Finite Group Databasebeta

Subgroups — #17958019

Order 4352.

298 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 152 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×#10587538	C₂	1	1	yes	maximal, characteristic
2	C₄×#10587538	C₂	1	1	yes	maximal, characteristic
3	#14543702	C₂	1	4	yes	maximal, complemented
4	C₂×#11762531	C₂	1	1	yes	maximal, characteristic
5	C₂×#10587538	C₄	1	2	yes	—
6	C₂×#10587538	C₄	1	2	yes	—
7	#11763697	C₂²	1	2	yes	—
8	#11763697	C₂²	1	2	yes	—
9	C₄×#63216	C₄	1	2	yes	—
10	C₄×#63216	C₄	1	2	yes	—
11	C₂×C₄×#3622	C₄	1	1	yes	characteristic
12	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
13	#11762531	C₄	1	2	yes	—
14	#11762531	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₁₇	C₄	1	1	yes	characteristic
16	C₂²×C₈×C₁₇	—	2	1	no	—
17	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
18	C₂²×#3622	—	2	1	no	—
19	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
20	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
21	#10587538	—	2	4	no	—
22	#10587538	C₈	1	8	yes	complemented
23	C₂×#63216	C₂×C₄	1	4	yes	—
24	C₂×#63216	C₂×C₄	1	4	yes	—
25	C₄×C₈×C₁₇	—	2	1	no	—
26	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
27	C₄×#3622	—	2	1	no	—
28	C₄×#3622	C₂×C₄	1	1	yes	characteristic
29	C₄×#3622	C₂×C₄	1	1	yes	characteristic
30	C₄×#3622	C₂×C₄	1	2	yes	—
31	C₄×#3622	C₂×C₄	1	1	yes	characteristic
32	C₄×#3622	C₂³	1	1	yes	characteristic
33	C₂²×C₄×C₁₇	C₄²	1	1	yes	characteristic
34	C₂²×C₄×C₁₇	M₁₆	1	1	yes	characteristic
35	C₂²×C₄×C₁₇	M₁₆	1	1	yes	characteristic
36	C₂×C₈×C₁₇	—	2	2	no	—
37	C₂×C₈×C₁₇	—	4	2	no	—
38	C₄²×C₁₇	C₄²	1	1	yes	characteristic
39	C₄²×C₁₇	M₁₆	1	1	yes	characteristic
40	C₄²×C₁₇	M₁₆	1	1	yes	characteristic
41	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
42	C₂×#3622	—	2	2	no	—
43	C₂×#3622	—	4	2	no	—
44	C₂×#3622	C₄²	1	1	yes	characteristic
45	C₂×#3622	C₄²	1	1	yes	characteristic
46	C₂×#3622	C₂×C₈	1	4	yes	—
47	C₂×#3622	M₁₆	1	4	yes	—
48	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
49	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
50	#63216	—	2	4	no	—
51	#63216	C₂×C₈	1	8	yes	complemented
52	#11404	—	17	1	no	maximal, Sylow
53	C₂³×C₁₇	#97	1	1	yes	characteristic
54	C₂×C₄×C₁₇	—	2	2	no	—
55	C₂×C₄×C₁₇	—	2	2	no	—
56	C₂×C₄×C₁₇	C₄×C₈	1	2	yes	—
57	C₂×C₄×C₁₇	#97	1	2	yes	—
58	C₂×C₄×C₁₇	#97	1	1	yes	characteristic
59	C₂×C₄×C₁₇	#97	1	1	yes	characteristic
60	C₂×C₄×C₁₇	#97	1	1	yes	characteristic
61	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
62	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
63	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
64	C₈×C₁₇	—	4	2	no	—
65	#3622	—	4	2	no	—
66	#3622	C₄×C₈	1	2	yes	—
67	#3622	#97	1	2	yes	—
68	#3622	C₂²×C₈	1	2	yes	—
69	#3622	C₂×M₁₆	1	2	yes	—
70	C₄×M₃₂	—	17	1	no	—
71	C₄×M₃₂	—	17	1	no	—
72	C₂×#322	—	17	1	no	—
73	#1313	—	17	4	no	—
74	C₂²×C₁₇	—	2	1	no	—
75	C₂²×C₁₇	—	2	1	no	—
76	C₂²×C₁₇	#322	1	2	yes	—
77	C₂²×C₁₇	C₂×#97	1	1	yes	characteristic
78	C₄×C₁₇	—	2	1	no	—
79	C₄×C₁₇	—	2	1	no	—
80	C₄×C₁₇	#322	1	2	yes	—
81	C₄×C₁₇	#347	1	2	yes	—
82	C₄×C₁₇	#347	1	2	yes	—
83	C₄×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
84	C₄×C₁₇	C₂×#97	1	1	yes	characteristic
85	C₂×M₃₂	—	17	2	no	—
86	C₂×M₃₂	—	17	2	no	—
87	C₂×C₄×C₈	—	17	1	no	—
88	C₂×C₄×C₈	—	17	1	no	—
89	C₂×C₄×C₈	#595	1	1	yes	characteristic
90	#346	—	17	2	no	—
91	#346	—	17	2	no	—
92	C₄×C₁₆	—	17	2	no	—
93	C₄×C₁₆	—	17	2	no	—
94	#322	—	17	2	no	—
95	#322	—	17	2	no	—
96	C₂×C₁₇	—	2	2	no	—
97	C₂×C₁₇	C₂×#322	1	1	yes	characteristic
98	C₂×C₁₇	C₂×#347	1	1	yes	characteristic
99	C₂×C₁₇	C₂×#347	1	1	yes	characteristic
100	C₂²×C₈	—	2	1	no	—
101	C₂²×C₈	—	17	1	no	—
102	C₂²×C₈	—	17	1	no	—
103	C₂²×C₈	—	34	1	no	—
104	C₂×C₄²	C₂×#595	1	1	yes	characteristic
105	M₃₂	—	17	8	no	—
106	M₃₂	—	34	4	no	—
107	C₂×C₁₆	—	17	4	no	—
108	C₂×C₁₆	—	17	4	no	—
109	C₄×C₈	—	2	1	no	—
110	C₄×C₈	—	17	1	no	—
111	C₄×C₈	—	17	1	no	—
112	C₄×C₈	—	17	1	no	—
113	C₄×C₈	—	17	1	no	—
114	C₄×C₈	—	17	2	no	—
115	C₄×C₈	—	34	1	no	—
116	C₄×C₈	C₂×#595	1	2	yes	—
117	C₁₇	#11404	1	1	yes	characteristic, complemented, Sylow
118	C₂²×C₄	#63243	1	1	yes	characteristic
119	C₂²×C₄	C₄×#595	1	1	yes	characteristic
120	C₂²×C₄	#63247	1	1	yes	characteristic
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	—	4	2	no	—
123	C₂×C₈	—	17	1	no	—
124	C₂×C₈	—	17	1	no	—
125	C₂×C₈	—	17	1	no	—
126	C₂×C₈	—	17	1	no	—
127	C₂×C₈	—	17	4	no	—
128	C₂×C₈	—	17	4	no	—
129	C₂×C₈	—	34	2	no	—
130	C₂×C₈	—	68	2	no	—
131	C₄²	#63243	1	1	yes	characteristic
132	C₄²	C₄×#595	1	1	yes	characteristic
133	C₄²	#63247	1	1	yes	characteristic
134	C₄²	C₂²×#595	1	1	yes	characteristic
135	C₁₆	—	17	8	no	—
136	C₁₆	—	34	4	no	—
137	C₂³	#10587542	1	1	yes	characteristic
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	C₈×#595	1	2	yes	—
141	C₂×C₄	#10587532	1	2	yes	—
142	C₂×C₄	#10587542	1	1	yes	characteristic
143	C₂×C₄	#10587542	1	1	yes	characteristic
144	C₂×C₄	#10587542	1	1	yes	characteristic
145	C₂×C₄	C₂×#63243	1	1	yes	characteristic
146	C₂×C₄	C₂×C₄×#595	1	1	yes	characteristic
147	C₂×C₄	C₂×#63247	1	1	yes	characteristic
148	C₈	—	4	2	no	—
149	C₈	—	17	2	no	—
150	C₈	—	17	2	no	—
151	C₈	—	17	2	no	—
152	C₈	—	17	2	no	—
153	C₈	—	68	2	no	—
154	C₂²	—	2	1	no	—
155	C₂²	—	2	1	no	—
156	C₂²	#11763651	1	2	yes	—
157	C₂²	C₂×#10587542	1	1	yes	characteristic
158	C₄	—	2	1	no	—
159	C₄	—	2	1	no	—
160	C₄	#11763651	1	2	yes	—
161	C₄	#11763701	1	2	yes	—
162	C₄	#11763702	1	2	yes	—
163	C₄	C₂×C₈×#595	1	1	yes	characteristic
164	C₄	C₂×#10587532	1	1	yes	characteristic
165	C₂	—	2	2	no	—
166	C₂	C₂×#11763651	1	1	yes	characteristic
167	C₂	C₂×#11763701	1	1	yes	characteristic
168	C₂	C₂×#11763702	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₄×#595
commutator	C₄×C₁₇	C₂×C₄×C₈
Frattini	C₄²	C₂²×#595
Fitting	C₂×C₄×C₈×C₁₇	C₄
socle	C₂²×C₁₇	C₂×#97
prime radical	C₂³×C₁₇	#97
cosocle	C₄×#3622	C₂³
hypercenter	C₂×C₄×C₈	#595
hypocenter	C₁₇	#11404
LCS	G ▷ C₄×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. #14543702	C₂	semidirect
22. #10587538	C₈	semidirect
51. #63216	C₂×C₈	semidirect
117. C₁₇	#11404	semidirect

Maximal subgroups

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

subgroup	index
C₄×#10587538	2
#14543702	2
C₂×#11762531	2
#11404	17