Finite Group Databasebeta

Subgroups — #17962829

Order 4352.

194 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 78 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	#14545135	C₂	1	1	yes	maximal, characteristic
2	C₂×#11763735	C₂	1	1	yes	maximal, characteristic
3	#14543946	C₂	1	2	yes	maximal, complemented
4	#14543945	C₂	1	2	yes	maximal, complemented
5	D₁₇×M₆₄	C₂	1	1	yes	maximal, characteristic
6	#11763819	—	2	1	no	—
7	C₂×#10587534	—	2	1	no	—
8	#11763736	—	2	1	no	—
9	#11763735	—	2	1	no	—
10	#11763735	C₂²	1	1	yes	characteristic
11	#11763735	C₂²	1	2	yes	—
12	#11763735	C₂²	1	1	yes	characteristic
13	C₂×C₁₆×D₁₇	C₂²	1	1	yes	characteristic
14	#11762643	C₄	1	1	yes	characteristic, complemented
15	#11762642	C₄	1	2	yes	complemented
16	C₃₂×D₁₇	C₂²	1	2	yes	—
17	C₁₇×M₆₄	C₄	1	1	yes	characteristic, complemented
18	#10587677	—	4	1	no	—
19	C₂×C₄⋊#595	—	4	1	no	—
20	C₂×C₈×D₁₇	D₄	1	1	yes	characteristic
21	#10587534	—	2	2	no	—
22	#10587534	—	2	1	no	—
23	#10587534	—	2	1	no	—
24	#10587534	—	4	1	no	—
25	#10587533	—	4	1	no	—
26	C₂×C₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
27	C₂×#63214	Q₈	1	1	yes	characteristic
28	C₁₆×D₁₇	C₂×C₄	1	1	yes	characteristic
29	C₁₆×D₁₇	D₄	1	1	yes	characteristic
30	C₁₆×D₁₇	Q₈	1	1	yes	characteristic
31	C₁₆×D₁₇	C₂³	1	1	yes	characteristic
32	C₁₇×C₃₂	C₂×C₄	1	2	yes	complemented
33	#10587465	C₂×C₄	1	2	yes	complemented
34	C₂²×#595	—	8	1	no	—
35	C₂×C₄×D₁₇	D₈	1	1	yes	characteristic
36	C₂²⋊#595	—	8	1	no	—
37	C₄⋊#595	—	4	2	no	—
38	C₄⋊#595	—	4	1	no	—
39	C₄⋊#595	—	4	1	no	—
40	C₄⋊#595	—	8	1	no	—
41	C₄×#595	—	8	1	no	—
42	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
43	C₂×#3622	Q₁₆	1	1	yes	characteristic
44	C₈×D₁₇	C₄⋊C₄	1	1	yes	characteristic
45	C₈×D₁₇	D₈	1	1	yes	characteristic
46	C₈×D₁₇	Q₁₆	1	1	yes	characteristic
47	C₈×D₁₇	C₂×D₄	1	1	yes	characteristic
48	C₁₆×C₁₇	C₄⋊C₄	1	1	yes	characteristic
49	C₁₆×C₁₇	C₂²×C₄	1	1	yes	characteristic
50	#63214	C₄⋊C₄	1	1	yes	characteristic
51	#63214	C₂×Q₈	1	1	yes	characteristic
52	#13634	—	17	1	no	maximal, Sylow
53	C₂²×D₁₇	D₁₆	1	1	yes	characteristic
54	C₂×#595	—	8	1	no	—
55	C₂×#595	—	8	1	no	—
56	C₂×#595	—	8	2	no	—
57	C₂×#595	—	16	2	no	—
58	C₂×C₄×C₁₇	#107	1	1	yes	characteristic
59	C₂×Dic₁₇	Q₃₂	1	1	yes	characteristic
60	C₄×D₁₇	#107	1	1	yes	characteristic
61	C₄×D₁₇	D₁₆	1	1	yes	characteristic
62	C₄×D₁₇	Q₃₂	1	1	yes	characteristic
63	C₄×D₁₇	C₂×D₈	1	1	yes	characteristic
64	C₈×C₁₇	#107	1	1	yes	characteristic
65	C₈×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
66	#3622	#107	1	1	yes	characteristic
67	#3622	C₂×Q₁₆	1	1	yes	characteristic
68	C₂×M₆₄	—	17	1	no	—
69	#2158	—	17	1	no	—
70	C₂×#366	—	17	1	no	—
71	#1424	—	17	2	no	—
72	#1423	—	17	2	no	—
73	C₂²×C₁₇	#366	1	1	yes	characteristic
74	C₂×D₁₇	—	2	1	no	—
75	C₂×D₁₇	—	2	1	no	—
76	C₂×D₁₇	#366	1	1	yes	characteristic
77	C₂×D₁₇	C₂×D₁₆	1	1	yes	characteristic
78	#595	—	16	1	no	—
79	#595	—	16	2	no	—
80	#595	—	16	1	no	—
81	C₄×C₁₇	#366	1	1	yes	characteristic
82	C₄×C₁₇	C₂×#107	1	1	yes	characteristic
83	Dic₁₇	#366	1	1	yes	characteristic
84	Dic₁₇	C₂×Q₃₂	1	1	yes	characteristic
85	C₂²×C₁₆	—	17	1	no	—
86	#427	—	34	1	no	—
87	C₂×#107	—	34	1	no	—
88	M₆₄	—	17	1	no	—
89	M₆₄	—	17	2	no	—
90	M₆₄	#595	1	1	yes	characteristic, complemented
91	C₂×C₃₂	—	17	2	no	—
92	#367	—	34	1	no	—
93	#366	—	17	2	no	—
94	#366	—	17	1	no	—
95	#366	—	17	1	no	—
96	#366	—	34	1	no	—
97	C₂×C₁₇	—	2	1	no	—
98	C₂×C₁₇	C₂×#366	1	1	yes	characteristic
99	D₁₇	—	2	1	no	—
100	D₁₇	#2263	1	1	yes	characteristic
101	D₁₇	#2264	1	1	yes	characteristic
102	C₂²×C₈	—	17	1	no	—
103	#117	—	68	1	no	—
104	C₂×C₄⋊C₄	—	68	1	no	—
105	C₂×C₁₆	—	17	1	no	—
106	C₂×C₁₆	—	17	1	no	—
107	C₂×C₁₆	—	17	1	no	—
108	C₂×C₁₆	—	17	1	no	—
109	C₂×C₁₆	—	17	1	no	—
110	C₂×C₁₆	C₂×#595	1	1	yes	characteristic
111	#107	—	34	2	no	—
112	#107	—	34	1	no	—
113	#107	—	34	1	no	—
114	#107	—	68	1	no	—
115	#106	—	68	1	no	—
116	C₃₂	—	17	2	no	—
117	C₃₂	C₂×#595	1	2	yes	complemented
118	C₁₇	#13634	1	1	yes	characteristic, complemented, Sylow
119	C₂²×C₄	—	17	1	no	—
120	C₂²×C₄	—	136	1	no	—
121	C₂×C₈	—	17	1	no	—
122	C₂×C₈	—	17	1	no	—
123	C₂×C₈	—	17	1	no	—
124	C₂×C₈	—	17	1	no	—
125	C₂×C₈	—	17	1	no	—
126	C₂×C₈	C₄⋊#595	1	1	yes	characteristic
127	C₄⋊C₄	—	68	1	no	—
128	C₄⋊C₄	—	68	1	no	—
129	C₄⋊C₄	—	68	2	no	—
130	C₄⋊C₄	—	136	1	no	—
131	C₂²⋊C₄	—	136	1	no	—
132	C₄²	—	136	1	no	—
133	C₁₆	—	17	1	no	—
134	C₁₆	—	17	1	no	—
135	C₁₆	C₄⋊#595	1	1	yes	characteristic
136	C₁₆	C₂²×#595	1	1	yes	characteristic
137	C₂³	—	17	1	no	—
138	C₂×C₄	—	272	2	no	—
139	C₂×C₄	—	17	1	no	—
140	C₂×C₄	—	17	1	no	—
141	C₂×C₄	—	17	1	no	—
142	C₂×C₄	—	17	1	no	—
143	C₂×C₄	—	17	1	no	—
144	C₂×C₄	—	136	1	no	—
145	C₂×C₄	—	136	1	no	—
146	C₂×C₄	—	136	2	no	—
147	C₂×C₄	#10587534	1	1	yes	characteristic
148	C₈	—	17	1	no	—
149	C₈	—	17	1	no	—
150	C₈	#10587534	1	1	yes	characteristic
151	C₈	C₂×C₄⋊#595	1	1	yes	characteristic
152	C₂²	—	17	1	no	—
153	C₂²	—	17	1	no	—
154	C₂²	—	34	1	no	—
155	C₂²	—	34	1	no	—
156	C₂²	#11763735	1	1	yes	characteristic
157	C₄	—	272	2	no	—
158	C₄	—	272	1	no	—
159	C₄	—	272	1	no	—
160	C₄	—	17	1	no	—
161	C₄	—	17	1	no	—
162	C₄	#11763735	1	1	yes	characteristic
163	C₄	C₂×#10587534	1	1	yes	characteristic
164	C₂	—	2	1	no	—
165	C₂	—	17	1	no	—
166	C₂	—	17	1	no	—
167	C₂	—	34	1	no	—
168	C₂	C₂×#11763735	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#11763735
commutator	C₁₆×C₁₇	C₂²×C₄
Frattini	C₁₆	C₂²×#595
Fitting	C₁₇×M₆₄	C₄
socle	C₂×C₁₇	C₂×#366
prime radical	C₂²×D₁₇	D₁₆
cosocle	C₁₆×D₁₇	C₂³
hypercenter	M₆₄	#595
hypocenter	C₁₇	#13634
LCS	G ▷ C₁₆×C₁₇ ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
3. #14543946	C₂	semidirect
4. #14543945	C₂	semidirect
14. #11762643	C₄	semidirect
15. #11762642	C₄	semidirect
17. C₁₇×M₆₄	C₄	semidirect
32. C₁₇×C₃₂	C₂×C₄	semidirect
33. #10587465	C₂×C₄	semidirect
90. M₆₄	#595	semidirect
117. C₃₂	C₂×#595	semidirect
118. C₁₇	#13634	semidirect

Maximal subgroups

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

subgroup	index
#14545135	2
C₂×#11763735	2
#14543946	2
#14543945	2
D₁₇×M₆₄	2
#13634	17