Finite Group Databasebeta

Subgroups — #17960126

Order 4352.

232 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 98 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	#14545116	C₂	1	1	yes	maximal, characteristic
3	#14543814	C₂	1	2	yes	maximal, complemented
4	#14543807	C₂	1	2	yes	maximal, complemented
5	#14527065	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587538	—	2	1	no	—
7	C₄⋊#63216	—	2	1	no	—
8	C₄⋊#63216	C₂²	1	1	yes	characteristic
9	C₄⋊#63216	C₂²	1	2	yes	—
10	C₄⋊#63216	C₂²	1	1	yes	characteristic
11	C₂²⋊#63216	—	2	1	no	—
12	#11763697	—	2	1	no	—
13	C₄×#63223	C₄	1	1	yes	characteristic
14	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
15	#11762531	C₄	1	2	yes	—
16	C₈×#3622	C₂²	1	2	yes	—
17	C₁₇×#431	C₄	1	1	yes	characteristic
18	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
19	C₂×#63223	C₈	1	2	yes	—
20	C₂²×#3622	D₄	1	1	yes	characteristic
21	C₂²×#3622	Q₈	1	1	yes	characteristic
22	#10587538	—	4	2	no	—
23	C₂×#63216	—	2	2	no	—
24	C₂×#63216	—	2	1	no	—
25	C₂×#63216	—	2	1	no	—
26	C₁₇×C₄⋊C₈	—	2	1	no	—
27	C₁₇×C₂²⋊C₈	—	2	1	no	—
28	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
29	#10587474	C₈	1	1	yes	characteristic
30	#10587474	C₈	1	1	yes	characteristic
31	C₄×#3622	C₂×C₄	1	1	yes	characteristic
32	C₄×#3622	C₂×C₄	1	2	yes	—
33	C₄×#3622	D₄	1	1	yes	characteristic
34	C₄×#3622	Q₈	1	1	yes	characteristic
35	C₄×#3622	C₂³	1	1	yes	characteristic
36	C₂²×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
38	C₂²×C₄×C₁₇	M₁₆	1	1	yes	characteristic
39	C₂×C₈×C₁₇	—	2	2	no	—
40	C₄²×C₁₇	C₄⋊C₄	1	1	yes	characteristic
41	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
42	C₄²×C₁₇	M₁₆	1	1	yes	characteristic
43	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
44	#63223	—	2	4	no	—
45	C₂×#3622	—	2	1	no	—
46	C₂×#3622	—	2	1	no	—
47	C₂×#3622	—	4	1	no	—
48	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
49	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
50	C₂×#3622	C₂×C₈	1	2	yes	—
51	C₂×#3622	C₂×C₈	1	2	yes	—
52	C₂×#3622	C₂×D₄	1	1	yes	characteristic
53	C₂×#3622	C₂×Q₈	1	1	yes	characteristic
54	#63216	—	4	1	no	—
55	#63216	—	4	2	no	—
56	#63216	—	4	1	no	—
57	#12017	—	17	1	no	maximal, Sylow
58	C₂³×C₁₇	C₄⋊C₈	1	1	yes	characteristic
59	C₂×C₄×C₁₇	—	2	1	no	—
60	C₂×C₄×C₁₇	—	2	1	no	—
61	C₂×C₄×C₁₇	—	2	2	no	—
62	C₂×C₄×C₁₇	—	4	1	no	—
63	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
64	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
65	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
66	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
68	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
69	C₈×C₁₇	—	4	2	no	—
70	#3622	—	2	1	no	—
71	#3622	—	2	1	no	—
72	#3622	—	2	1	no	—
73	#3622	—	2	2	no	—
74	#3622	M₃₂	1	4	yes	complemented
75	#3622	AGL(1,ℤ/8)	1	1	yes	characteristic
76	#3622	#137	1	1	yes	characteristic
77	#2163	—	17	1	no	—
78	#2150	—	17	1	no	—
79	C₈×M₁₆	—	17	1	no	—
80	#1370	—	17	2	no	—
81	#1367	—	17	2	no	—
82	C₂²×C₁₇	—	2	1	no	—
83	C₂²×C₁₇	—	2	1	no	—
84	C₂²×C₁₇	—	2	1	no	—
85	C₂²×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
86	C₄×C₁₇	—	2	1	no	—
87	C₄×C₁₇	—	2	1	no	—
88	C₄×C₁₇	—	2	1	no	—
89	C₄×C₁₇	—	2	1	no	—
90	C₄×C₁₇	#428	1	1	yes	characteristic
91	C₄×C₁₇	#430	1	1	yes	characteristic
92	C₄×C₁₇	C₂×M₃₂	1	2	yes	—
93	C₂×M₃₂	—	34	1	no	—
94	#431	#595	1	1	yes	characteristic
95	C₄×M₁₆	—	17	1	no	—
96	C₂×C₄×C₈	—	17	1	no	—
97	C₄⋊C₁₆	—	17	1	no	—
98	C₄⋊C₁₆	—	17	2	no	—
99	C₄⋊C₁₆	—	17	1	no	—
100	C₄⋊C₁₆	—	34	1	no	—
101	C₂²⋊C₁₆	—	34	1	no	—
102	#346	—	34	1	no	—
103	#322	—	17	2	no	—
104	C₈²	—	17	2	no	—
105	C₂×C₁₇	—	4	1	no	—
106	C₂×C₁₇	#1565	1	1	yes	characteristic
107	C₂×C₁₇	#2150	1	1	yes	characteristic
108	C₂×C₁₇	#2153	1	1	yes	characteristic
109	C₂×M₁₆	—	17	2	no	—
110	C₂²×C₈	—	17	1	no	—
111	C₂²×C₈	—	17	1	no	—
112	C₂×C₄²	C₂×#595	1	1	yes	characteristic
113	M₃₂	—	68	2	no	—
114	C₂×C₁₆	—	34	1	no	—
115	C₂×C₁₆	—	34	1	no	—
116	C₂×C₁₆	—	34	2	no	—
117	C₄⋊C₈	—	2	1	no	—
118	C₂²⋊C₈	—	2	1	no	—
119	#97	—	17	1	no	—
120	#97	—	17	1	no	—
121	C₄×C₈	—	17	1	no	—
122	C₄×C₈	—	17	1	no	—
123	C₄×C₈	—	17	1	no	—
124	C₄×C₈	—	17	2	no	—
125	C₄×C₈	—	17	1	no	—
126	C₄×C₈	C₂×#595	1	2	yes	—
127	C₁₇	#12017	1	1	yes	characteristic, complemented, Sylow
128	C₂²×C₄	#63242	1	1	yes	characteristic
129	C₂²×C₄	C₄⋊#595	1	1	yes	characteristic
130	C₂²×C₄	#63247	1	1	yes	characteristic
131	M₁₆	—	34	4	no	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	17	2	no	—
134	C₂×C₈	—	17	1	no	—
135	C₂×C₈	—	17	2	no	—
136	C₂×C₈	—	17	1	no	—
137	C₂×C₈	—	17	1	no	—
138	C₂×C₈	—	17	1	no	—
139	C₂×C₈	—	34	1	no	—
140	C₂×C₈	—	34	1	no	—
141	C₂×C₈	—	68	1	no	—
142	C₄²	#63242	1	1	yes	characteristic
143	C₄²	C₄⋊#595	1	1	yes	characteristic
144	C₄²	#63247	1	1	yes	characteristic
145	C₄²	C₂²×#595	1	1	yes	characteristic
146	C₁₆	—	68	2	no	—
147	C₁₆	—	68	1	no	—
148	C₁₆	—	68	1	no	—
149	C₂³	#10587544	1	1	yes	characteristic
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	4	1	no	—
154	C₂×C₄	#10587544	1	1	yes	characteristic
155	C₂×C₄	#10587544	1	1	yes	characteristic
156	C₂×C₄	#10587544	1	1	yes	characteristic
157	C₂×C₄	C₂×#63242	1	1	yes	characteristic
158	C₂×C₄	C₂×C₄⋊#595	1	1	yes	characteristic
159	C₂×C₄	C₂×#63247	1	1	yes	characteristic
160	C₈	—	4	2	no	—
161	C₈	—	17	1	no	—
162	C₈	—	17	1	no	—
163	C₈	—	17	4	no	—
164	C₈	—	34	1	no	—
165	C₈	—	34	1	no	—
166	C₈	—	34	1	no	—
167	C₈	—	34	2	no	—
168	C₂²	—	2	1	no	—
169	C₂²	—	2	1	no	—
170	C₂²	—	2	1	no	—
171	C₂²	C₂×#10587544	1	1	yes	characteristic
172	C₄	—	2	1	no	—
173	C₄	—	2	1	no	—
174	C₄	—	2	1	no	—
175	C₄	—	2	1	no	—
176	C₄	#11763820	1	1	yes	characteristic
177	C₄	#11763824	1	1	yes	characteristic
178	C₄	#11763865	1	2	yes	—
179	C₂	—	4	1	no	—
180	C₂	#14544337	1	1	yes	characteristic
181	C₂	#14545118	1	1	yes	characteristic
182	C₂	#14545127	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄⋊#595
commutator	C₂×C₄×C₁₇	C₂²×C₈
Frattini	C₄²	C₂²×#595
Fitting	C₁₇×#431	C₄
socle	C₂²×C₁₇	C₂×C₄⋊C₈
prime radical	C₂³×C₁₇	C₄⋊C₈
cosocle	C₄×#3622	C₂³
hypercenter	#431	#595
hypocenter	C₁₇	#12017
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

Complements

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

N	H ≅ G/N	type
3. #14543814	C₂	semidirect
4. #14543807	C₂	semidirect
74. #3622	M₃₂	semidirect
127. C₁₇	#12017	semidirect

Maximal subgroups

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

subgroup	index
#14545145	2
#14545116	2
#14543814	2
#14543807	2
#14527065	2
#12017	17