Finite Group Databasebeta

Subgroups — #17958031

Order 4352.

282 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 144 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	C₂×#11763697	C₂	1	1	yes	maximal, characteristic
3	#14543702	C₂	1	4	yes	maximal, complemented
4	#14527068	C₂	1	1	yes	maximal, characteristic
5	C₂×#10587538	C₄	1	2	yes	—
6	C₂²×#63216	C₄	1	2	yes	—
7	#11763697	C₄	1	2	yes	—
8	#11763697	C₂²	1	2	yes	—
9	#11763697	C₂²	1	2	yes	—
10	C₄×#63216	C₄	1	2	yes	—
11	#11762862	C₄	1	1	yes	characteristic
12	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
13	#11762531	C₂²	1	2	yes	—
14	C₈×#3622	C₄	1	2	yes	—
15	C₁₇×#431	C₄	1	1	yes	characteristic
16	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
17	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
18	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
19	#10587538	C₈	1	8	yes	complemented
20	C₂×#63216	—	2	4	no	—
21	C₂×#63216	C₂×C₄	1	4	yes	—
22	C₂×#63216	C₂×C₄	1	4	yes	—
23	C₁₇×C₄⋊C₈	—	2	1	no	—
24	C₁₇×C₂²⋊C₈	—	2	1	no	—
25	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
26	C₂²⋊#3622	—	2	1	no	—
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	C₄²×C₁₇	C₄²	1	1	yes	characteristic
38	C₄²×C₁₇	M₁₆	1	1	yes	characteristic
39	C₄²×C₁₇	M₁₆	1	1	yes	characteristic
40	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×#3622	—	2	2	no	—
42	C₂×#3622	C₄²	1	1	yes	characteristic
43	C₂×#3622	C₄²	1	1	yes	characteristic
44	C₂×#3622	C₂×C₈	1	4	yes	—
45	C₂×#3622	M₁₆	1	4	yes	—
46	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
47	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
48	#63216	—	2	4	no	—
49	#63216	C₂×C₈	1	8	yes	—
50	#11406	—	17	1	no	maximal, Sylow
51	C₂³×C₁₇	#97	1	1	yes	characteristic
52	C₂×C₄×C₁₇	—	4	1	no	—
53	C₂×C₄×C₁₇	—	4	1	no	—
54	C₂×C₄×C₁₇	C₄×C₈	1	2	yes	—
55	C₂×C₄×C₁₇	#97	1	2	yes	—
56	C₂×C₄×C₁₇	#97	1	1	yes	characteristic
57	C₂×C₄×C₁₇	#97	1	1	yes	characteristic
58	C₂×C₄×C₁₇	#97	1	1	yes	characteristic
59	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
60	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
61	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
62	C₈×C₁₇	—	4	2	no	—
63	#3622	—	4	2	no	—
64	#3622	C₄×C₈	1	2	yes	—
65	#3622	#97	1	2	yes	—
66	#3622	C₂²×C₈	1	2	yes	—
67	#3622	C₂×M₁₆	1	2	yes	—
68	C₄×M₃₂	—	17	1	no	—
69	C₂×#346	—	17	1	no	—
70	#1450	—	17	1	no	—
71	#1313	—	17	4	no	—
72	C₂²×C₁₇	—	2	1	no	—
73	C₂²×C₁₇	—	2	1	no	—
74	C₂²×C₁₇	#322	1	2	yes	—
75	C₂²×C₁₇	C₂×#97	1	1	yes	characteristic
76	C₄×C₁₇	—	2	1	no	—
77	C₄×C₁₇	—	2	1	no	—
78	C₄×C₁₇	—	2	1	no	—
79	C₄×C₁₇	—	2	1	no	—
80	C₄×C₁₇	#322	1	2	yes	—
81	C₄×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
82	C₄×C₁₇	C₂×#97	1	1	yes	characteristic
83	C₂×M₃₂	—	17	2	no	—
84	C₂²×C₁₆	—	17	2	no	—
85	#431	—	17	1	no	—
86	#431	#595	1	1	yes	characteristic
87	C₂×C₄×C₈	—	17	1	no	—
88	#346	—	17	2	no	—
89	#346	—	17	2	no	—
90	#346	—	17	2	no	—
91	C₄×C₁₆	—	17	2	no	—
92	#322	—	17	2	no	—
93	C₈²	—	17	2	no	—
94	C₂×C₁₇	—	2	2	no	—
95	C₂×C₁₇	C₂×#322	1	1	yes	characteristic
96	C₂×C₁₇	#2110	1	1	yes	characteristic
97	C₂×C₁₇	#2110	1	1	yes	characteristic
98	C₂²×C₈	—	17	1	no	—
99	C₂²×C₈	—	17	1	no	—
100	C₂×C₄²	C₂×#595	1	1	yes	characteristic
101	M₃₂	—	17	8	no	—
102	C₂×C₁₆	—	17	4	no	—
103	C₂×C₁₆	—	17	4	no	—
104	C₂×C₁₆	—	34	4	no	—
105	C₄⋊C₈	—	2	1	no	—
106	C₄⋊C₈	—	34	1	no	—
107	C₂²⋊C₈	—	2	1	no	—
108	C₂²⋊C₈	—	34	1	no	—
109	C₄×C₈	—	17	1	no	—
110	C₄×C₈	—	17	1	no	—
111	C₄×C₈	—	17	1	no	—
112	C₄×C₈	—	17	1	no	—
113	C₄×C₈	—	17	2	no	—
114	C₄×C₈	C₂×#595	1	2	yes	—
115	C₁₇	#11406	1	1	yes	characteristic, complemented, Sylow
116	C₂²×C₄	#63243	1	1	yes	characteristic
117	C₂²×C₄	C₄×#595	1	1	yes	characteristic
118	C₂²×C₄	#63247	1	1	yes	characteristic
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	17	1	no	—
121	C₂×C₈	—	17	1	no	—
122	C₂×C₈	—	17	1	no	—
123	C₂×C₈	—	17	1	no	—
124	C₂×C₈	—	17	4	no	—
125	C₂×C₈	—	17	4	no	—
126	C₂×C₈	—	34	2	no	—
127	C₄²	#63243	1	1	yes	characteristic
128	C₄²	C₄×#595	1	1	yes	characteristic
129	C₄²	#63247	1	1	yes	characteristic
130	C₄²	C₂²×#595	1	1	yes	characteristic
131	C₁₆	—	17	8	no	—
132	C₁₆	—	34	4	no	—
133	C₂³	#10587542	1	1	yes	characteristic
134	C₂×C₄	—	4	1	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	C₈×#595	1	2	yes	—
137	C₂×C₄	#10587532	1	2	yes	—
138	C₂×C₄	#10587542	1	1	yes	characteristic
139	C₂×C₄	#10587542	1	1	yes	characteristic
140	C₂×C₄	#10587542	1	1	yes	characteristic
141	C₂×C₄	C₂×#63243	1	1	yes	characteristic
142	C₂×C₄	C₂×C₄×#595	1	1	yes	characteristic
143	C₂×C₄	C₂×#63247	1	1	yes	characteristic
144	C₈	—	4	2	no	—
145	C₈	—	17	2	no	—
146	C₈	—	17	2	no	—
147	C₈	—	17	2	no	—
148	C₈	—	17	2	no	—
149	C₈	—	68	2	no	—
150	C₂²	—	2	1	no	—
151	C₂²	—	2	1	no	—
152	C₂²	#11763651	1	2	yes	—
153	C₂²	C₂×#10587542	1	1	yes	characteristic
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	—	2	1	no	—
157	C₄	—	2	1	no	—
158	C₄	#11763651	1	2	yes	—
159	C₄	C₂×C₈×#595	1	1	yes	characteristic
160	C₄	C₂×#10587532	1	1	yes	characteristic
161	C₂	—	2	2	no	—
162	C₂	C₂×#11763651	1	1	yes	characteristic
163	C₂	#14545005	1	1	yes	characteristic
164	C₂	#14545007	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄×#595
commutator	C₄×C₁₇	C₂×C₄×C₈
Frattini	C₄²	C₂²×#595
Fitting	C₁₇×#431	C₄
socle	C₂²×C₁₇	C₂×#97
prime radical	C₂³×C₁₇	#97
cosocle	C₄×#3622	C₂³
hypercenter	#431	#595
hypocenter	C₁₇	#11406
LCS	G ▷ C₄×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ #431

Complements

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

N	H ≅ G/N	type
3. #14543702	C₂	semidirect
19. #10587538	C₈	semidirect
115. C₁₇	#11406	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
C₂×#11763697	2
#14543702	2
#14527068	2
#11406	17