Finite Group Databasebeta

Subgroups — #17960174

Order 4352.

222 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 88 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	#14544341	C₂	1	1	yes	maximal, characteristic
2	C₂×#11763672	C₂	1	1	yes	maximal, characteristic
3	#14543816	C₂	1	2	yes	maximal, complemented
4	#14543815	C₂	1	2	yes	maximal, complemented
5	#14530580	C₂	1	1	yes	maximal, characteristic
6	#11763812	—	2	1	no	—
7	C₂×C₄⋊#3624	—	2	1	no	—
8	#11763673	—	2	1	no	—
9	#11763672	—	2	1	no	—
10	#11763672	C₂²	1	1	yes	characteristic
11	#11763672	C₂²	1	2	yes	—
12	#11763672	C₂²	1	1	yes	characteristic
13	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
14	#11762627	C₄	1	2	yes	—
15	C₂²⋊#63214	C₄	1	1	yes	characteristic
16	C₁₆×Dic₁₇	C₂²	1	2	yes	—
17	C₁₇×C₂²⋊C₁₆	C₄	1	1	yes	characteristic
18	C₂×#63247	—	4	1	no	—
19	C₂²×#3624	—	4	1	no	—
20	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₁₇	D₄	1	1	yes	characteristic
22	C₂²×#3622	Q₈	1	1	yes	characteristic
23	#10587544	—	4	1	no	—
24	C₄⋊#3624	—	2	2	no	—
25	C₄⋊#3624	—	2	1	no	—
26	C₄⋊#3624	—	2	1	no	—
27	C₄⋊#3624	—	4	1	no	—
28	C₂×C₁₆×C₁₇	C₂×C₄	1	2	yes	—
29	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
30	C₈×Dic₁₇	D₄	1	1	yes	characteristic
31	C₈×Dic₁₇	Q₈	1	1	yes	characteristic
32	C₈×Dic₁₇	C₂³	1	1	yes	characteristic
33	C₂×#63214	C₂×C₄	1	2	yes	—
34	C₂²×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
35	C₂²×Dic₁₇	D₈	1	1	yes	characteristic
36	C₂²×Dic₁₇	Q₁₆	1	1	yes	characteristic
37	#63247	—	8	2	no	—
38	C₂×#3624	—	4	2	no	—
39	C₂×#3624	—	4	1	no	—
40	C₂×#3624	—	4	1	no	—
41	C₂×#3624	—	8	2	no	—
42	C₂×C₈×C₁₇	—	2	2	no	—
43	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
45	C₄×Dic₁₇	C₄⋊C₄	1	1	yes	characteristic
46	C₄×Dic₁₇	D₈	1	1	yes	characteristic
47	C₄×Dic₁₇	Q₁₆	1	1	yes	characteristic
48	C₄×Dic₁₇	C₂×D₄	1	1	yes	characteristic
49	C₂×#3622	—	4	1	no	—
50	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
51	C₂×#3622	C₂×Q₈	1	1	yes	characteristic
52	C₁₆×C₁₇	—	2	2	no	—
53	#63214	—	2	2	no	—
54	#12032	—	17	1	no	maximal, Sylow
55	C₂³×C₁₇	#107	1	1	yes	characteristic
56	C₂×C₄×C₁₇	—	2	1	no	—
57	C₂×C₄×C₁₇	—	2	1	no	—
58	C₂×C₄×C₁₇	#107	1	1	yes	characteristic
59	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂×Dic₁₇	—	4	1	no	—
61	C₂×Dic₁₇	#107	1	1	yes	characteristic
62	C₂×Dic₁₇	#107	1	1	yes	characteristic
63	C₂×Dic₁₇	D₁₆	1	2	yes	—
64	C₂×Dic₁₇	Q₃₂	1	2	yes	—
65	C₂×Dic₁₇	C₂×D₈	1	1	yes	characteristic
66	C₂×Dic₁₇	C₂×Q₁₆	1	1	yes	characteristic
67	#3624	—	8	2	no	—
68	#3624	—	8	1	no	—
69	#3624	—	8	1	no	—
70	C₈×C₁₇	—	2	1	no	—
71	C₈×C₁₇	#131	1	2	yes	—
72	#3622	—	2	1	no	—
73	#3622	—	2	1	no	—
74	#2162	—	17	1	no	—
75	#1567	—	17	1	no	—
76	C₂×#335	—	17	1	no	—
77	#1371	—	17	2	no	—
78	#1371	—	17	2	no	—
79	C₂²×C₁₇	—	2	1	no	—
80	C₂²×C₁₇	—	2	1	no	—
81	C₂²×C₁₇	#366	1	2	yes	—
82	C₂²×C₁₇	C₂×#107	1	1	yes	characteristic
83	C₄×C₁₇	—	2	1	no	—
84	C₄×C₁₇	#424	1	1	yes	characteristic
85	C₄×C₁₇	#430	1	1	yes	characteristic
86	Dic₁₇	—	2	1	no	—
87	Dic₁₇	—	2	1	no	—
88	Dic₁₇	#366	1	2	yes	—
89	Dic₁₇	C₂×D₁₆	1	1	yes	characteristic
90	Dic₁₇	C₂×Q₃₂	1	1	yes	characteristic
91	#423	—	34	1	no	—
92	C₂×C₄⋊C₈	—	34	1	no	—
93	C₂×C₄×C₈	—	17	1	no	—
94	C₄⋊C₁₆	—	17	2	no	—
95	C₂²⋊C₁₆	—	17	1	no	—
96	C₂²⋊C₁₆	#595	1	1	yes	characteristic
97	C₄×C₁₆	—	17	2	no	—
98	#335	—	17	2	no	—
99	#335	—	17	1	no	—
100	#335	—	17	1	no	—
101	#335	—	34	1	no	—
102	#335	—	34	1	no	—
103	C₂×C₁₇	—	2	2	no	—
104	C₂×C₁₇	#1571	1	1	yes	characteristic
105	C₂×C₁₇	C₂×#366	1	1	yes	characteristic
106	C₂×C₁₇	#2161	1	1	yes	characteristic
107	C₂×M₁₆	—	68	1	no	—
108	C₂²×C₈	—	17	1	no	—
109	C₂²×C₈	—	68	1	no	—
110	C₂²×C₈	C₂×#595	1	1	yes	characteristic
111	C₂×C₄²	—	17	1	no	—
112	C₂×C₁₆	—	17	2	no	—
113	C₂×C₁₆	C₂×#595	1	2	yes	—
114	C₄⋊C₈	—	34	2	no	—
115	C₄⋊C₈	—	34	1	no	—
116	C₄⋊C₈	—	34	1	no	—
117	C₄⋊C₈	—	68	1	no	—
118	C₄⋊C₈	—	68	1	no	—
119	C₄×C₈	—	17	1	no	—
120	C₄×C₈	—	17	1	no	—
121	C₄×C₈	—	17	1	no	—
122	C₄×C₈	—	17	1	no	—
123	C₁₇	#12032	1	1	yes	characteristic, complemented, Sylow
124	C₂²×C₄	—	17	1	no	—
125	C₂²×C₄	—	17	1	no	—
126	C₂²×C₄	C₄⋊#595	1	1	yes	characteristic
127	M₁₆	—	136	2	no	—
128	C₂×C₈	—	2	2	no	—
129	C₂×C₈	—	17	1	no	—
130	C₂×C₈	—	17	1	no	—
131	C₂×C₈	—	68	1	no	—
132	C₂×C₈	—	68	2	no	—
133	C₂×C₈	—	68	1	no	—
134	C₂×C₈	—	68	1	no	—
135	C₂×C₈	—	136	2	no	—
136	C₂×C₈	C₄⋊#595	1	1	yes	characteristic
137	C₂×C₈	C₂²×#595	1	1	yes	characteristic
138	C₄²	—	17	1	no	—
139	C₄²	—	17	1	no	—
140	C₄²	—	17	1	no	—
141	C₄²	—	17	1	no	—
142	C₁₆	—	2	2	no	—
143	C₁₆	—	34	2	no	—
144	C₂³	#10587534	1	1	yes	characteristic
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	17	2	no	—
148	C₂×C₄	—	17	2	no	—
149	C₂×C₄	—	17	1	no	—
150	C₂×C₄	—	17	1	no	—
151	C₂×C₄	—	17	1	no	—
152	C₂×C₄	—	17	1	no	—
153	C₂×C₄	—	68	1	no	—
154	C₂×C₄	#10587534	1	1	yes	characteristic
155	C₂×C₄	C₂×C₄⋊#595	1	1	yes	characteristic
156	C₈	—	2	1	no	—
157	C₈	—	34	1	no	—
158	C₈	—	34	1	no	—
159	C₈	—	136	2	no	—
160	C₈	—	136	1	no	—
161	C₈	—	136	1	no	—
162	C₈	#10587678	1	2	yes	—
163	C₂²	—	2	1	no	—
164	C₂²	—	2	1	no	—
165	C₂²	#11763735	1	2	yes	—
166	C₂²	C₂×#10587534	1	1	yes	characteristic
167	C₄	—	2	1	no	—
168	C₄	—	17	1	no	—
169	C₄	—	17	2	no	—
170	C₄	—	17	1	no	—
171	C₄	—	34	1	no	—
172	C₄	—	34	1	no	—
173	C₄	#11763815	1	1	yes	characteristic
174	C₄	#11763824	1	1	yes	characteristic
175	C₂	—	2	2	no	—
176	C₂	#14544353	1	1	yes	characteristic
177	C₂	C₂×#11763735	1	1	yes	characteristic
178	C₂	#14545140	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#10587534
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×#595
Fitting	C₁₇×C₂²⋊C₁₆	C₄
socle	C₂²×C₁₇	C₂×#107
prime radical	C₂³×C₁₇	#107
cosocle	C₈×Dic₁₇	C₂³
hypercenter	C₂²⋊C₁₆	#595
hypocenter	C₁₇	#12032
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²⋊C₁₆

Complements

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

N	H ≅ G/N	type
3. #14543816	C₂	semidirect
4. #14543815	C₂	semidirect
123. C₁₇	#12032	semidirect

Maximal subgroups

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

subgroup	index
#14544341	2
C₂×#11763672	2
#14543816	2
#14543815	2
#14530580	2
#12032	17