Subgroups — #17187411

Order 4352.

590 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14530591	C₂	1	8	yes	maximal, complemented
2	#14530540	C₂	1	2	yes	maximal, complemented
3	#14530395	C₂	1	2	yes	maximal
4	C₄×#10587483	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₇×#405	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587483	C₂²	1	4	yes	—
7	C₄⋊#63214	C₂²	1	8	yes	complemented
8	C₂²⋊#63214	C₂²	1	8	yes	—
9	#11762585	C₂²	1	4	yes	complemented
10	C₄×#63214	C₂²	1	4	yes	complemented
11	C₂²×M₁₆×C₁₇	C₄	1	1	yes	characteristic
12	C₂³×C₈×C₁₇	C₂²	1	1	yes	characteristic
13	C₂×C₁₇×#117	C₄	1	1	yes	characteristic
14	C₁₇×#405	C₄	1	4	yes	—
15	C₁₇×#405	C₂²	1	4	yes	—
16	C₂×C₁₇×#97	C₄	1	1	yes	characteristic
17	C₂×C₁₇×#97	C₄	1	1	yes	characteristic
18	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
20	C₂³×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
21	C₂×M₁₆×C₁₇	—	2	4	no	—
22	C₂×M₁₆×C₁₇	C₂×C₄	1	4	yes	—
23	C₂²×C₈×C₁₇	—	2	4	no	—
24	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₇	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₇	C₂³	1	2	yes	—
29	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
30	C₁₇×#117	C₈	1	4	yes	—
31	C₁₇×#117	C₂×C₄	1	4	yes	—
32	C₂×C₄⋊C₄×C₁₇	C₈	1	2	yes	—
33	C₂×C₂²⋊C₄×C₁₇	C₈	1	2	yes	—
34	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
36	C₁₇×#97	C₂×C₄	1	4	yes	—
37	C₁₇×#97	C₂×C₄	1	4	yes	—
38	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
40	C₄×C₈×C₁₇	C₂³	1	2	yes	—
41	C₄×C₈×C₁₇	C₂³	1	2	yes	—
42	#10587483	—	2	8	no	—
43	C₂×#63214	C₂³	1	8	yes	—
44	C₂⁴×C₁₇	C₂×C₈	1	1	yes	characteristic
45	C₂²×C₄×C₁₇	—	2	2	no	—
46	C₂²×C₄×C₁₇	—	2	2	no	—
47	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
48	C₂²×C₄×C₁₇	C₂×C₈	1	2	yes	—
49	C₂²×C₄×C₁₇	C₂×C₈	1	2	yes	—
50	C₂²×C₄×C₁₇	C₂×C₈	1	2	yes	—
51	C₂²×C₄×C₁₇	C₂²×C₄	1	2	yes	—
52	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
53	M₁₆×C₁₇	—	2	8	no	—
54	C₂×C₈×C₁₇	—	2	2	no	—
55	C₂×C₈×C₁₇	—	2	2	no	—
56	C₂×C₈×C₁₇	—	2	4	no	—
57	C₂×C₈×C₁₇	—	2	4	no	—
58	C₂×C₈×C₁₇	M₁₆	1	4	yes	—
59	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
60	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
61	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
63	C₂×C₈×C₁₇	#41	1	4	yes	—
64	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
65	C₄⋊C₄×C₁₇	C₂×C₈	1	8	yes	—
66	C₂²⋊C₄×C₁₇	C₂×C₈	1	8	yes	—
67	C₄²×C₁₇	C₂×C₈	1	2	yes	—
68	C₄²×C₁₇	C₂×C₈	1	2	yes	—
69	C₄²×C₁₇	C₂²×C₄	1	2	yes	—
70	C₄²×C₁₇	C₂²×C₄	1	2	yes	—
71	#63214	—	2	8	no	—
72	#33419	—	17	1	no	maximal, Sylow
73	C₂³×C₁₇	—	2	2	no	—
74	C₂³×C₁₇	—	4	2	no	—
75	C₂³×C₁₇	C₂²×C₈	1	2	yes	—
76	C₂³×C₁₇	C₂²×C₈	1	1	yes	characteristic
77	C₂×C₄×C₁₇	—	2	2	no	—
78	C₂×C₄×C₁₇	—	2	2	no	—
79	C₂×C₄×C₁₇	—	2	2	no	—
80	C₂×C₄×C₁₇	—	2	4	no	—
81	C₂×C₄×C₁₇	—	4	2	no	—
82	C₂×C₄×C₁₇	—	4	2	no	—
83	C₂×C₄×C₁₇	#117	1	2	yes	—
84	C₂×C₄×C₁₇	C₂²×C₈	1	4	yes	—
85	C₂×C₄×C₁₇	C₂²×C₈	1	4	yes	—
86	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
87	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
88	C₂×C₄×C₁₇	C₂×M₁₆	1	2	yes	—
89	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
90	C₈×C₁₇	—	2	2	no	—
91	C₈×C₁₇	—	2	2	no	—
92	C₈×C₁₇	#117	1	2	yes	—
93	C₈×C₁₇	C₂×M₁₆	1	4	yes	—
94	C₈×C₁₇	C₂×#41	1	2	yes	—
95	C₂×#405	D₁₇	1	1	yes	characteristic
96	#2164	—	17	8	no	—
97	#2150	—	17	2	no	—
98	#2112	—	17	2	no	—
99	C₄×M₃₂	—	17	2	no	—
100	C₂²×C₁₇	—	2	2	no	—
101	C₂²×C₁₇	—	2	4	no	—
102	C₂²×C₁₇	—	2	2	no	—
103	C₂²×C₁₇	—	4	4	no	—
104	C₂²×C₁₇	#431	1	2	yes	—
105	C₂²×C₁₇	C₂³×C₈	1	1	yes	characteristic
106	C₄×C₁₇	—	2	4	no	—
107	C₄×C₁₇	—	2	4	no	—
108	C₄×C₁₇	—	2	2	no	—
109	C₄×C₁₇	#431	1	2	yes	—
110	C₄×C₁₇	C₂×#117	1	1	yes	characteristic
111	C₄×C₁₇	C₂²×M₁₆	1	1	yes	characteristic
112	C₂²×M₁₆	Dic₁₇	1	1	yes	characteristic
113	C₂³×C₈	C₂×D₁₇	1	1	yes	characteristic
114	C₂×#117	Dic₁₇	1	1	yes	characteristic
115	C₂×M₃₂	—	17	4	no	—
116	#405	Dic₁₇	1	4	yes	—
117	#405	C₂×D₁₇	1	4	yes	—
118	C₂×#97	Dic₁₇	1	1	yes	characteristic
119	C₂×#97	Dic₁₇	1	1	yes	characteristic
120	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
121	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
122	C₄⋊C₁₆	—	17	8	no	—
123	C₂²⋊C₁₆	—	17	8	no	—
124	#346	—	17	4	no	—
125	C₄×C₁₆	—	17	4	no	—
126	C₂×C₁₇	—	2	2	no	—
127	C₂×C₁₇	—	4	2	no	—
128	C₂×C₁₇	C₂×#431	1	1	yes	characteristic
129	C₂×C₁₇	#3407	1	2	yes	—
130	C₂³×C₄	C₂×Dic₁₇	1	1	yes	characteristic
131	C₂×M₁₆	—	2	4	no	—
132	C₂×M₁₆	C₂×Dic₁₇	1	4	yes	—
133	C₂²×C₈	—	2	4	no	—
134	C₂²×C₈	C₂×Dic₁₇	1	1	yes	characteristic
135	C₂²×C₈	C₂×Dic₁₇	1	2	yes	—
136	C₂²×C₈	C₂×Dic₁₇	1	1	yes	characteristic
137	C₂²×C₈	C₂×Dic₁₇	1	1	yes	characteristic
138	C₂²×C₈	C₂²×D₁₇	1	2	yes	—
139	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
140	#117	#3622	1	4	yes	—
141	#117	C₂×Dic₁₇	1	4	yes	—
142	C₂×C₄⋊C₄	#3622	1	2	yes	—
143	C₂×C₂²⋊C₄	#3622	1	2	yes	—
144	C₂×C₄²	C₂×Dic₁₇	1	1	yes	characteristic
145	C₂×C₄²	C₂×Dic₁₇	1	1	yes	characteristic
146	M₃₂	—	34	8	no	—
147	C₂×C₁₆	—	17	8	no	—
148	#97	C₂×Dic₁₇	1	4	yes	—
149	#97	C₂×Dic₁₇	1	4	yes	—
150	C₄×C₈	C₂×Dic₁₇	1	2	yes	—
151	C₄×C₈	C₂×Dic₁₇	1	2	yes	—
152	C₄×C₈	C₂²×D₁₇	1	2	yes	—
153	C₄×C₈	C₂²×D₁₇	1	2	yes	—
154	C₁₇	#33419	1	1	yes	characteristic, complemented, Sylow
155	C₂⁴	C₂×#3622	1	1	yes	characteristic
156	C₂²×C₄	—	2	2	no	—
157	C₂²×C₄	—	2	2	no	—
158	C₂²×C₄	C₂×#3622	1	1	yes	characteristic
159	C₂²×C₄	C₂×#3622	1	2	yes	—
160	C₂²×C₄	C₂×#3622	1	2	yes	—
161	C₂²×C₄	C₂×#3622	1	2	yes	—
162	C₂²×C₄	C₂²×Dic₁₇	1	2	yes	—
163	C₂²×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
164	M₁₆	—	2	8	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	2	4	no	—
168	C₂×C₈	—	2	4	no	—
169	C₂×C₈	#63223	1	4	yes	—
170	C₂×C₈	#63252	1	4	yes	—
171	C₂×C₈	C₂²×Dic₁₇	1	2	yes	—
172	C₂×C₈	C₂²×Dic₁₇	1	2	yes	—
173	C₂×C₈	C₂²×Dic₁₇	1	2	yes	—
174	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
175	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
176	C₄⋊C₄	C₂×#3622	1	8	yes	—
177	C₂²⋊C₄	C₂×#3622	1	8	yes	—
178	C₄²	C₂×#3622	1	2	yes	—
179	C₄²	C₂×#3622	1	2	yes	—
180	C₄²	C₂²×Dic₁₇	1	2	yes	—
181	C₄²	C₂²×Dic₁₇	1	2	yes	—
182	C₁₆	—	34	8	no	—
183	C₂³	—	2	2	no	—
184	C₂³	—	4	2	no	—
185	C₂³	C₂²×#3622	1	2	yes	—
186	C₂³	C₂²×#3622	1	1	yes	characteristic
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	4	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	C₂²×#3622	1	4	yes	—
194	C₂×C₄	C₂²×#3622	1	4	yes	—
195	C₂×C₄	C₂²×#3622	1	2	yes	—
196	C₂×C₄	C₂²×#3622	1	1	yes	characteristic
197	C₂×C₄	C₂×#63223	1	2	yes	—
198	C₂×C₄	#10587612	1	2	yes	—
199	C₂×C₄	C₂³×Dic₁₇	1	1	yes	characteristic
200	C₈	—	2	2	no	—
201	C₈	—	2	2	no	—
202	C₈	C₂×#63223	1	4	yes	—
203	C₈	#10587612	1	2	yes	—
204	C₈	C₂×#63252	1	2	yes	—
205	C₂²	—	2	2	no	—
206	C₂²	—	2	4	no	—
207	C₂²	—	2	2	no	—
208	C₂²	—	4	4	no	—
209	C₂²	#11762862	1	2	yes	—
210	C₂²	C₂³×#3622	1	1	yes	characteristic
211	C₄	—	2	4	no	—
212	C₄	—	2	4	no	—
213	C₄	—	2	2	no	—
214	C₄	#11762862	1	2	yes	—
215	C₄	C₂×#10587612	1	1	yes	characteristic
216	C₄	C₂²×#63223	1	1	yes	characteristic
217	C₂	—	2	2	no	—
218	C₂	—	4	2	no	—
219	C₂	C₂×#11762862	1	1	yes	characteristic
220	C₂	#14541887	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₁₇
commutator, socle	C₂²×C₁₇	C₂³×C₈
Fitting	C₂×C₁₇×#405	C₂
prime radical	C₂⁴×C₁₇	C₂×C₈
cosocle	C₂×C₈×C₁₇	C₂⁴
hypercenter	C₂×#405	D₁₇
hypocenter	C₁₇	#33419
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
1. #14530591	C₂	semidirect
2. #14530540	C₂	semidirect
4. C₄×#10587483	C₂	semidirect
7. C₄⋊#63214	C₂²	semidirect
9. #11762585	C₂²	semidirect
10. C₄×#63214	C₂²	semidirect
154. C₁₇	#33419	semidirect

Maximal subgroups

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

subgroup	index
#14530591	2
#14530540	2
#14530395	2
C₄×#10587483	2
C₂×C₁₇×#405	2
#33419	17