Finite Group Databasebeta

Subgroups — #16948704

Order 4352.

188 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 63 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	#14531057	C₂	1	1	yes	maximal, characteristic
2	#14530689	C₂	1	1	yes	maximal, characteristic
3	#14528189	C₂	1	1	yes	maximal, characteristic
4	#14528185	C₂	1	1	yes	maximal, characteristic
5	#14526857	C₂	1	1	yes	maximal, characteristic
6	#14526841	C₂	1	1	yes	maximal, characteristic
7	C₁₇×#1365	C₂	1	1	yes	maximal, characteristic
8	#11763214	C₂²	1	1	yes	characteristic
9	#11763105	—	2	1	no	—
10	#11763101	—	2	1	no	—
11	C₈×Dic₃₄	C₂²	1	1	yes	characteristic
12	#11762634	—	2	1	no	—
13	#11762634	—	2	1	no	—
14	#11762627	—	2	1	no	—
15	#11762584	—	2	1	no	—
16	C₄×#63214	C₂²	1	1	yes	characteristic
17	#11762562	C₂²	1	1	yes	characteristic
18	#11762558	—	2	1	no	—
19	#11762557	—	2	1	no	—
20	C₁₇×#500	C₂²	1	1	yes	characteristic
21	C₁₇×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
22	C₁₇×#335	C₂²	1	1	yes	characteristic
23	C₁₇×C₄⋊Q₈	—	2	1	no	—
24	#10587555	—	2	1	no	—
25	C₄×Dic₃₄	D₄	1	1	yes	characteristic
26	C₂×C₁₆×C₁₇	—	2	1	no	—
27	C₁₇×#107	—	2	1	no	—
28	C₁₇×#107	—	2	1	no	—
29	C₁₇×C₄⋊C₈	—	2	1	no	—
30	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
31	#10587489	—	2	1	no	—
32	#10587489	—	2	1	no	—
33	#10587489	—	4	1	no	—
34	#10587487	—	4	1	no	—
35	#10587485	—	2	1	no	—
36	C₈×Dic₁₇	—	2	1	no	—
37	C₂×#63214	D₄	1	1	yes	characteristic
38	C₂×#63214	D₄	1	1	yes	characteristic
39	#10587481	—	4	1	no	—
40	#10587478	—	4	1	no	—
41	C₄⋊#3622	—	2	1	no	—
42	C₄⋊#3622	D₄	1	1	yes	characteristic
43	C₂×Q₈×C₁₇	—	4	1	no	—
44	C₂×Dic₃₄	—	4	1	no	—
45	C₂×Dic₃₄	D₈	1	1	yes	characteristic
46	C₂×C₈×C₁₇	—	4	1	no	—
47	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
49	C₄⋊C₄×C₁₇	—	4	1	no	—
50	C₄⋊C₄×C₁₇	—	4	1	no	—
51	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
52	C₄⋊Dic₁₇	—	4	1	no	—
53	C₄⋊Dic₁₇	—	4	1	no	—
54	C₄⋊Dic₁₇	D₈	1	1	yes	characteristic
55	#63225	—	2	1	no	—
56	C₄×Dic₁₇	—	2	1	no	—
57	C₂×#3622	—	2	1	no	—
58	C₂×#3622	—	4	1	no	—
59	C₁₆×C₁₇	—	4	1	no	—
60	#63214	—	2	1	no	—
61	#63214	Q₁₆	1	2	yes	—
62	#12461	—	17	1	no	maximal, Sylow
63	Q₈×C₁₇	—	16	1	no	—
64	C₂×C₄×C₁₇	—	8	1	no	—
65	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
66	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄×C₁₇	#135	1	1	yes	characteristic
68	C₂×Dic₁₇	—	2	1	no	—
69	C₂×Dic₁₇	—	2	1	no	—
70	C₂×Dic₁₇	—	8	1	no	—
71	Dic₃₄	—	2	1	no	—
72	Dic₃₄	—	2	1	no	—
73	Dic₃₄	—	16	1	no	—
74	C₈×C₁₇	—	2	1	no	—
75	C₈×C₁₇	—	8	1	no	—
76	C₈×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
77	C₈×C₁₇	#137	1	1	yes	characteristic
78	#3622	—	4	1	no	—
79	#3622	—	8	1	no	—
80	#2252	—	17	1	no	—
81	#2183	—	17	1	no	—
82	#1676	—	17	1	no	—
83	#1676	—	17	1	no	—
84	#1372	—	17	1	no	—
85	#1365	—	17	1	no	—
86	#1365	D₁₇	1	1	yes	characteristic
87	C₂²×C₁₇	#466	1	1	yes	characteristic
88	C₄×C₁₇	—	2	1	no	—
89	C₄×C₁₇	—	16	1	no	—
90	C₄×C₁₇	#462	1	1	yes	characteristic
91	C₄×C₁₇	#473	1	1	yes	characteristic
92	C₄×C₁₇	#510	1	1	yes	characteristic
93	C₄×C₁₇	#510	1	1	yes	characteristic
94	Dic₁₇	—	4	1	no	—
95	Dic₁₇	—	4	1	no	—
96	Dic₁₇	—	16	1	no	—
97	#500	—	17	1	no	—
98	#500	C₂×D₁₇	1	1	yes	characteristic
99	#477	—	34	1	no	—
100	#477	—	34	1	no	—
101	C₈×Q₈	—	17	1	no	—
102	#366	—	34	1	no	—
103	#366	—	34	1	no	—
104	C₄⋊C₁₆	—	34	1	no	—
105	C₄⋊C₁₆	C₂×D₁₇	1	1	yes	characteristic
106	#346	—	34	1	no	—
107	C₄×C₁₆	—	17	1	no	—
108	#335	—	17	1	no	—
109	#335	C₂×D₁₇	1	1	yes	characteristic
110	#333	—	34	1	no	—
111	#333	—	34	1	no	—
112	C₂×C₁₇	#1674	1	1	yes	characteristic
113	C₂×C₁₇	#2219	1	1	yes	characteristic
114	C₂×C₁₇	#2221	1	1	yes	characteristic
115	C₄⋊Q₈	—	2	1	no	—
116	C₄⋊Q₈	—	34	1	no	—
117	C₄×Q₈	—	17	1	no	—
118	C₂×C₁₆	—	2	1	no	—
119	C₂×C₁₆	—	17	1	no	—
120	C₂×C₁₆	—	17	1	no	—
121	#107	—	2	1	no	—
122	#107	—	2	1	no	—
123	#107	—	34	1	no	—
124	#107	—	34	1	no	—
125	#107	—	68	1	no	—
126	#107	—	68	1	no	—
127	C₄⋊C₈	—	2	1	no	—
128	C₄⋊C₈	—	17	1	no	—
129	C₄⋊C₈	—	34	1	no	—
130	C₄⋊C₈	—	34	1	no	—
131	#103	—	68	1	no	—
132	#103	—	68	1	no	—
133	C₄×C₈	—	34	1	no	—
134	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
135	C₁₇	#12461	1	1	yes	characteristic, complemented, Sylow
136	C₂×Q₈	—	4	1	no	—
137	C₂×Q₈	—	17	1	no	—
138	C₂×Q₈	—	68	1	no	—
139	C₂×C₈	—	4	1	no	—
140	C₂×C₈	—	34	1	no	—
141	C₂×C₈	—	68	1	no	—
142	C₂×C₈	#63252	1	1	yes	characteristic
143	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
144	C₄⋊C₄	—	4	1	no	—
145	C₄⋊C₄	—	4	1	no	—
146	C₄⋊C₄	—	17	1	no	—
147	C₄⋊C₄	—	34	1	no	—
148	C₄⋊C₄	—	68	1	no	—
149	C₄⋊C₄	—	68	1	no	—
150	C₄²	—	34	1	no	—
151	C₄²	D₄×D₁₇	1	1	yes	characteristic
152	C₁₆	—	4	1	no	—
153	C₁₆	—	17	2	no	—
154	C₁₆	—	34	1	no	—
155	Q₈	—	16	1	no	—
156	Q₈	—	272	1	no	—
157	Q₈	—	34	1	no	—
158	Q₈	—	34	1	no	—
159	C₂×C₄	—	8	1	no	—
160	C₂×C₄	—	34	1	no	—
161	C₂×C₄	—	34	1	no	—
162	C₂×C₄	—	136	1	no	—
163	C₂×C₄	#10587570	1	1	yes	characteristic
164	C₂×C₄	D₈×D₁₇	1	1	yes	characteristic
165	C₂×C₄	#10587602	1	1	yes	characteristic
166	C₈	—	2	1	no	—
167	C₈	—	8	1	no	—
168	C₈	—	68	1	no	—
169	C₈	—	136	1	no	—
170	C₈	#10587601	1	1	yes	characteristic
171	C₈	Q₁₆×D₁₇	1	1	yes	characteristic
172	C₂²	#11763044	1	1	yes	characteristic
173	C₄	—	2	1	no	—
174	C₄	—	16	1	no	—
175	C₄	—	272	1	no	—
176	C₄	—	68	1	no	—
177	C₄	—	68	1	no	—
178	C₄	#11763018	1	1	yes	characteristic
179	C₄	#11763083	1	1	yes	characteristic
180	C₄	#11763248	1	1	yes	characteristic
181	C₄	#11763251	1	1	yes	characteristic
182	C₂	#14528177	1	1	yes	characteristic
183	C₂	#14530891	1	1	yes	characteristic
184	C₂	#14530903	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11763044
commutator, cosocle	C₄×C₈×C₁₇	C₂³
Frattini	C₄×C₈	C₂²×D₁₇
Fitting	C₁₇×#1365	C₂
socle, prime radical	C₂²×C₁₇	#466
hypercenter	#1365	D₁₇
hypocenter	C₁₇	#12461
LCS	G ▷ C₄×C₈×C₁₇ ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1365

Complements

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

N	H ≅ G/N	type
135. C₁₇	#12461	semidirect

Maximal subgroups

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

subgroup	index
#14531057	2
#14530689	2
#14528189	2
#14528185	2
#14526857	2
#14526841	2
C₁₇×#1365	2
#12461	17