Finite Group Databasebeta

Subgroups — #17958073

Order 4352.

250 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 108 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×#11763697	C₂	1	1	yes	maximal, characteristic
3	#14543702	C₂	1	2	yes	maximal, complemented
4	#14543702	C₂	1	2	yes	maximal, complemented
5	#14527071	C₂	1	1	yes	maximal, characteristic
6	C₂²×#63216	C₄	1	2	yes	—
7	C₄⋊#63216	C₄	1	2	yes	—
8	C₂²⋊#63216	C₄	1	2	yes	—
9	#11763697	C₄	1	2	yes	—
10	#11763697	C₂²	1	1	yes	characteristic
11	#11763697	C₂²	1	1	yes	characteristic
12	#11763697	C₂²	1	2	yes	—
13	#11762862	C₄	1	1	yes	characteristic
14	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
15	#11762531	C₄	1	2	yes	—
16	#11762531	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	C₂×C₄	1	1	yes	characteristic
21	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
22	C₂×#63216	—	2	4	no	—
23	C₂×#63216	C₂×C₄	1	4	yes	—
24	C₂×#63216	C₂×C₄	1	2	yes	—
25	C₂×#63216	C₂×C₄	1	2	yes	—
26	C₁₇×#97	—	2	1	no	—
27	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
28	C₂²⋊#3622	—	2	1	no	—
29	C₄⋊#3622	—	2	1	no	—
30	C₄×#3622	C₂×C₄	1	1	yes	characteristic
31	C₄×#3622	C₂×C₄	1	1	yes	characteristic
32	C₄×#3622	C₂×C₄	1	2	yes	—
33	C₄×#3622	C₂×C₄	1	1	yes	characteristic
34	C₄×#3622	C₂³	1	1	yes	characteristic
35	C₂²×C₄×C₁₇	C₄²	1	1	yes	characteristic
36	C₂²×C₄×C₁₇	M₁₆	1	1	yes	characteristic
37	C₂²×C₄×C₁₇	M₁₆	1	1	yes	characteristic
38	M₁₆×C₁₇	—	4	2	no	—
39	C₂×C₈×C₁₇	—	2	2	no	—
40	C₄²×C₁₇	C₄²	1	1	yes	characteristic
41	C₄²×C₁₇	M₁₆	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	2	no	—
45	C₂×#3622	—	2	2	no	—
46	C₂×#3622	—	2	2	no	—
47	C₂×#3622	C₄²	1	1	yes	characteristic
48	C₂×#3622	C₄²	1	1	yes	characteristic
49	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
50	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
51	#63216	—	2	2	no	—
52	#63216	—	2	4	no	—
53	#63216	M₁₆	1	4	yes	complemented
54	#11413	—	17	1	no	maximal, Sylow
55	C₂³×C₁₇	#97	1	1	yes	characteristic
56	C₂×C₄×C₁₇	—	2	2	no	—
57	C₂×C₄×C₁₇	—	2	1	no	—
58	C₂×C₄×C₁₇	—	2	1	no	—
59	C₂×C₄×C₁₇	—	4	1	no	—
60	C₂×C₄×C₁₇	#97	1	1	yes	characteristic
61	C₂×C₄×C₁₇	#97	1	1	yes	characteristic
62	C₂×C₄×C₁₇	#97	1	1	yes	characteristic
63	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
64	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
65	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
66	C₈×C₁₇	—	4	2	no	—
67	#3622	—	2	1	no	—
68	#3622	—	2	1	no	—
69	#3622	—	4	2	no	—
70	#3622	C₂×M₁₆	1	2	yes	—
71	#3622	#131	1	2	yes	—
72	#2163	—	17	1	no	—
73	C₂×#346	—	17	1	no	—
74	#1451	—	17	1	no	—
75	#1313	—	17	2	no	—
76	#1313	—	17	2	no	—
77	C₂²×C₁₇	—	2	1	no	—
78	C₂²×C₁₇	—	2	1	no	—
79	C₂²×C₁₇	#347	1	2	yes	—
80	C₂²×C₁₇	C₂×#97	1	1	yes	characteristic
81	C₄×C₁₇	—	2	1	no	—
82	C₄×C₁₇	—	2	1	no	—
83	C₄×C₁₇	—	2	1	no	—
84	C₄×C₁₇	—	2	1	no	—
85	C₄×C₁₇	—	2	1	no	—
86	C₄×C₁₇	C₄×M₁₆	1	1	yes	characteristic
87	C₄×C₁₇	#405	1	1	yes	characteristic
88	C₂²×C₁₆	—	17	2	no	—
89	#431	—	17	1	no	—
90	C₄×M₁₆	#595	1	1	yes	characteristic
91	C₂×C₄×C₈	—	17	1	no	—
92	C₄⋊C₁₆	—	17	2	no	—
93	C₂²⋊C₁₆	—	17	2	no	—
94	#346	—	17	1	no	—
95	#346	—	17	1	no	—
96	#346	—	17	2	no	—
97	#346	—	17	2	no	—
98	#322	—	17	2	no	—
99	#322	—	17	2	no	—
100	C₂×C₁₇	—	2	2	no	—
101	C₂×C₁₇	#1451	1	1	yes	characteristic
102	C₂×C₁₇	C₂×#347	1	1	yes	characteristic
103	C₂×C₁₇	#2110	1	1	yes	characteristic
104	C₂×M₁₆	—	2	1	no	—
105	C₂²×C₈	—	17	1	no	—
106	C₂²×C₈	—	17	1	no	—
107	C₂×C₄²	C₂×#595	1	1	yes	characteristic
108	C₂×C₁₆	—	17	2	no	—
109	C₂×C₁₆	—	17	2	no	—
110	C₂×C₁₆	—	17	4	no	—
111	C₂×C₁₆	—	34	4	no	—
112	C₄⋊C₈	—	34	1	no	—
113	C₂²⋊C₈	—	34	1	no	—
114	#97	—	2	1	no	—
115	C₄×C₈	—	17	1	no	—
116	C₄×C₈	—	17	1	no	—
117	C₄×C₈	—	17	1	no	—
118	C₄×C₈	—	17	1	no	—
119	C₄×C₈	—	17	2	no	—
120	C₄×C₈	C₂×#595	1	2	yes	—
121	C₁₇	#11413	1	1	yes	characteristic, complemented, Sylow
122	C₂²×C₄	#63243	1	1	yes	characteristic
123	C₂²×C₄	C₄×#595	1	1	yes	characteristic
124	C₂²×C₄	#63247	1	1	yes	characteristic
125	M₁₆	—	4	2	no	—
126	C₂×C₈	—	2	2	no	—
127	C₂×C₈	—	17	1	no	—
128	C₂×C₈	—	17	1	no	—
129	C₂×C₈	—	17	1	no	—
130	C₂×C₈	—	17	1	no	—
131	C₂×C₈	—	34	2	no	—
132	C₂×C₈	—	34	2	no	—
133	C₂×C₈	—	34	2	no	—
134	C₄²	#63243	1	1	yes	characteristic
135	C₄²	C₄×#595	1	1	yes	characteristic
136	C₄²	#63247	1	1	yes	characteristic
137	C₄²	C₂²×#595	1	1	yes	characteristic
138	C₁₆	—	17	4	no	—
139	C₁₆	—	34	2	no	—
140	C₁₆	—	34	4	no	—
141	C₂³	#10587542	1	1	yes	characteristic
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	4	1	no	—
146	C₂×C₄	#10587542	1	1	yes	characteristic
147	C₂×C₄	#10587542	1	1	yes	characteristic
148	C₂×C₄	#10587542	1	1	yes	characteristic
149	C₂×C₄	C₂×#63243	1	1	yes	characteristic
150	C₂×C₄	C₂×C₄×#595	1	1	yes	characteristic
151	C₂×C₄	C₂×#63247	1	1	yes	characteristic
152	C₈	—	4	2	no	—
153	C₈	—	17	2	no	—
154	C₈	—	17	2	no	—
155	C₈	—	34	1	no	—
156	C₈	—	34	1	no	—
157	C₈	—	68	2	no	—
158	C₂²	—	2	1	no	—
159	C₂²	—	2	1	no	—
160	C₂²	#11763701	1	2	yes	—
161	C₂²	C₂×#10587542	1	1	yes	characteristic
162	C₄	—	2	1	no	—
163	C₄	—	2	1	no	—
164	C₄	—	2	1	no	—
165	C₄	—	2	1	no	—
166	C₄	—	2	1	no	—
167	C₄	M₁₆×#595	1	1	yes	characteristic
168	C₄	#11763775	1	1	yes	characteristic
169	C₂	—	2	2	no	—
170	C₂	#14543991	1	1	yes	characteristic
171	C₂	C₂×#11763701	1	1	yes	characteristic
172	C₂	#14545007	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄×#595
commutator	C₂×C₄×C₁₇	C₂×C₄²
Frattini	C₄²	C₂²×#595
Fitting	C₄×M₁₆×C₁₇	C₄
socle	C₂²×C₁₇	C₂×#97
prime radical	C₂³×C₁₇	#97
cosocle	C₄×#3622	C₂³
hypercenter	C₄×M₁₆	#595
hypocenter	C₁₇	#11413
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
3. #14543702	C₂	semidirect
4. #14543702	C₂	semidirect
53. #63216	M₁₆	semidirect
121. C₁₇	#11413	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×#11763697	2
#14543702	2
#14527071	2
#11413	17