Subgroups — #16952541

Order 4352.

510 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11763189	C₂	1	1	yes	maximal, characteristic
2	C₂×#11762612	C₂	1	4	yes	maximal
3	C₂×C₄×#63214	C₂	1	1	yes	maximal, characteristic
4	C₁₇×#1845	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₆₈	—	2	1	no	—
6	C₂²×Dic₆₈	C₂²	1	2	yes	—
7	C₂×#10587555	—	2	1	no	—
8	C₂²×#63214	C₂²	1	2	yes	—
9	#11763189	—	2	2	no	—
10	#11763189	C₄	1	4	yes	complemented
11	#11762612	—	2	8	no	—
12	C₄×#63214	—	2	2	no	—
13	C₂×C₁₇×#107	—	2	1	no	—
14	C₂×C₁₇×#107	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
16	C₁₇×#384	—	2	1	no	—
17	C₂²×Dic₃₄	—	2	2	no	—
18	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
21	C₂×C₄⋊C₄×C₁₇	—	4	2	no	—
22	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
23	C₂×C₄⋊Dic₁₇	—	4	2	no	—
24	C₂×Dic₆₈	—	2	4	no	—
25	C₂×Dic₆₈	—	2	2	no	—
26	C₂×Dic₆₈	—	2	4	no	—
27	C₂×Dic₆₈	—	2	4	no	—
28	C₂×Dic₆₈	—	4	2	no	—
29	C₂×Dic₆₈	—	4	2	no	—
30	C₂×Dic₆₈	C₂×C₄	1	4	yes	—
31	#10587555	—	2	4	no	—
32	#10587555	—	4	2	no	—
33	C₁₇×#107	—	2	4	no	—
34	C₁₇×#107	—	4	2	no	—
35	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
36	C₄×C₈×C₁₇	D₄	1	2	yes	—
37	C₂×#63214	—	2	2	no	—
38	C₂×#63214	—	2	2	no	—
39	C₂×#63214	D₄	1	4	yes	—
40	C₂²×C₄×C₁₇	—	4	2	no	—
41	C₂²×C₄×C₁₇	D₈	1	2	yes	—
42	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
43	C₂²×Dic₁₇	—	4	2	no	—
44	C₂×Dic₃₄	—	2	4	no	—
45	C₂×Dic₃₄	—	4	2	no	—
46	C₂×Dic₃₄	—	4	2	no	—
47	C₂×Dic₃₄	—	4	2	no	—
48	C₂×Dic₃₄	—	4	2	no	—
49	C₂×Dic₃₄	—	4	2	no	—
50	C₂×C₈×C₁₇	C₂²⋊C₄	1	2	yes	—
51	C₂×C₈×C₁₇	D₈	1	2	yes	—
52	C₂×C₈×C₁₇	SD₁₆	1	2	yes	—
53	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
55	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
56	C₂×C₈×C₁₇	#41	1	2	yes	—
57	C₄⋊C₄×C₁₇	—	4	2	no	—
58	C₄⋊C₄×C₁₇	—	4	2	no	—
59	C₄⋊C₄×C₁₇	—	8	4	no	—
60	C₄²×C₁₇	C₂²⋊C₄	1	2	yes	—
61	C₄²×C₁₇	SD₁₆	1	2	yes	—
62	C₄⋊Dic₁₇	—	4	8	no	—
63	Dic₆₈	—	2	8	no	—
64	Dic₆₈	—	4	8	no	—
65	Dic₆₈	—	4	4	no	—
66	#63214	—	2	4	no	—
67	#13203	—	17	1	no	maximal, Sylow
68	C₂³×C₁₇	C₂×D₈	1	1	yes	characteristic
69	C₂×C₄×C₁₇	—	8	2	no	—
70	C₂×C₄×C₁₇	—	8	4	no	—
71	C₂×C₄×C₁₇	#102	1	4	yes	—
72	C₂×C₄×C₁₇	SD₃₂	1	4	yes	—
73	C₂×C₄×C₁₇	Q₃₂	1	4	yes	—
74	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₁₇	#124	1	1	yes	characteristic
76	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
77	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
79	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
80	C₂×Dic₁₇	—	4	4	no	—
81	C₂×Dic₁₇	—	8	2	no	—
82	C₂×Dic₁₇	—	8	2	no	—
83	Dic₃₄	—	4	4	no	—
84	Dic₃₄	—	4	4	no	—
85	Dic₃₄	—	8	2	no	—
86	Dic₃₄	—	8	2	no	—
87	C₈×C₁₇	#102	1	4	yes	—
88	C₈×C₁₇	C₄×D₄	1	2	yes	—
89	C₈×C₁₇	#121	1	2	yes	—
90	C₂×#494	—	17	1	no	—
91	C₂×#358	—	17	4	no	—
92	C₂×C₄×C₁₆	—	17	1	no	—
93	#1845	D₁₇	1	1	yes	characteristic
94	C₂²×C₁₇	C₂×#102	1	1	yes	characteristic
95	C₂²×C₁₇	#486	1	1	yes	characteristic
96	C₂²×C₁₇	#493	1	1	yes	characteristic
97	C₂²×C₁₇	C₂×SD₃₂	1	2	yes	—
98	C₂²×C₁₇	C₂×Q₃₂	1	2	yes	—
99	C₄×C₁₇	—	16	2	no	—
100	C₄×C₁₇	#358	1	8	yes	complemented
101	C₄×C₁₇	#390	1	1	yes	characteristic
102	C₄×C₁₇	C₂×#102	1	1	yes	characteristic
103	C₄×C₁₇	#486	1	1	yes	characteristic
104	C₄×C₁₇	#492	1	1	yes	characteristic
105	Dic₁₇	—	8	4	no	—
106	C₂²×Q₁₆	—	17	2	no	—
107	C₂²×Q₁₆	—	34	1	no	—
108	C₂×C₄⋊Q₈	—	34	1	no	—
109	C₂²×C₁₆	—	17	2	no	—
110	#494	—	17	4	no	—
111	#494	—	34	2	no	—
112	C₂×#107	—	2	1	no	—
113	C₂×#107	C₂×D₁₇	1	2	yes	—
114	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
115	#384	—	2	1	no	—
116	#358	—	34	8	no	—
117	C₄×C₁₆	—	34	2	no	—
118	C₂×C₁₇	#1957	1	1	yes	characteristic
119	C₂×C₁₇	C₂×#358	1	2	yes	—
120	C₂×C₁₇	#2241	1	2	yes	—
121	C₂×C₁₇	#2247	1	1	yes	characteristic
122	C₂×C₁₇	#2248	1	1	yes	characteristic
123	C₂²×Q₈	—	34	2	no	—
124	C₂×Q₁₆	—	17	4	no	—
125	C₂×Q₁₆	—	34	2	no	—
126	C₂×Q₁₆	—	34	4	no	—
127	C₂×Q₁₆	—	34	4	no	—
128	C₂×Q₁₆	—	34	4	no	—
129	C₂×Q₁₆	—	68	2	no	—
130	C₂×Q₁₆	—	68	2	no	—
131	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
132	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
133	C₄⋊Q₈	—	34	4	no	—
134	C₄⋊Q₈	—	68	2	no	—
135	C₂×C₄⋊C₄	—	2	2	no	—
136	C₂×C₄⋊C₄	—	4	2	no	—
137	C₂×C₄⋊C₄	—	68	2	no	—
138	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
139	C₂×C₁₆	—	17	4	no	—
140	C₂×C₁₆	—	34	2	no	—
141	C₂×C₁₆	—	34	2	no	—
142	#107	—	2	4	no	—
143	#107	—	4	2	no	—
144	C₄×C₈	C₄×D₁₇	1	2	yes	—
145	C₄×C₈	D₆₈	1	2	yes	—
146	C₁₇	#13203	1	1	yes	characteristic, complemented, Sylow
147	C₂×Q₈	—	34	4	no	—
148	C₂×Q₈	—	68	2	no	—
149	C₂×Q₈	—	68	2	no	—
150	C₂×Q₈	—	68	2	no	—
151	C₂×Q₈	—	68	2	no	—
152	C₂×Q₈	—	68	2	no	—
153	C₂²×C₄	—	4	2	no	—
154	C₂²×C₄	—	68	2	no	—
155	C₂²×C₄	#63228	1	2	yes	—
156	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
157	Q₁₆	—	34	8	no	—
158	Q₁₆	—	68	8	no	—
159	Q₁₆	—	68	4	no	—
160	C₂×C₈	#63227	1	2	yes	—
161	C₂×C₈	#63228	1	2	yes	—
162	C₂×C₈	#63230	1	2	yes	—
163	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
164	C₂×C₈	C₂×D₆₈	1	1	yes	characteristic
165	C₂×C₈	D₄×D₁₇	1	2	yes	—
166	C₂×C₈	#63256	1	2	yes	—
167	C₄⋊C₄	—	4	2	no	—
168	C₄⋊C₄	—	4	2	no	—
169	C₄⋊C₄	—	8	4	no	—
170	C₄⋊C₄	—	68	8	no	—
171	C₄²	#63227	1	2	yes	—
172	C₄²	#63230	1	2	yes	—
173	C₁₆	—	34	4	no	—
174	C₂³	C₂×#63228	1	1	yes	characteristic
175	Q₈	—	68	4	no	—
176	Q₈	—	68	4	no	—
177	Q₈	—	136	2	no	—
178	Q₈	—	136	2	no	—
179	C₂×C₄	—	8	4	no	—
180	C₂×C₄	—	8	2	no	—
181	C₂×C₄	—	68	4	no	—
182	C₂×C₄	—	136	2	no	—
183	C₂×C₄	—	136	2	no	—
184	C₂×C₄	#10587480	1	4	yes	—
185	C₂×C₄	#10587498	1	4	yes	—
186	C₂×C₄	#10587500	1	4	yes	—
187	C₂×C₄	C₂×#63227	1	1	yes	characteristic
188	C₂×C₄	C₂×#63228	1	1	yes	characteristic
189	C₂×C₄	#10587626	1	1	yes	characteristic
190	C₂×C₄	C₂×#63230	1	1	yes	characteristic
191	C₂×C₄	C₂×#63230	1	1	yes	characteristic
192	C₂×C₄	#10587633	1	1	yes	characteristic
193	C₈	#10587480	1	4	yes	—
194	C₈	#10587579	1	2	yes	—
195	C₈	#10587581	1	2	yes	—
196	C₂²	C₂×#10587480	1	1	yes	characteristic
197	C₂²	#11763157	1	1	yes	characteristic
198	C₂²	#11763185	1	1	yes	characteristic
199	C₂²	C₂×#10587498	1	2	yes	—
200	C₂²	C₂×#10587500	1	2	yes	—
201	C₄	—	16	2	no	—
202	C₄	—	136	4	no	—
203	C₄	#11762612	1	8	yes	complemented
204	C₄	#11762710	1	1	yes	characteristic
205	C₄	C₂×#10587480	1	1	yes	characteristic
206	C₄	#11763157	1	1	yes	characteristic
207	C₄	#11763183	1	1	yes	characteristic
208	C₂	#14529589	1	1	yes	characteristic
209	C₂	C₂×#11762612	1	2	yes	—
210	C₂	#14531013	1	2	yes	—
211	C₂	#14531036	1	1	yes	characteristic
212	C₂	#14531039	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#63228
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₁₇×#1845	C₂
socle, prime radical	C₂³×C₁₇	C₂×D₈
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	#1845	D₁₇
hypocenter	C₁₇	#13203
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1845

Complements

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

N	H ≅ G/N	type
10. #11763189	C₄	semidirect
100. C₄×C₁₇	#358	semidirect
146. C₁₇	#13203	semidirect
203. C₄	#11762612	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11763189	2
C₂×#11762612	2
C₂×C₄×#63214	2
C₁₇×#1845	2
#13203	17