Finite Group Databasebeta

Subgroups — #17959845

Order 4352.

274 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 60 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	#14545153	C₂	1	2	yes	maximal
2	#14543791	C₂	1	2	yes	maximal, complemented
3	#14543790	C₂	1	2	yes	maximal, complemented
4	#14539166	C₂	1	1	yes	maximal, characteristic
5	#11763732	—	2	2	no	—
6	#11763705	—	2	2	no	—
7	#11763701	C₂²	1	4	yes	—
8	C₂×#10587555	C₄	1	1	yes	characteristic
9	#11763179	C₄	1	1	yes	characteristic
10	#11763179	C₄	1	1	yes	characteristic
11	#11763179	C₂²	1	1	yes	characteristic
12	#11763179	C₂²	1	1	yes	characteristic
13	#11763161	—	2	2	no	—
14	#11762817	—	2	2	no	—
15	C₄×#63223	C₂²	1	1	yes	characteristic
16	C₁₇×#533	C₄	1	1	yes	characteristic
17	C₂²×Dic₃₄	—	2	1	no	—
18	C₁₇×C₄⋊Q₈	—	2	1	no	—
19	C₁₇×#125	C₂×C₄	1	1	yes	characteristic
20	C₁₇×#125	C₂×C₄	1	1	yes	characteristic
21	C₁₇×C₂²⋊Q₈	—	4	1	no	—
22	C₄×Q₈×C₁₇	—	4	1	no	—
23	C₁₇×#117	—	2	1	no	—
24	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄⋊Dic₁₇	—	4	1	no	—
26	C₂×#63223	D₄	1	2	yes	—
27	#10587555	—	2	1	no	—
28	#10587555	—	4	1	no	—
29	#10587555	C₂×C₄	1	1	yes	characteristic
30	#10587555	C₂×C₄	1	1	yes	characteristic
31	#10587538	—	2	4	no	—
32	#10587481	—	2	2	no	—
33	#10587481	—	2	2	no	—
34	#10587481	—	2	2	no	—
35	#10587481	—	2	2	no	—
36	#10587474	C₂×C₄	1	1	yes	characteristic
37	#10587474	C₂³	1	1	yes	characteristic
38	C₄×#3622	D₄	1	2	yes	—
39	C₂×Q₈×C₁₇	—	4	1	no	—
40	C₂²×C₄×C₁₇	—	2	1	no	—
41	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
42	C₂²×Dic₁₇	—	4	1	no	—
43	C₂×Dic₃₄	—	2	1	no	—
44	C₂×Dic₃₄	—	2	1	no	—
45	C₂×Dic₃₄	—	4	4	no	—
46	C₂×Dic₃₄	—	4	1	no	—
47	C₄⋊C₄×C₁₇	—	2	1	no	—
48	C₄⋊C₄×C₁₇	—	2	1	no	—
49	C₄⋊C₄×C₁₇	—	4	1	no	—
50	C₄⋊C₄×C₁₇	—	4	1	no	—
51	C₄⋊C₄×C₁₇	—	4	1	no	—
52	C₂²⋊C₄×C₁₇	—	4	1	no	—
53	C₄²×C₁₇	—	2	1	no	—
54	C₄²×C₁₇	—	4	1	no	—
55	C₄²×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
56	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
57	C₄⋊Dic₁₇	—	4	2	no	—
58	C₄⋊Dic₁₇	—	4	1	no	—
59	C₄⋊Dic₁₇	—	4	1	no	—
60	#63223	—	4	2	no	—
61	C₂×#3622	C₂²⋊C₄	1	2	yes	—
62	C₂×#3622	C₂×D₄	1	2	yes	—
63	#63216	—	4	4	no	—
64	#11916	—	17	1	no	maximal, Sylow
65	C₂³×C₁₇	#100	1	1	yes	characteristic
66	Q₈×C₁₇	—	8	2	no	—
67	C₂×C₄×C₁₇	—	2	1	no	—
68	C₂×C₄×C₁₇	—	2	2	no	—
69	C₂×C₄×C₁₇	—	2	1	no	—
70	C₂×C₄×C₁₇	—	4	2	no	—
71	C₂×C₄×C₁₇	—	4	1	no	—
72	C₂×C₄×C₁₇	—	4	1	no	—
73	C₂×C₄×C₁₇	#100	1	1	yes	characteristic
74	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×Dic₁₇	—	4	1	no	—
76	C₂×Dic₁₇	—	4	1	no	—
77	C₂×Dic₁₇	—	8	2	no	—
78	Dic₃₄	—	4	2	no	—
79	Dic₃₄	—	4	2	no	—
80	Dic₃₄	—	8	2	no	—
81	#3622	—	2	2	no	—
82	#3622	—	2	2	no	—
83	#3139	—	17	1	no	—
84	#2165	—	17	2	no	—
85	#1356	—	17	2	no	—
86	#1356	—	17	2	no	—
87	C₂²×C₁₇	—	2	1	no	—
88	C₂²×C₁₇	—	4	1	no	—
89	C₂²×C₁₇	C₂×#100	1	1	yes	characteristic
90	C₄×C₁₇	—	2	2	no	—
91	C₄×C₁₇	—	2	1	no	—
92	C₄×C₁₇	—	4	1	no	—
93	C₄×C₁₇	—	8	1	no	—
94	C₄×C₁₇	—	8	1	no	—
95	C₄×C₁₇	#421	1	2	yes	—
96	Dic₁₇	—	8	1	no	—
97	Dic₁₇	—	8	1	no	—
98	#533	#595	1	1	yes	characteristic
99	C₂×C₄⋊Q₈	—	17	1	no	—
100	#491	—	17	1	no	—
101	#491	—	17	1	no	—
102	#491	—	17	1	no	—
103	#491	—	17	1	no	—
104	#487	—	34	2	no	—
105	#419	—	34	2	no	—
106	C₄×M₁₆	—	17	1	no	—
107	#364	—	34	2	no	—
108	#349	—	34	2	no	—
109	#347	—	17	4	no	—
110	C₂×C₁₇	—	2	1	no	—
111	C₂×C₁₇	—	4	1	no	—
112	C₂×C₁₇	#1533	1	1	yes	characteristic
113	C₂²×Q₈	—	34	1	no	—
114	C₂×M₁₆	—	17	2	no	—
115	C₄⋊Q₈	—	2	1	no	—
116	C₄⋊Q₈	—	17	1	no	—
117	C₄⋊Q₈	—	17	1	no	—
118	C₄⋊Q₈	—	34	1	no	—
119	C₄⋊Q₈	—	68	1	no	—
120	#125	C₂×#595	1	1	yes	characteristic
121	#125	C₂×#595	1	1	yes	characteristic
122	C₂²⋊Q₈	—	4	1	no	—
123	C₄×Q₈	—	4	1	no	—
124	#117	—	2	1	no	—
125	C₂×C₄⋊C₄	—	68	1	no	—
126	C₂×C₄²	C₂×#595	1	1	yes	characteristic
127	M₃₂	—	34	4	no	—
128	#103	—	34	2	no	—
129	#103	—	34	2	no	—
130	#103	—	34	2	no	—
131	#103	—	34	2	no	—
132	#97	—	17	1	no	—
133	#97	—	17	1	no	—
134	C₄×C₈	—	17	2	no	—
135	C₁₇	#11916	1	1	yes	characteristic, complemented, Sylow
136	C₂×Q₈	—	4	1	no	—
137	C₂×Q₈	—	34	1	no	—
138	C₂×Q₈	—	34	1	no	—
139	C₂×Q₈	—	68	4	no	—
140	C₂×Q₈	—	68	1	no	—
141	C₂²×C₄	—	2	1	no	—
142	C₂²×C₄	—	68	1	no	—
143	C₂²×C₄	C₂²⋊#595	1	1	yes	characteristic
144	M₁₆	—	68	2	no	—
145	C₂×C₈	—	17	2	no	—
146	C₂×C₈	—	17	2	no	—
147	C₄⋊C₄	—	2	1	no	—
148	C₄⋊C₄	—	2	1	no	—
149	C₄⋊C₄	—	4	1	no	—
150	C₄⋊C₄	—	4	1	no	—
151	C₄⋊C₄	—	4	1	no	—
152	C₄⋊C₄	—	68	1	no	—
153	C₄⋊C₄	—	68	1	no	—
154	C₄⋊C₄	—	68	2	no	—
155	C₂²⋊C₄	—	4	1	no	—
156	C₄²	—	2	1	no	—
157	C₄²	—	4	1	no	—
158	C₄²	C₂²⋊#595	1	1	yes	characteristic
159	C₄²	C₂²×#595	1	1	yes	characteristic
160	C₁₆	—	68	4	no	—
161	C₂³	#10587553	1	1	yes	characteristic
162	Q₈	—	8	2	no	—
163	Q₈	—	68	2	no	—
164	Q₈	—	68	2	no	—
165	Q₈	—	136	2	no	—
166	C₂×C₄	—	2	1	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	—	4	2	no	—
170	C₂×C₄	—	4	1	no	—
171	C₂×C₄	—	4	1	no	—
172	C₂×C₄	—	68	1	no	—
173	C₂×C₄	—	68	1	no	—
174	C₂×C₄	—	136	2	no	—
175	C₂×C₄	#10587553	1	1	yes	characteristic
176	C₂×C₄	C₂×C₂²⋊#595	1	1	yes	characteristic
177	C₈	—	34	2	no	—
178	C₈	—	34	2	no	—
179	C₂²	—	2	1	no	—
180	C₂²	—	4	1	no	—
181	C₂²	C₂×#10587553	1	1	yes	characteristic
182	C₄	—	2	2	no	—
183	C₄	—	2	1	no	—
184	C₄	—	4	1	no	—
185	C₄	—	8	1	no	—
186	C₄	—	8	1	no	—
187	C₄	—	136	1	no	—
188	C₄	—	136	1	no	—
189	C₄	#11763808	1	2	yes	—
190	C₂	—	2	1	no	—
191	C₂	—	4	1	no	—
192	C₂	#14544265	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#14544265
commutator	C₄²×C₁₇	C₂²×C₄
Frattini	C₄²	C₂²×#595
Fitting	C₁₇×#533	C₄
socle	C₂×C₁₇	#1533
prime radical	C₂³×C₁₇	#100
cosocle	#10587474	C₂³
hypercenter	#533	#595
hypocenter	C₁₇	#11916
LCS	G ▷ C₄²×C₁₇ ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

Complements

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

N	H ≅ G/N	type
2. #14543791	C₂	semidirect
3. #14543790	C₂	semidirect
135. C₁₇	#11916	semidirect

Maximal subgroups

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

subgroup	index
#14545153	2
#14543791	2
#14543790	2
#14539166	2
#11916	17