Finite Group Databasebeta

Subgroups — #17962233

Order 4352.

262 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 84 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	C₂×#11763738	C₂	1	1	yes	maximal, characteristic
2	#14544992	C₂	1	1	yes	maximal, characteristic
3	#14544904	C₂	1	2	yes	maximal
4	#14543863	C₂	1	2	yes	maximal
5	D₁₇×#368	C₂	1	1	yes	maximal, characteristic
6	#11763865	C₂²	1	1	yes	characteristic
7	#11763824	—	2	2	no	—
8	C₂×#10587536	C₂²	1	2	yes	—
9	#11763772	C₂²	1	1	yes	characteristic
10	#11763738	—	2	1	no	—
11	#11763738	—	2	1	no	—
12	#11763696	—	2	1	no	—
13	C₁₆×#595	—	2	1	no	—
14	#11763692	—	2	2	no	—
15	C₂×C₁₆×D₁₇	C₂²	1	1	yes	characteristic
16	#108×D₁₇	C₂²	1	2	yes	—
17	#11762639	C₄	1	1	yes	characteristic, complemented
18	#11762638	C₄	1	2	yes	complemented
19	C₁₇×#368	C₄	1	1	yes	characteristic, complemented
20	#10587677	D₄	1	1	yes	characteristic
21	D₁₇⋊M₁₆	D₄	1	1	yes	characteristic
22	C₂×#63243	—	2	2	no	—
23	M₁₆×D₁₇	—	2	2	no	—
24	C₂×C₈×D₁₇	C₂³	1	1	yes	characteristic
25	#10587538	—	2	1	no	—
26	#10587536	—	2	2	no	—
27	#10587536	—	2	2	no	—
28	#10587536	—	4	2	no	—
29	#10587535	—	4	2	no	—
30	#10587532	—	2	1	no	—
31	C₈×#595	—	2	1	no	—
32	#10587530	Q₈	1	2	yes	—
33	#10587529	—	2	1	no	—
34	C₂×C₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
35	C₁₇×#108	C₂×C₄	1	2	yes	—
36	#10587493	—	2	2	no	—
37	#10587490	C₂×C₄	1	2	yes	—
38	C₂×#63214	C₂×C₄	1	1	yes	characteristic
39	C₁₆×D₁₇	D₄	1	1	yes	characteristic
40	C₁₆×D₁₇	D₄	1	1	yes	characteristic
41	C₁₆×D₁₇	Q₈	1	1	yes	characteristic
42	C₁₆×D₁₇	Q₈	1	1	yes	characteristic
43	C₂×C₄×D₁₇	C₂×D₄	1	1	yes	characteristic
44	C₂²⋊#595	Q₁₆	1	2	yes	—
45	#63247	—	2	1	no	—
46	C₂×#3624	—	4	2	no	—
47	C₄⋊#595	—	2	1	no	—
48	C₄×#595	—	2	1	no	—
49	#63243	—	4	2	no	—
50	#63243	—	4	2	no	—
51	#63243	D₈	1	2	yes	—
52	#63242	—	2	1	no	—
53	M₁₆×C₁₇	—	2	2	no	—
54	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
55	#63223	—	2	2	no	—
56	C₂×#3622	#41	1	1	yes	characteristic
57	#63218	—	4	2	no	—
58	C₈×D₁₇	—	4	2	no	—
59	C₈×D₁₇	C₂×D₄	1	1	yes	characteristic
60	C₈×D₁₇	C₂×Q₈	1	1	yes	characteristic
61	C₈×D₁₇	C₂×Q₈	1	1	yes	characteristic
62	C₈×D₁₇	#41	1	1	yes	characteristic
63	#63216	—	2	2	no	—
64	C₁₆×C₁₇	C₄⋊C₄	1	1	yes	characteristic
65	C₁₆×C₁₇	C₄⋊C₄	1	1	yes	characteristic
66	#63214	C₄⋊C₄	1	1	yes	characteristic
67	#63214	C₄⋊C₄	1	1	yes	characteristic
68	#13173	—	17	1	no	maximal, Sylow
69	C₂²×D₁₇	C₂×Q₁₆	1	1	yes	characteristic
70	C₂×#595	—	2	2	no	—
71	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
72	C₂×Dic₁₇	#135	1	1	yes	characteristic
73	C₄×D₁₇	#121	1	1	yes	characteristic
74	C₄×D₁₇	C₄⋊Q₈	1	1	yes	characteristic
75	C₄×D₁₇	C₂×D₈	1	1	yes	characteristic
76	C₄×D₁₇	#135	1	1	yes	characteristic
77	#3624	—	2	2	no	—
78	#3624	—	8	2	no	—
79	C₈×C₁₇	—	4	2	no	—
80	C₈×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₈×C₁₇	C₄×Q₈	1	1	yes	characteristic
82	#3622	—	4	2	no	—
83	#3622	C₂²⋊Q₈	1	1	yes	characteristic
84	#3622	#125	1	1	yes	characteristic
85	C₂×#368	—	17	1	no	—
86	C₂×#368	—	17	1	no	—
87	#2108	—	17	1	no	—
88	#1953	—	17	2	no	—
89	#1388	—	17	2	no	—
90	C₂²×C₁₇	C₄×Q₁₆	1	1	yes	characteristic
91	C₂×D₁₇	—	4	1	no	—
92	C₂×D₁₇	#467	1	1	yes	characteristic
93	C₂×D₁₇	#500	1	1	yes	characteristic
94	#595	—	4	2	no	—
95	C₄×C₁₇	#389	1	1	yes	characteristic
96	C₄×C₁₇	C₄×D₈	1	1	yes	characteristic
97	Dic₁₇	#466	1	1	yes	characteristic
98	Dic₁₇	#499	1	1	yes	characteristic
99	C₂×M₃₂	—	17	1	no	—
100	C₂²×C₁₆	—	17	1	no	—
101	#430	—	34	2	no	—
102	C₂×#108	—	17	2	no	—
103	C₂×#108	—	17	2	no	—
104	#405	—	17	1	no	—
105	#368	—	17	1	no	—
106	#368	—	17	2	no	—
107	#368	—	34	1	no	—
108	#368	—	34	1	no	—
109	#368	#595	1	1	yes	characteristic, complemented
110	#346	—	34	1	no	—
111	C₄×C₁₆	—	34	1	no	—
112	#344	—	34	2	no	—
113	C₂×C₁₇	—	2	1	no	—
114	C₂×C₁₇	#1942	1	1	yes	characteristic
115	D₁₇	—	2	1	no	—
116	D₁₇	—	2	1	no	—
117	C₂×M₁₆	—	17	1	no	—
118	C₂×M₁₆	—	34	2	no	—
119	C₂×M₁₆	—	34	2	no	—
120	C₂²×C₈	—	17	1	no	—
121	#117	—	17	1	no	—
122	M₃₂	—	17	2	no	—
123	M₃₂	—	34	1	no	—
124	C₂×C₁₆	—	17	1	no	—
125	C₂×C₁₆	—	17	1	no	—
126	C₂×C₁₆	—	17	1	no	—
127	C₂×C₁₆	—	17	1	no	—
128	C₂×C₁₆	—	17	1	no	—
129	C₂×C₁₆	—	34	1	no	—
130	C₂×C₁₆	C₂×#595	1	1	yes	characteristic
131	#108	—	17	2	no	—
132	#108	—	34	2	no	—
133	#108	—	34	2	no	—
134	#108	—	34	2	no	—
135	#108	—	68	2	no	—
136	#108	—	68	2	no	—
137	#108	C₂×#595	1	2	yes	—
138	#97	—	34	1	no	—
139	C₄×C₈	—	34	1	no	—
140	C₁₇	#13173	1	1	yes	characteristic, complemented, Sylow
141	C₂²×C₄	—	17	1	no	—
142	M₁₆	—	2	2	no	—
143	M₁₆	—	17	2	no	—
144	M₁₆	—	34	1	no	—
145	M₁₆	—	34	2	no	—
146	M₁₆	—	68	2	no	—
147	M₁₆	—	68	2	no	—
148	M₁₆	—	68	2	no	—
149	C₂×C₈	—	17	1	no	—
150	C₂×C₈	—	17	1	no	—
151	C₂×C₈	—	17	1	no	—
152	C₂×C₈	—	17	1	no	—
153	C₂×C₈	—	17	1	no	—
154	C₂×C₈	—	34	1	no	—
155	C₂×C₈	—	68	2	no	—
156	C₂×C₈	—	68	2	no	—
157	C₂×C₈	C₂²×#595	1	1	yes	characteristic
158	C₄⋊C₄	—	34	1	no	—
159	C₂²⋊C₄	—	17	2	no	—
160	C₄²	—	34	1	no	—
161	C₁₆	—	17	1	no	—
162	C₁₆	—	17	1	no	—
163	C₁₆	—	34	2	no	—
164	C₁₆	C₄⋊#595	1	1	yes	characteristic
165	C₁₆	C₄⋊#595	1	1	yes	characteristic
166	C₂³	—	17	1	no	—
167	C₂×C₄	—	17	1	no	—
168	C₂×C₄	—	17	1	no	—
169	C₂×C₄	—	17	1	no	—
170	C₂×C₄	—	17	1	no	—
171	C₂×C₄	—	17	1	no	—
172	C₂×C₄	—	34	2	no	—
173	C₂×C₄	D₄×#595	1	1	yes	characteristic
174	C₈	—	4	2	no	—
175	C₈	—	17	1	no	—
176	C₈	—	17	1	no	—
177	C₈	—	34	2	no	—
178	C₈	—	68	2	no	—
179	C₈	—	136	2	no	—
180	C₈	C₂×C₄⋊#595	1	1	yes	characteristic
181	C₈	Q₈×#595	1	1	yes	characteristic
182	C₂²	—	17	1	no	—
183	C₂²	—	17	1	no	—
184	C₂²	—	68	1	no	—
185	C₂²	Q₁₆×#595	1	1	yes	characteristic
186	C₄	—	17	1	no	—
187	C₄	—	17	1	no	—
188	C₄	—	68	2	no	—
189	C₄	#11763760	1	1	yes	characteristic
190	C₄	D₈×#595	1	1	yes	characteristic
191	C₂	—	2	1	no	—
192	C₂	—	34	1	no	—
193	C₂	—	34	1	no	—
194	C₂	#14544888	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#14544888
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×#595
Fitting	C₁₇×#368	C₄
socle	C₂×C₁₇	#1942
prime radical	C₂²×D₁₇	C₂×Q₁₆
cosocle	C₂×C₈×D₁₇	C₂³
hypercenter	#368	#595
hypocenter	C₁₇	#13173
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #368

Complements

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

N	H ≅ G/N	type
17. #11762639	C₄	semidirect
18. #11762638	C₄	semidirect
19. C₁₇×#368	C₄	semidirect
109. #368	#595	semidirect
140. C₁₇	#13173	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11763738	2
#14544992	2
#14544904	2
#14543863	2
D₁₇×#368	2
#13173	17