	yes	characteristic
30	C₈×Dic₁₇	Q₈	1	1	yes	characteristic
31	#10587483	—	2	1	no	—
32	C₄×#3622	C₂×C₄	1	1	yes	characteristic
33	#10587469	—	4	1	no	—
34	C₁₆×D₁₇	—	4	1	no	—
35	C₂²×#595	—	4	2	no	—
36	C₂×C₄×D₁₇	—	2	1	no	—
37	C₂×C₄×D₁₇	C₂²⋊C₄	1	1	yes	characteristic
38	C₄⋊#595	—	4	2	no	—
39	C₄⋊#595	—	4	2	no	—
40	C₄⋊#595	—	8	4	no	—
41	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
43	C₄²×C₁₇	C₄²	1	1	yes	characteristic
44	C₄×Dic₁₇	—	2	1	no	—
45	C₄×Dic₁₇	C₄⋊C₄	1	1	yes	characteristic
46	C₂×#3622	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
48	C₈×D₁₇	D₈	1	1	yes	characteristic
49	C₈×D₁₇	D₈	1	1	yes	characteristic
50	C₈×D₁₇	SD₁₆	1	1	yes	characteristic
51	C₈×D₁₇	SD₁₆	1	1	yes	characteristic
52	C₈×D₁₇	SD₁₆	1	1	yes	characteristic
53	C₈×D₁₇	SD₁₆	1	1	yes	characteristic
54	C₈×D₁₇	Q₁₆	1	1	yes	characteristic
55	C₈×D₁₇	Q₁₆	1	1	yes	characteristic
56	#63216	—	4	2	no	—
57	C₁₆×C₁₇	—	4	1	no	—
58	#63214	—	4	1	no	—
59	#7268	—	17	1	no	maximal, Sylow
60	C₂²×D₁₇	#100	1	1	yes	characteristic
61	C₂×#595	—	8	4	no	—
62	C₂×#595	—	8	2	no	—
63	C₂×C₄×C₁₇	—	2	1	no	—
64	C₂×C₄×C₁₇	#95	1	1	yes	characteristic
65	C₂×Dic₁₇	—	2	1	no	—
66	C₂×Dic₁₇	#101	1	1	yes	characteristic
67	C₄×D₁₇	—	2	4	no	—
68	C₄×D₁₇	#102	1	1	yes	characteristic
69	C₄×D₁₇	#102	1	1	yes	characteristic
70	C₄×D₁₇	#103	1	1	yes	characteristic
71	C₄×D₁₇	#103	1	1	yes	characteristic
72	C₈×C₁₇	#102	1	1	yes	characteristic
73	C₈×C₁₇	#103	1	1	yes	characteristic
74	C₈×C₁₇	#106	1	1	yes	characteristic
75	C₈×C₁₇	#107	1	1	yes	characteristic
76	#3622	#102	1	1	yes	characteristic
77	#3622	#103	1	1	yes	characteristic
78	#3622	#106	1	1	yes	characteristic
79	#3622	#107	1	1	yes	characteristic
80	C₂×#364	—	17	1	no	—
81	C₂×#364	—	17	1	no	—
82	#1844	—	17	1	no	—
83	C₂²×C₁₇	#343	1	1	yes	characteristic
84	C₂×D₁₇	—	2	2	no	—
85	C₂×D₁₇	#331	1	1	yes	characteristic
86	C₂×D₁₇	#333	1	1	yes	characteristic
87	#595	—	16	2	no	—
88	C₄×C₁₇	—	2	2	no	—
89	C₄×C₁₇	#340	1	1	yes	characteristic
90	C₄×C₁₇	#340	1	1	yes	characteristic
91	Dic₁₇	—	2	2	no	—
92	Dic₁₇	#332	1	1	yes	characteristic
93	Dic₁₇	#332	1	1	yes	characteristic
94	C₂×M₃₂	—	34	1	no	—
95	C₂×M₃₂	—	34	1	no	—
96	C₂×#106	—	34	1	no	—
97	C₂×#106	—	34	1	no	—
98	C₂×C₄×C₈	—	17	1	no	—
99	#384	—	17	2	no	—
100	#364	—	17	2	no	—
101	#364	—	17	1	no	—
102	#364	—	34	1	no	—
103	#364	—	34	1	no	—
104	#364	#595	1	1	yes	characteristic, complemented
105	C₂×C₁₇	—	2	1	no	—
106	C₂×C₁₇	#1296	1	1	yes	characteristic
107	D₁₇	—	2	1	no	—
108	D₁₇	#1359	1	2	yes	—
109	C₂²×C₈	—	17	1	no	—
110	C₂²×C₈	—	17	1	no	—
111	C₂×C₄⋊C₄	—	34	2	no	—
112	C₂×C₄⋊C₄	—	68	2	no	—
113	C₂×C₄²	—	17	1	no	—
114	M₃₂	—	2	1	no	—
115	M₃₂	—	34	2	no	—
116	M₃₂	—	34	1	no	—
117	M₃₂	—	68	1	no	—
118	M₃₂	—	68	1	no	—
119	C₂×C₁₆	—	68	1	no	—
120	C₂×C₁₆	—	68	1	no	—
121	#106	—	68	1	no	—
122	#106	—	68	1	no	—
123	#106	—	68	1	no	—
124	#106	—	68	1	no	—
125	C₄×C₈	—	17	1	no	—
126	C₄×C₈	—	17	1	no	—
127	C₄×C₈	—	17	1	no	—
128	C₄×C₈	C₂×#595	1	1	yes	characteristic
129	C₁₇	#7268	1	1	yes	characteristic, complemented, Sylow
130	C₂²×C₄	—	17	1	no	—
131	C₂²×C₄	—	34	1	no	—
132	C₂²×C₄	—	68	2	no	—
133	C₂×C₈	—	17	1	no	—
134	C₂×C₈	—	17	1	no	—
135	C₂×C₈	—	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	C₄⋊#595	1	1	yes	characteristic
144	C₂×C₈	C₂²⋊#595	1	1	yes	characteristic
145	C₄⋊C₄	—	68	2	no	—
146	C₄⋊C₄	—	68	2	no	—
147	C₄⋊C₄	—	136	4	no	—
148	C₄²	—	17	1	no	—
149	C₄²	—	34	1	no	—
150	C₄²	C₄×#595	1	1	yes	characteristic
151	C₁₆	—	4	1	no	—
152	C₁₆	—	68	2	no	—
153	C₁₆	—	68	1	no	—
154	C₂³	—	17	1	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	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	—	34	4	no	—
162	C₂×C₄	—	34	1	no	—
163	C₂×C₄	—	136	2	no	—
164	C₂×C₄	—	136	4	no	—
165	C₂×C₄	(C₂×#595)⋊C₄	1	1	yes	characteristic
166	C₈	—	17	1	no	—
167	C₈	—	17	1	no	—
168	C₈	—	17	1	no	—
169	C₈	—	17	1	no	—
170	C₈	#10587533	1	1	yes	characteristic
171	C₈	#10587534	1	1	yes	characteristic
172	C₈	#10587547	1	1	yes	characteristic
173	C₈	#10587549	1	1	yes	characteristic
174	C₂²	—	17	1	no	—
175	C₂²	—	17	1	no	—
176	C₂²	—	34	2	no	—
177	C₂²	#11763688	1	1	yes	characteristic
178	C₄	—	2	2	no	—
179	C₄	—	272	2	no	—
180	C₄	—	17	1	no	—
181	C₄	—	17	1	no	—
182	C₄	—	34	2	no	—
183	C₄	#11763682	1	1	yes	characteristic
184	C₄	#11763683	1	1	yes	characteristic
185	C₂	—	2	1	no	—
186	C₂	—	17	2	no	—
187	C₂	—	34	1	no	—
188	C₂	#14543663	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#14543663
commutator	C₄²×C₁₇	C₄²
Frattini	C₄×C₈	C₂×#595
Fitting	C₁₇×#364	C₄
socle	C₂×C₁₇	#1296
prime radical	C₂²×D₁₇	#100
cosocle	C₄×C₈×D₁₇	C₂²
hypercenter	#364	#595
hypocenter	C₁₇	#7268
LCS	G ▷ C₄²×C₁₇ ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

Complements

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

N	H ≅ G/N	type
8. #11763732	C₄	semidirect
12. #11762630	C₄	semidirect
14. C₁₇×#364	C₄	semidirect
104. #364	#595	semidirect
129. C₁₇	#7268	semidirect

Maximal subgroups

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

subgroup	index
#14545123	2
#14544719	2
D₁₇×#364	2
#7268	17