Finite Group Databasebeta

Subgroups — #17960307

Order 4352.

204 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 66 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	#14545176	C₂	1	1	yes	maximal, characteristic, complemented
2	#14545174	C₂	1	1	yes	maximal, characteristic, complemented
3	#14543812	C₂	1	1	yes	maximal, characteristic, complemented
4	#14543749	C₂	1	1	yes	maximal, characteristic, complemented
5	#14543748	C₂	1	1	yes	maximal, characteristic, complemented
6	#14543700	C₂	1	1	yes	maximal, characteristic, complemented
7	#14530667	C₂	1	1	yes	maximal, characteristic
8	#11763868	—	2	1	no	—
9	#11763867	—	2	1	no	—
10	#11763731	C₂²	1	1	yes	characteristic
11	#11763731	C₂²	1	1	yes	characteristic
12	#11763708	—	2	1	no	—
13	#11763707	—	2	1	no	—
14	#11763698	C₂²	1	1	yes	characteristic
15	#11763650	C₂²	1	1	yes	characteristic
16	#11763239	C₄	1	1	yes	characteristic
17	#11762926	C₂²	1	1	yes	characteristic
18	#11762925	C₂²	1	1	yes	characteristic
19	#11762638	C₄	1	1	yes	characteristic
20	#11762617	C₄	1	1	yes	characteristic
21	#11762584	C₂²	1	1	yes	characteristic
22	C₁₇×#359	C₄	1	1	yes	characteristic
23	C₁₇×#135	C₂×C₄	1	1	yes	characteristic
24	#10587634	—	2	1	no	—
25	#10587593	—	2	1	no	—
26	#10587590	C₂×C₄	1	1	yes	characteristic
27	C₄×#3624	—	2	1	no	—
28	#10587538	—	2	1	no	—
29	#10587538	—	2	1	no	—
30	#10587538	—	4	1	no	—
31	C₂×#63216	—	4	1	no	—
32	C₂×#63216	D₄	1	1	yes	characteristic
33	C₂×#63216	D₄	1	1	yes	characteristic
34	#10587530	—	4	1	no	—
35	#10587529	—	4	1	no	—
36	C₂×C₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
37	C₁₇×#108	C₂×C₄	1	1	yes	characteristic
38	#10587508	—	2	1	no	—
39	#10587500	C₈	1	1	yes	characteristic, complemented
40	#10587499	C₈	1	1	yes	characteristic, complemented
41	#10587498	C₈	1	1	yes	characteristic, complemented
42	#10587497	C₈	1	1	yes	characteristic, complemented
43	#10587496	—	2	1	no	—
44	#10587490	C₂×C₄	1	1	yes	characteristic
45	C₈×Dic₁₇	C₂³	1	1	yes	characteristic
46	C₂×#63214	C₂×C₄	1	1	yes	characteristic
47	#41×C₁₇	—	2	1	no	—
48	#63252	—	2	1	no	—
49	C₂×#3624	—	4	1	no	—
50	Q₁₆×C₁₇	C₂×C₈	1	1	yes	characteristic
51	SD₁₆×C₁₇	—	2	1	no	—
52	D₈×C₁₇	C₂×C₈	1	1	yes	characteristic
53	M₁₆×C₁₇	—	2	1	no	—
54	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
55	C₄×Dic₁₇	#41	1	1	yes	characteristic
56	#63223	—	2	1	no	—
57	#63223	—	4	1	no	—
58	C₂×#3622	—	4	1	no	—
59	C₂×#3622	C₂×D₄	1	1	yes	characteristic
60	Dic₆₈	C₂×C₈	1	1	yes	characteristic
61	D₁₃₆	C₂×C₈	1	1	yes	characteristic
62	#63219	—	2	1	no	—
63	#63218	—	4	1	no	—
64	C₈×D₁₇	—	4	1	no	—
65	#63216	—	2	1	no	—
66	#63216	—	2	1	no	—
67	#63216	—	4	1	no	—
68	#63216	—	4	1	no	—
69	C₁₆×C₁₇	—	2	1	no	—
70	#63214	C₂×C₈	1	1	yes	characteristic
71	#63214	C₂×C₈	1	1	yes	characteristic
72	#12068	—	17	1	no	maximal, Sylow
73	Q₈×C₁₇	—	2	1	no	—
74	D₄×C₁₇	—	2	1	no	—
75	D₄×C₁₇	—	4	1	no	—
76	C₂×C₄×C₁₇	—	4	1	no	—
77	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
78	C₂²⋊D₁₇	—	4	1	no	—
79	C₂×Dic₁₇	—	2	1	no	—
80	D₆₈	—	2	1	no	—
81	C₄×D₁₇	—	4	1	no	—
82	Dic₃₄	—	2	1	no	—
83	#3624	—	8	1	no	—
84	C₈×C₁₇	—	4	1	no	—
85	C₈×C₁₇	C₂²×C₈	1	1	yes	characteristic
86	C₈×C₁₇	#131	1	1	yes	characteristic
87	#3622	—	4	1	no	—
88	#3622	AGL(1,ℤ/8)	1	1	yes	characteristic
89	#3622	AGL(1,ℤ/8)	1	1	yes	characteristic
90	#2179	—	17	1	no	—
91	#2171	—	17	1	no	—
92	#2171	—	17	1	no	—
93	#1369	—	17	1	no	—
94	#1336	—	17	1	no	—
95	#1336	—	17	1	no	—
96	#1313	—	17	1	no	—
97	C₂²×C₁₇	—	4	1	no	—
98	C₂²×C₁₇	#444	1	1	yes	characteristic
99	C₂×D₁₇	—	4	1	no	—
100	C₄×C₁₇	—	4	1	no	—
101	C₄×C₁₇	C₈×D₄	1	1	yes	characteristic
102	C₄×C₁₇	#442	1	1	yes	characteristic
103	Dic₁₇	—	4	1	no	—
104	Dic₁₇	—	4	1	no	—
105	#508	—	17	1	no	—
106	#504	—	34	1	no	—
107	#504	—	34	1	no	—
108	#443	—	17	1	no	—
109	#443	—	17	1	no	—
110	#368	—	17	1	no	—
111	#364	—	17	1	no	—
112	#364	—	17	1	no	—
113	#359	—	17	1	no	—
114	#359	#595	1	1	yes	characteristic
115	#350	—	34	1	no	—
116	#350	—	34	1	no	—
117	#346	—	17	1	no	—
118	#346	—	17	1	no	—
119	#322	—	17	1	no	—
120	C₂×C₁₇	—	2	1	no	—
121	C₂×C₁₇	—	8	1	no	—
122	C₂×C₁₇	#1580	1	1	yes	characteristic
123	D₁₇	—	8	1	no	—
124	#135	—	17	1	no	—
125	#135	C₂×#595	1	1	yes	characteristic
126	#131	—	34	1	no	—
127	#131	—	34	1	no	—
128	Q₃₂	—	17	1	no	—
129	SD₃₂	—	17	1	no	—
130	SD₃₂	—	17	1	no	—
131	D₁₆	—	17	1	no	—
132	M₃₂	—	34	1	no	—
133	M₃₂	—	34	1	no	—
134	M₃₂	—	68	1	no	—
135	M₃₂	—	68	1	no	—
136	C₂×C₁₆	—	17	1	no	—
137	C₂×C₁₆	—	17	1	no	—
138	C₂×C₁₆	—	17	1	no	—
139	C₂×C₁₆	—	68	1	no	—
140	C₂×C₁₆	—	68	1	no	—
141	C₂×C₁₆	C₂×#595	1	1	yes	characteristic
142	#108	—	17	1	no	—
143	#108	C₂×#595	1	1	yes	characteristic
144	#104	—	34	1	no	—
145	#104	—	34	1	no	—
146	C₄×C₈	—	17	1	no	—
147	C₄×C₈	—	34	1	no	—
148	C₁₇	#12068	1	1	yes	characteristic, complemented, Sylow
149	#41	—	2	1	no	—
150	#41	—	34	1	no	—
151	Q₁₆	—	17	1	no	—
152	Q₁₆	#63242	1	1	yes	characteristic
153	SD₁₆	—	2	1	no	—
154	SD₁₆	—	34	1	no	—
155	D₈	—	17	1	no	—
156	D₈	#63242	1	1	yes	characteristic
157	M₁₆	—	2	1	no	—
158	M₁₆	—	34	1	no	—
159	M₁₆	—	68	1	no	—
160	M₁₆	—	68	1	no	—
161	C₂×C₈	—	17	1	no	—
162	C₂×C₈	—	68	1	no	—
163	C₂×C₈	—	68	1	no	—
164	C₂×C₈	—	68	1	no	—
165	C₂×C₈	C₂²×#595	1	1	yes	characteristic
166	C₄²	—	17	1	no	—
167	C₁₆	—	2	1	no	—
168	C₁₆	—	17	1	no	—
169	C₁₆	—	17	1	no	—
170	C₁₆	—	34	1	no	—
171	C₁₆	—	34	1	no	—
172	C₁₆	—	68	1	no	—
173	C₁₆	—	68	1	no	—
174	Q₈	—	2	1	no	—
175	Q₈	—	34	1	no	—
176	D₄	—	2	1	no	—
177	D₄	—	4	1	no	—
178	D₄	—	34	1	no	—
179	D₄	—	68	1	no	—
180	C₂×C₄	—	4	1	no	—
181	C₂×C₄	—	34	1	no	—
182	C₂×C₄	—	68	1	no	—
183	C₂×C₄	D₄×#595	1	1	yes	characteristic
184	C₈	—	4	1	no	—
185	C₈	—	17	1	no	—
186	C₈	—	17	1	no	—
187	C₈	—	68	1	no	—
188	C₈	—	136	1	no	—
189	C₈	C₂×#63242	1	1	yes	characteristic
190	C₈	#10587678	1	1	yes	characteristic
191	C₂²	—	4	1	no	—
192	C₂²	—	68	1	no	—
193	C₂²	#11763854	1	1	yes	characteristic
194	C₄	—	4	1	no	—
195	C₄	—	68	1	no	—
196	C₄	—	68	1	no	—
197	C₄	#11763836	1	1	yes	characteristic
198	C₄	#11763850	1	1	yes	characteristic
199	C₂	—	2	1	no	—
200	C₂	—	8	1	no	—
201	C₂	—	136	1	no	—
202	C₂	#14544381	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#11763850
commutator	C₈×C₁₇	C₂²×C₈
Frattini	C₂×C₈	C₂²×#595
Fitting	C₁₇×#359	C₄
socle	C₂×C₁₇	#1580
prime radical	#11763239	C₄
cosocle	C₈×Dic₁₇	C₂³
hypercenter	#359	#595
hypocenter	C₁₇	#12068
LCS	G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #359

Complements

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

N	H ≅ G/N	type
1. #14545176	C₂	semidirect
2. #14545174	C₂	semidirect
3. #14543812	C₂	semidirect
4. #14543749	C₂	semidirect
5. #14543748	C₂	semidirect
6. #14543700	C₂	semidirect
39. #10587500	C₈	semidirect
40. #10587499	C₈	semidirect
41. #10587498	C₈	semidirect
42. #10587497	C₈	semidirect
148. C₁₇	#12068	semidirect

Maximal subgroups

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

subgroup	index
#14545176	2
#14545174	2
#14543812	2
#14543749	2
#14543748	2
#14543700	2
#14530667	2
#12068	17