Finite Group Databasebeta

Subgroups — #17981204

Order 4352.

246 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 91 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	C₄×#10587670	C₂	1	1	yes	maximal, characteristic
2	C₂×#11763903	C₂	1	1	yes	maximal, characteristic
3	#14543989	C₂	1	1	yes	maximal, characteristic
4	C₂×#10587670	C₄	1	2	yes	—
5	C₂²×#63241	C₄	1	2	yes	—
6	#11763903	—	2	1	no	—
7	#11763903	—	2	1	no	—
8	#11763903	—	2	1	no	—
9	C₄×#63241	—	2	1	no	—
10	#11763826	C₄	1	1	yes	characteristic
11	C₂×C₄×#3624	C₂²	1	1	yes	characteristic
12	#11763651	—	2	1	no	—
13	C₈×#3624	—	2	1	no	—
14	#11762859	C₄	1	1	yes	characteristic
15	C₂²×#3624	C₂×C₄	1	1	yes	characteristic
16	C₂²×#3624	C₂×C₄	1	1	yes	characteristic
17	#10587670	—	2	2	no	—
18	#10587670	C₈	1	4	yes	complemented
19	C₂×#63241	—	2	2	no	—
20	C₂×#63241	—	2	2	no	—
21	C₂×#63241	—	2	2	no	—
22	C₂×#63241	C₈	1	4	yes	complemented
23	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
24	C₂²⋊#3624	C₈	1	2	yes	—
25	#10587544	—	2	1	no	—
26	C₄×#3624	—	2	1	no	—
27	C₄×#3624	D₄	1	1	yes	characteristic
28	C₄×#3624	D₄	1	1	yes	characteristic
29	C₄×#3624	D₄	1	1	yes	characteristic
30	C₄×#3624	Q₈	1	1	yes	characteristic
31	C₁₇×C₂²⋊C₈	C₈	1	1	yes	characteristic
32	C₂²⋊#3622	C₈	1	1	yes	characteristic
33	#10587485	—	2	1	no	—
34	C₈×Dic₁₇	—	2	1	no	—
35	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
36	C₂²×Dic₁₇	C₄²	1	1	yes	characteristic
37	C₂²×Dic₁₇	C₂×C₈	1	1	yes	characteristic
38	C₂×#3624	—	2	2	no	—
39	C₂×#3624	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×#3624	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×#3624	C₄⋊C₄	1	1	yes	characteristic
42	C₂×#3624	C₄⋊C₄	1	1	yes	characteristic
43	C₂×#3624	C₂×C₈	1	2	yes	—
44	C₂×#3624	C₂×C₈	1	2	yes	—
45	C₂×#3624	M₁₆	1	2	yes	—
46	C₂×#3624	M₁₆	1	2	yes	—
47	#63241	—	2	4	no	—
48	#63241	—	2	4	no	—
49	C₂×C₈×C₁₇	—	2	1	no	—
50	C₄×Dic₁₇	C₂²⋊C₄	1	1	yes	characteristic
51	C₄×Dic₁₇	C₄⋊C₄	1	1	yes	characteristic
52	C₄×Dic₁₇	M₁₆	1	1	yes	characteristic
53	C₄×Dic₁₇	M₁₆	1	1	yes	characteristic
54	C₂×#3622	—	2	1	no	—
55	#7140	—	17	1	no	maximal, Sylow
56	C₂³×C₁₇	C₄×C₈	1	1	yes	characteristic
57	C₂×C₄×C₁₇	—	4	1	no	—
58	C₂×C₄×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
59	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
60	C₂×Dic₁₇	—	4	1	no	—
61	C₂×Dic₁₇	#95	1	1	yes	characteristic
62	C₂×Dic₁₇	C₄×C₈	1	1	yes	characteristic
63	C₂×Dic₁₇	C₄×C₈	1	1	yes	characteristic
64	C₂×Dic₁₇	#97	1	1	yes	characteristic
65	C₂×Dic₁₇	#97	1	1	yes	characteristic
66	C₂×Dic₁₇	#97	1	1	yes	characteristic
67	C₂×Dic₁₇	C₂²⋊C₈	1	1	yes	characteristic
68	C₂×Dic₁₇	C₄⋊C₈	1	1	yes	characteristic
69	#3624	—	4	2	no	—
70	#3624	C₂²⋊C₈	1	2	yes	—
71	#3624	C₂²⋊C₈	1	2	yes	—
72	#3624	C₄⋊C₈	1	2	yes	—
73	#3624	C₄⋊C₈	1	2	yes	—
74	C₈×C₁₇	—	4	1	no	—
75	#3622	—	4	1	no	—
76	C₄×M₃₂	—	17	1	no	—
77	C₂×#346	—	17	1	no	—
78	#1450	—	17	1	no	—
79	C₂²×C₁₇	—	2	1	no	—
80	C₂²×C₁₇	—	2	1	no	—
81	C₂²×C₁₇	C₈²	1	1	yes	characteristic
82	C₂²×C₁₇	#322	1	1	yes	characteristic
83	C₂²×C₁₇	#336	1	1	yes	characteristic
84	C₄×C₁₇	—	2	1	no	—
85	C₄×C₁₇	—	2	1	no	—
86	Dic₁₇	—	2	1	no	—
87	Dic₁₇	—	2	1	no	—
88	Dic₁₇	#322	1	1	yes	characteristic
89	Dic₁₇	#322	1	1	yes	characteristic
90	Dic₁₇	#336	1	1	yes	characteristic
91	Dic₁₇	#336	1	1	yes	characteristic
92	C₂×M₃₂	—	17	2	no	—
93	C₂²×C₁₆	—	17	2	no	—
94	#431	—	17	1	no	—
95	#431	—	17	1	no	—
96	C₂×C₄×C₈	—	17	1	no	—
97	#346	—	34	1	no	—
98	#346	—	34	1	no	—
99	#346	—	34	1	no	—
100	C₄×C₁₆	—	34	1	no	—
101	#322	—	34	1	no	—
102	C₈²	—	34	1	no	—
103	C₂×C₁₇	—	2	1	no	—
104	C₂×C₁₇	—	2	1	no	—
105	C₂×C₁₇	#1273	1	1	yes	characteristic
106	C₂×C₁₇	#1378	1	1	yes	characteristic
107	C₂×C₁₇	#1378	1	1	yes	characteristic
108	C₂²×C₈	—	17	1	no	—
109	C₂²×C₈	—	17	1	no	—
110	C₂×C₄²	—	17	1	no	—
111	M₃₂	—	17	4	no	—
112	M₃₂	—	34	2	no	—
113	C₂×C₁₆	—	17	4	no	—
114	C₂×C₁₆	—	34	2	no	—
115	C₂×C₁₆	—	34	2	no	—
116	C₂×C₁₆	—	34	2	no	—
117	C₄⋊C₈	—	34	1	no	—
118	C₄⋊C₈	—	34	1	no	—
119	C₂²⋊C₈	—	17	2	no	—
120	C₂²⋊C₈	—	17	1	no	—
121	C₂²⋊C₈	#3633	1	1	yes	characteristic
122	C₄×C₈	—	17	1	no	—
123	C₄×C₈	—	17	1	no	—
124	C₄×C₈	—	17	1	no	—
125	C₄×C₈	—	17	1	no	—
126	C₄×C₈	—	34	1	no	—
127	C₄×C₈	—	34	1	no	—
128	C₁₇	#7140	1	1	yes	characteristic, complemented, Sylow
129	C₂²×C₄	—	17	1	no	—
130	C₂²×C₄	—	17	1	no	—
131	C₂²×C₄	C₂×#3633	1	1	yes	characteristic
132	C₂×C₈	—	2	1	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	2	no	—
138	C₂×C₈	—	17	2	no	—
139	C₂×C₈	—	17	2	no	—
140	C₂×C₈	—	17	2	no	—
141	C₂×C₈	—	34	2	no	—
142	C₂×C₈	—	34	1	no	—
143	C₄²	—	17	1	no	—
144	C₄²	—	17	1	no	—
145	C₄²	—	17	1	no	—
146	C₄²	—	17	1	no	—
147	C₁₆	—	34	4	no	—
148	C₁₆	—	34	4	no	—
149	C₂³	C₄×#3633	1	1	yes	characteristic
150	C₂×C₄	—	4	1	no	—
151	C₂×C₄	—	17	1	no	—
152	C₂×C₄	—	17	1	no	—
153	C₂×C₄	—	17	1	no	—
154	C₂×C₄	—	17	1	no	—
155	C₂×C₄	—	17	1	no	—
156	C₂×C₄	—	17	1	no	—
157	C₂×C₄	—	17	1	no	—
158	C₂×C₄	—	17	1	no	—
159	C₂×C₄	—	68	1	no	—
160	C₂×C₄	C₄⋊#3633	1	1	yes	characteristic
161	C₂×C₄	C₂²⋊#3633	1	1	yes	characteristic
162	C₈	—	4	1	no	—
163	C₈	—	17	2	no	—
164	C₈	—	17	2	no	—
165	C₈	—	17	2	no	—
166	C₈	—	17	2	no	—
167	C₈	—	68	1	no	—
168	C₈	—	68	2	no	—
169	C₂²	—	2	1	no	—
170	C₂²	—	2	1	no	—
171	C₂²	C₈×#3633	1	1	yes	characteristic
172	C₂²	#11763893	1	1	yes	characteristic
173	C₂²	(C₂×#3633)⋊C₄	1	1	yes	characteristic
174	C₄	—	2	1	no	—
175	C₄	—	2	1	no	—
176	C₄	—	17	1	no	—
177	C₄	—	17	1	no	—
178	C₄	—	17	1	no	—
179	C₄	—	17	1	no	—
180	C₄	—	34	1	no	—
181	C₄	—	34	1	no	—
182	C₂	—	2	1	no	—
183	C₂	—	2	1	no	—
184	C₂	(C₂×#3633)⋊C₈	1	1	yes	characteristic
185	C₂	#14545862	1	1	yes	characteristic
186	C₂	#14545863	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	(C₂×#3633)⋊C₄
commutator	C₂²×C₁₇	C₈²
Frattini	C₂²×C₄	C₂×#3633
Fitting	C₁₇×C₂²⋊C₈	C₈
socle	C₂²×C₁₇	#336
prime radical	C₂³×C₁₇	C₄×C₈
cosocle	C₂×C₄×#3624	C₂²
hypercenter	C₂²⋊C₈	#3633
hypocenter	C₁₇	#7140
LCS	G ▷ C₂²×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
18. #10587670	C₈	semidirect
22. C₂×#63241	C₈	semidirect
128. C₁₇	#7140	semidirect

Maximal subgroups

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

subgroup	index
C₄×#10587670	2
C₂×#11763903	2
#14543989	2
#7140	17