Subgroups — #14541153

Order 2176.

342 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763499	C₂	1	2	yes	maximal
2	#11763211	C₂	1	1	yes	maximal, characteristic, complemented
3	#11763151	C₂	1	2	yes	maximal, complemented
4	#11763140	C₂	1	2	yes	maximal, complemented
5	#11763117	C₂	1	2	yes	maximal, complemented
6	#11763050	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763050	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×Dic₆₈	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
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	#10587621	—	2	2	no	—
16	#10587612	C₂²	1	2	yes	—
17	C₂×C₄⋊Dic₁₇	C₂²	1	2	yes	—
18	#10587610	C₂²	1	2	yes	—
19	#10587610	C₂²	1	2	yes	—
20	C₂×Dic₆₈	C₂²	1	1	yes	characteristic
21	#10587575	—	2	2	no	—
22	#10587565	—	2	2	no	—
23	#10587563	—	2	2	no	—
24	#10587556	C₂²	1	2	yes	—
25	C₄×Dic₃₄	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	#10587489	C₂²	1	1	yes	characteristic
31	#10587489	C₂²	1	1	yes	characteristic
32	#10587489	C₂²	1	1	yes	characteristic
33	#10587489	C₂²	1	2	yes	—
34	#10587488	C₂²	1	2	yes	—
35	#10587488	C₂²	1	2	yes	—
36	#10587487	C₂²	1	2	yes	—
37	#10587487	C₂²	1	2	yes	—
38	#10587487	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	C₂²×Dic₁₇	—	2	2	no	—
43	C₂×Dic₃₄	C₂³	1	2	yes	—
44	C₂×C₈×C₁₇	—	2	2	no	—
45	C₂×C₈×C₁₇	—	2	2	no	—
46	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
47	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
48	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
49	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
50	C₄⋊C₄×C₁₇	D₄	1	1	yes	characteristic
51	C₂²⋊C₄×C₁₇	D₄	1	1	yes	characteristic
52	C₂²⋊C₄×C₁₇	D₄	1	1	yes	characteristic
53	C₄²×C₁₇	C₂³	1	1	yes	characteristic
54	C₂²⋊Dic₁₇	—	2	2	no	—
55	C₂²⋊Dic₁₇	—	2	2	no	—
56	C₂²⋊Dic₁₇	—	2	2	no	—
57	C₄⋊Dic₁₇	—	2	2	no	—
58	C₄⋊Dic₁₇	C₂³	1	2	yes	—
59	C₄⋊Dic₁₇	C₂³	1	2	yes	—
60	C₄⋊Dic₁₇	C₂³	1	2	yes	—
61	#63225	—	2	2	no	—
62	#63225	—	2	2	no	—
63	#63225	—	2	2	no	—
64	#63225	—	2	2	no	—
65	#63225	—	2	2	no	—
66	C₄×Dic₁₇	—	2	2	no	—
67	C₄×Dic₁₇	—	2	2	no	—
68	Dic₆₈	—	2	2	no	—
69	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
70	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
71	D₄×C₁₇	—	2	2	no	—
72	C₂×C₄×C₁₇	—	2	1	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	C₂×D₄	1	1	yes	characteristic
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	C₂⁴	1	1	yes	characteristic
81	C₂×Dic₁₇	—	2	2	no	—
82	C₂×Dic₁₇	—	2	2	no	—
83	C₂×Dic₁₇	—	2	2	no	—
84	C₂×Dic₁₇	—	2	2	no	—
85	C₂×Dic₁₇	—	4	4	no	—
86	Dic₃₄	—	2	2	no	—
87	Dic₃₄	—	2	2	no	—
88	C₈×C₁₇	—	2	1	no	—
89	C₈×C₁₇	—	2	1	no	—
90	C₈×C₁₇	—	2	1	no	—
91	C₈×C₁₇	#41	1	2	yes	—
92	#3336	—	17	1	no	maximal, Sylow
93	C₂²×C₁₇	—	2	1	no	—
94	C₂²×C₁₇	—	2	1	no	—
95	C₂²×C₁₇	—	2	1	no	—
96	C₂²×C₁₇	—	2	1	no	—
97	C₂²×C₁₇	—	2	1	no	—
98	C₂²×C₁₇	#135	1	2	yes	—
99	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
100	C₄×C₁₇	—	2	1	no	—
101	C₄×C₁₇	—	2	1	no	—
102	C₄×C₁₇	—	2	1	no	—
103	C₄×C₁₇	—	2	1	no	—
104	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
105	C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
106	Dic₁₇	—	4	2	no	—
107	Dic₁₇	—	4	2	no	—
108	Dic₁₇	—	4	2	no	—
109	Dic₁₇	—	4	2	no	—
110	#552	—	17	2	no	—
111	#499	—	17	1	no	—
112	#485	—	17	2	no	—
113	#483	—	17	2	no	—
114	#479	—	17	2	no	—
115	#467	—	17	1	no	—
116	#467	—	17	1	no	—
117	C₄×Q₁₆	—	17	1	no	—
118	C₈×D₄	D₁₇	1	1	yes	characteristic
119	#427	—	17	1	no	—
120	C₂×#107	—	17	1	no	—
121	C₂×C₁₇	—	2	2	no	—
122	C₂×C₁₇	—	4	1	no	—
123	C₂×C₁₇	#547	1	1	yes	characteristic
124	C₂×C₁₇	C₂×#135	1	1	yes	characteristic
125	C₂×C₁₇	#578	1	1	yes	characteristic
126	C₂×Q₁₆	—	17	1	no	—
127	C₂²×C₈	C₂×D₁₇	1	1	yes	characteristic
128	C₂²×C₈	C₂×D₁₇	1	1	yes	characteristic
129	#126	—	34	2	no	—
130	#125	—	17	2	no	—
131	#125	—	34	2	no	—
132	#123	—	34	2	no	—
133	C₂²⋊Q₈	—	17	2	no	—
134	C₂²⋊Q₈	—	17	2	no	—
135	C₄×Q₈	—	17	2	no	—
136	C₄×D₄	C₂×D₁₇	1	1	yes	characteristic
137	#117	—	17	2	no	—
138	#117	—	34	2	no	—
139	C₂×C₄⋊C₄	—	17	2	no	—
140	#107	—	17	1	no	—
141	#107	—	17	1	no	—
142	#107	—	17	2	no	—
143	#107	—	17	1	no	—
144	#106	—	17	2	no	—
145	#106	—	17	2	no	—
146	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
147	#103	—	17	2	no	—
148	#103	—	17	2	no	—
149	#103	—	17	2	no	—
150	C₂²⋊C₈	C₂×D₁₇	1	1	yes	characteristic
151	C₂²⋊C₈	C₂×D₁₇	1	1	yes	characteristic
152	C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
153	C₁₇	#3336	1	1	yes	characteristic, complemented, Sylow
154	C₂×Q₈	—	17	2	no	—
155	C₂×D₄	D₆₈	1	1	yes	characteristic
156	C₂²×C₄	—	34	2	no	—
157	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
158	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
159	Q₁₆	—	34	2	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
163	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
164	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
165	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
166	C₄⋊C₄	—	17	2	no	—
167	C₄⋊C₄	—	17	2	no	—
168	C₄⋊C₄	—	17	2	no	—
169	C₄⋊C₄	—	34	2	no	—
170	C₄⋊C₄	—	34	2	no	—
171	C₄⋊C₄	—	34	2	no	—
172	C₄⋊C₄	—	34	2	no	—
173	C₄⋊C₄	—	34	2	no	—
174	C₄⋊C₄	—	34	2	no	—
175	C₄⋊C₄	D₆₈	1	1	yes	characteristic
176	C₂²⋊C₄	—	34	2	no	—
177	C₂²⋊C₄	—	34	2	no	—
178	C₂²⋊C₄	—	34	2	no	—
179	C₂²⋊C₄	D₆₈	1	1	yes	characteristic
180	C₂²⋊C₄	D₆₈	1	1	yes	characteristic
181	C₄²	—	34	2	no	—
182	C₄²	—	34	2	no	—
183	C₄²	C₂²×D₁₇	1	1	yes	characteristic
184	C₂³	C₂×D₆₈	1	1	yes	characteristic
185	C₂³	C₂×D₆₈	1	1	yes	characteristic
186	Q₈	—	34	2	no	—
187	Q₈	—	34	2	no	—
188	D₄	—	2	2	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	34	2	no	—
194	C₂×C₄	—	34	2	no	—
195	C₂×C₄	—	34	2	no	—
196	C₂×C₄	—	34	2	no	—
197	C₂×C₄	—	68	4	no	—
198	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
199	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
200	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
201	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
202	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
203	C₈	—	2	1	no	—
204	C₈	—	2	1	no	—
205	C₈	—	2	1	no	—
206	C₈	#63254	1	2	yes	—
207	C₂²	—	2	1	no	—
208	C₂²	—	2	1	no	—
209	C₂²	—	2	1	no	—
210	C₂²	—	2	1	no	—
211	C₂²	—	2	1	no	—
212	C₂²	#10587590	1	2	yes	—
213	C₂²	C₂²×D₆₈	1	1	yes	characteristic
214	C₄	—	2	1	no	—
215	C₄	—	2	1	no	—
216	C₄	—	2	1	no	—
217	C₄	—	2	1	no	—
218	C₄	—	68	2	no	—
219	C₄	—	68	2	no	—
220	C₄	—	68	2	no	—
221	C₄	—	68	2	no	—
222	C₄	C₂×#63254	1	1	yes	characteristic
223	C₄	#10587697	1	1	yes	characteristic
224	C₂	—	2	2	no	—
225	C₂	—	4	1	no	—
226	C₂	#11763463	1	1	yes	characteristic
227	C₂	C₂×#10587590	1	1	yes	characteristic
228	C₂	#11763626	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
prime radical	C₂×D₄×C₁₇	D₄
hypercenter	C₈×D₄	D₁₇
hypocenter	C₁₇	#3336
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
2. #11763211	C₂	semidirect
3. #11763151	C₂	semidirect
4. #11763140	C₂	semidirect
5. #11763117	C₂	semidirect
6. #11763050	C₂	semidirect
7. #11763050	C₂	semidirect
8. C₄×Dic₆₈	C₂	semidirect
9. #11762847	C₂	semidirect
10. C₂×#10587489	C₂	semidirect
153. C₁₇	#3336	semidirect

Maximal subgroups

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

subgroup	index
#11763499	2
#11763211	2
#11763151	2
#11763140	2
#11763117	2
#11763050	2
C₄×Dic₆₈	2
#11762847	2
C₂×#10587489	2
C₈×D₄×C₁₇	2
#3336	17