Finite Group Databasebeta

Subgroups — #17948535

Order 4352.

230 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 104 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	#14545116	C₂	1	2	yes	maximal
2	C₂×#11762531	C₂	1	1	yes	maximal, characteristic
3	C₂×#10587538	—	2	2	no	—
4	C₄⋊#63216	—	2	2	no	—
5	C₄⋊#63216	C₄	1	4	yes	—
6	C₂×C₄×#3622	C₄	1	1	yes	characteristic
7	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
8	#11762531	—	2	2	no	—
9	C₂×C₄×C₈×C₁₇	C₄	1	1	yes	characteristic
10	C₂²×C₈×C₁₇	—	2	1	no	—
11	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
12	C₂²×#3622	C₈	1	2	yes	—
13	C₂²×#3622	D₄	1	1	yes	characteristic
14	C₂²×#3622	Q₈	1	1	yes	characteristic
15	#10587538	—	4	2	no	—
16	#10587538	—	4	2	no	—
17	C₂×#63216	—	2	4	no	—
18	C₄×C₈×C₁₇	—	2	1	no	—
19	C₄×C₈×C₁₇	C₈	1	2	yes	—
20	C₄×#3622	—	2	1	no	—
21	C₄×#3622	—	2	1	no	—
22	C₄×#3622	C₂×C₄	1	2	yes	—
23	C₄×#3622	D₄	1	2	yes	—
24	C₂²×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
25	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
26	C₂²×C₄×C₁₇	M₁₆	1	1	yes	characteristic
27	C₂×C₈×C₁₇	—	2	2	no	—
28	C₂×C₈×C₁₇	—	4	2	no	—
29	C₄²×C₁₇	C₄²	1	1	yes	characteristic
30	C₄²×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
31	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
32	C₄²×C₁₇	M₁₆	1	1	yes	characteristic
33	C₂×#3622	—	2	1	no	—
34	C₂×#3622	—	2	1	no	—
35	C₂×#3622	—	2	2	no	—
36	C₂×#3622	—	4	2	no	—
37	C₂×#3622	C₂²⋊C₄	1	2	yes	—
38	C₂×#3622	C₄⋊C₄	1	2	yes	—
39	C₂×#3622	D₈	1	1	yes	characteristic
40	C₂×#3622	SD₁₆	1	2	yes	—
41	C₂×#3622	Q₁₆	1	1	yes	characteristic
42	#63216	—	4	4	no	—
43	#7152	—	17	1	no	maximal, Sylow
44	C₂³×C₁₇	C₄⋊C₈	1	1	yes	characteristic
45	C₂×C₄×C₁₇	—	2	2	no	—
46	C₂×C₄×C₁₇	—	2	2	no	—
47	C₂×C₄×C₁₇	#95	1	1	yes	characteristic
48	C₂×C₄×C₁₇	C₄×C₈	1	1	yes	characteristic
49	C₂×C₄×C₁₇	#97	1	1	yes	characteristic
50	C₂×C₄×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
51	C₂×C₄×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
52	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
53	C₂×C₄×C₁₇	#106	1	1	yes	characteristic
54	C₂×C₄×C₁₇	#107	1	1	yes	characteristic
55	C₂×C₄×C₁₇	#108	1	1	yes	characteristic
56	C₂×C₄×C₁₇	#108	1	1	yes	characteristic
57	C₈×C₁₇	—	4	2	no	—
58	#3622	—	2	2	no	—
59	#3622	—	4	2	no	—
60	#3622	#102	1	2	yes	—
61	#3622	#103	1	2	yes	—
62	#2150	—	17	2	no	—
63	C₂×#322	—	17	1	no	—
64	C₂²×C₁₇	—	2	1	no	—
65	C₂²×C₁₇	—	2	1	no	—
66	C₂²×C₁₇	#334	1	1	yes	characteristic
67	C₂²×C₁₇	#335	1	1	yes	characteristic
68	C₂²×C₁₇	#336	1	1	yes	characteristic
69	C₄×C₁₇	—	2	1	no	—
70	C₄×C₁₇	—	2	1	no	—
71	C₄×C₁₇	#334	1	1	yes	characteristic
72	C₄×C₁₇	#335	1	1	yes	characteristic
73	C₄×C₁₇	#340	1	1	yes	characteristic
74	C₄×C₁₇	#341	1	1	yes	characteristic
75	C₄×C₁₇	#347	1	2	yes	—
76	C₄×C₁₇	#349	1	2	yes	—
77	C₂×M₃₂	—	34	2	no	—
78	C₂×C₄×C₈	—	17	1	no	—
79	C₂×C₄×C₈	—	17	1	no	—
80	C₂×C₄×C₈	#595	1	1	yes	characteristic
81	C₄⋊C₁₆	—	17	4	no	—
82	C₄⋊C₁₆	—	34	2	no	—
83	#322	—	34	2	no	—
84	C₂×C₁₇	—	2	1	no	—
85	C₂×C₁₇	—	2	1	no	—
86	C₂×C₁₇	#1277	1	1	yes	characteristic
87	C₂×C₁₇	#1375	1	1	yes	characteristic
88	C₂×C₁₇	#1377	1	1	yes	characteristic
89	C₂²×C₈	—	2	1	no	—
90	C₂²×C₈	—	17	1	no	—
91	C₂²×C₈	—	17	1	no	—
92	C₂²×C₈	—	17	2	no	—
93	C₂×C₄²	C₂×#595	1	1	yes	characteristic
94	M₃₂	—	68	2	no	—
95	M₃₂	—	68	2	no	—
96	C₂×C₁₆	—	34	4	no	—
97	C₄×C₈	—	2	1	no	—
98	C₄×C₈	—	17	2	no	—
99	C₄×C₈	—	17	2	no	—
100	C₄×C₈	—	34	1	no	—
101	C₄×C₈	—	34	1	no	—
102	C₄×C₈	#3624	1	2	yes	—
103	C₁₇	#7152	1	1	yes	characteristic, complemented, Sylow
104	C₂²×C₄	#63242	1	1	yes	characteristic
105	C₂²×C₄	C₄⋊#595	1	1	yes	characteristic
106	C₂²×C₄	#63247	1	1	yes	characteristic
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	4	2	no	—
109	C₂×C₈	—	17	2	no	—
110	C₂×C₈	—	17	1	no	—
111	C₂×C₈	—	17	1	no	—
112	C₂×C₈	—	17	2	no	—
113	C₂×C₈	—	17	2	no	—
114	C₂×C₈	—	34	1	no	—
115	C₂×C₈	—	34	1	no	—
116	C₂×C₈	—	34	2	no	—
117	C₂×C₈	—	68	2	no	—
118	C₄²	#63243	1	1	yes	characteristic
119	C₄²	C₄×#595	1	1	yes	characteristic
120	C₄²	C₂×#3624	1	1	yes	characteristic
121	C₄²	C₂²⋊#595	1	1	yes	characteristic
122	C₁₆	—	68	4	no	—
123	C₂³	#10587544	1	1	yes	characteristic
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	#10587533	1	1	yes	characteristic
127	C₂×C₄	#10587534	1	1	yes	characteristic
128	C₂×C₄	#10587535	1	1	yes	characteristic
129	C₂×C₄	#10587536	1	1	yes	characteristic
130	C₂×C₄	C₄×#3624	1	1	yes	characteristic
131	C₂×C₄	C₄⋊#3624	1	1	yes	characteristic
132	C₂×C₄	#10587542	1	1	yes	characteristic
133	C₂×C₄	#10587543	1	1	yes	characteristic
134	C₂×C₄	(C₂×#595)⋊C₄	1	1	yes	characteristic
135	C₂×C₄	C₂²⋊#3624	1	1	yes	characteristic
136	C₈	—	4	2	no	—
137	C₈	—	17	2	no	—
138	C₈	—	17	2	no	—
139	C₈	—	34	2	no	—
140	C₈	—	68	2	no	—
141	C₂²	—	2	1	no	—
142	C₂²	—	2	1	no	—
143	C₂²	#11763671	1	1	yes	characteristic
144	C₂²	#11763673	1	1	yes	characteristic
145	C₂²	#11763675	1	1	yes	characteristic
146	C₄	—	2	1	no	—
147	C₄	—	2	1	no	—
148	C₄	#11763670	1	1	yes	characteristic
149	C₄	#11763672	1	1	yes	characteristic
150	C₄	#11763683	1	1	yes	characteristic
151	C₄	#11763685	1	1	yes	characteristic
152	C₄	#11763701	1	2	yes	—
153	C₄	#11763706	1	2	yes	—
154	C₂	—	2	1	no	—
155	C₂	—	2	1	no	—
156	C₂	#14543614	1	1	yes	characteristic
157	C₂	#14543825	1	1	yes	characteristic
158	C₂	#14543832	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	C₂×C₄×C₈×C₁₇	C₄
socle	C₂²×C₁₇	#336
prime radical	C₂³×C₁₇	C₄⋊C₈
cosocle	C₂×C₄×#3622	C₂²
hypercenter	C₂×C₄×C₈	#595
hypocenter	C₁₇	#7152
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
103. C₁₇	#7152	semidirect

Maximal subgroups

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

subgroup	index
#14545116	2
C₂×#11762531	2
#7152	17