Subgroups — #14545538

Order 2176.

492 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41×#595	C₂	1	2	yes	maximal, complemented
2	#11763850	C₂	1	2	yes	maximal, complemented
3	#11763849	C₂	1	2	yes	maximal, complemented
4	#11763848	C₂	1	4	yes	maximal, complemented
5	#11763819	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763799	C₂	1	2	yes	maximal, complemented
7	C₂×#10587532	C₂	1	1	yes	maximal, characteristic, complemented
8	#135×D₁₇	C₂	1	1	yes	maximal, characteristic
9	#41×D₁₇	C₂²	1	2	yes	—
10	Q₈×#595	C₂²	1	4	yes	—
11	D₄×#595	—	2	4	no	—
12	D₄×#595	C₂²	1	4	yes	—
13	#10587677	—	2	2	no	—
14	#10587677	C₂²	1	2	yes	—
15	C₂×C₄×#595	—	2	2	no	—
16	C₂×C₄×#595	C₂²	1	1	yes	characteristic
17	C₂×#63242	C₂²	1	1	yes	characteristic
18	C₁₇×#135	C₄	1	1	yes	characteristic, complemented
19	#10587636	C₄	1	2	yes	complemented
20	#10587605	C₄	1	1	yes	characteristic, complemented
21	Q₁₆×D₁₇	C₂²	1	1	yes	characteristic
22	#10587602	C₄	1	2	yes	complemented
23	SD₁₆×D₁₇	C₂²	1	2	yes	—
24	#10587598	C₄	1	1	yes	characteristic, complemented
25	D₈×D₁₇	C₂²	1	1	yes	characteristic
26	#10587590	C₄	1	1	yes	characteristic, complemented
27	C₂×C₈×D₁₇	C₂²	1	1	yes	characteristic
28	#10587549	C₂²	1	4	yes	—
29	#10587547	C₂²	1	4	yes	—
30	#10587534	C₂²	1	2	yes	—
31	#10587533	C₂²	1	2	yes	—
32	#10587532	C₂²	1	2	yes	—
33	#10587532	C₂²	1	2	yes	—
34	C₂²×#595	—	2	4	no	—
35	C₂²×#595	D₄	1	2	yes	—
36	#41×C₁₇	C₂×C₄	1	2	yes	complemented
37	#63256	C₂×C₄	1	2	yes	complemented
38	Q₈×D₁₇	C₂³	1	2	yes	—
39	#63254	—	2	2	no	—
40	#63254	C₂×C₄	1	2	yes	complemented
41	D₄×D₁₇	—	2	2	no	—
42	D₄×D₁₇	C₂³	1	2	yes	—
43	#63252	—	2	2	no	—
44	#63252	C₂×C₄	1	2	yes	complemented
45	C₂×C₄×D₁₇	—	2	2	no	—
46	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
47	C₂²⋊#595	—	2	2	no	—
48	C₂²⋊#595	—	2	4	no	—
49	C₂×#3624	—	2	1	no	—
50	C₄⋊#595	—	2	4	no	—
51	C₄⋊#595	C₂³	1	4	yes	—
52	C₄×#595	—	2	4	no	—
53	C₄×#595	—	2	2	no	—
54	C₄×#595	D₄	1	2	yes	—
55	C₄×#595	C₂³	1	2	yes	—
56	#63242	—	2	1	no	—
57	#63242	C₂³	1	2	yes	—
58	Q₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic, complemented
59	SD₁₆×C₁₇	C₂×C₄	1	2	yes	complemented
60	D₈×C₁₇	C₂×C₄	1	1	yes	characteristic, complemented
61	C₂×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic, complemented
62	#63231	C₂×C₄	1	2	yes	complemented
63	#63230	C₂×C₄	1	2	yes	complemented
64	#63229	C₂×C₄	1	2	yes	complemented
65	#63228	C₂×C₄	1	2	yes	complemented
66	C₂×#3622	C₂×C₄	1	1	yes	characteristic, complemented
67	Dic₆₈	C₂×C₄	1	1	yes	characteristic, complemented
68	D₁₃₆	C₂×C₄	1	1	yes	characteristic, complemented
69	#63219	C₂×C₄	1	2	yes	complemented
70	C₈×D₁₇	C₂×C₄	1	1	yes	characteristic, complemented
71	C₈×D₁₇	C₂×C₄	1	1	yes	characteristic, complemented
72	C₈×D₁₇	C₂³	1	1	yes	characteristic
73	C₈×D₁₇	C₂³	1	1	yes	characteristic
74	C₂²×D₁₇	—	2	2	no	—
75	C₂²×D₁₇	C₂×D₄	1	1	yes	characteristic
76	C₂×#595	—	2	4	no	—
77	C₂×#595	—	4	2	no	—
78	C₂×#595	—	4	8	no	—
79	C₂×#595	C₂×D₄	1	4	yes	—
80	Q₈×C₁₇	C₂²×C₄	1	2	yes	—
81	D₄×C₁₇	—	2	2	no	—
82	D₄×C₁₇	C₂²×C₄	1	2	yes	—
83	C₂×C₄×C₁₇	—	2	2	no	—
84	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
85	C₂²⋊D₁₇	—	2	2	no	—
86	C₂²⋊D₁₇	—	2	2	no	—
87	C₂²⋊D₁₇	—	2	2	no	—
88	C₂×Dic₁₇	—	2	2	no	—
89	C₂×Dic₁₇	#41	1	1	yes	characteristic
90	D₆₈	—	2	2	no	—
91	D₆₈	C₂²×C₄	1	2	yes	—
92	C₄×D₁₇	—	2	2	no	—
93	C₄×D₁₇	—	2	2	no	—
94	C₄×D₁₇	—	2	2	no	—
95	C₄×D₁₇	C₂²×C₄	1	1	yes	characteristic
96	C₄×D₁₇	C₂×D₄	1	1	yes	characteristic
97	C₄×D₁₇	#41	1	1	yes	characteristic
98	C₄×D₁₇	C₂⁴	1	1	yes	characteristic
99	Dic₃₄	—	2	2	no	—
100	Dic₃₄	C₂²×C₄	1	2	yes	—
101	#3624	—	2	2	no	—
102	C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
103	C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
104	#3622	C₂²×C₄	1	1	yes	characteristic
105	#3622	C₂²×C₄	1	1	yes	characteristic
106	#2943	—	17	1	no	maximal, Sylow
107	C₂²×C₁₇	—	2	2	no	—
108	C₂²×C₁₇	C₄×D₄	1	1	yes	characteristic
109	C₂×D₁₇	—	2	2	no	—
110	C₂×D₁₇	—	2	2	no	—
111	C₂×D₁₇	—	4	4	no	—
112	C₂×D₁₇	C₄×D₄	1	1	yes	characteristic
113	C₂×D₁₇	C₂²×D₄	1	1	yes	characteristic
114	#595	—	2	4	no	—
115	#595	—	4	4	no	—
116	C₄×C₁₇	—	2	2	no	—
117	C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
118	C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
119	Dic₁₇	—	2	2	no	—
120	Dic₁₇	C₄×D₄	1	1	yes	characteristic
121	Dic₁₇	C₂×#41	1	1	yes	characteristic
122	C₂×#135	—	17	1	no	—
123	C₄×#41	—	17	2	no	—
124	#442	—	17	2	no	—
125	#441	—	17	2	no	—
126	#440	—	17	4	no	—
127	#427	—	17	1	no	—
128	#416	—	17	2	no	—
129	C₂×#97	—	17	1	no	—
130	C₂×C₁₇	—	2	1	no	—
131	C₂×C₁₇	—	4	2	no	—
132	C₂×C₁₇	C₂×C₄×D₄	1	1	yes	characteristic
133	D₁₇	—	2	1	no	—
134	D₁₇	—	4	2	no	—
135	D₁₇	#575	1	2	yes	—
136	C₂×#41	—	17	2	no	—
137	#135	—	17	2	no	—
138	#135	—	17	1	no	—
139	#135	—	17	2	no	—
140	#135	—	17	1	no	—
141	#135	—	17	1	no	—
142	#135	#595	1	1	yes	characteristic, complemented
143	C₂×Q₁₆	—	17	1	no	—
144	C₂×SD₁₆	—	17	2	no	—
145	C₂×D₈	—	17	1	no	—
146	C₂²×C₈	—	17	1	no	—
147	C₂²×C₈	—	17	1	no	—
148	C₄×Q₈	—	17	4	no	—
149	C₄×D₄	—	17	4	no	—
150	C₄×D₄	—	34	4	no	—
151	#117	—	17	2	no	—
152	#117	—	34	2	no	—
153	C₂×C₄²	—	17	1	no	—
154	C₂×C₄²	—	34	2	no	—
155	#107	—	17	2	no	—
156	#106	—	17	2	no	—
157	#103	—	17	4	no	—
158	#102	—	17	4	no	—
159	#97	—	17	2	no	—
160	#97	—	17	2	no	—
161	C₁₇	#2943	1	1	yes	characteristic, complemented, Sylow
162	#41	—	17	2	no	—
163	#41	—	17	2	no	—
164	#41	—	17	2	no	—
165	#41	—	34	2	no	—
166	#41	—	34	2	no	—
167	#41	C₂×#595	1	2	yes	complemented
168	C₂×Q₈	—	17	2	no	—
169	C₂×D₄	—	17	2	no	—
170	C₂×D₄	—	34	2	no	—
171	C₂²×C₄	—	17	1	no	—
172	C₂²×C₄	—	17	2	no	—
173	C₂²×C₄	—	34	2	no	—
174	C₂²×C₄	—	34	4	no	—
175	Q₁₆	—	17	2	no	—
176	Q₁₆	—	17	1	no	—
177	Q₁₆	C₂×#595	1	1	yes	characteristic, complemented
178	SD₁₆	—	17	2	no	—
179	SD₁₆	—	17	2	no	—
180	SD₁₆	—	17	2	no	—
181	SD₁₆	C₂×#595	1	2	yes	complemented
182	D₈	—	17	2	no	—
183	D₈	—	17	1	no	—
184	D₈	C₂×#595	1	1	yes	characteristic, complemented
185	C₂×C₈	—	17	1	no	—
186	C₂×C₈	—	17	1	no	—
187	C₂×C₈	—	17	1	no	—
188	C₂×C₈	—	17	1	no	—
189	C₂×C₈	—	17	1	no	—
190	C₂×C₈	—	17	2	no	—
191	C₂×C₈	—	34	1	no	—
192	C₂×C₈	—	34	1	no	—
193	C₂×C₈	C₂×#595	1	1	yes	characteristic, complemented
194	C₄⋊C₄	—	17	4	no	—
195	C₄⋊C₄	—	34	4	no	—
196	C₂²⋊C₄	—	34	4	no	—
197	C₂²⋊C₄	—	34	2	no	—
198	C₄²	—	17	2	no	—
199	C₄²	—	17	2	no	—
200	C₄²	—	34	2	no	—
201	C₄²	—	34	4	no	—
202	C₂³	—	17	1	no	—
203	C₂³	—	34	2	no	—
204	Q₈	—	17	2	no	—
205	Q₈	—	34	2	no	—
206	Q₈	C₂²×#595	1	2	yes	—
207	D₄	—	2	2	no	—
208	D₄	—	17	2	no	—
209	D₄	—	34	2	no	—
210	D₄	—	34	2	no	—
211	D₄	—	34	2	no	—
212	D₄	—	34	2	no	—
213	D₄	C₂²×#595	1	2	yes	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	17	1	no	—
216	C₂×C₄	—	17	1	no	—
217	C₂×C₄	—	17	1	no	—
218	C₂×C₄	—	17	1	no	—
219	C₂×C₄	—	17	1	no	—
220	C₂×C₄	—	17	4	no	—
221	C₂×C₄	—	34	2	no	—
222	C₂×C₄	—	34	2	no	—
223	C₂×C₄	—	34	2	no	—
224	C₂×C₄	—	34	2	no	—
225	C₂×C₄	—	34	4	no	—
226	C₂×C₄	—	68	2	no	—
227	C₂×C₄	—	68	8	no	—
228	C₂×C₄	C₂²×#595	1	1	yes	characteristic
229	C₈	—	17	1	no	—
230	C₈	—	17	1	no	—
231	C₈	—	34	2	no	—
232	C₈	C₂²×#595	1	1	yes	characteristic
233	C₈	C₂²×#595	1	1	yes	characteristic
234	C₂²	—	2	2	no	—
235	C₂²	—	17	1	no	—
236	C₂²	—	17	1	no	—
237	C₂²	—	34	2	no	—
238	C₂²	—	34	2	no	—
239	C₂²	—	68	4	no	—
240	C₂²	D₄×#595	1	1	yes	characteristic
241	C₄	—	2	2	no	—
242	C₄	—	17	1	no	—
243	C₄	—	17	1	no	—
244	C₄	—	34	4	no	—
245	C₄	—	34	2	no	—
246	C₄	—	68	4	no	—
247	C₄	D₄×#595	1	1	yes	characteristic
248	C₄	C₂³×#595	1	1	yes	characteristic
249	C₂	—	2	1	no	—
250	C₂	—	4	2	no	—
251	C₂	—	17	2	no	—
252	C₂	—	34	1	no	—
253	C₂	—	68	2	no	—
254	C₂	C₂×D₄×#595	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄×#595
commutator	C₄×C₁₇	C₂³×C₄
Frattini	C₄	C₂³×#595
Fitting	C₁₇×#135	C₄
socle	C₂×C₁₇	C₂×C₄×D₄
prime radical	#135×D₁₇	C₂
cosocle	C₄×D₁₇	C₂⁴
hypercenter	#135	#595
hypocenter	C₁₇	#2943
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #135

Complements

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

N	H ≅ G/N	type
1. #41×#595	C₂	semidirect
2. #11763850	C₂	semidirect
3. #11763849	C₂	semidirect
4. #11763848	C₂	semidirect
5. #11763819	C₂	semidirect
6. #11763799	C₂	semidirect
7. C₂×#10587532	C₂	semidirect
18. C₁₇×#135	C₄	semidirect
19. #10587636	C₄	semidirect
20. #10587605	C₄	semidirect
22. #10587602	C₄	semidirect
24. #10587598	C₄	semidirect
26. #10587590	C₄	semidirect
36. #41×C₁₇	C₂×C₄	semidirect
37. #63256	C₂×C₄	semidirect
40. #63254	C₂×C₄	semidirect
44. #63252	C₂×C₄	semidirect
58. Q₁₆×C₁₇	C₂×C₄	semidirect
59. SD₁₆×C₁₇	C₂×C₄	semidirect
60. D₈×C₁₇	C₂×C₄	semidirect
61. C₂×C₈×C₁₇	C₂×C₄	semidirect
62. #63231	C₂×C₄	semidirect
63. #63230	C₂×C₄	semidirect
64. #63229	C₂×C₄	semidirect
65. #63228	C₂×C₄	semidirect
66. C₂×#3622	C₂×C₄	semidirect
67. Dic₆₈	C₂×C₄	semidirect
68. D₁₃₆	C₂×C₄	semidirect
69. #63219	C₂×C₄	semidirect
70. C₈×D₁₇	C₂×C₄	semidirect
71. C₈×D₁₇	C₂×C₄	semidirect
142. #135	#595	semidirect
161. C₁₇	#2943	semidirect
167. #41	C₂×#595	semidirect
177. Q₁₆	C₂×#595	semidirect
181. SD₁₆	C₂×#595	semidirect
184. D₈	C₂×#595	semidirect
193. C₂×C₈	C₂×#595	semidirect

Maximal subgroups

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

subgroup	index
#41×#595	2
#11763850	2
#11763849	2
#11763848	2
#11763819	2
#11763799	2
C₂×#10587532	2
#135×D₁₇	2
#2943	17