Finite Group Databasebeta

Subgroups — #17962889

Order 4352.

182 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 50 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	#14545187	C₂	1	1	yes	maximal, characteristic
2	#14545181	C₂	1	1	yes	maximal, characteristic, complemented
3	#14543949	C₂	1	1	yes	maximal, characteristic, complemented
4	#14543940	C₂	1	1	yes	maximal, characteristic, complemented
5	#14543938	C₂	1	1	yes	maximal, characteristic
6	#14543893	C₂	1	1	yes	maximal, characteristic, complemented
7	#14531103	C₂	1	1	yes	maximal, characteristic
8	#11763853	—	2	1	no	—
9	Q₁₆×#595	—	2	1	no	—
10	#11763742	C₂²	1	1	yes	characteristic
11	#11763738	C₂²	1	1	yes	characteristic
12	#11763736	C₂²	1	1	yes	characteristic
13	#11763724	—	2	1	no	—
14	#11763722	—	2	1	no	—
15	#11763696	C₂²	1	1	yes	characteristic
16	#11763242	C₂²	1	1	yes	characteristic
17	Q₃₂×D₁₇	C₂²	1	1	yes	characteristic
18	#11762649	C₄	1	1	yes	characteristic, complemented
19	#11762648	C₄	1	1	yes	characteristic, complemented
20	#11762646	C₄	1	1	yes	characteristic, complemented
21	#11762642	C₂²	1	1	yes	characteristic
22	C₁₇×Q₆₄	C₄	1	1	yes	characteristic, complemented
23	Q₈×#595	—	4	1	no	—
24	#10587680	—	4	1	no	—
25	#10587605	—	2	1	no	—
26	Q₁₆×D₁₇	—	2	1	no	—
27	#10587550	—	4	1	no	—
28	#10587549	—	4	1	no	—
29	#10587536	—	2	1	no	—
30	#10587534	—	2	1	no	—
31	C₈×#595	D₄	1	1	yes	characteristic
32	#10587529	D₄	1	1	yes	characteristic
33	C₁₇×Q₃₂	C₂×C₄	1	1	yes	characteristic
34	C₁₇×Q₃₂	C₂×C₄	1	1	yes	characteristic
35	#10587500	—	2	1	no	—
36	#10587499	—	2	1	no	—
37	Dic₁₃₆	C₂×C₄	1	1	yes	characteristic
38	D₂₇₂	C₂×C₄	1	1	yes	characteristic
39	C₁₆×D₁₇	C₂³	1	1	yes	characteristic
40	#10587467	—	2	1	no	—
41	C₁₇×C₃₂	C₂×C₄	1	1	yes	characteristic
42	#10587465	C₂×C₄	1	1	yes	characteristic
43	#63256	—	4	1	no	—
44	Q₈×D₁₇	—	4	1	no	—
45	C₄⋊#595	—	4	1	no	—
46	C₄⋊#595	—	8	1	no	—
47	C₄×#595	—	8	1	no	—
48	C₄×#595	D₈	1	1	yes	characteristic
49	#63243	—	4	1	no	—
50	#63243	—	8	1	no	—
51	#63242	—	8	1	no	—
52	#63242	D₈	1	1	yes	characteristic
53	Q₁₆×C₁₇	—	2	1	no	—
54	Q₁₆×C₁₇	—	2	1	no	—
55	#63231	—	4	1	no	—
56	#63230	—	4	1	no	—
57	Dic₆₈	—	2	1	no	—
58	D₁₃₆	—	2	1	no	—
59	C₈×D₁₇	C₂×D₄	1	1	yes	characteristic
60	#63216	—	2	1	no	—
61	C₁₆×C₁₇	C₂²×C₄	1	1	yes	characteristic
62	#63214	#41	1	1	yes	characteristic
63	#13657	—	17	1	no	maximal, Sylow
64	C₂×#595	—	2	1	no	—
65	C₂×#595	—	8	1	no	—
66	Q₈×C₁₇	—	4	1	no	—
67	Q₈×C₁₇	—	4	1	no	—
68	D₆₈	—	4	1	no	—
69	D₆₈	—	8	1	no	—
70	C₄×D₁₇	—	8	1	no	—
71	C₄×D₁₇	—	8	1	no	—
72	C₄×D₁₇	C₂×D₈	1	1	yes	characteristic
73	Dic₃₄	—	4	1	no	—
74	Dic₃₄	—	8	1	no	—
75	#3624	—	2	1	no	—
76	#3624	—	8	1	no	—
77	C₈×C₁₇	C₄×D₄	1	1	yes	characteristic
78	#3622	#135	1	1	yes	characteristic
79	#2264	—	17	1	no	—
80	#2179	—	17	1	no	—
81	#2176	—	17	1	no	—
82	#1426	—	17	1	no	—
83	#1420	—	17	1	no	—
84	#1418	—	17	1	no	—
85	#1398	—	17	1	no	—
86	C₂×D₁₇	—	8	1	no	—
87	C₂×D₁₇	#509	1	1	yes	characteristic
88	#595	—	4	1	no	—
89	#595	—	16	1	no	—
90	C₄×C₁₇	—	8	1	no	—
91	C₄×C₁₇	—	8	1	no	—
92	C₄×C₁₇	C₄×D₈	1	1	yes	characteristic
93	Dic₁₇	—	8	1	no	—
94	Dic₁₇	#509	1	1	yes	characteristic
95	#508	—	17	1	no	—
96	C₂×Q₃₂	—	17	1	no	—
97	#443	—	34	1	no	—
98	C₄×Q₁₆	—	34	1	no	—
99	Q₆₄	—	17	1	no	—
100	Q₆₄	#595	1	1	yes	characteristic, complemented
101	SD₆₄	—	17	1	no	—
102	SD₆₄	—	17	1	no	—
103	M₆₄	—	17	1	no	—
104	M₆₄	—	17	1	no	—
105	#368	—	17	1	no	—
106	#367	—	17	1	no	—
107	#359	—	34	1	no	—
108	#358	—	34	1	no	—
109	#346	—	17	1	no	—
110	C₂×C₁₇	#2177	1	1	yes	characteristic
111	D₁₇	—	2	1	no	—
112	D₁₇	—	16	1	no	—
113	#135	—	34	1	no	—
114	C₂×Q₁₆	—	34	1	no	—
115	#131	—	68	1	no	—
116	C₄×Q₈	—	68	1	no	—
117	Q₃₂	—	17	1	no	—
118	Q₃₂	—	34	1	no	—
119	Q₃₂	C₂×#595	1	1	yes	characteristic
120	Q₃₂	C₂×#595	1	1	yes	characteristic
121	SD₃₂	—	34	1	no	—
122	D₁₆	—	17	1	no	—
123	C₂×C₁₆	—	17	1	no	—
124	C₂×C₁₆	—	17	1	no	—
125	#108	—	34	1	no	—
126	#107	—	34	1	no	—
127	#104	—	68	1	no	—
128	#103	—	68	1	no	—
129	C₄×C₈	—	17	1	no	—
130	C₃₂	—	17	1	no	—
131	C₃₂	—	34	1	no	—
132	C₃₂	C₂×#595	1	1	yes	characteristic
133	C₁₇	#13657	1	1	yes	characteristic, complemented, Sylow
134	#41	—	68	1	no	—
135	C₂×Q₈	—	68	1	no	—
136	Q₁₆	—	2	1	no	—
137	Q₁₆	—	2	1	no	—
138	Q₁₆	—	34	1	no	—
139	Q₁₆	—	68	1	no	—
140	SD₁₆	—	68	1	no	—
141	D₈	—	34	1	no	—
142	M₁₆	—	68	1	no	—
143	M₁₆	—	136	1	no	—
144	C₂×C₈	—	17	1	no	—
145	C₂×C₈	—	17	1	no	—
146	C₂×C₈	—	136	1	no	—
147	C₄⋊C₄	—	68	1	no	—
148	C₄⋊C₄	—	136	1	no	—
149	C₄²	—	17	1	no	—
150	C₄²	—	136	1	no	—
151	C₁₆	—	17	1	no	—
152	C₁₆	—	34	1	no	—
153	C₁₆	C₂²×#595	1	1	yes	characteristic
154	Q₈	—	4	1	no	—
155	Q₈	—	4	1	no	—
156	Q₈	—	68	1	no	—
157	Q₈	—	136	1	no	—
158	D₄	—	68	1	no	—
159	D₄	—	136	1	no	—
160	C₂×C₄	—	17	1	no	—
161	C₂×C₄	—	34	1	no	—
162	C₂×C₄	—	136	1	no	—
163	C₂×C₄	—	136	1	no	—
164	C₂×C₄	—	136	1	no	—
165	C₈	—	17	1	no	—
166	C₈	—	34	1	no	—
167	C₈	—	136	1	no	—
168	C₈	D₄×#595	1	1	yes	characteristic
169	C₂²	—	17	1	no	—
170	C₂²	—	136	1	no	—
171	C₄	—	8	1	no	—
172	C₄	—	8	1	no	—
173	C₄	—	272	1	no	—
174	C₄	—	17	1	no	—
175	C₄	—	68	1	no	—
176	C₄	—	136	1	no	—
177	C₄	D₈×#595	1	1	yes	characteristic
178	C₂	—	272	1	no	—
179	C₂	—	34	1	no	—
180	C₂	#14545182	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#14545182
commutator	C₁₆×C₁₇	C₂²×C₄
Frattini	C₁₆	C₂²×#595
Fitting	C₁₇×Q₆₄	C₄
socle	C₂×C₁₇	#2177
prime radical	#11763242	C₂²
cosocle	C₁₆×D₁₇	C₂³
hypercenter	Q₆₄	#595
hypocenter	C₁₇	#13657
LCS	G ▷ C₁₆×C₁₇ ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ Q₆₄

Complements

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

N	H ≅ G/N	type
2. #14545181	C₂	semidirect
3. #14543949	C₂	semidirect
4. #14543940	C₂	semidirect
6. #14543893	C₂	semidirect
18. #11762649	C₄	semidirect
19. #11762648	C₄	semidirect
20. #11762646	C₄	semidirect
22. C₁₇×Q₆₄	C₄	semidirect
100. Q₆₄	#595	semidirect
133. C₁₇	#13657	semidirect

Maximal subgroups

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

subgroup	index
#14545187	2
#14545181	2
#14543949	2
#14543940	2
#14543938	2
#14543893	2
#14531103	2
#13657	17