Finite Group Databasebeta

Subgroups — #17948528

Order 4352.

254 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 80 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	#14545145	C₂	1	1	yes	maximal, characteristic
2	#14545141	C₂	1	1	yes	maximal, characteristic
3	#14528471	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#63216	C₄	1	2	yes	complemented
5	C₄⋊#63216	C₄	1	2	yes	complemented
6	C₂²⋊#63216	—	2	1	no	—
7	C₂²⋊#63216	—	2	1	no	—
8	#11763697	—	2	1	no	—
9	C₄×#63216	—	2	1	no	—
10	C₂×#10587474	—	2	1	no	—
11	C₂×#10587474	C₄	1	1	yes	characteristic
12	C₂×C₄×#3622	—	2	1	no	—
13	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
14	C₁₇×#376	C₄	1	1	yes	characteristic
15	C₂×C₄²×C₁₇	—	2	1	no	—
16	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
17	C₂²×#3622	—	2	1	no	—
18	C₂²×#3622	D₄	1	1	yes	characteristic
19	C₂²×#3622	D₄	1	1	yes	characteristic
20	C₂×#63216	—	2	2	no	—
21	C₂×#63216	—	2	2	no	—
22	C₁₇×#95	—	2	1	no	—
23	C₁₇×#95	C₈	1	2	yes	—
24	#10587474	—	2	4	no	—
25	#10587474	—	2	1	no	—
26	#10587474	C₈	1	2	yes	—
27	C₄×#3622	—	2	4	no	—
28	C₄×#3622	C₂×C₄	1	1	yes	characteristic
29	C₄×#3622	C₂×C₄	1	1	yes	characteristic
30	C₄×#3622	D₄	1	1	yes	characteristic
31	C₄×#3622	Q₈	1	1	yes	characteristic
32	C₂²×C₄×C₁₇	—	2	2	no	—
33	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
35	C₂²×C₄×C₁₇	M₁₆	1	1	yes	characteristic
36	C₄²×C₁₇	—	2	2	no	—
37	C₄²×C₁₇	—	2	2	no	—
38	C₄²×C₁₇	C₄²	1	1	yes	characteristic
39	C₄²×C₁₇	C₄⋊C₄	1	1	yes	characteristic
40	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
41	C₄²×C₁₇	M₁₆	1	1	yes	characteristic
42	C₂×#3622	—	2	2	no	—
43	C₂×#3622	—	2	2	no	—
44	C₂×#3622	—	2	2	no	—
45	C₂×#3622	—	2	2	no	—
46	C₂×#3622	—	2	2	no	—
47	C₂×#3622	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×#3622	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
50	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
51	#63216	—	4	2	no	—
52	#63216	—	4	2	no	—
53	#7150	—	17	1	no	maximal, Sylow
54	C₂³×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
55	C₂×C₄×C₁₇	—	2	2	no	—
56	C₂×C₄×C₁₇	—	2	1	no	—
57	C₂×C₄×C₁₇	—	2	1	no	—
58	C₂×C₄×C₁₇	—	2	4	no	—
59	C₂×C₄×C₁₇	—	4	1	no	—
60	C₂×C₄×C₁₇	—	4	2	no	—
61	C₂×C₄×C₁₇	—	4	2	no	—
62	C₂×C₄×C₁₇	#95	1	1	yes	characteristic
63	C₂×C₄×C₁₇	C₄×C₈	1	1	yes	characteristic
64	C₂×C₄×C₁₇	#97	1	1	yes	characteristic
65	C₂×C₄×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
66	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
67	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
68	#3622	—	2	4	no	—
69	#3622	—	2	1	no	—
70	#3622	—	2	1	no	—
71	#3622	#104	1	2	yes	—
72	#3622	#104	1	2	yes	—
73	#2163	—	17	1	no	—
74	#2162	—	17	1	no	—
75	#1726	—	17	1	no	—
76	C₂²×C₁₇	—	2	1	no	—
77	C₂²×C₁₇	—	2	1	no	—
78	C₂²×C₁₇	#336	1	1	yes	characteristic
79	C₂²×C₁₇	#350	1	2	yes	—
80	C₄×C₁₇	—	2	1	no	—
81	C₄×C₁₇	—	2	1	no	—
82	C₄×C₁₇	—	2	1	no	—
83	C₄×C₁₇	—	2	1	no	—
84	C₄×C₁₇	—	2	1	no	—
85	C₄×C₁₇	—	4	4	no	—
86	C₄×C₁₇	#339	1	1	yes	characteristic
87	C₄×C₁₇	#339	1	1	yes	characteristic
88	C₂×#97	—	17	1	no	—
89	C₂×#97	—	34	1	no	—
90	C₂×C₄×C₈	—	17	1	no	—
91	C₂×C₄×C₈	—	34	1	no	—
92	#376	#595	1	1	yes	characteristic
93	C₄⋊C₁₆	—	17	2	no	—
94	C₄⋊C₁₆	—	17	2	no	—
95	C₂²⋊C₁₆	—	34	1	no	—
96	C₂²⋊C₁₆	—	34	1	no	—
97	#346	—	34	1	no	—
98	C₄×C₁₆	—	34	1	no	—
99	C₂×C₁₇	—	2	2	no	—
100	C₂×C₁₇	#1276	1	1	yes	characteristic
101	C₂×C₁₇	#1378	1	1	yes	characteristic
102	C₂×C₁₇	#1379	1	1	yes	characteristic
103	C₂²×C₈	—	17	1	no	—
104	C₂²×C₈	—	17	1	no	—
105	C₂²×C₈	—	34	1	no	—
106	C₂×C₄²	—	2	1	no	—
107	C₂×C₄²	C₂×#595	1	1	yes	characteristic
108	C₂×C₁₆	—	34	2	no	—
109	C₂×C₁₆	—	34	2	no	—
110	#97	—	17	2	no	—
111	#97	—	34	4	no	—
112	#97	—	34	1	no	—
113	C₄×C₈	—	17	1	no	—
114	C₄×C₈	—	17	1	no	—
115	C₄×C₈	—	17	1	no	—
116	C₄×C₈	—	17	1	no	—
117	C₄×C₈	—	34	4	no	—
118	#95	—	2	1	no	—
119	#95	#3624	1	2	yes	—
120	C₁₇	#7150	1	1	yes	characteristic, complemented, Sylow
121	C₂²×C₄	—	2	2	no	—
122	C₂²×C₄	C₂×#3624	1	1	yes	characteristic
123	C₂²×C₄	#63247	1	1	yes	characteristic
124	C₂²×C₄	C₂²⋊#595	1	1	yes	characteristic
125	C₂×C₈	—	17	1	no	—
126	C₂×C₈	—	17	1	no	—
127	C₂×C₈	—	17	1	no	—
128	C₂×C₈	—	17	1	no	—
129	C₂×C₈	—	34	2	no	—
130	C₂×C₈	—	34	2	no	—
131	C₂×C₈	—	34	2	no	—
132	C₂×C₈	—	34	2	no	—
133	C₂×C₈	—	34	2	no	—
134	C₄²	—	2	2	no	—
135	C₄²	—	2	2	no	—
136	C₄²	#63242	1	1	yes	characteristic
137	C₄²	#63243	1	1	yes	characteristic
138	C₄²	C₄×#595	1	1	yes	characteristic
139	C₄²	C₄⋊#595	1	1	yes	characteristic
140	C₁₆	—	68	2	no	—
141	C₁₆	—	68	2	no	—
142	C₂³	C₂²⋊#3624	1	1	yes	characteristic
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	4	no	—
147	C₂×C₄	—	4	1	no	—
148	C₂×C₄	—	4	2	no	—
149	C₂×C₄	—	4	2	no	—
150	C₂×C₄	C₄×#3624	1	1	yes	characteristic
151	C₂×C₄	C₄⋊#3624	1	1	yes	characteristic
152	C₂×C₄	#10587542	1	1	yes	characteristic
153	C₂×C₄	#10587543	1	1	yes	characteristic
154	C₂×C₄	#10587544	1	1	yes	characteristic
155	C₂×C₄	(C₂×#595)⋊C₄	1	1	yes	characteristic
156	C₈	—	17	2	no	—
157	C₈	—	17	2	no	—
158	C₈	—	34	4	no	—
159	C₈	—	34	1	no	—
160	C₈	—	34	1	no	—
161	C₂²	—	2	1	no	—
162	C₂²	—	2	1	no	—
163	C₂²	#11763675	1	1	yes	characteristic
164	C₂²	#11763707	1	2	yes	—
165	C₄	—	2	1	no	—
166	C₄	—	2	1	no	—
167	C₄	—	2	1	no	—
168	C₄	—	2	1	no	—
169	C₄	—	2	1	no	—
170	C₄	—	4	4	no	—
171	C₄	#11763680	1	1	yes	characteristic
172	C₄	#11763680	1	1	yes	characteristic
173	C₂	—	2	2	no	—
174	C₂	#14543612	1	1	yes	characteristic
175	C₂	#14543834	1	1	yes	characteristic
176	C₂	#14543838	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	(C₂×#595)⋊C₄
commutator	C₂×C₄×C₁₇	C₄×C₈
Frattini	C₂×C₄²	C₂×#595
Fitting	C₁₇×#376	C₄
socle	C₂²×C₁₇	#336
prime radical	C₂³×C₁₇	C₂²⋊C₈
cosocle	C₂×C₄×#3622	C₂²
hypercenter	#376	#595
hypocenter	C₁₇	#7150
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #376

Complements

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

N	H ≅ G/N	type
4. C₄⋊#63216	C₄	semidirect
5. C₄⋊#63216	C₄	semidirect
120. C₁₇	#7150	semidirect

Maximal subgroups

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

subgroup	index
#14545145	2
#14545141	2
#14528471	2
#7150	17