Subgroups — #17961013

Order 4352.

374 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11763696	C₂	1	2	yes	maximal
2	C₄×#10587529	C₂	1	1	yes	maximal, characteristic
3	C₈×C₄⋊#595	C₂	1	1	yes	maximal, characteristic
4	#14543821	C₂	1	2	yes	maximal
5	D₁₇×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587529	C₂²	1	1	yes	characteristic
7	C₂×#10587529	C₂²	1	1	yes	characteristic
8	#11763810	C₄	1	1	yes	characteristic
9	C₂×C₈×#595	C₂²	1	2	yes	—
10	C₄×C₄⋊#595	C₄	1	1	yes	characteristic
11	#11763696	—	2	4	no	—
12	#11763695	C₄	1	2	yes	complemented
13	C₄×#63216	C₄	1	2	yes	complemented
14	#11763674	C₄	1	2	yes	—
15	C₂×C₁₆×D₁₇	C₂²	1	2	yes	—
16	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
17	C₄⋊#63214	C₄	1	1	yes	characteristic, complemented
18	#11762627	C₄	1	2	yes	complemented
19	C₁₇×C₄⋊C₁₆	C₄	1	1	yes	characteristic, complemented
20	C₂×C₄⋊#595	C₈	1	2	yes	—
21	C₂×C₄×#595	C₂×C₄	1	2	yes	—
22	C₂×#63242	C₂×C₄	1	2	yes	—
23	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×D₁₇	C₂³	1	1	yes	characteristic
25	C₄²×D₁₇	C₂×C₄	1	1	yes	characteristic
26	(C₂×#595)⋊C₄	C₈	1	2	yes	—
27	#10587544	C₈	1	2	yes	—
28	C₄⋊#3624	C₈	1	2	yes	—
29	C₂×#63216	C₂×C₄	1	2	yes	—
30	C₂×#63216	C₂×C₄	1	2	yes	—
31	C₈×#595	—	2	2	no	—
32	C₈×#595	—	2	2	no	—
33	#10587529	—	2	1	no	—
34	#10587529	—	2	1	no	—
35	#10587529	D₄	1	2	yes	—
36	#10587529	Q₈	1	2	yes	—
37	C₂×C₁₆×C₁₇	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
39	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
40	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
41	C₂×#63214	C₂×C₄	1	2	yes	—
42	C₄×#3622	C₂×C₄	1	1	yes	characteristic
43	C₁₆×D₁₇	—	2	2	no	—
44	C₁₆×D₁₇	—	2	2	no	—
45	C₂²×#595	C₂×C₈	1	4	yes	—
46	C₂×C₄×D₁₇	C₂×C₈	1	1	yes	characteristic
47	C₂×C₄×D₁₇	C₂×C₈	1	1	yes	characteristic
48	C₂×C₄×D₁₇	C₂²×C₄	1	1	yes	characteristic
49	C₂×#3624	C₂×C₈	1	4	yes	—
50	C₄⋊#595	—	2	4	no	—
51	C₄×#595	—	2	2	no	—
52	C₄×#595	—	2	2	no	—
53	#63242	—	2	2	no	—
54	#63242	—	2	2	no	—
55	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
56	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
57	C₄²×C₁₇	C₄²	1	1	yes	characteristic
58	C₄×Dic₁₇	C₄²	1	1	yes	characteristic
59	C₄×Dic₁₇	C₂×C₈	1	1	yes	characteristic
60	C₄×Dic₁₇	C₂×C₈	1	1	yes	characteristic
61	C₂×#3622	C₄²	1	1	yes	characteristic
62	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
63	C₈×D₁₇	—	2	1	no	—
64	C₈×D₁₇	—	2	1	no	—
65	C₈×D₁₇	C₂×D₄	1	1	yes	characteristic
66	C₈×D₁₇	C₂×Q₈	1	1	yes	characteristic
67	C₈×D₁₇	#41	1	1	yes	characteristic
68	C₈×D₁₇	#41	1	1	yes	characteristic
69	#63216	—	2	2	no	—
70	#63216	C₄⋊C₄	1	4	yes	complemented
71	C₁₆×C₁₇	—	2	2	no	—
72	#63214	—	2	2	no	—
73	#12662	—	17	1	no	maximal, Sylow
74	C₂²×D₁₇	C₂²×C₈	1	1	yes	characteristic
75	C₂×#595	—	2	4	no	—
76	C₂×#595	—	2	4	no	—
77	C₂×#595	—	2	4	no	—
78	C₂×C₄×C₁₇	C₄×C₈	1	1	yes	characteristic
79	C₂×C₄×C₁₇	C₄×C₈	1	1	yes	characteristic
80	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
81	C₂×Dic₁₇	C₄×C₈	1	1	yes	characteristic
82	C₂×Dic₁₇	C₄×C₈	1	1	yes	characteristic
83	C₂×Dic₁₇	C₂²×C₈	1	1	yes	characteristic
84	C₄×D₁₇	—	2	2	no	—
85	C₄×D₁₇	M₃₂	1	4	yes	—
86	C₄×D₁₇	C₄×D₄	1	1	yes	characteristic
87	C₄×D₁₇	C₄×Q₈	1	1	yes	characteristic
88	C₄×D₁₇	#131	1	1	yes	characteristic
89	C₄×D₁₇	#131	1	1	yes	characteristic
90	#3624	—	2	4	no	—
91	C₈×C₁₇	—	2	1	no	—
92	C₈×C₁₇	C₄×D₄	1	1	yes	characteristic
93	C₈×C₁₇	C₄×Q₈	1	1	yes	characteristic
94	#3622	—	2	1	no	—
95	#3622	C₂×C₄⋊C₄	1	1	yes	characteristic
96	#3622	#117	1	1	yes	characteristic
97	C₂×C₄⋊C₁₆	—	17	1	no	—
98	C₂×#346	—	17	2	no	—
99	C₂×C₄×C₁₆	—	17	1	no	—
100	C₈×C₄⋊C₄	—	17	1	no	—
101	#1374	—	17	2	no	—
102	C₂²×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
103	C₂×D₁₇	C₈×D₄	1	1	yes	characteristic
104	C₂×D₁₇	C₈×Q₈	1	1	yes	characteristic
105	C₂×D₁₇	C₂×M₃₂	1	2	yes	—
106	C₂×D₁₇	#504	1	2	yes	—
107	#595	—	4	4	no	—
108	C₄×C₁₇	—	2	1	no	—
109	C₄×C₁₇	#346	1	2	yes	complemented
110	C₄×C₁₇	C₄×C₄⋊C₄	1	1	yes	characteristic
111	C₄×C₁₇	#405	1	1	yes	characteristic
112	Dic₁₇	—	2	1	no	—
113	Dic₁₇	#346	1	2	yes	complemented
114	Dic₁₇	C₈×D₄	1	1	yes	characteristic
115	Dic₁₇	C₈×Q₈	1	1	yes	characteristic
116	C₂²×C₁₆	—	17	1	no	—
117	C₂²×C₁₆	—	17	1	no	—
118	C₂²×C₁₆	—	17	2	no	—
119	C₂×C₄⋊C₈	—	17	1	no	—
120	C₂×C₄×C₈	—	17	1	no	—
121	C₂×C₄×C₈	—	17	2	no	—
122	C₄×C₄⋊C₄	—	17	1	no	—
123	C₄⋊C₁₆	—	17	2	no	—
124	C₄⋊C₁₆	—	17	1	no	—
125	C₄⋊C₁₆	#595	1	1	yes	characteristic, complemented
126	#346	—	34	4	no	—
127	C₄×C₁₆	—	17	2	no	—
128	C₄×C₁₆	—	17	2	no	—
129	#336	—	17	2	no	—
130	C₂×C₁₇	C₈×C₄⋊C₄	1	1	yes	characteristic
131	C₂×C₁₇	C₂×#346	1	1	yes	characteristic
132	C₂×C₁₇	#2108	1	1	yes	characteristic
133	D₁₇	#2169	1	2	yes	—
134	D₁₇	#2183	1	2	yes	—
135	C₂²×C₈	—	17	1	no	—
136	C₂²×C₈	—	17	2	no	—
137	C₂²×C₈	—	17	1	no	—
138	C₂×C₄⋊C₄	—	17	2	no	—
139	C₂×C₄²	—	17	1	no	—
140	C₂×C₄²	—	17	2	no	—
141	C₂×C₁₆	—	17	2	no	—
142	C₂×C₁₆	—	17	2	no	—
143	C₂×C₁₆	—	17	2	no	—
144	C₂×C₁₆	—	17	2	no	—
145	C₂×C₁₆	—	17	2	no	—
146	C₂×C₁₆	—	34	2	no	—
147	C₂×C₁₆	—	34	2	no	—
148	C₂×C₁₆	—	34	1	no	—
149	C₂×C₁₆	—	34	1	no	—
150	C₂×C₁₆	C₂×#595	1	2	yes	—
151	C₄⋊C₈	—	17	2	no	—
152	C₄⋊C₈	—	17	2	no	—
153	C₄×C₈	—	17	1	no	—
154	C₄×C₈	—	17	1	no	—
155	C₄×C₈	—	17	1	no	—
156	C₄×C₈	—	34	2	no	—
157	C₄×C₈	—	34	2	no	—
158	C₄×C₈	C₂×#595	1	1	yes	characteristic
159	#95	—	17	2	no	—
160	C₁₇	#12662	1	1	yes	characteristic, complemented, Sylow
161	C₂²×C₄	—	17	1	no	—
162	C₂²×C₄	—	17	1	no	—
163	C₂²×C₄	—	17	1	no	—
164	C₂²×C₄	—	17	4	no	—
165	C₂×C₈	—	17	1	no	—
166	C₂×C₈	—	17	1	no	—
167	C₂×C₈	—	17	1	no	—
168	C₂×C₈	—	17	1	no	—
169	C₂×C₈	—	17	1	no	—
170	C₂×C₈	—	17	4	no	—
171	C₂×C₈	—	17	1	no	—
172	C₂×C₈	—	34	1	no	—
173	C₂×C₈	—	34	1	no	—
174	C₂×C₈	—	34	2	no	—
175	C₂×C₈	—	34	2	no	—
176	C₂×C₈	C₄×#595	1	1	yes	characteristic
177	C₂×C₈	C₂²×#595	1	1	yes	characteristic
178	C₄⋊C₄	—	34	4	no	—
179	C₄²	—	17	1	no	—
180	C₄²	—	17	1	no	—
181	C₄²	—	17	1	no	—
182	C₄²	—	34	2	no	—
183	C₄²	—	34	2	no	—
184	C₄²	C₄×#595	1	1	yes	characteristic
185	C₁₆	—	2	2	no	—
186	C₁₆	—	17	4	no	—
187	C₁₆	—	34	2	no	—
188	C₁₆	—	34	2	no	—
189	C₂³	—	17	1	no	—
190	C₂×C₄	—	17	1	no	—
191	C₂×C₄	—	17	1	no	—
192	C₂×C₄	—	17	1	no	—
193	C₂×C₄	—	17	1	no	—
194	C₂×C₄	—	17	1	no	—
195	C₂×C₄	—	17	1	no	—
196	C₂×C₄	—	17	4	no	—
197	C₂×C₄	—	17	1	no	—
198	C₂×C₄	—	34	2	no	—
199	C₂×C₄	—	34	4	no	—
200	C₂×C₄	—	34	4	no	—
201	C₂×C₄	—	34	4	no	—
202	C₂×C₄	C₈×#595	1	1	yes	characteristic
203	C₂×C₄	C₈×#595	1	1	yes	characteristic
204	C₂×C₄	C₂×C₄×#595	1	1	yes	characteristic
205	C₈	—	2	1	no	—
206	C₈	—	17	1	no	—
207	C₈	—	17	1	no	—
208	C₈	—	34	4	no	—
209	C₈	—	34	1	no	—
210	C₈	D₄×#595	1	1	yes	characteristic
211	C₈	Q₈×#595	1	1	yes	characteristic
212	C₂²	—	17	2	no	—
213	C₂²	—	17	2	no	—
214	C₂²	—	17	1	no	—
215	C₂²	—	17	1	no	—
216	C₂²	C₂×C₈×#595	1	1	yes	characteristic
217	C₄	—	2	1	no	—
218	C₄	—	17	2	no	—
219	C₄	—	17	1	no	—
220	C₄	—	17	1	no	—
221	C₄	—	34	1	no	—
222	C₄	—	68	4	no	—
223	C₄	#11763696	1	2	yes	complemented
224	C₄	C₄⋊C₄×#595	1	1	yes	characteristic
225	C₄	#11763775	1	1	yes	characteristic
226	C₂	—	17	2	no	—
227	C₂	—	17	2	no	—
228	C₂	#14544577	1	1	yes	characteristic
229	C₂	C₂×#11763696	1	1	yes	characteristic
230	C₂	#14544996	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×#595
commutator, socle	C₂²×C₁₇	C₂×C₄×C₈
Fitting	C₁₇×C₄⋊C₁₆	C₄
prime radical	C₂²×D₁₇	C₂²×C₈
cosocle	C₂×C₈×D₁₇	C₂³
hypercenter	C₄⋊C₁₆	#595
hypocenter	C₁₇	#12662
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₈ ◁ C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
12. #11763695	C₄	semidirect
13. C₄×#63216	C₄	semidirect
17. C₄⋊#63214	C₄	semidirect
18. #11762627	C₄	semidirect
19. C₁₇×C₄⋊C₁₆	C₄	semidirect
70. #63216	C₄⋊C₄	semidirect
109. C₄×C₁₇	#346	semidirect
113. Dic₁₇	#346	semidirect
125. C₄⋊C₁₆	#595	semidirect
160. C₁₇	#12662	semidirect
223. C₄	#11763696	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11763696	2
C₄×#10587529	2
C₈×C₄⋊#595	2
#14543821	2
D₁₇×C₄⋊C₁₆	2
#12662	17