Finite Group Databasebeta

Subgroups — #17960688

Order 4352.

258 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 92 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	#14544994	C₂	1	1	yes	maximal, characteristic
2	#14544506	C₂	1	1	yes	maximal, characteristic
3	#14543840	C₂	1	2	yes	maximal
4	#14543836	C₂	1	2	yes	maximal
5	#14530387	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587530	C₂²	1	1	yes	characteristic
7	C₂×#10587529	C₂²	1	1	yes	characteristic
8	#11763772	C₂²	1	2	yes	—
9	#11763698	C₄	1	1	yes	characteristic, complemented
10	#11763697	C₄	1	1	yes	characteristic, complemented
11	#11763695	C₄	1	1	yes	characteristic, complemented
12	C₄×#63216	C₄	1	1	yes	characteristic, complemented
13	#11763678	C₄	1	1	yes	characteristic
14	#11763677	C₄	1	1	yes	characteristic
15	#11763676	C₄	1	2	yes	—
16	M₃₂×D₁₇	C₂²	1	2	yes	—
17	#11762766	C₂²	1	1	yes	characteristic
18	#11762587	C₄	1	1	yes	characteristic, complemented
19	#11762586	C₄	1	2	yes	complemented
20	C₁₇×#347	C₄	1	1	yes	characteristic, complemented
21	#10587677	C₂×C₄	1	2	yes	—
22	D₁₇⋊M₁₆	C₂×C₄	1	2	yes	—
23	C₂×#63218	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×D₁₇	C₂³	1	1	yes	characteristic
25	#10587558	C₂×C₄	1	1	yes	characteristic
26	C₂×#63216	C₂×C₄	1	1	yes	characteristic
27	C₂×#63216	C₂×C₄	1	1	yes	characteristic
28	C₂×#63216	C₂×C₄	1	1	yes	characteristic
29	C₂×#63216	C₂×C₄	1	1	yes	characteristic
30	#10587532	—	2	2	no	—
31	C₈×#595	—	2	2	no	—
32	#10587530	—	2	2	no	—
33	#10587529	—	2	2	no	—
34	C₁₇×M₃₂	C₂×C₄	1	2	yes	—
35	C₁₇×#97	C₂×C₄	1	1	yes	characteristic
36	#10587486	C₂×C₄	1	1	yes	characteristic
37	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
38	#10587483	C₂×C₄	1	2	yes	—
39	C₄×#3622	C₂×C₄	1	1	yes	characteristic
40	#10587469	—	2	2	no	—
41	C₁₆×D₁₇	—	2	2	no	—
42	C₂×C₄×D₁₇	C₂²×C₄	1	1	yes	characteristic
43	C₂²⋊#595	—	2	2	no	—
44	#63247	—	2	2	no	—
45	C₄⋊#595	—	2	2	no	—
46	C₄×#595	—	2	2	no	—
47	#63243	—	2	2	no	—
48	#63242	—	2	2	no	—
49	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
50	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
51	C₄²×C₁₇	C₄²	1	1	yes	characteristic
52	#63227	—	2	1	no	—
53	#63225	—	2	1	no	—
54	C₄×Dic₁₇	C₄²	1	1	yes	characteristic
55	C₂×#3622	C₄²	1	1	yes	characteristic
56	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
57	#63218	—	2	2	no	—
58	C₈×D₁₇	#41	1	2	yes	—
59	C₈×D₁₇	#41	1	2	yes	—
60	#63216	—	2	1	no	—
61	#63216	—	2	1	no	—
62	#63216	—	2	1	no	—
63	#63216	—	2	1	no	—
64	C₁₆×C₁₇	—	2	2	no	—
65	#63214	—	2	2	no	—
66	#12597	—	17	1	no	maximal, Sylow
67	C₂²×D₁₇	#131	1	1	yes	characteristic
68	C₂×#595	—	4	2	no	—
69	C₂×C₄×C₁₇	—	2	1	no	—
70	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
71	C₂×Dic₁₇	—	2	1	no	—
72	C₂×Dic₁₇	#131	1	1	yes	characteristic
73	C₄×D₁₇	#117	1	2	yes	—
74	C₄×D₁₇	#131	1	2	yes	—
75	#3624	—	4	2	no	—
76	C₈×C₁₇	—	2	1	no	—
77	C₈×C₁₇	#117	1	2	yes	—
78	#3622	—	2	1	no	—
79	#3622	#117	1	1	yes	characteristic
80	#3622	#117	1	1	yes	characteristic
81	#2110	—	17	1	no	—
82	#2108	—	17	1	no	—
83	#1747	—	17	1	no	—
84	#1379	—	17	2	no	—
85	#1378	—	17	2	no	—
86	C₂²×C₁₇	#405	1	1	yes	characteristic
87	C₂×D₁₇	—	4	1	no	—
88	C₂×D₁₇	#433	1	2	yes	—
89	#595	—	8	2	no	—
90	C₄×C₁₇	—	4	1	no	—
91	C₄×C₁₇	#376	1	1	yes	characteristic
92	C₄×C₁₇	#405	1	1	yes	characteristic
93	Dic₁₇	—	4	1	no	—
94	Dic₁₇	#433	1	2	yes	—
95	C₂×M₃₂	—	17	1	no	—
96	C₂×M₃₂	—	17	2	no	—
97	C₂²×C₁₆	—	17	1	no	—
98	#405	—	17	1	no	—
99	#405	—	17	2	no	—
100	#347	—	17	2	no	—
101	#347	—	17	1	no	—
102	#347	#595	1	1	yes	characteristic, complemented
103	#346	—	17	1	no	—
104	#346	—	17	1	no	—
105	C₄×C₁₆	—	17	1	no	—
106	C₄×C₁₆	—	17	1	no	—
107	#338	—	17	1	no	—
108	H(3,ℤ/4)	—	17	2	no	—
109	H(3,ℤ/4)	—	17	1	no	—
110	C₂×C₁₇	—	2	1	no	—
111	C₂×C₁₇	#1746	1	1	yes	characteristic
112	D₁₇	—	2	2	no	—
113	C₂×M₁₆	—	17	1	no	—
114	C₂×M₁₆	—	17	2	no	—
115	C₂²×C₈	—	17	1	no	—
116	#117	—	17	1	no	—
117	#117	—	17	2	no	—
118	M₃₂	—	17	2	no	—
119	M₃₂	—	34	2	no	—
120	M₃₂	—	34	2	no	—
121	M₃₂	C₂×#595	1	2	yes	—
122	C₂×C₁₆	—	17	1	no	—
123	C₂×C₁₆	—	17	1	no	—
124	C₂×C₁₆	—	17	1	no	—
125	C₂×C₁₆	—	17	1	no	—
126	C₂×C₁₆	—	34	2	no	—
127	C₂×C₁₆	—	34	2	no	—
128	#97	—	17	1	no	—
129	#97	—	34	2	no	—
130	#97	C₂×#595	1	1	yes	characteristic
131	C₄×C₈	—	17	1	no	—
132	C₄×C₈	—	17	1	no	—
133	C₄×C₈	—	34	2	no	—
134	C₁₇	#12597	1	1	yes	characteristic, complemented, Sylow
135	C₂²×C₄	—	17	1	no	—
136	M₁₆	—	34	2	no	—
137	M₁₆	—	34	2	no	—
138	M₁₆	—	34	2	no	—
139	C₂×C₈	—	17	1	no	—
140	C₂×C₈	—	17	2	no	—
141	C₂×C₈	—	17	2	no	—
142	C₂×C₈	—	17	1	no	—
143	C₂×C₈	—	34	2	no	—
144	C₂×C₈	C₄×#595	1	1	yes	characteristic
145	C₂×C₈	C₂²×#595	1	1	yes	characteristic
146	C₄⋊C₄	—	34	1	no	—
147	C₄⋊C₄	—	34	2	no	—
148	C₂²⋊C₄	—	34	1	no	—
149	C₂²⋊C₄	—	34	2	no	—
150	C₄²	—	17	1	no	—
151	C₄²	—	34	2	no	—
152	C₄²	C₄×#595	1	1	yes	characteristic
153	C₁₆	—	2	2	no	—
154	C₁₆	—	34	1	no	—
155	C₁₆	—	34	1	no	—
156	C₁₆	—	34	1	no	—
157	C₁₆	—	34	1	no	—
158	C₁₆	—	34	2	no	—
159	C₂³	—	17	1	no	—
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	17	1	no	—
162	C₂×C₄	—	17	2	no	—
163	C₂×C₄	—	17	2	no	—
164	C₂×C₄	—	34	1	no	—
165	C₂×C₄	—	68	2	no	—
166	C₂×C₄	C₂×C₄×#595	1	1	yes	characteristic
167	C₈	—	2	1	no	—
168	C₈	—	17	1	no	—
169	C₈	—	17	1	no	—
170	C₈	—	34	1	no	—
171	C₈	—	68	2	no	—
172	C₈	#10587677	1	2	yes	—
173	C₂²	—	17	2	no	—
174	C₂²	—	68	1	no	—
175	C₂²	#11763775	1	1	yes	characteristic
176	C₄	—	4	1	no	—
177	C₄	—	17	2	no	—
178	C₄	—	68	1	no	—
179	C₄	—	136	2	no	—
180	C₄	#11763745	1	1	yes	characteristic
181	C₄	#11763775	1	1	yes	characteristic
182	C₂	—	2	1	no	—
183	C₂	—	34	2	no	—
184	C₂	#14544504	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#11763745
commutator	C₂×C₄×C₁₇	C₂×C₄²
Frattini	C₂×C₈	C₂²×#595
Fitting	C₁₇×#347	C₄
socle	C₂×C₁₇	#1746
prime radical	C₂²×D₁₇	#131
cosocle	C₂×C₈×D₁₇	C₂³
hypercenter	#347	#595
hypocenter	C₁₇	#12597
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄ ◁ #97 ◁ #347

Complements

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

N	H ≅ G/N	type
9. #11763698	C₄	semidirect
10. #11763697	C₄	semidirect
11. #11763695	C₄	semidirect
12. C₄×#63216	C₄	semidirect
18. #11762587	C₄	semidirect
19. #11762586	C₄	semidirect
20. C₁₇×#347	C₄	semidirect
102. #347	#595	semidirect
134. C₁₇	#12597	semidirect

Maximal subgroups

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

subgroup	index
#14544994	2
#14544506	2
#14543840	2
#14543836	2
#14530387	2
#12597	17