Subgroups — #17189730

Order 4352.

834 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11763241	C₂	1	4	yes	maximal, complemented
2	#14539212	C₂	1	2	yes	maximal, complemented
3	#14531045	C₂	1	4	yes	maximal, complemented
4	#14531040	C₂	1	2	yes	maximal, complemented
5	#14531037	C₂	1	1	yes	maximal, characteristic, complemented
6	#14531034	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄×C₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10587590	—	2	2	no	—
9	C₂×#10587590	C₂²	1	4	yes	—
10	#11763359	—	2	2	no	—
11	#11763241	—	2	16	no	—
12	C₂×Dic₁₃₆	C₂²	1	4	yes	—
13	C₂×#10587471	C₂²	1	8	yes	—
14	C₂×D₂₇₂	C₂²	1	4	yes	—
15	#11763193	—	2	2	no	—
16	#11763193	C₂²	1	4	yes	—
17	#11763189	C₂²	1	2	yes	—
18	#11763186	C₂²	1	2	yes	—
19	#11763182	—	2	2	no	—
20	C₂²×C₁₆×C₁₇	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
22	C₄×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
23	C₄×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
24	C₄×C₁₆×C₁₇	C₂²	1	2	yes	—
25	C₂×#63252	—	2	4	no	—
26	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
28	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
29	#10587616	—	4	4	no	—
30	C₂×Dic₆₈	—	2	2	no	—
31	C₂×Dic₆₈	C₂³	1	4	yes	—
32	#10587590	—	2	16	no	—
33	#10587590	—	4	8	no	—
34	C₂×D₁₃₆	—	2	2	no	—
35	C₂×D₁₃₆	C₂³	1	4	yes	—
36	C₂×#63219	—	2	4	no	—
37	C₂×#63219	—	2	4	no	—
38	#10587561	—	2	4	no	—
39	#10587560	—	2	2	no	—
40	C₄×D₆₈	—	4	4	no	—
41	#10587555	—	2	2	no	—
42	C₂×C₁₆×C₁₇	D₄	1	8	yes	complemented
43	C₂×C₁₆×C₁₇	C₂³	1	2	yes	—
44	C₂×C₁₆×C₁₇	C₂³	1	2	yes	—
45	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
46	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
47	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
48	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
49	Dic₁₃₆	—	2	8	no	—
50	#10587471	—	2	16	no	—
51	D₂₇₂	—	2	8	no	—
52	C₂²×C₄×C₁₇	D₈	1	2	yes	—
53	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
54	C₂×C₂²⋊D₁₇	—	4	4	no	—
55	#63252	—	4	8	no	—
56	#63252	—	4	8	no	—
57	C₂×D₆₈	—	2	4	no	—
58	C₂×D₆₈	—	4	4	no	—
59	C₂×C₄×D₁₇	—	4	4	no	—
60	C₂×Dic₃₄	—	2	4	no	—
61	C₂×C₈×C₁₇	D₈	1	4	yes	—
62	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₁₇	C₂×D₄	1	4	yes	—
64	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
67	C₄²×C₁₇	D₈	1	2	yes	—
68	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
69	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
70	#63227	—	4	8	no	—
71	C₄⋊Dic₁₇	—	4	4	no	—
72	Dic₆₈	—	2	8	no	—
73	Dic₆₈	—	4	4	no	—
74	D₁₃₆	—	2	8	no	—
75	D₁₃₆	—	4	4	no	—
76	#63219	—	4	8	no	—
77	#63219	—	4	8	no	—
78	C₁₆×C₁₇	C₂×D₄	1	4	yes	—
79	C₁₆×C₁₇	C₂×D₄	1	4	yes	—
80	#33713	—	17	1	no	maximal, Sylow
81	C₂³×C₁₇	C₂×D₈	1	1	yes	characteristic
82	C₂²×D₁₇	—	4	4	no	—
83	C₂×C₄×C₁₇	—	2	4	no	—
84	C₂×C₄×C₁₇	#127	1	2	yes	—
85	C₂×C₄×C₁₇	C₂×D₈	1	2	yes	—
86	C₂×C₄×C₁₇	C₂×D₈	1	2	yes	—
87	C₂×C₄×C₁₇	C₂×D₈	1	2	yes	—
88	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
89	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
90	C₂²⋊D₁₇	—	8	8	no	—
91	C₂×Dic₁₇	—	4	4	no	—
92	D₆₈	—	4	4	no	—
93	D₆₈	—	4	4	no	—
94	D₆₈	—	8	8	no	—
95	C₄×D₁₇	—	8	8	no	—
96	Dic₃₄	—	4	4	no	—
97	Dic₃₄	—	4	4	no	—
98	C₈×C₁₇	#127	1	2	yes	—
99	C₈×C₁₇	C₂×D₈	1	2	yes	—
100	C₈×C₁₇	C₂×D₈	1	2	yes	—
101	C₈×C₁₇	C₂²×D₄	1	2	yes	—
102	C₂×#508	—	17	4	no	—
103	#3147	—	17	2	no	—
104	#2249	—	17	4	no	—
105	#2248	—	17	2	no	—
106	#2247	—	17	1	no	—
107	#2246	—	17	1	no	—
108	C₂×C₄×C₁₆	D₁₇	1	1	yes	characteristic, complemented
109	C₂²×C₁₇	—	2	2	no	—
110	C₂²×C₁₇	#493	1	2	yes	—
111	C₂²×C₁₇	C₂²×D₈	1	1	yes	characteristic
112	C₂×D₁₇	—	8	4	no	—
113	C₂×D₁₇	—	8	8	no	—
114	C₄×C₁₇	—	2	2	no	—
115	C₄×C₁₇	#493	1	2	yes	—
116	C₄×C₁₇	#508	1	4	yes	complemented
117	C₄×C₁₇	C₂×#127	1	1	yes	characteristic
118	C₄×C₁₇	C₂²×D₈	1	1	yes	characteristic
119	Dic₁₇	—	8	4	no	—
120	C₂×#135	—	17	4	no	—
121	C₂×#135	—	34	2	no	—
122	#532	—	34	2	no	—
123	#508	—	34	16	no	—
124	C₂×Q₃₂	—	17	4	no	—
125	C₂×SD₃₂	—	17	8	no	—
126	C₂×D₁₆	—	17	4	no	—
127	C₂²×C₁₆	C₂×D₁₇	1	2	yes	—
128	#495	—	17	4	no	—
129	#495	—	34	2	no	—
130	#494	—	17	2	no	—
131	#493	—	17	2	no	—
132	#492	—	34	2	no	—
133	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
134	C₄×C₁₆	C₂×D₁₇	1	1	yes	characteristic
135	C₄×C₁₆	C₂×D₁₇	1	1	yes	characteristic
136	C₄×C₁₆	C₂×D₁₇	1	2	yes	—
137	C₂×C₁₇	—	2	2	no	—
138	C₂×C₁₇	C₂×#493	1	1	yes	characteristic
139	C₂×C₁₇	C₂×#508	1	2	yes	—
140	D₁₇	—	16	4	no	—
141	C₂×#41	—	34	4	no	—
142	#135	—	34	16	no	—
143	#135	—	68	8	no	—
144	C₂×Q₁₆	—	17	4	no	—
145	C₂×Q₁₆	—	34	2	no	—
146	C₂×SD₁₆	—	34	4	no	—
147	C₂×SD₁₆	—	34	4	no	—
148	C₂×D₈	—	17	4	no	—
149	C₂×D₈	—	34	2	no	—
150	C₂²×C₈	D₆₈	1	1	yes	characteristic
151	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
152	C₄⋊Q₈	—	34	2	no	—
153	#127	—	34	2	no	—
154	#124	—	34	4	no	—
155	#121	—	68	4	no	—
156	C₄×D₄	—	68	4	no	—
157	C₂×C₄²	D₆₈	1	1	yes	characteristic
158	Q₃₂	—	34	8	no	—
159	SD₃₂	—	34	16	no	—
160	D₁₆	—	34	8	no	—
161	C₂×C₁₆	D₆₈	1	8	yes	complemented
162	C₂×C₁₆	C₂²×D₁₇	1	2	yes	—
163	C₂×C₁₆	C₂²×D₁₇	1	2	yes	—
164	C₄×C₈	D₆₈	1	1	yes	characteristic
165	C₄×C₈	D₆₈	1	1	yes	characteristic
166	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
167	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
168	C₁₇	#33713	1	1	yes	characteristic, complemented, Sylow
169	#41	—	68	8	no	—
170	#41	—	68	8	no	—
171	C₂×Q₈	—	34	4	no	—
172	C₂×D₄	—	34	4	no	—
173	C₂×D₄	—	68	4	no	—
174	C₂×D₄	—	68	4	no	—
175	C₂²×C₄	—	68	4	no	—
176	C₂²×C₄	D₁₃₆	1	2	yes	—
177	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
178	Q₁₆	—	34	8	no	—
179	Q₁₆	—	68	4	no	—
180	SD₁₆	—	68	8	no	—
181	SD₁₆	—	68	8	no	—
182	D₈	—	34	8	no	—
183	D₈	—	68	4	no	—
184	C₂×C₈	D₁₃₆	1	4	yes	—
185	C₂×C₈	C₂×D₆₈	1	4	yes	—
186	C₂×C₈	C₂×D₆₈	1	1	yes	characteristic
187	C₂×C₈	C₂×D₆₈	1	1	yes	characteristic
188	C₂×C₈	C₂×D₆₈	1	1	yes	characteristic
189	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
190	C₄⋊C₄	—	68	4	no	—
191	C₂²⋊C₄	—	68	8	no	—
192	C₄²	D₁₃₆	1	2	yes	—
193	C₄²	C₂×D₆₈	1	1	yes	characteristic
194	C₄²	C₂×D₆₈	1	1	yes	characteristic
195	C₁₆	C₂×D₆₈	1	4	yes	—
196	C₁₆	C₂×D₆₈	1	4	yes	—
197	C₂³	—	68	4	no	—
198	C₂³	C₂×D₁₃₆	1	1	yes	characteristic
199	Q₈	—	68	4	no	—
200	Q₈	—	68	4	no	—
201	D₄	—	68	4	no	—
202	D₄	—	68	4	no	—
203	D₄	—	136	8	no	—
204	D₄	—	136	8	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	68	4	no	—
207	C₂×C₄	—	136	8	no	—
208	C₂×C₄	#10587560	1	2	yes	—
209	C₂×C₄	C₂×D₁₃₆	1	2	yes	—
210	C₂×C₄	C₂×D₁₃₆	1	2	yes	—
211	C₂×C₄	C₂×D₁₃₆	1	2	yes	—
212	C₂×C₄	C₂×D₁₃₆	1	1	yes	characteristic
213	C₂×C₄	C₂²×D₆₈	1	1	yes	characteristic
214	C₈	#10587560	1	2	yes	—
215	C₈	C₂×D₁₃₆	1	2	yes	—
216	C₈	C₂×D₁₃₆	1	2	yes	—
217	C₈	C₂²×D₆₈	1	2	yes	—
218	C₂²	—	2	2	no	—
219	C₂²	—	136	8	no	—
220	C₂²	—	136	4	no	—
221	C₂²	#11763186	1	2	yes	—
222	C₂²	C₂²×D₁₃₆	1	1	yes	characteristic
223	C₄	—	2	2	no	—
224	C₄	—	136	4	no	—
225	C₄	#11763186	1	2	yes	—
226	C₄	#11763241	1	4	yes	complemented
227	C₄	C₂×#10587560	1	1	yes	characteristic
228	C₄	C₂²×D₁₃₆	1	1	yes	characteristic
229	C₂	—	2	2	no	—
230	C₂	—	272	4	no	—
231	C₂	C₂×#11763186	1	1	yes	characteristic
232	C₂	C₂×#11763241	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₁₃₆
commutator, cosocle	C₂×C₈×C₁₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₇
Fitting	C₂×C₄×C₁₆×C₁₇	C₂
socle	C₂²×C₁₇	C₂²×D₈
hypercenter	C₂×C₄×C₁₆	D₁₇
hypocenter	C₁₇	#33713
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#11763241	C₂	semidirect
2. #14539212	C₂	semidirect
3. #14531045	C₂	semidirect
4. #14531040	C₂	semidirect
5. #14531037	C₂	semidirect
6. #14531034	C₂	semidirect
7. C₂×C₄×C₁₆×C₁₇	C₂	semidirect
42. C₂×C₁₆×C₁₇	D₄	semidirect
108. C₂×C₄×C₁₆	D₁₇	semidirect
116. C₄×C₁₇	#508	semidirect
161. C₂×C₁₆	D₆₈	semidirect
168. C₁₇	#33713	semidirect
226. C₄	#11763241	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11763241	2
#14539212	2
#14531045	2
#14531040	2
#14531037	2
#14531034	2
C₂×C₄×C₁₆×C₁₇	2
#33713	17