Finite Group Databasebeta

Subgroups — #16955253

Order 4352.

186 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 55 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	#14530867	C₂	1	1	yes	maximal, characteristic
2	D₁₇×#366	C₂	1	1	yes	maximal, characteristic, complemented
3	#14526986	C₂	1	1	yes	maximal, characteristic, complemented
4	#14526985	C₂	1	1	yes	maximal, characteristic, complemented
5	#14526962	C₂	1	1	yes	maximal, characteristic, complemented
6	#14526921	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₇×#1415	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₆×D₁₇	C₂²	1	1	yes	characteristic
9	#11763042	—	2	1	no	—
10	#107×D₁₇	—	2	1	no	—
11	C₂×#10587465	C₂²	1	1	yes	characteristic
12	#11762635	C₂²	1	1	yes	characteristic
13	#11762635	C₂²	1	1	yes	characteristic
14	#11762634	—	2	1	no	—
15	#11762610	—	2	1	no	—
16	C₂×C₁₇×D₁₆	C₂²	1	1	yes	characteristic
17	C₂×C₁₇×C₃₂	C₂²	1	1	yes	characteristic
18	C₁₇×#366	C₂²	1	1	yes	characteristic
19	C₂×D₈×C₁₇	—	2	1	no	—
20	#10587624	—	4	1	no	—
21	C₂×C₈×D₁₇	D₄	1	1	yes	characteristic
22	C₄⋊C₄×D₁₇	—	4	1	no	—
23	C₁₇×D₁₆	—	2	1	no	—
24	C₁₇×D₁₆	—	2	1	no	—
25	C₂×C₁₆×C₁₇	C₂³	1	1	yes	characteristic
26	C₁₇×#107	—	2	1	no	—
27	#10587503	—	4	1	no	—
28	#10587489	—	2	1	no	—
29	#10587489	—	2	1	no	—
30	C₂×#63214	D₄	1	1	yes	characteristic
31	#10587478	—	4	1	no	—
32	C₁₆×D₁₇	—	2	1	no	—
33	C₁₆×D₁₇	—	2	1	no	—
34	C₁₇×C₃₂	—	2	1	no	—
35	#10587465	—	2	1	no	—
36	C₂×D₄×C₁₇	—	4	1	no	—
37	C₂×C₂²⋊D₁₇	—	8	1	no	—
38	C₂×C₄×D₁₇	—	8	1	no	—
39	C₂×C₄×D₁₇	D₈	1	1	yes	characteristic
40	D₈×C₁₇	—	4	1	no	—
41	D₈×C₁₇	—	4	1	no	—
42	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
43	C₄⋊C₄×C₁₇	—	4	1	no	—
44	C₂²⋊Dic₁₇	—	8	1	no	—
45	C₄⋊Dic₁₇	—	4	1	no	—
46	C₄⋊Dic₁₇	—	4	1	no	—
47	#63225	—	8	1	no	—
48	C₂×#3622	D₈	1	1	yes	characteristic
49	C₈×D₁₇	—	2	1	no	—
50	C₈×D₁₇	—	2	1	no	—
51	C₁₆×C₁₇	#41	1	1	yes	characteristic
52	C₁₆×C₁₇	#41	1	1	yes	characteristic
53	#63214	—	2	1	no	—
54	#13712	—	17	1	no	maximal, Sylow
55	C₂³×C₁₇	—	8	1	no	—
56	C₂²×D₁₇	D₁₆	1	1	yes	characteristic
57	D₄×C₁₇	—	8	1	no	—
58	D₄×C₁₇	—	8	1	no	—
59	C₂×C₄×C₁₇	—	8	1	no	—
60	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
61	C₂²⋊D₁₇	—	16	2	no	—
62	C₂×Dic₁₇	—	8	1	no	—
63	C₂×Dic₁₇	—	8	1	no	—
64	C₂×Dic₁₇	D₁₆	1	1	yes	characteristic
65	C₄×D₁₇	—	2	1	no	—
66	C₄×D₁₇	—	2	1	no	—
67	C₄×D₁₇	—	16	2	no	—
68	C₈×C₁₇	#123	1	1	yes	characteristic
69	C₈×C₁₇	#137	1	1	yes	characteristic
70	#3622	—	2	1	no	—
71	#2215	—	17	1	no	—
72	C₂×#366	—	17	1	no	—
73	#1424	—	17	1	no	—
74	#1424	—	17	1	no	—
75	#1415	—	17	1	no	—
76	#1415	D₁₇	1	1	yes	characteristic, complemented
77	C₂²⋊C₃₂	—	17	1	no	—
78	C₂²×C₁₇	—	16	1	no	—
79	C₂²×C₁₇	—	16	1	no	—
80	C₂²×C₁₇	—	16	1	no	—
81	C₂²×C₁₇	C₂×D₁₆	1	1	yes	characteristic
82	C₂×D₁₇	—	2	1	no	—
83	C₂×D₁₇	—	2	1	no	—
84	C₂×D₁₇	SD₆₄	1	2	yes	—
85	C₄×C₁₇	—	16	1	no	—
86	C₄×C₁₇	#480	1	1	yes	characteristic
87	C₄×C₁₇	#510	1	1	yes	characteristic
88	Dic₁₇	—	2	1	no	—
89	Dic₁₇	—	16	1	no	—
90	Dic₁₇	—	16	1	no	—
91	C₂×D₁₆	C₂×D₁₇	1	1	yes	characteristic
92	C₂²×C₁₆	—	17	1	no	—
93	#466	—	34	1	no	—
94	C₂×#107	—	34	1	no	—
95	C₂×C₃₂	—	17	1	no	—
96	C₂×C₃₂	C₂×D₁₇	1	1	yes	characteristic
97	#366	—	17	1	no	—
98	#366	—	17	1	no	—
99	#366	—	34	1	no	—
100	#366	C₂×D₁₇	1	1	yes	characteristic
101	#357	—	34	1	no	—
102	C₂×C₁₇	—	32	1	no	—
103	C₂×C₁₇	#2232	1	1	yes	characteristic
104	C₂×C₁₇	C₂×SD₆₄	1	1	yes	characteristic
105	C₂×C₁₇	#2263	1	1	yes	characteristic
106	D₁₇	—	2	2	no	—
107	C₂×D₈	—	2	1	no	—
108	C₂²×C₈	—	17	1	no	—
109	#121	—	68	1	no	—
110	C₂×C₄⋊C₄	—	68	1	no	—
111	D₁₆	—	2	1	no	—
112	D₁₆	—	2	1	no	—
113	C₂×C₁₆	—	17	1	no	—
114	C₂×C₁₆	—	34	1	no	—
115	C₂×C₁₆	—	34	1	no	—
116	C₂×C₁₆	C₂²×D₁₇	1	1	yes	characteristic
117	#107	—	2	1	no	—
118	#107	—	34	1	no	—
119	#107	—	34	1	no	—
120	#107	—	68	1	no	—
121	#102	—	68	1	no	—
122	C₃₂	—	2	1	no	—
123	C₃₂	—	34	1	no	—
124	C₁₇	#13712	1	1	yes	characteristic, complemented, Sylow
125	C₂×D₄	—	4	1	no	—
126	C₂×D₄	—	136	1	no	—
127	C₂²×C₄	—	17	1	no	—
128	C₂²×C₄	—	136	1	no	—
129	D₈	—	4	1	no	—
130	D₈	—	4	1	no	—
131	C₂×C₈	—	17	1	no	—
132	C₂×C₈	—	34	1	no	—
133	C₂×C₈	—	34	1	no	—
134	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
135	C₄⋊C₄	—	4	1	no	—
136	C₄⋊C₄	—	68	1	no	—
137	C₄⋊C₄	—	68	1	no	—
138	C₄⋊C₄	—	136	1	no	—
139	C₂²⋊C₄	—	136	1	no	—
140	C₁₆	—	34	1	no	—
141	C₁₆	#63252	1	1	yes	characteristic
142	C₁₆	#63254	1	1	yes	characteristic
143	C₂³	—	8	1	no	—
144	C₂³	—	17	1	no	—
145	D₄	—	8	1	no	—
146	D₄	—	8	1	no	—
147	D₄	—	272	2	no	—
148	C₂×C₄	—	8	1	no	—
149	C₂×C₄	—	272	2	no	—
150	C₂×C₄	—	17	1	no	—
151	C₂×C₄	—	34	1	no	—
152	C₂×C₄	—	34	1	no	—
153	C₂×C₄	—	136	1	no	—
154	C₂×C₄	—	136	1	no	—
155	C₂×C₄	D₈×D₁₇	1	1	yes	characteristic
156	C₈	—	34	1	no	—
157	C₈	#10587569	1	1	yes	characteristic
158	C₈	#10587601	1	1	yes	characteristic
159	C₂²	—	16	1	no	—
160	C₂²	—	16	1	no	—
161	C₂²	—	16	1	no	—
162	C₂²	—	17	2	no	—
163	C₂²	—	34	1	no	—
164	C₂²	—	34	1	no	—
165	C₂²	D₁₆×D₁₇	1	1	yes	characteristic
166	C₄	—	16	1	no	—
167	C₄	—	272	1	no	—
168	C₄	—	272	1	no	—
169	C₄	—	34	1	no	—
170	C₄	#11763121	1	1	yes	characteristic
171	C₄	#11763248	1	1	yes	characteristic
172	C₂	—	32	1	no	—
173	C₂	—	34	2	no	—
174	C₂	#14530963	1	1	yes	characteristic
175	C₂	D₁₇×SD₆₄	1	1	yes	characteristic
176	C₂	#14531095	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₁₆×D₁₇
commutator, cosocle	C₂×C₁₆×C₁₇	C₂³
Frattini	C₂×C₁₆	C₂²×D₁₇
Fitting	C₁₇×#1415	C₂
socle	C₂²×C₁₇	C₂×D₁₆
prime radical	#14530867	C₂
hypercenter	#1415	D₁₇
hypocenter	C₁₇	#13712
LCS	G ▷ C₂×C₁₆×C₁₇ ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1415

Complements

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

N	H ≅ G/N	type
2. D₁₇×#366	C₂	semidirect
3. #14526986	C₂	semidirect
4. #14526985	C₂	semidirect
5. #14526962	C₂	semidirect
6. #14526921	C₂	semidirect
7. C₁₇×#1415	C₂	semidirect
76. #1415	D₁₇	semidirect
124. C₁₇	#13712	semidirect

Maximal subgroups

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

subgroup	index
#14530867	2
D₁₇×#366	2
#14526986	2
#14526985	2
#14526962	2
#14526921	2
C₁₇×#1415	2
#13712	17