Finite Group Databasebeta

Subgroups — #17981226

Order 4352.

254 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 63 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	#14545987	C₂	1	1	yes	maximal, characteristic
2	#14545983	C₂	1	1	yes	maximal, characteristic
3	#14544442	C₂	1	1	yes	maximal, characteristic
4	#11763912	C₄	1	2	yes	complemented
5	C₄⋊#63241	C₄	1	2	yes	complemented
6	#11763906	—	2	1	no	—
7	#11763906	—	2	1	no	—
8	#11763903	—	2	1	no	—
9	C₄×#63241	—	2	1	no	—
10	C₂×#10587532	—	2	1	no	—
11	C₄×#63242	C₂²	1	1	yes	characteristic
12	C₂×C₈×#595	—	2	1	no	—
13	#11763745	C₄	1	1	yes	characteristic
14	#97×D₁₇	C₄	1	1	yes	characteristic
15	C₂×C₄×#595	—	2	1	no	—
16	C₂×#63242	D₄	1	1	yes	characteristic
17	C₂×#63242	D₄	1	1	yes	characteristic
18	C₂×#63241	—	2	2	no	—
19	C₂×#63241	—	2	2	no	—
20	C₂×C₈×D₁₇	—	2	1	no	—
21	C₄²×D₁₇	C₂×C₄	1	1	yes	characteristic
22	(C₂×#595)⋊C₄	—	2	1	no	—
23	(C₂×#595)⋊C₄	C₈	1	2	yes	—
24	C₄×#3624	C₂×C₄	1	1	yes	characteristic
25	C₄×#3624	C₂×C₄	1	1	yes	characteristic
26	C₄×#3624	D₄	1	1	yes	characteristic
27	C₄×#3624	Q₈	1	1	yes	characteristic
28	#10587532	—	2	4	no	—
29	C₈×#595	—	2	4	no	—
30	C₁₇×#97	C₈	1	1	yes	characteristic
31	#10587486	—	2	1	no	—
32	#10587474	C₈	1	1	yes	characteristic
33	C₂²×#595	—	2	2	no	—
34	C₂×C₄×D₁₇	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×C₄×D₁₇	C₂×C₈	1	1	yes	characteristic
36	C₂×C₄×D₁₇	M₁₆	1	1	yes	characteristic
37	C₂×#3624	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×#3624	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×#3624	C₄⋊C₄	1	1	yes	characteristic
40	C₂×#3624	C₄⋊C₄	1	1	yes	characteristic
41	C₄×#595	—	2	2	no	—
42	C₄×#595	—	2	2	no	—
43	#63242	—	2	2	no	—
44	#63242	—	2	2	no	—
45	#63241	—	4	2	no	—
46	#63241	—	4	2	no	—
47	C₂×C₈×C₁₇	—	2	1	no	—
48	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
49	C₄×Dic₁₇	C₄²	1	1	yes	characteristic
50	C₄×Dic₁₇	C₄⋊C₄	1	1	yes	characteristic
51	C₄×Dic₁₇	M₁₆	1	1	yes	characteristic
52	C₂×#3622	—	2	1	no	—
53	C₈×D₁₇	—	2	2	no	—
54	C₈×D₁₇	—	2	2	no	—
55	#7150	—	17	1	no	maximal, Sylow
56	C₂²×D₁₇	C₂²⋊C₈	1	1	yes	characteristic
57	C₂×#595	—	2	4	no	—
58	C₂×#595	—	4	2	no	—
59	C₂×#595	—	4	2	no	—
60	C₂×C₄×C₁₇	C₄×C₈	1	1	yes	characteristic
61	C₂×C₄×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
62	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
63	C₂×Dic₁₇	#95	1	1	yes	characteristic
64	C₂×Dic₁₇	#97	1	1	yes	characteristic
65	C₂×Dic₁₇	C₄⋊C₈	1	1	yes	characteristic
66	C₄×D₁₇	—	2	2	no	—
67	C₄×D₁₇	—	2	1	no	—
68	C₄×D₁₇	—	2	1	no	—
69	C₄×D₁₇	—	4	1	no	—
70	#3624	—	2	1	no	—
71	#3624	—	2	1	no	—
72	#3624	#104	1	2	yes	—
73	#3624	#104	1	2	yes	—
74	C₈×C₁₇	—	2	2	no	—
75	#3622	—	2	2	no	—
76	#2163	—	17	1	no	—
77	#2162	—	17	1	no	—
78	#1726	—	17	1	no	—
79	C₂²×C₁₇	#336	1	1	yes	characteristic
80	C₂×D₁₇	—	2	1	no	—
81	C₂×D₁₇	—	2	1	no	—
82	C₂×D₁₇	#350	1	2	yes	—
83	#595	—	4	4	no	—
84	C₄×C₁₇	—	2	1	no	—
85	C₄×C₁₇	—	2	1	no	—
86	C₄×C₁₇	—	2	1	no	—
87	Dic₁₇	—	2	1	no	—
88	Dic₁₇	—	2	1	no	—
89	Dic₁₇	#339	1	1	yes	characteristic
90	Dic₁₇	#339	1	1	yes	characteristic
91	C₂×#97	—	17	1	no	—
92	C₂×#97	—	34	1	no	—
93	C₂×C₄×C₈	—	17	1	no	—
94	C₂×C₄×C₈	—	34	1	no	—
95	#376	—	17	1	no	—
96	C₄⋊C₁₆	—	17	2	no	—
97	C₄⋊C₁₆	—	17	2	no	—
98	C₂²⋊C₁₆	—	34	1	no	—
99	C₂²⋊C₁₆	—	34	1	no	—
100	#346	—	34	1	no	—
101	C₄×C₁₆	—	34	1	no	—
102	C₂×C₁₇	#1276	1	1	yes	characteristic
103	C₂×C₁₇	#1378	1	1	yes	characteristic
104	C₂×C₁₇	#1379	1	1	yes	characteristic
105	D₁₇	—	2	2	no	—
106	C₂²×C₈	—	17	1	no	—
107	C₂²×C₈	—	17	1	no	—
108	C₂²×C₈	—	34	1	no	—
109	C₂×C₄²	—	17	1	no	—
110	C₂×C₄²	—	34	1	no	—
111	C₂×C₁₆	—	34	2	no	—
112	C₂×C₁₆	—	34	2	no	—
113	#97	—	17	1	no	—
114	#97	—	34	1	no	—
115	#97	—	34	4	no	—
116	#97	#3633	1	1	yes	characteristic
117	C₄×C₈	—	17	1	no	—
118	C₄×C₈	—	17	1	no	—
119	C₄×C₈	—	17	1	no	—
120	C₄×C₈	—	17	1	no	—
121	C₄×C₈	—	34	4	no	—
122	#95	—	17	2	no	—
123	#95	—	34	1	no	—
124	C₁₇	#7150	1	1	yes	characteristic, complemented, Sylow
125	C₂²×C₄	—	17	1	no	—
126	C₂²×C₄	—	17	1	no	—
127	C₂²×C₄	—	17	1	no	—
128	C₂²×C₄	—	34	2	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	17	1	no	—
131	C₂×C₈	—	17	1	no	—
132	C₂×C₈	—	17	1	no	—
133	C₂×C₈	—	17	1	no	—
134	C₂×C₈	—	34	2	no	—
135	C₂×C₈	—	34	2	no	—
136	C₂×C₈	—	34	2	no	—
137	C₂×C₈	—	34	2	no	—
138	C₂×C₈	—	34	1	no	—
139	C₄²	—	17	1	no	—
140	C₄²	—	17	1	no	—
141	C₄²	—	17	1	no	—
142	C₄²	—	34	2	no	—
143	C₄²	—	34	2	no	—
144	C₄²	C₂×#3633	1	1	yes	characteristic
145	C₁₆	—	68	2	no	—
146	C₁₆	—	68	2	no	—
147	C₂³	—	17	1	no	—
148	C₂×C₄	—	17	1	no	—
149	C₂×C₄	—	17	1	no	—
150	C₂×C₄	—	17	1	no	—
151	C₂×C₄	—	34	1	no	—
152	C₂×C₄	—	34	1	no	—
153	C₂×C₄	—	34	2	no	—
154	C₂×C₄	—	34	4	no	—
155	C₂×C₄	—	68	1	no	—
156	C₂×C₄	—	68	2	no	—
157	C₂×C₄	—	68	2	no	—
158	C₂×C₄	C₄×#3633	1	1	yes	characteristic
159	C₂×C₄	C₄⋊#3633	1	1	yes	characteristic
160	C₂×C₄	C₂²⋊#3633	1	1	yes	characteristic
161	C₈	—	2	2	no	—
162	C₈	—	17	2	no	—
163	C₈	—	17	2	no	—
164	C₈	—	34	2	no	—
165	C₈	—	34	1	no	—
166	C₈	—	34	1	no	—
167	C₂²	—	17	2	no	—
168	C₂²	—	34	1	no	—
169	C₂²	—	34	1	no	—
170	C₂²	(C₂×#3633)⋊C₄	1	1	yes	characteristic
171	C₄	—	2	1	no	—
172	C₄	—	2	1	no	—
173	C₄	—	2	1	no	—
174	C₄	—	17	1	no	—
175	C₄	—	17	1	no	—
176	C₄	—	34	1	no	—
177	C₄	—	34	1	no	—
178	C₄	—	68	4	no	—
179	C₂	—	34	2	no	—
180	C₂	#14545789	1	1	yes	characteristic
181	C₂	#14545863	1	1	yes	characteristic
182	C₂	#14545865	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	(C₂×#3633)⋊C₄
commutator	C₂×C₄×C₁₇	C₄×C₈
Frattini	C₄²	C₂×#3633
Fitting	C₁₇×#97	C₈
socle	C₂²×C₁₇	#336
prime radical	C₂²×D₁₇	C₂²⋊C₈
cosocle	C₄×#63242	C₂²
hypercenter	#97	#3633
hypocenter	C₁₇	#7150
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

Complements

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

N	H ≅ G/N	type
4. #11763912	C₄	semidirect
5. C₄⋊#63241	C₄	semidirect
124. C₁₇	#7150	semidirect

Maximal subgroups

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

subgroup	index
#14545987	2
#14545983	2
#14544442	2
#7150	17