Finite Group Databasebeta

Subgroups — #16948973

Order 4352.

200 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 61 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	#14530933	C₂	1	1	yes	maximal, characteristic
2	#14530927	C₂	1	1	yes	maximal, characteristic
3	#14528393	C₂	1	1	yes	maximal, characteristic
4	#14527680	C₂	1	1	yes	maximal, characteristic
5	#14526847	C₂	1	1	yes	maximal, characteristic
6	#14526840	C₂	1	1	yes	maximal, characteristic
7	C₁₇×#1365	C₂	1	1	yes	maximal, characteristic
8	#11763214	C₂²	1	1	yes	characteristic
9	#11763161	—	2	1	no	—
10	#11763159	—	2	1	no	—
11	#11762936	—	2	1	no	—
12	C₄×Dic₆₈	C₂²	1	1	yes	characteristic
13	#11762637	—	2	1	no	—
14	#11762634	—	2	1	no	—
15	C₄⋊#63214	C₂²	1	1	yes	characteristic
16	#11762614	—	2	1	no	—
17	#11762612	—	2	1	no	—
18	#11762539	—	2	1	no	—
19	C₈×#3622	C₂²	1	1	yes	characteristic
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	C₂×Dic₆₈	D₄	1	1	yes	characteristic
25	#10587555	—	2	1	no	—
26	C₄×Dic₃₄	—	2	1	no	—
27	C₂×C₁₆×C₁₇	—	2	1	no	—
28	C₁₇×#107	—	2	1	no	—
29	C₁₇×#107	—	2	1	no	—
30	C₁₇×C₄⋊C₈	—	2	1	no	—
31	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
32	#10587506	—	4	2	no	—
33	#10587489	—	2	1	no	—
34	#10587489	—	2	1	no	—
35	#10587489	D₄	1	1	yes	characteristic
36	#10587487	—	2	1	no	—
37	#10587486	—	4	1	no	—
38	#10587485	—	4	1	no	—
39	C₂×#63214	—	2	1	no	—
40	#10587481	—	4	1	no	—
41	#10587481	—	4	1	no	—
42	C₄×#3622	D₄	1	1	yes	characteristic
43	C₄×#3622	D₄	1	1	yes	characteristic
44	C₂×Q₈×C₁₇	—	4	1	no	—
45	C₂×Dic₃₄	—	2	1	no	—
46	C₂×Dic₃₄	—	4	1	no	—
47	C₂×C₈×C₁₇	—	4	1	no	—
48	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
50	C₄⋊C₄×C₁₇	—	4	1	no	—
51	C₄⋊C₄×C₁₇	—	4	1	no	—
52	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
53	C₄⋊Dic₁₇	—	2	1	no	—
54	C₄⋊Dic₁₇	—	4	1	no	—
55	C₄⋊Dic₁₇	—	4	1	no	—
56	#63225	—	4	1	no	—
57	C₄×Dic₁₇	—	4	1	no	—
58	C₂×#3622	—	2	1	no	—
59	C₂×#3622	D₈	1	1	yes	characteristic
60	C₂×#3622	D₈	1	1	yes	characteristic
61	Dic₆₈	—	2	1	no	—
62	Dic₆₈	—	2	1	no	—
63	C₁₆×C₁₇	—	4	1	no	—
64	#63214	—	4	1	no	—
65	#12502	—	17	1	no	maximal, Sylow
66	Q₈×C₁₇	—	8	2	no	—
67	C₂×C₄×C₁₇	—	8	1	no	—
68	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
69	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
70	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
71	C₂×Dic₁₇	—	4	1	no	—
72	C₂×Dic₁₇	—	4	1	no	—
73	C₂×Dic₁₇	—	8	1	no	—
74	Dic₃₄	—	4	1	no	—
75	Dic₃₄	—	4	1	no	—
76	Dic₃₄	—	8	1	no	—
77	Dic₃₄	—	8	1	no	—
78	C₈×C₁₇	—	2	1	no	—
79	C₈×C₁₇	—	8	1	no	—
80	C₈×C₁₇	#137	1	1	yes	characteristic
81	C₈×C₁₇	#137	1	1	yes	characteristic
82	#3622	—	2	1	no	—
83	#3622	—	2	1	no	—
84	#3622	—	4	1	no	—
85	#2227	—	17	1	no	—
86	#2225	—	17	1	no	—
87	#1710	—	17	1	no	—
88	#1591	—	17	1	no	—
89	#1367	—	17	1	no	—
90	#1365	D₁₇	1	1	yes	characteristic
91	#1364	—	17	1	no	—
92	C₂²×C₁₇	#459	1	1	yes	characteristic
93	C₄×C₁₇	—	2	1	no	—
94	C₄×C₁₇	—	16	1	no	—
95	C₄×C₁₇	#470	1	1	yes	characteristic
96	C₄×C₁₇	#473	1	1	yes	characteristic
97	C₄×C₁₇	#508	1	1	yes	characteristic
98	C₄×C₁₇	#510	1	1	yes	characteristic
99	Dic₁₇	—	8	1	no	—
100	Dic₁₇	—	8	1	no	—
101	Dic₁₇	—	16	1	no	—
102	#500	—	17	1	no	—
103	#500	C₂×D₁₇	1	1	yes	characteristic
104	#487	—	34	1	no	—
105	#487	—	34	1	no	—
106	#446	—	34	1	no	—
107	C₄×Q₁₆	—	17	1	no	—
108	#367	—	34	1	no	—
109	#366	—	34	1	no	—
110	C₄⋊C₁₆	—	17	1	no	—
111	C₄⋊C₁₆	C₂×D₁₇	1	1	yes	characteristic
112	#358	—	34	1	no	—
113	#358	—	34	1	no	—
114	#335	C₂×D₁₇	1	1	yes	characteristic
115	Q₈⋊C₈	—	34	1	no	—
116	C₈²	—	17	1	no	—
117	C₂×C₁₇	#1689	1	1	yes	characteristic
118	C₂×C₁₇	#2211	1	1	yes	characteristic
119	C₂×C₁₇	#2212	1	1	yes	characteristic
120	C₂×Q₁₆	—	17	1	no	—
121	C₄⋊Q₈	—	2	1	no	—
122	C₄⋊Q₈	—	34	1	no	—
123	C₄×Q₈	—	34	1	no	—
124	C₂×C₁₆	—	2	1	no	—
125	C₂×C₁₆	—	34	1	no	—
126	#107	—	2	1	no	—
127	#107	—	2	1	no	—
128	#107	—	17	1	no	—
129	#107	—	34	1	no	—
130	#107	—	34	1	no	—
131	C₄⋊C₈	—	2	1	no	—
132	C₄⋊C₈	—	68	1	no	—
133	#103	—	34	1	no	—
134	#103	—	68	1	no	—
135	#103	—	68	1	no	—
136	#103	—	68	2	no	—
137	#97	—	68	1	no	—
138	C₄×C₈	—	17	1	no	—
139	C₄×C₈	—	17	1	no	—
140	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
141	C₁₇	#12502	1	1	yes	characteristic, complemented, Sylow
142	C₂×Q₈	—	4	1	no	—
143	C₂×Q₈	—	34	1	no	—
144	C₂×Q₈	—	68	1	no	—
145	Q₁₆	—	34	1	no	—
146	Q₁₆	—	34	1	no	—
147	C₂×C₈	—	4	1	no	—
148	C₂×C₈	—	17	1	no	—
149	C₂×C₈	—	17	1	no	—
150	C₂×C₈	—	34	1	no	—
151	C₂×C₈	#63252	1	1	yes	characteristic
152	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
153	C₄⋊C₄	—	4	1	no	—
154	C₄⋊C₄	—	4	1	no	—
155	C₄⋊C₄	—	34	1	no	—
156	C₄⋊C₄	—	68	1	no	—
157	C₄⋊C₄	—	68	1	no	—
158	C₄⋊C₄	—	68	1	no	—
159	C₄²	—	68	1	no	—
160	C₄²	D₄×D₁₇	1	1	yes	characteristic
161	C₁₆	—	4	1	no	—
162	C₁₆	—	68	1	no	—
163	Q₈	—	8	2	no	—
164	Q₈	—	68	1	no	—
165	Q₈	—	68	1	no	—
166	Q₈	—	136	1	no	—
167	Q₈	—	136	1	no	—
168	C₂×C₄	—	8	1	no	—
169	C₂×C₄	—	68	1	no	—
170	C₂×C₄	—	68	1	no	—
171	C₂×C₄	—	136	1	no	—
172	C₂×C₄	#10587570	1	1	yes	characteristic
173	C₂×C₄	D₈×D₁₇	1	1	yes	characteristic
174	C₂×C₄	#10587600	1	1	yes	characteristic
175	C₈	—	2	1	no	—
176	C₈	—	8	1	no	—
177	C₈	—	34	1	no	—
178	C₈	—	34	1	no	—
179	C₈	—	68	1	no	—
180	C₈	#10587601	1	1	yes	characteristic
181	C₈	#10587604	1	1	yes	characteristic
182	C₂²	#11763003	1	1	yes	characteristic
183	C₄	—	2	1	no	—
184	C₄	—	16	1	no	—
185	C₄	—	272	1	no	—
186	C₄	—	136	1	no	—
187	C₄	—	136	1	no	—
188	C₄	#11763066	1	1	yes	characteristic
189	C₄	#11763083	1	1	yes	characteristic
190	C₄	#11763240	1	1	yes	characteristic
191	C₄	#11763251	1	1	yes	characteristic
192	C₂	#14528272	1	1	yes	characteristic
193	C₂	#14530849	1	1	yes	characteristic
194	C₂	#14530852	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11763003
commutator, cosocle	C₄×C₈×C₁₇	C₂³
Frattini	C₄×C₈	C₂²×D₁₇
Fitting	C₁₇×#1365	C₂
socle, prime radical	C₂²×C₁₇	#459
hypercenter	#1365	D₁₇
hypocenter	C₁₇	#12502
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
141. C₁₇	#12502	semidirect

Maximal subgroups

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

subgroup	index
#14530933	2
#14530927	2
#14528393	2
#14527680	2
#14526847	2
#14526840	2
C₁₇×#1365	2
#12502	17