Subgroups — #17961031

Order 4352.

318 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14545117	C₂	1	1	yes	maximal, characteristic
2	C₂×#11763700	C₂	1	2	yes	maximal
3	#14544578	C₂	1	1	yes	maximal, characteristic
4	#14543824	C₂	1	2	yes	maximal
5	C₄×#10587469	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587530	C₂²	1	2	yes	—
7	#11763810	C₄	1	1	yes	characteristic
8	C₂×#10587532	C₂²	1	2	yes	—
9	C₄×C₄⋊#595	C₄	1	1	yes	characteristic
10	#11763730	C₄	1	2	yes	complemented
11	C₄⋊#63216	C₄	1	2	yes	complemented
12	#11763700	—	2	4	no	—
13	#11763674	C₄	1	2	yes	—
14	C₂×#10587469	C₂²	1	1	yes	characteristic
15	C₂×#10587469	C₂²	1	1	yes	characteristic
16	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
17	#11762584	C₄	1	1	yes	characteristic, complemented
18	#11762584	C₄	1	1	yes	characteristic, complemented
19	C₄×#63214	C₄	1	1	yes	characteristic, complemented
20	C₄×C₁₆×C₁₇	C₄	1	1	yes	characteristic, complemented
21	C₂×C₄⋊#595	—	2	1	no	—
22	C₂×C₄×#595	C₂×C₄	1	2	yes	—
23	C₂×#63242	C₂×C₄	1	2	yes	—
24	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×D₁₇	C₂³	1	1	yes	characteristic
26	C₄²×D₁₇	C₂×C₄	1	1	yes	characteristic
27	(C₂×#595)⋊C₄	—	2	1	no	—
28	#10587544	—	2	1	no	—
29	C₄⋊#3624	—	2	1	no	—
30	C₂×#63216	C₂×C₄	1	4	yes	—
31	#10587532	—	2	2	no	—
32	#10587532	—	2	2	no	—
33	#10587530	—	2	2	no	—
34	#10587530	—	2	2	no	—
35	C₂×C₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
38	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
39	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
40	C₂×#63214	C₂×C₄	1	1	yes	characteristic
41	C₂×#63214	C₂×C₄	1	1	yes	characteristic
42	C₄×#3622	C₂×C₄	1	1	yes	characteristic
43	#10587469	—	2	1	no	—
44	#10587469	—	2	1	no	—
45	#10587469	D₄	1	2	yes	—
46	#10587469	Q₈	1	2	yes	—
47	C₂²×#595	—	2	2	no	—
48	C₂×C₄×D₁₇	M₁₆	1	1	yes	characteristic
49	C₂×C₄×D₁₇	M₁₆	1	1	yes	characteristic
50	C₂×C₄×D₁₇	C₂²×C₄	1	1	yes	characteristic
51	C₂×#3624	—	2	2	no	—
52	C₄⋊#595	—	4	2	no	—
53	C₄×#595	—	2	2	no	—
54	C₄×#595	—	2	2	no	—
55	#63242	—	2	2	no	—
56	#63242	—	2	2	no	—
57	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
58	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
59	C₄²×C₁₇	C₄²	1	1	yes	characteristic
60	C₄×Dic₁₇	C₄²	1	1	yes	characteristic
61	C₄×Dic₁₇	M₁₆	1	1	yes	characteristic
62	C₄×Dic₁₇	M₁₆	1	1	yes	characteristic
63	C₂×#3622	C₄²	1	1	yes	characteristic
64	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
65	C₈×D₁₇	—	2	1	no	—
66	C₈×D₁₇	—	2	1	no	—
67	C₈×D₁₇	C₂×D₄	1	1	yes	characteristic
68	C₈×D₁₇	C₂×Q₈	1	1	yes	characteristic
69	C₈×D₁₇	#41	1	1	yes	characteristic
70	C₈×D₁₇	#41	1	1	yes	characteristic
71	#63216	—	2	4	no	—
72	C₁₆×C₁₇	—	2	1	no	—
73	C₁₆×C₁₇	C₄⋊C₄	1	2	yes	complemented
74	#63214	—	2	1	no	—
75	#63214	C₄⋊C₄	1	2	yes	complemented
76	#12666	—	17	1	no	maximal, Sylow
77	C₂²×D₁₇	C₂×M₁₆	1	1	yes	characteristic
78	C₂×#595	—	4	2	no	—
79	C₂×#595	—	4	2	no	—
80	C₂×#595	—	4	2	no	—
81	C₂×C₄×C₁₇	#97	1	1	yes	characteristic
82	C₂×C₄×C₁₇	#97	1	1	yes	characteristic
83	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
84	C₂×Dic₁₇	#97	1	1	yes	characteristic
85	C₂×Dic₁₇	#97	1	1	yes	characteristic
86	C₂×Dic₁₇	C₂×M₁₆	1	1	yes	characteristic
87	C₄×D₁₇	—	2	2	no	—
88	C₄×D₁₇	—	4	1	no	—
89	C₄×D₁₇	C₄×D₄	1	1	yes	characteristic
90	C₄×D₁₇	C₄×Q₈	1	1	yes	characteristic
91	C₄×D₁₇	#131	1	1	yes	characteristic
92	C₄×D₁₇	#131	1	1	yes	characteristic
93	#3624	—	4	2	no	—
94	C₈×C₁₇	—	2	1	no	—
95	C₈×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₈×C₁₇	#117	1	1	yes	characteristic
97	#3622	—	2	1	no	—
98	#3622	C₄×D₄	1	1	yes	characteristic
99	#3622	C₄×Q₈	1	1	yes	characteristic
100	#2150	—	17	1	no	—
101	C₂×#347	—	17	2	no	—
102	C₄×M₃₂	—	17	1	no	—
103	#1770	—	17	1	no	—
104	#1375	—	17	2	no	—
105	C₂²×C₁₇	C₂×#97	1	1	yes	characteristic
106	C₂×D₁₇	—	2	1	no	—
107	C₂×D₁₇	—	2	1	no	—
108	C₂×D₁₇	#436	1	1	yes	characteristic
109	C₂×D₁₇	#446	1	1	yes	characteristic
110	#595	—	8	2	no	—
111	C₄×C₁₇	—	2	1	no	—
112	C₄×C₁₇	#347	1	2	yes	complemented
113	C₄×C₁₇	C₄×C₄⋊C₄	1	1	yes	characteristic
114	C₄×C₁₇	#405	1	1	yes	characteristic
115	Dic₁₇	—	2	1	no	—
116	Dic₁₇	—	2	1	no	—
117	Dic₁₇	#436	1	1	yes	characteristic
118	Dic₁₇	#446	1	1	yes	characteristic
119	C₂×M₃₂	—	17	2	no	—
120	C₂×M₃₂	—	17	1	no	—
121	C₂×M₃₂	—	17	1	no	—
122	C₂×C₄⋊C₈	—	17	1	no	—
123	C₂×#97	—	17	2	no	—
124	C₂×C₄×C₈	—	17	1	no	—
125	C₄×C₄⋊C₄	—	17	1	no	—
126	C₄⋊C₁₆	—	17	2	no	—
127	C₄⋊C₁₆	—	17	2	no	—
128	#347	—	34	4	no	—
129	#346	—	17	1	no	—
130	#346	—	17	1	no	—
131	C₄×C₁₆	—	17	1	no	—
132	C₄×C₁₆	#595	1	1	yes	characteristic, complemented
133	#336	—	17	2	no	—
134	C₂×C₁₇	#1770	1	1	yes	characteristic
135	C₂×C₁₇	C₂×#347	1	1	yes	characteristic
136	C₂×C₁₇	#2110	1	1	yes	characteristic
137	D₁₇	—	2	1	no	—
138	D₁₇	—	2	1	no	—
139	C₂²×C₈	—	17	1	no	—
140	C₂²×C₈	—	17	1	no	—
141	C₂²×C₈	—	17	2	no	—
142	C₂×C₄⋊C₄	—	34	1	no	—
143	C₂×C₄²	—	17	1	no	—
144	C₂×C₄²	—	17	2	no	—
145	M₃₂	—	17	2	no	—
146	M₃₂	—	17	2	no	—
147	M₃₂	—	34	1	no	—
148	M₃₂	—	34	1	no	—
149	M₃₂	—	34	2	no	—
150	M₃₂	—	34	2	no	—
151	C₂×C₁₆	—	17	1	no	—
152	C₂×C₁₆	—	17	4	no	—
153	C₂×C₁₆	—	17	1	no	—
154	C₂×C₁₆	C₂×#595	1	1	yes	characteristic
155	C₂×C₁₆	C₂×#595	1	1	yes	characteristic
156	C₄⋊C₈	—	34	1	no	—
157	C₄⋊C₈	—	34	1	no	—
158	#97	—	34	2	no	—
159	#97	—	34	2	no	—
160	C₄×C₈	—	17	1	no	—
161	C₄×C₈	—	17	1	no	—
162	C₄×C₈	—	17	1	no	—
163	C₄×C₈	C₂×#595	1	1	yes	characteristic
164	#95	—	34	1	no	—
165	C₁₇	#12666	1	1	yes	characteristic, complemented, Sylow
166	C₂²×C₄	—	17	1	no	—
167	C₂²×C₄	—	17	1	no	—
168	C₂²×C₄	—	17	1	no	—
169	C₂²×C₄	—	34	2	no	—
170	C₂×C₈	—	17	1	no	—
171	C₂×C₈	—	17	1	no	—
172	C₂×C₈	—	17	1	no	—
173	C₂×C₈	—	17	1	no	—
174	C₂×C₈	—	17	1	no	—
175	C₂×C₈	—	17	1	no	—
176	C₂×C₈	—	34	1	no	—
177	C₂×C₈	—	34	1	no	—
178	C₂×C₈	—	34	2	no	—
179	C₂×C₈	—	34	2	no	—
180	C₂×C₈	—	34	2	no	—
181	C₂×C₈	C₄×#595	1	1	yes	characteristic
182	C₂×C₈	C₂²×#595	1	1	yes	characteristic
183	C₄⋊C₄	—	68	2	no	—
184	C₄²	—	17	1	no	—
185	C₄²	—	17	1	no	—
186	C₄²	—	17	1	no	—
187	C₄²	—	34	2	no	—
188	C₄²	—	34	2	no	—
189	C₄²	C₄×#595	1	1	yes	characteristic
190	C₁₆	—	2	1	no	—
191	C₁₆	—	17	2	no	—
192	C₁₆	—	34	4	no	—
193	C₁₆	—	34	1	no	—
194	C₁₆	C₄⋊#595	1	2	yes	complemented
195	C₂³	—	17	1	no	—
196	C₂×C₄	—	17	1	no	—
197	C₂×C₄	—	17	1	no	—
198	C₂×C₄	—	17	1	no	—
199	C₂×C₄	—	17	1	no	—
200	C₂×C₄	—	17	1	no	—
201	C₂×C₄	—	17	1	no	—
202	C₂×C₄	—	17	1	no	—
203	C₂×C₄	—	34	2	no	—
204	C₂×C₄	—	68	1	no	—
205	C₂×C₄	—	68	2	no	—
206	C₂×C₄	—	68	2	no	—
207	C₂×C₄	—	68	2	no	—
208	C₂×C₄	#10587532	1	1	yes	characteristic
209	C₂×C₄	#10587532	1	1	yes	characteristic
210	C₂×C₄	C₂×C₄×#595	1	1	yes	characteristic
211	C₈	—	2	1	no	—
212	C₈	—	17	1	no	—
213	C₈	—	17	1	no	—
214	C₈	—	34	1	no	—
215	C₈	—	68	2	no	—
216	C₈	C₂×C₄⋊#595	1	1	yes	characteristic
217	C₈	#10587677	1	1	yes	characteristic
218	C₂²	—	17	1	no	—
219	C₂²	—	17	1	no	—
220	C₂²	—	34	1	no	—
221	C₂²	—	34	1	no	—
222	C₂²	C₂×#10587532	1	1	yes	characteristic
223	C₄	—	2	1	no	—
224	C₄	—	17	1	no	—
225	C₄	—	17	1	no	—
226	C₄	—	34	1	no	—
227	C₄	—	34	1	no	—
228	C₄	—	136	2	no	—
229	C₄	#11763700	1	2	yes	complemented
230	C₄	C₄×C₄⋊#595	1	1	yes	characteristic
231	C₄	#11763772	1	1	yes	characteristic
232	C₂	—	34	1	no	—
233	C₂	—	34	1	no	—
234	C₂	#14544578	1	1	yes	characteristic
235	C₂	C₂×#11763700	1	1	yes	characteristic
236	C₂	#14545003	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄×#595
commutator	C₂×C₄×C₁₇	C₂×C₄²
Frattini	C₂×C₈	C₂²×#595
Fitting	C₄×C₁₆×C₁₇	C₄
socle	C₂²×C₁₇	C₂×#97
prime radical	C₂²×D₁₇	C₂×M₁₆
cosocle	C₂×C₈×D₁₇	C₂³
hypercenter	C₄×C₁₆	#595
hypocenter	C₁₇	#12666
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
10. #11763730	C₄	semidirect
11. C₄⋊#63216	C₄	semidirect
17. #11762584	C₄	semidirect
18. #11762584	C₄	semidirect
19. C₄×#63214	C₄	semidirect
20. C₄×C₁₆×C₁₇	C₄	semidirect
73. C₁₆×C₁₇	C₄⋊C₄	semidirect
75. #63214	C₄⋊C₄	semidirect
112. C₄×C₁₇	#347	semidirect
132. C₄×C₁₆	#595	semidirect
165. C₁₇	#12666	semidirect
194. C₁₆	C₄⋊#595	semidirect
229. C₄	#11763700	semidirect

Maximal subgroups

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

subgroup	index
#14545117	2
C₂×#11763700	2
#14544578	2
#14543824	2
C₄×#10587469	2
#12666	17