Subgroups — #14541187

Order 2176.

366 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763505	C₂	1	2	yes	maximal, complemented
2	#11763211	C₂	1	1	yes	maximal, characteristic, complemented
3	#11763134	C₂	1	2	yes	maximal, complemented
4	#11763128	C₂	1	2	yes	maximal, complemented
5	#11763110	C₂	1	2	yes	maximal, complemented
6	#11763045	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763045	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×D₁₃₆	C₂	1	1	yes	maximal, characteristic, complemented
9	#11762847	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#10587489	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₄×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₈×C₁₇	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₁₇	C₂²	1	1	yes	characteristic
14	C₄×D₄×C₁₇	C₂²	1	1	yes	characteristic
15	#10587625	—	2	2	no	—
16	#10587616	C₂²	1	2	yes	—
17	#10587616	C₂²	1	2	yes	—
18	#10587612	C₂²	1	2	yes	—
19	C₂×C₄⋊Dic₁₇	C₂²	1	2	yes	—
20	C₂×D₁₃₆	C₂²	1	1	yes	characteristic
21	#10587584	—	2	2	no	—
22	#10587569	—	2	2	no	—
23	#10587567	—	2	2	no	—
24	C₄×D₆₈	C₂²	1	2	yes	—
25	#10587556	C₂²	1	2	yes	—
26	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
27	C₁₇×C₂²⋊C₈	C₂²	1	1	yes	characteristic
28	C₁₇×C₂²⋊C₈	C₂²	1	1	yes	characteristic
29	C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
30	#10587492	C₂²	1	2	yes	—
31	#10587492	C₂²	1	2	yes	—
32	#10587492	C₂²	1	2	yes	—
33	#10587489	C₂²	1	1	yes	characteristic
34	#10587489	C₂²	1	1	yes	characteristic
35	#10587489	C₂²	1	1	yes	characteristic
36	#10587489	C₂²	1	2	yes	—
37	#10587488	C₂²	1	2	yes	—
38	#10587488	C₂²	1	2	yes	—
39	C₂×D₄×C₁₇	D₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
42	C₂×C₂²⋊D₁₇	—	2	2	no	—
43	C₂×C₂²⋊D₁₇	—	2	2	no	—
44	C₂²×Dic₁₇	—	2	2	no	—
45	C₂×D₆₈	C₂³	1	2	yes	—
46	C₂×C₄×D₁₇	—	2	2	no	—
47	C₂×C₈×C₁₇	—	2	2	no	—
48	C₂×C₈×C₁₇	—	2	2	no	—
49	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
50	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
51	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
52	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
53	C₄⋊C₄×C₁₇	D₄	1	1	yes	characteristic
54	C₂²⋊C₄×C₁₇	D₄	1	1	yes	characteristic
55	C₂²⋊C₄×C₁₇	D₄	1	1	yes	characteristic
56	C₄²×C₁₇	C₂³	1	1	yes	characteristic
57	C₂²⋊Dic₁₇	—	2	2	no	—
58	#63227	—	2	2	no	—
59	#63227	—	2	2	no	—
60	#63227	—	2	2	no	—
61	C₄⋊Dic₁₇	—	2	2	no	—
62	C₄⋊Dic₁₇	C₂³	1	2	yes	—
63	C₄⋊Dic₁₇	C₂³	1	2	yes	—
64	C₄⋊Dic₁₇	C₂³	1	2	yes	—
65	#63225	—	2	2	no	—
66	#63225	—	2	2	no	—
67	C₄×Dic₁₇	—	2	2	no	—
68	D₁₃₆	—	2	2	no	—
69	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
70	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×D₁₇	—	2	2	no	—
72	D₄×C₁₇	—	2	2	no	—
73	C₂×C₄×C₁₇	—	2	1	no	—
74	C₂×C₄×C₁₇	—	2	1	no	—
75	C₂×C₄×C₁₇	—	2	1	no	—
76	C₂×C₄×C₁₇	—	2	1	no	—
77	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
78	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
79	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
80	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
82	C₂²⋊D₁₇	—	4	4	no	—
83	C₂²⋊D₁₇	—	8	2	no	—
84	C₂×Dic₁₇	—	2	2	no	—
85	C₂×Dic₁₇	—	2	2	no	—
86	C₂×Dic₁₇	—	2	2	no	—
87	C₂×Dic₁₇	—	8	2	no	—
88	D₆₈	—	2	2	no	—
89	D₆₈	—	2	2	no	—
90	C₄×D₁₇	—	8	2	no	—
91	C₈×C₁₇	—	2	1	no	—
92	C₈×C₁₇	—	2	1	no	—
93	C₈×C₁₇	—	2	1	no	—
94	C₈×C₁₇	#41	1	2	yes	—
95	#3339	—	17	1	no	maximal, Sylow
96	C₂²×C₁₇	—	2	1	no	—
97	C₂²×C₁₇	—	2	1	no	—
98	C₂²×C₁₇	—	2	1	no	—
99	C₂²×C₁₇	—	2	1	no	—
100	C₂²×C₁₇	—	2	1	no	—
101	C₂²×C₁₇	#135	1	2	yes	—
102	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
103	C₂×D₁₇	—	4	2	no	—
104	C₂×D₁₇	—	4	2	no	—
105	C₂×D₁₇	—	4	2	no	—
106	C₄×C₁₇	—	2	1	no	—
107	C₄×C₁₇	—	2	1	no	—
108	C₄×C₁₇	—	2	1	no	—
109	C₄×C₁₇	—	2	1	no	—
110	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
111	C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
112	Dic₁₇	—	4	2	no	—
113	Dic₁₇	—	4	2	no	—
114	Dic₁₇	—	4	2	no	—
115	#553	—	17	2	no	—
116	#499	—	17	1	no	—
117	#482	—	17	2	no	—
118	#481	—	17	2	no	—
119	#478	—	17	2	no	—
120	#466	—	17	1	no	—
121	#466	—	17	1	no	—
122	C₄×D₈	—	17	1	no	—
123	C₈×D₄	D₁₇	1	1	yes	characteristic, complemented
124	#427	—	17	1	no	—
125	C₂×#107	—	17	1	no	—
126	C₂×C₁₇	—	2	2	no	—
127	C₂×C₁₇	—	4	1	no	—
128	C₂×C₁₇	#547	1	1	yes	characteristic
129	C₂×C₁₇	C₂×#135	1	1	yes	characteristic
130	C₂×C₁₇	#576	1	1	yes	characteristic
131	D₁₇	—	8	2	no	—
132	C₂×D₈	—	17	1	no	—
133	C₂²×C₈	C₂×D₁₇	1	1	yes	characteristic
134	C₂²×C₈	C₂×D₁₇	1	1	yes	characteristic
135	#126	—	34	2	no	—
136	#125	—	17	2	no	—
137	#123	—	34	2	no	—
138	#121	—	17	2	no	—
139	#121	—	17	2	no	—
140	#121	—	34	2	no	—
141	C₄×D₄	—	17	2	no	—
142	C₄×D₄	—	34	2	no	—
143	C₄×D₄	C₂×D₁₇	1	1	yes	characteristic
144	#117	—	17	2	no	—
145	C₂×C₄⋊C₄	—	17	2	no	—
146	#107	—	17	1	no	—
147	#107	—	17	1	no	—
148	#107	—	17	2	no	—
149	#107	—	17	1	no	—
150	#106	—	17	2	no	—
151	#106	—	17	2	no	—
152	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
153	#102	—	17	2	no	—
154	#102	—	17	2	no	—
155	#102	—	17	2	no	—
156	C₂²⋊C₈	C₂×D₁₇	1	1	yes	characteristic
157	C₂²⋊C₈	C₂×D₁₇	1	1	yes	characteristic
158	C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
159	C₁₇	#3339	1	1	yes	characteristic, complemented, Sylow
160	C₂×D₄	—	17	2	no	—
161	C₂×D₄	—	34	2	no	—
162	C₂×D₄	—	34	2	no	—
163	C₂×D₄	D₆₈	1	1	yes	characteristic
164	C₂²×C₄	—	34	2	no	—
165	C₂²×C₄	—	34	2	no	—
166	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
167	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
168	D₈	—	34	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
172	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
173	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
174	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
175	C₄⋊C₄	—	17	2	no	—
176	C₄⋊C₄	—	17	2	no	—
177	C₄⋊C₄	—	17	2	no	—
178	C₄⋊C₄	—	34	2	no	—
179	C₄⋊C₄	—	34	2	no	—
180	C₄⋊C₄	—	34	2	no	—
181	C₄⋊C₄	D₆₈	1	1	yes	characteristic
182	C₂²⋊C₄	—	34	2	no	—
183	C₂²⋊C₄	—	34	2	no	—
184	C₂²⋊C₄	—	34	2	no	—
185	C₂²⋊C₄	—	34	2	no	—
186	C₂²⋊C₄	D₆₈	1	1	yes	characteristic
187	C₂²⋊C₄	D₆₈	1	1	yes	characteristic
188	C₄²	—	34	2	no	—
189	C₄²	C₂²×D₁₇	1	1	yes	characteristic
190	C₂³	—	34	2	no	—
191	C₂³	C₂×D₆₈	1	1	yes	characteristic
192	C₂³	C₂×D₆₈	1	1	yes	characteristic
193	D₄	—	2	2	no	—
194	D₄	—	34	2	no	—
195	D₄	—	34	2	no	—
196	D₄	—	68	4	no	—
197	D₄	—	136	2	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	34	2	no	—
203	C₂×C₄	—	34	2	no	—
204	C₂×C₄	—	34	2	no	—
205	C₂×C₄	—	136	2	no	—
206	C₂×C₄	—	136	2	no	—
207	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
208	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
209	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
210	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
211	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
212	C₈	—	2	1	no	—
213	C₈	—	2	1	no	—
214	C₈	—	2	1	no	—
215	C₈	#63254	1	2	yes	—
216	C₂²	—	2	1	no	—
217	C₂²	—	2	1	no	—
218	C₂²	—	2	1	no	—
219	C₂²	—	2	1	no	—
220	C₂²	—	2	1	no	—
221	C₂²	—	68	2	no	—
222	C₂²	—	68	2	no	—
223	C₂²	—	68	2	no	—
224	C₂²	#10587590	1	2	yes	—
225	C₂²	C₂²×D₆₈	1	1	yes	characteristic
226	C₄	—	2	1	no	—
227	C₄	—	2	1	no	—
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	68	2	no	—
231	C₄	—	68	2	no	—
232	C₄	—	68	2	no	—
233	C₄	C₂×#63254	1	1	yes	characteristic
234	C₄	#10587697	1	1	yes	characteristic
235	C₂	—	2	2	no	—
236	C₂	—	4	1	no	—
237	C₂	—	136	2	no	—
238	C₂	#11763463	1	1	yes	characteristic
239	C₂	C₂×#10587590	1	1	yes	characteristic
240	C₂	#11763614	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₆₈
commutator, cosocle	C₂×C₄×C₁₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₇
Fitting	C₈×D₄×C₁₇	C₂
socle	C₂²×C₁₇	C₂²×D₄
hypercenter	C₈×D₄	D₁₇
hypocenter	C₁₇	#3339
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄

Complements

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

N	H ≅ G/N	type
1. #11763505	C₂	semidirect
2. #11763211	C₂	semidirect
3. #11763134	C₂	semidirect
4. #11763128	C₂	semidirect
5. #11763110	C₂	semidirect
6. #11763045	C₂	semidirect
7. #11763045	C₂	semidirect
8. C₄×D₁₃₆	C₂	semidirect
9. #11762847	C₂	semidirect
10. C₂×#10587489	C₂	semidirect
11. C₈×D₄×C₁₇	C₂	semidirect
123. C₈×D₄	D₁₇	semidirect
159. C₁₇	#3339	semidirect

Maximal subgroups

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

subgroup	index
#11763505	2
#11763211	2
#11763134	2
#11763128	2
#11763110	2
#11763045	2
C₄×D₁₃₆	2
#11762847	2
C₂×#10587489	2
C₈×D₄×C₁₇	2
#3339	17