Finite Group Databasebeta

Subgroups — #17953803

Order 4352.

216 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 58 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	#14543651	C₂	1	1	yes	maximal, characteristic
2	#14543651	C₂	1	1	yes	maximal, characteristic
3	#14543651	C₂	1	1	yes	maximal, characteristic
4	#14543644	C₂	1	1	yes	maximal, characteristic
5	#14543644	C₂	1	1	yes	maximal, characteristic
6	#14543644	C₂	1	1	yes	maximal, characteristic
7	#14527088	C₂	1	1	yes	maximal, characteristic
8	#11763832	C₂²	1	1	yes	characteristic
9	#11763832	C₂²	1	1	yes	characteristic
10	#11763832	C₂²	1	1	yes	characteristic
11	#11763754	C₂²	1	1	yes	characteristic
12	C₄×#63218	C₂²	1	1	yes	characteristic
13	C₄×#63218	C₂²	1	1	yes	characteristic
14	C₄×#63218	C₂²	1	1	yes	characteristic
15	#11762531	C₄	1	1	yes	characteristic, complemented
16	#11762531	C₄	1	1	yes	characteristic, complemented
17	C₈×#3622	C₄	1	1	yes	characteristic, complemented
18	C₁₇×#322	C₄	1	1	yes	characteristic, complemented
19	C₂×#63218	—	2	1	no	—
20	C₂×#63218	—	2	1	no	—
21	C₂×#63218	—	2	1	no	—
22	C₄²×D₁₇	C₂³	1	1	yes	characteristic
23	(C₂×#595)⋊C₄	—	2	1	no	—
24	(C₂×#595)⋊C₄	—	2	1	no	—
25	(C₂×#595)⋊C₄	—	2	1	no	—
26	#10587544	—	2	1	no	—
27	#10587544	—	2	1	no	—
28	#10587544	—	2	1	no	—
29	#10587543	—	2	1	no	—
30	#10587543	—	2	1	no	—
31	#10587543	—	2	1	no	—
32	#10587542	—	2	1	no	—
33	#10587542	—	2	1	no	—
34	#10587542	—	2	1	no	—
35	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
38	#10587486	—	2	1	no	—
39	#10587486	—	2	1	no	—
40	#10587486	—	2	1	no	—
41	C₄×#3622	C₂×C₄	1	1	yes	characteristic
42	C₄×#3622	C₂×C₄	1	1	yes	characteristic
43	C₄×#3622	C₂×C₄	1	1	yes	characteristic
44	C₂²×#595	—	4	1	no	—
45	C₂×C₄×D₁₇	#41	1	1	yes	characteristic
46	C₂×C₄×D₁₇	#41	1	1	yes	characteristic
47	C₂×C₄×D₁₇	#41	1	1	yes	characteristic
48	C₂×#3624	—	4	1	no	—
49	C₂×#3624	—	4	1	no	—
50	C₂×#3624	—	4	1	no	—
51	C₂×C₈×C₁₇	—	2	1	no	—
52	C₂×C₈×C₁₇	—	2	1	no	—
53	C₂×C₈×C₁₇	—	2	1	no	—
54	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
55	C₄×Dic₁₇	#41	1	1	yes	characteristic
56	C₄×Dic₁₇	#41	1	1	yes	characteristic
57	C₄×Dic₁₇	#41	1	1	yes	characteristic
58	C₂×#3622	—	2	1	no	—
59	C₂×#3622	—	2	1	no	—
60	C₂×#3622	—	2	1	no	—
61	#63218	—	4	2	no	—
62	#63218	—	4	2	no	—
63	#63218	—	4	2	no	—
64	#8803	—	17	1	no	maximal, Sylow
65	C₂²×D₁₇	#126	1	1	yes	characteristic
66	C₂×#595	—	8	1	no	—
67	C₂×#595	—	8	1	no	—
68	C₂×#595	—	8	1	no	—
69	C₂×C₄×C₁₇	#117	1	1	yes	characteristic
70	C₂×C₄×C₁₇	#117	1	1	yes	characteristic
71	C₂×C₄×C₁₇	#117	1	1	yes	characteristic
72	C₂×Dic₁₇	#126	1	1	yes	characteristic
73	C₂×Dic₁₇	#126	1	1	yes	characteristic
74	C₂×Dic₁₇	#126	1	1	yes	characteristic
75	C₄×D₁₇	—	4	1	no	—
76	C₄×D₁₇	—	4	1	no	—
77	C₄×D₁₇	—	4	1	no	—
78	#3624	—	8	1	no	—
79	#3624	—	8	1	no	—
80	#3624	—	8	1	no	—
81	C₈×C₁₇	—	2	2	no	—
82	C₈×C₁₇	—	4	1	no	—
83	C₈×C₁₇	—	4	1	no	—
84	#3622	—	2	2	no	—
85	#3622	—	2	2	no	—
86	#3622	—	4	1	no	—
87	#1455	—	17	1	no	—
88	#1291	—	17	1	no	—
89	#1291	—	17	1	no	—
90	#1291	—	17	1	no	—
91	#1288	—	17	1	no	—
92	#1288	—	17	1	no	—
93	#1288	—	17	1	no	—
94	C₂²×C₁₇	#383	1	1	yes	characteristic
95	C₂×D₁₇	—	2	1	no	—
96	C₂×D₁₇	—	2	1	no	—
97	C₂×D₁₇	—	2	1	no	—
98	#595	—	16	1	no	—
99	C₄×C₁₇	—	2	1	no	—
100	C₄×C₁₇	—	2	1	no	—
101	C₄×C₁₇	—	2	1	no	—
102	Dic₁₇	—	2	1	no	—
103	Dic₁₇	—	2	1	no	—
104	Dic₁₇	—	2	1	no	—
105	#432	—	17	1	no	—
106	#432	—	17	1	no	—
107	#432	—	17	1	no	—
108	C₄×M₁₆	—	17	1	no	—
109	C₄×M₁₆	—	17	1	no	—
110	C₄×M₁₆	—	17	1	no	—
111	#383	—	17	1	no	—
112	#322	—	17	1	no	—
113	#322	—	17	1	no	—
114	#322	#595	1	1	yes	characteristic, complemented
115	C₈²	—	17	1	no	—
116	C₂×C₁₇	#1841	1	1	yes	characteristic
117	C₂×C₁₇	#1841	1	1	yes	characteristic
118	C₂×C₁₇	#1841	1	1	yes	characteristic
119	D₁₇	—	4	1	no	—
120	C₂×M₁₆	—	34	1	no	—
121	C₂×M₁₆	—	34	1	no	—
122	C₂×M₁₆	—	34	1	no	—
123	C₂×C₄²	—	17	1	no	—
124	C₄⋊C₈	—	34	1	no	—
125	C₄⋊C₈	—	34	1	no	—
126	C₄⋊C₈	—	34	1	no	—
127	C₂²⋊C₈	—	34	1	no	—
128	C₂²⋊C₈	—	34	1	no	—
129	C₂²⋊C₈	—	34	1	no	—
130	#97	—	34	1	no	—
131	#97	—	34	1	no	—
132	#97	—	34	1	no	—
133	#97	—	34	1	no	—
134	#97	—	34	1	no	—
135	#97	—	34	1	no	—
136	C₄×C₈	—	17	1	no	—
137	C₄×C₈	—	17	1	no	—
138	C₄×C₈	—	17	1	no	—
139	C₄×C₈	C₂×#595	1	1	yes	characteristic
140	C₄×C₈	C₂×#595	1	1	yes	characteristic
141	C₄×C₈	C₂×#595	1	1	yes	characteristic
142	#95	—	34	1	no	—
143	#95	—	34	1	no	—
144	#95	—	34	1	no	—
145	C₁₇	#8803	1	1	yes	characteristic, complemented, Sylow
146	C₂²×C₄	—	17	1	no	—
147	C₂²×C₄	—	17	1	no	—
148	C₂²×C₄	—	17	1	no	—
149	C₂²×C₄	—	68	1	no	—
150	M₁₆	—	68	2	no	—
151	M₁₆	—	68	2	no	—
152	M₁₆	—	68	2	no	—
153	C₂×C₈	—	2	1	no	—
154	C₂×C₈	—	2	1	no	—
155	C₂×C₈	—	2	1	no	—
156	C₂×C₈	—	34	1	no	—
157	C₂×C₈	—	34	1	no	—
158	C₂×C₈	—	34	1	no	—
159	C₂×C₈	—	68	1	no	—
160	C₂×C₈	—	68	1	no	—
161	C₂×C₈	—	68	1	no	—
162	C₄²	—	17	1	no	—
163	C₄²	—	17	1	no	—
164	C₄²	—	17	1	no	—
165	C₄²	C₂²×#595	1	1	yes	characteristic
166	C₂³	—	17	1	no	—
167	C₂×C₄	—	17	1	no	—
168	C₂×C₄	—	17	1	no	—
169	C₂×C₄	—	17	1	no	—
170	C₂×C₄	—	68	1	no	—
171	C₂×C₄	—	68	1	no	—
172	C₂×C₄	—	68	1	no	—
173	C₂×C₄	—	136	1	no	—
174	C₂×C₄	—	136	1	no	—
175	C₂×C₄	—	136	1	no	—
176	C₂×C₄	#10587677	1	1	yes	characteristic
177	C₂×C₄	#10587677	1	1	yes	characteristic
178	C₂×C₄	#10587677	1	1	yes	characteristic
179	C₈	—	2	2	no	—
180	C₈	—	4	1	no	—
181	C₈	—	4	1	no	—
182	C₈	—	34	2	no	—
183	C₈	—	34	2	no	—
184	C₈	—	68	1	no	—
185	C₈	—	136	1	no	—
186	C₈	—	136	1	no	—
187	C₈	—	136	1	no	—
188	C₂²	—	34	1	no	—
189	C₂²	—	34	1	no	—
190	C₂²	—	34	1	no	—
191	C₂²	#11763754	1	1	yes	characteristic
192	C₄	—	2	1	no	—
193	C₄	—	2	1	no	—
194	C₄	—	2	1	no	—
195	C₄	—	272	1	no	—
196	C₄	—	34	1	no	—
197	C₄	—	34	1	no	—
198	C₄	—	34	1	no	—
199	C₂	—	68	1	no	—
200	C₂	#14544714	1	1	yes	characteristic
201	C₂	#14544715	1	1	yes	characteristic
202	C₂	#14544715	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11763754
commutator	C₄²×C₁₇	C₂²×C₄
Frattini	C₄²	C₂²×#595
Fitting	C₁₇×#322	C₄
socle	C₂²×C₁₇	#383
prime radical	C₂²×D₁₇	#126
cosocle	C₄²×D₁₇	C₂³
hypercenter	#322	#595
hypocenter	C₁₇	#8803
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322

Complements

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

N	H ≅ G/N	type
15. #11762531	C₄	semidirect
16. #11762531	C₄	semidirect
17. C₈×#3622	C₄	semidirect
18. C₁₇×#322	C₄	semidirect
114. #322	#595	semidirect
145. C₁₇	#8803	semidirect

Maximal subgroups

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

subgroup	index
#14543651	2
#14543644	2
#14527088	2
#8803	17