Finite Group Databasebeta

Subgroups — #17960086

Order 4352.

242 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 116 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	1	yes	maximal, characteristic
2	C₂×C₄⋊#63216	C₂	1	1	yes	maximal, characteristic
3	#14543814	C₂	1	4	yes	maximal, complemented
4	#14527068	C₂	1	1	yes	maximal, characteristic
5	C₂×#10587538	—	2	1	no	—
6	C₂²×#63216	—	2	1	no	—
7	C₄⋊#63216	—	2	1	no	—
8	C₄⋊#63216	—	2	1	no	—
9	C₄⋊#63216	C₂²	1	2	yes	—
10	C₄⋊#63216	C₂²	1	2	yes	—
11	#11762862	C₄	1	1	yes	characteristic
12	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
13	#11762531	C₄	1	2	yes	—
14	C₈×#3622	C₂²	1	2	yes	—
15	C₁₇×#431	C₄	1	1	yes	characteristic
16	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
17	C₂²×#3622	D₄	1	1	yes	characteristic
18	C₂²×#3622	Q₈	1	1	yes	characteristic
19	#10587538	—	4	2	no	—
20	C₂×#63216	—	2	2	no	—
21	C₂×#63216	—	2	2	no	—
22	C₂×#63216	—	4	2	no	—
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	C₈	1	2	yes	—
27	C₄⋊#3622	C₈	1	2	yes	—
28	C₄×#3622	C₂×C₄	1	1	yes	characteristic
29	C₄×#3622	C₂×C₄	1	2	yes	—
30	C₄×#3622	D₄	1	1	yes	characteristic
31	C₄×#3622	Q₈	1	1	yes	characteristic
32	C₄×#3622	C₂³	1	1	yes	characteristic
33	C₂²×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
35	C₂²×C₄×C₁₇	M₁₆	1	1	yes	characteristic
36	C₂×C₈×C₁₇	—	2	2	no	—
37	C₄²×C₁₇	C₄⋊C₄	1	1	yes	characteristic
38	C₄²×C₁₇	C₂×C₈	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	C₄⋊C₄	1	1	yes	characteristic
43	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
44	C₂×#3622	C₂×C₈	1	4	yes	—
45	C₂×#3622	D₈	1	2	yes	—
46	C₂×#3622	Q₁₆	1	2	yes	—
47	C₂×#3622	C₂×D₄	1	1	yes	characteristic
48	C₂×#3622	C₂×Q₈	1	1	yes	characteristic
49	#63216	—	4	2	no	—
50	#63216	—	4	2	no	—
51	#12005	—	17	1	no	maximal, Sylow
52	C₂³×C₁₇	C₄⋊C₈	1	1	yes	characteristic
53	C₂×C₄×C₁₇	—	2	2	no	—
54	C₂×C₄×C₁₇	—	4	1	no	—
55	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
56	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
57	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
58	C₂×C₄×C₁₇	#107	1	2	yes	—
59	C₂×C₄×C₁₇	#108	1	2	yes	—
60	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
62	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
63	C₈×C₁₇	—	4	2	no	—
64	#3622	—	2	1	no	—
65	#3622	—	2	1	no	—
66	#3622	—	2	4	no	—
67	#3622	#107	1	2	yes	—
68	#3622	C₂×D₈	1	1	yes	characteristic
69	#3622	C₂×Q₁₆	1	1	yes	characteristic
70	#2150	—	17	1	no	—
71	C₂×C₄⋊C₁₆	—	17	1	no	—
72	#1450	—	17	1	no	—
73	#1370	—	17	4	no	—
74	C₂²×C₁₇	—	2	1	no	—
75	C₂²×C₁₇	—	2	1	no	—
76	C₂²×C₁₇	#335	1	2	yes	—
77	C₂²×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
78	C₄×C₁₇	—	2	1	no	—
79	C₄×C₁₇	—	2	1	no	—
80	C₄×C₁₇	—	2	1	no	—
81	C₄×C₁₇	#335	1	2	yes	—
82	C₄×C₁₇	C₂×#107	1	1	yes	characteristic
83	C₄×C₁₇	C₂×#108	1	1	yes	characteristic
84	C₄×C₁₇	#504	1	2	yes	—
85	C₂×M₃₂	—	34	1	no	—
86	C₂²×C₁₆	—	34	1	no	—
87	#431	—	17	1	no	—
88	#431	#595	1	1	yes	characteristic
89	C₂×C₄×C₈	—	17	1	no	—
90	C₄⋊C₁₆	—	17	2	no	—
91	C₄⋊C₁₆	—	17	2	no	—
92	C₄⋊C₁₆	—	34	1	no	—
93	C₄⋊C₁₆	—	34	1	no	—
94	#322	—	17	2	no	—
95	C₈²	—	17	2	no	—
96	C₂×C₁₇	—	2	2	no	—
97	C₂×C₁₇	C₂×#335	1	1	yes	characteristic
98	C₂×C₁₇	#2151	1	1	yes	characteristic
99	C₂×C₁₇	#2153	1	1	yes	characteristic
100	C₂²×C₈	—	17	1	no	—
101	C₂²×C₈	—	17	1	no	—
102	C₂×C₄²	C₂×#595	1	1	yes	characteristic
103	M₃₂	—	68	2	no	—
104	C₂×C₁₆	—	34	2	no	—
105	C₂×C₁₆	—	34	2	no	—
106	C₂×C₁₆	—	68	2	no	—
107	C₄⋊C₈	—	2	1	no	—
108	C₄⋊C₈	—	17	2	no	—
109	C₂²⋊C₈	—	2	1	no	—
110	C₂²⋊C₈	—	17	2	no	—
111	C₄×C₈	—	17	1	no	—
112	C₄×C₈	—	17	1	no	—
113	C₄×C₈	—	17	1	no	—
114	C₄×C₈	—	17	2	no	—
115	C₄×C₈	—	17	1	no	—
116	C₄×C₈	C₂×#595	1	2	yes	—
117	C₁₇	#12005	1	1	yes	characteristic, complemented, Sylow
118	C₂²×C₄	#63242	1	1	yes	characteristic
119	C₂²×C₄	C₄⋊#595	1	1	yes	characteristic
120	C₂²×C₄	#63247	1	1	yes	characteristic
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	—	17	1	no	—
123	C₂×C₈	—	17	1	no	—
124	C₂×C₈	—	17	2	no	—
125	C₂×C₈	—	17	2	no	—
126	C₂×C₈	—	17	4	no	—
127	C₂×C₈	—	17	1	no	—
128	C₂×C₈	—	17	1	no	—
129	C₂×C₈	—	34	2	no	—
130	C₄²	#63242	1	1	yes	characteristic
131	C₄²	C₄⋊#595	1	1	yes	characteristic
132	C₄²	#63247	1	1	yes	characteristic
133	C₄²	C₂²×#595	1	1	yes	characteristic
134	C₁₆	—	68	2	no	—
135	C₁₆	—	68	2	no	—
136	C₂³	#10587544	1	1	yes	characteristic
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	4	1	no	—
139	C₂×C₄	#10587534	1	2	yes	—
140	C₂×C₄	#10587535	1	2	yes	—
141	C₂×C₄	#10587544	1	1	yes	characteristic
142	C₂×C₄	#10587544	1	1	yes	characteristic
143	C₂×C₄	#10587544	1	1	yes	characteristic
144	C₂×C₄	C₂×#63242	1	1	yes	characteristic
145	C₂×C₄	C₂×C₄⋊#595	1	1	yes	characteristic
146	C₂×C₄	C₂×#63247	1	1	yes	characteristic
147	C₈	—	4	2	no	—
148	C₈	—	17	1	no	—
149	C₈	—	17	2	no	—
150	C₈	—	17	1	no	—
151	C₈	—	34	1	no	—
152	C₈	—	34	1	no	—
153	C₈	—	34	4	no	—
154	C₂²	—	2	1	no	—
155	C₂²	—	2	1	no	—
156	C₂²	#11763673	1	2	yes	—
157	C₂²	C₂×#10587544	1	1	yes	characteristic
158	C₄	—	2	1	no	—
159	C₄	—	2	1	no	—
160	C₄	—	2	1	no	—
161	C₄	#11763673	1	2	yes	—
162	C₄	C₂×#10587534	1	1	yes	characteristic
163	C₄	C₂×#10587535	1	1	yes	characteristic
164	C₄	#11763868	1	2	yes	—
165	C₂	—	2	2	no	—
166	C₂	C₂×#11763673	1	1	yes	characteristic
167	C₂	#14545121	1	1	yes	characteristic
168	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	#12005
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
117. C₁₇	#12005	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×C₄⋊#63216	2
#14543814	2
#14527068	2
#12005	17