Finite Group Databasebeta

Subgroups — #17960123

Order 4352.

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

normal subgroups only

This group has 238 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	C₂×C₄⋊#63216	C₂	1	1	yes	maximal, characteristic
3	#14543814	C₂	1	2	yes	maximal, complemented
4	#14543807	C₂	1	2	yes	maximal, complemented
5	M₁₆×#3622	C₂	1	1	yes	maximal, characteristic
6	C₂²×#63216	—	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	#11762862	C₄	1	1	yes	characteristic
14	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
15	#11762530	C₄	1	2	yes	—
16	C₈×#3622	C₂²	1	2	yes	—
17	C₄×M₁₆×C₁₇	C₄	1	1	yes	characteristic
18	C₂×M₁₆×C₁₇	—	2	1	no	—
19	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×#3622	D₄	1	1	yes	characteristic
21	C₂²×#3622	Q₈	1	1	yes	characteristic
22	C₂×#63216	—	2	2	no	—
23	C₂×#63216	—	2	1	no	—
24	C₂×#63216	—	2	1	no	—
25	C₂×#63216	—	4	2	no	—
26	C₁₇×#97	—	2	1	no	—
27	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
28	C₂²⋊#3622	C₈	1	2	yes	—
29	C₄⋊#3622	C₈	1	2	yes	—
30	C₄×#3622	C₂×C₄	1	1	yes	characteristic
31	C₄×#3622	C₂×C₄	1	2	yes	—
32	C₄×#3622	D₄	1	1	yes	characteristic
33	C₄×#3622	Q₈	1	1	yes	characteristic
34	C₄×#3622	C₂³	1	1	yes	characteristic
35	C₂²×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
37	C₂²×C₄×C₁₇	M₁₆	1	1	yes	characteristic
38	M₁₆×C₁₇	—	2	4	no	—
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	C₂×#3622	—	2	1	no	—
45	C₂×#3622	—	2	1	no	—
46	C₂×#3622	—	4	1	no	—
47	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
48	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
49	C₂×#3622	C₂×C₈	1	4	yes	—
50	C₂×#3622	C₂×D₄	1	1	yes	characteristic
51	C₂×#3622	C₂×Q₈	1	1	yes	characteristic
52	#63216	—	4	1	no	—
53	#63216	—	4	2	no	—
54	#63216	—	4	1	no	—
55	#12016	—	17	1	no	maximal, Sylow
56	C₂³×C₁₇	C₄⋊C₈	1	1	yes	characteristic
57	C₂×C₄×C₁₇	—	2	2	no	—
58	C₂×C₄×C₁₇	—	2	1	no	—
59	C₂×C₄×C₁₇	—	2	1	no	—
60	C₂×C₄×C₁₇	—	2	2	no	—
61	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
62	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
63	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
64	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
66	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
67	C₈×C₁₇	—	2	4	no	—
68	#3622	—	2	1	no	—
69	#3622	—	2	1	no	—
70	#3622	—	2	1	no	—
71	#3622	—	2	4	no	—
72	#3622	AGL(1,ℤ/8)	1	1	yes	characteristic
73	#3622	#137	1	1	yes	characteristic
74	#2163	—	17	1	no	—
75	C₂×C₄⋊C₁₆	—	17	1	no	—
76	C₈×M₁₆	—	17	1	no	—
77	#1370	—	17	2	no	—
78	#1367	—	17	2	no	—
79	C₂²×C₁₇	—	2	1	no	—
80	C₂²×C₁₇	—	2	1	no	—
81	C₂²×C₁₇	#364	1	2	yes	—
82	C₂²×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
83	C₄×C₁₇	—	2	1	no	—
84	C₄×C₁₇	—	2	1	no	—
85	C₄×C₁₇	—	2	1	no	—
86	C₄×C₁₇	—	2	1	no	—
87	C₄×C₁₇	#428	1	1	yes	characteristic
88	C₄×C₁₇	#430	1	1	yes	characteristic
89	C₄×C₁₇	#504	1	2	yes	—
90	C₂²×C₁₆	—	34	1	no	—
91	#431	—	17	1	no	—
92	C₄×M₁₆	#595	1	1	yes	characteristic
93	C₂×C₄×C₈	—	17	1	no	—
94	C₄⋊C₁₆	—	17	1	no	—
95	C₄⋊C₁₆	—	17	2	no	—
96	C₄⋊C₁₆	—	17	1	no	—
97	C₄⋊C₁₆	—	34	1	no	—
98	C₂²⋊C₁₆	—	34	1	no	—
99	#346	—	34	1	no	—
100	#322	—	17	2	no	—
101	C₈²	—	17	2	no	—
102	C₂×C₁₇	—	2	2	no	—
103	C₂×C₁₇	#1565	1	1	yes	characteristic
104	C₂×C₁₇	#2151	1	1	yes	characteristic
105	C₂×C₁₇	C₂×#364	1	1	yes	characteristic
106	C₂×M₁₆	—	2	1	no	—
107	C₂²×C₈	—	17	1	no	—
108	C₂²×C₈	—	17	1	no	—
109	C₂×C₄²	C₂×#595	1	1	yes	characteristic
110	C₂×C₁₆	—	34	1	no	—
111	C₂×C₁₆	—	34	1	no	—
112	C₂×C₁₆	—	34	2	no	—
113	C₂×C₁₆	—	68	2	no	—
114	C₄⋊C₈	—	17	2	no	—
115	C₂²⋊C₈	—	17	2	no	—
116	#97	—	2	1	no	—
117	C₄×C₈	—	17	1	no	—
118	C₄×C₈	—	17	1	no	—
119	C₄×C₈	—	17	1	no	—
120	C₄×C₈	—	17	2	no	—
121	C₄×C₈	—	17	1	no	—
122	C₄×C₈	C₂×#595	1	2	yes	—
123	C₁₇	#12016	1	1	yes	characteristic, complemented, Sylow
124	C₂²×C₄	#63242	1	1	yes	characteristic
125	C₂²×C₄	C₄⋊#595	1	1	yes	characteristic
126	C₂²×C₄	#63247	1	1	yes	characteristic
127	M₁₆	—	2	4	no	—
128	C₂×C₈	—	2	2	no	—
129	C₂×C₈	—	17	1	no	—
130	C₂×C₈	—	17	4	no	—
131	C₂×C₈	—	17	1	no	—
132	C₂×C₈	—	17	1	no	—
133	C₂×C₈	—	17	1	no	—
134	C₂×C₈	—	34	1	no	—
135	C₂×C₈	—	34	1	no	—
136	C₂×C₈	—	68	1	no	—
137	C₄²	#63242	1	1	yes	characteristic
138	C₄²	C₄⋊#595	1	1	yes	characteristic
139	C₄²	#63247	1	1	yes	characteristic
140	C₄²	C₂²×#595	1	1	yes	characteristic
141	C₁₆	—	68	2	no	—
142	C₁₆	—	68	1	no	—
143	C₁₆	—	68	1	no	—
144	C₂³	#10587544	1	1	yes	characteristic
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	#10587544	1	1	yes	characteristic
150	C₂×C₄	#10587544	1	1	yes	characteristic
151	C₂×C₄	#10587544	1	1	yes	characteristic
152	C₂×C₄	C₂×#63242	1	1	yes	characteristic
153	C₂×C₄	C₂×C₄⋊#595	1	1	yes	characteristic
154	C₂×C₄	C₂×#63247	1	1	yes	characteristic
155	C₈	—	2	4	no	—
156	C₈	—	17	1	no	—
157	C₈	—	17	1	no	—
158	C₈	—	34	1	no	—
159	C₈	—	34	1	no	—
160	C₈	—	34	1	no	—
161	C₈	—	34	4	no	—
162	C₂²	—	2	1	no	—
163	C₂²	—	2	1	no	—
164	C₂²	#11763732	1	2	yes	—
165	C₂²	C₂×#10587544	1	1	yes	characteristic
166	C₄	—	2	1	no	—
167	C₄	—	2	1	no	—
168	C₄	—	2	1	no	—
169	C₄	—	2	1	no	—
170	C₄	#11763820	1	1	yes	characteristic
171	C₄	#11763824	1	1	yes	characteristic
172	C₄	#11763868	1	2	yes	—
173	C₂	—	2	2	no	—
174	C₂	#14544337	1	1	yes	characteristic
175	C₂	#14545121	1	1	yes	characteristic
176	C₂	C₂×#11763732	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄⋊#595
commutator	C₂×C₄×C₁₇	C₂²×C₈
Frattini	C₄²	C₂²×#595
Fitting	C₄×M₁₆×C₁₇	C₄
socle	C₂²×C₁₇	C₂×C₄⋊C₈
prime radical	C₂³×C₁₇	C₄⋊C₈
cosocle	C₄×#3622	C₂³
hypercenter	C₄×M₁₆	#595
hypocenter	C₁₇	#12016
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
3. #14543814	C₂	semidirect
4. #14543807	C₂	semidirect
123. C₁₇	#12016	semidirect

Maximal subgroups

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

subgroup	index
#14545145	2
C₂×C₄⋊#63216	2
#14543814	2
#14543807	2
M₁₆×#3622	2
#12016	17