Subgroups — #16934159

Order 4352.

510 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14529746	C₂	1	2	yes	maximal
2	#14529587	C₂	1	2	yes	maximal
3	#14528941	C₂	1	1	yes	maximal, characteristic
4	C₂×#11762530	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₇×#500	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587488	—	2	2	no	—
7	C₂×#10587506	—	2	4	no	—
8	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
9	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
10	#11762679	C₂²	1	2	yes	—
11	#11762571	—	2	4	no	—
12	#11762530	—	2	2	no	—
13	C₂×C₁₇×C₄⋊Q₈	C₂²	1	2	yes	—
14	C₁₇×#500	—	2	2	no	—
15	C₁₇×#500	C₄	1	4	yes	complemented
16	C₂×C₁₇×#107	—	2	2	no	—
17	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
18	C₂²×Q₈×C₁₇	—	2	2	no	—
19	C₂²×C₈×C₁₇	D₄	1	2	yes	—
20	C₁₇×C₄⋊Q₈	—	2	2	no	—
21	C₁₇×C₄⋊Q₈	—	2	2	no	—
22	C₁₇×C₄⋊Q₈	—	2	2	no	—
23	C₁₇×C₄⋊Q₈	C₂×C₄	1	4	yes	—
24	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
25	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
26	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
27	C₂×C₄⋊Dic₁₇	—	2	2	no	—
28	C₂×C₄⋊Dic₁₇	—	2	2	no	—
29	C₂×C₄⋊Dic₁₇	—	2	2	no	—
30	C₂²×#3622	D₄	1	2	yes	—
31	C₂²×#3622	D₄	1	2	yes	—
32	C₁₇×#107	—	2	8	no	—
33	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₁₇	D₄	1	2	yes	—
35	#10587506	—	4	8	no	—
36	#10587488	—	4	4	no	—
37	C₄×#3622	—	2	1	no	—
38	C₄×#3622	—	2	1	no	—
39	C₄×#3622	—	2	1	no	—
40	C₄×#3622	—	2	1	no	—
41	C₂×Q₈×C₁₇	—	2	4	no	—
42	C₂×Q₈×C₁₇	—	4	2	no	—
43	C₂×Q₈×C₁₇	—	4	4	no	—
44	C₂×Q₈×C₁₇	—	4	4	no	—
45	C₂²×C₄×C₁₇	—	4	2	no	—
46	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
49	C₂²×Dic₁₇	—	4	2	no	—
50	C₂×C₈×C₁₇	C₂²⋊C₄	1	4	yes	—
51	C₂×C₈×C₁₇	SD₁₆	1	4	yes	—
52	C₂×C₈×C₁₇	Q₁₆	1	4	yes	—
53	C₄⋊C₄×C₁₇	—	2	4	no	—
54	C₄⋊C₄×C₁₇	—	2	4	no	—
55	C₄⋊C₄×C₁₇	—	4	2	no	—
56	C₄⋊C₄×C₁₇	—	4	2	no	—
57	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
58	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
59	C₄²×C₁₇	#41	1	1	yes	characteristic
60	C₄²×C₁₇	#41	1	1	yes	characteristic
61	C₄⋊Dic₁₇	—	4	4	no	—
62	C₄⋊Dic₁₇	—	4	4	no	—
63	C₄⋊Dic₁₇	—	4	4	no	—
64	C₂×#3622	—	2	2	no	—
65	C₂×#3622	—	2	2	no	—
66	C₂×#3622	—	2	4	no	—
67	C₂×#3622	—	2	4	no	—
68	#9836	—	17	1	no	maximal, Sylow
69	C₂³×C₁₇	#127	1	1	yes	characteristic
70	Q₈×C₁₇	—	4	8	no	—
71	Q₈×C₁₇	—	8	4	no	—
72	C₂×C₄×C₁₇	—	4	4	no	—
73	C₂×C₄×C₁₇	—	8	2	no	—
74	C₂×C₄×C₁₇	—	8	2	no	—
75	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
77	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
78	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
79	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
80	C₂×C₄×C₁₇	#124	1	1	yes	characteristic
81	C₂×C₄×C₁₇	C₂×SD₁₆	1	2	yes	—
82	C₂×C₄×C₁₇	C₂×Q₁₆	1	2	yes	—
83	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	4	yes	—
84	C₂×C₄×C₁₇	#137	1	4	yes	—
85	C₂×Dic₁₇	—	8	2	no	—
86	C₂×Dic₁₇	—	8	2	no	—
87	C₂×Dic₁₇	—	8	2	no	—
88	C₈×C₁₇	#103	1	8	yes	complemented
89	#3622	—	4	2	no	—
90	#3622	—	4	2	no	—
91	C₂×#500	D₁₇	1	1	yes	characteristic
92	#1990	—	17	2	no	—
93	#1956	—	17	2	no	—
94	#1844	—	17	1	no	—
95	C₂×#322	—	17	1	no	—
96	C₂²×C₁₇	#390	1	1	yes	characteristic
97	C₂²×C₁₇	#492	1	1	yes	characteristic
98	C₂²×C₁₇	#494	1	1	yes	characteristic
99	C₂²×C₁₇	#496	1	2	yes	—
100	C₂²×C₁₇	#497	1	2	yes	—
101	C₄×C₁₇	—	8	4	no	—
102	C₄×C₁₇	C₂×#103	1	2	yes	—
103	C₄×C₁₇	#441	1	2	yes	—
104	C₄×C₁₇	#442	1	2	yes	—
105	C₄×C₁₇	#469	1	2	yes	—
106	C₄×C₁₇	#470	1	2	yes	—
107	C₄×C₁₇	#487	1	2	yes	—
108	Dic₁₇	—	16	2	no	—
109	C₂×C₄⋊Q₈	C₂×D₁₇	1	2	yes	—
110	#500	—	2	2	no	—
111	#500	Dic₁₇	1	4	yes	complemented
112	C₂×#107	—	2	2	no	—
113	C₂×#106	—	34	2	no	—
114	C₂×#103	—	34	4	no	—
115	C₂×C₄×C₈	—	17	1	no	—
116	C₂×C₄×C₈	—	17	1	no	—
117	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
118	#384	—	17	2	no	—
119	#340	—	34	4	no	—
120	#322	—	34	2	no	—
121	C₂×C₁₇	#1714	1	2	yes	—
122	C₂×C₁₇	#1723	1	2	yes	—
123	C₂×C₁₇	#1956	1	1	yes	characteristic
124	C₂×C₁₇	#1971	1	1	yes	characteristic
125	C₂×C₁₇	#1972	1	1	yes	characteristic
126	C₂²×Q₈	—	2	2	no	—
127	C₂²×C₈	—	17	2	no	—
128	C₂²×C₈	—	17	2	no	—
129	C₂²×C₈	C₂²⋊D₁₇	1	2	yes	—
130	C₄⋊Q₈	—	2	2	no	—
131	C₄⋊Q₈	—	2	2	no	—
132	C₄⋊Q₈	—	2	2	no	—
133	C₄⋊Q₈	C₂×Dic₁₇	1	4	yes	—
134	C₂×C₄⋊C₄	—	2	2	no	—
135	C₂×C₄⋊C₄	—	2	2	no	—
136	C₂×C₄⋊C₄	—	34	2	no	—
137	C₂×C₄⋊C₄	—	34	2	no	—
138	C₂×C₄⋊C₄	—	34	2	no	—
139	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
140	#107	—	2	8	no	—
141	#106	—	68	4	no	—
142	#103	—	68	8	no	—
143	C₄×C₈	—	34	1	no	—
144	C₄×C₈	—	34	1	no	—
145	C₄×C₈	—	34	1	no	—
146	C₄×C₈	—	34	1	no	—
147	C₄×C₈	C₂×Dic₁₇	1	2	yes	—
148	C₄×C₈	C₂²⋊D₁₇	1	2	yes	—
149	C₁₇	#9836	1	1	yes	characteristic, complemented, Sylow
150	C₂×Q₈	—	2	4	no	—
151	C₂×Q₈	—	4	2	no	—
152	C₂×Q₈	—	4	4	no	—
153	C₂×Q₈	—	4	4	no	—
154	C₂²×C₄	—	4	2	no	—
155	C₂²×C₄	—	68	2	no	—
156	C₂²×C₄	D₄×D₁₇	1	1	yes	characteristic
157	C₂²×C₄	D₄×D₁₇	1	1	yes	characteristic
158	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
159	C₂×C₈	—	34	4	no	—
160	C₂×C₈	—	34	4	no	—
161	C₂×C₈	—	34	2	no	—
162	C₂×C₈	—	34	2	no	—
163	C₂×C₈	#63230	1	4	yes	—
164	C₂×C₈	#63231	1	4	yes	—
165	C₂×C₈	C₂²⋊Dic₁₇	1	4	yes	—
166	C₄⋊C₄	—	2	4	no	—
167	C₄⋊C₄	—	2	4	no	—
168	C₄⋊C₄	—	4	2	no	—
169	C₄⋊C₄	—	4	2	no	—
170	C₄⋊C₄	—	68	4	no	—
171	C₄⋊C₄	—	68	4	no	—
172	C₄⋊C₄	—	68	4	no	—
173	C₄²	#63254	1	1	yes	characteristic
174	C₄²	#63254	1	1	yes	characteristic
175	C₄²	C₂²×Dic₁₇	1	1	yes	characteristic
176	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
177	C₂³	#10587626	1	1	yes	characteristic
178	Q₈	—	4	8	no	—
179	Q₈	—	8	4	no	—
180	C₂×C₄	—	4	4	no	—
181	C₂×C₄	—	8	2	no	—
182	C₂×C₄	—	8	2	no	—
183	C₂×C₄	—	136	2	no	—
184	C₂×C₄	—	136	2	no	—
185	C₂×C₄	—	136	2	no	—
186	C₂×C₄	#10587597	1	4	yes	—
187	C₂×C₄	#10587604	1	4	yes	—
188	C₂×C₄	D₄×Dic₁₇	1	1	yes	characteristic
189	C₂×C₄	D₄×Dic₁₇	1	1	yes	characteristic
190	C₂×C₄	#10587622	1	1	yes	characteristic
191	C₂×C₄	#10587624	1	1	yes	characteristic
192	C₂×C₄	#10587624	1	1	yes	characteristic
193	C₂×C₄	C₂×#63230	1	2	yes	—
194	C₂×C₄	C₂×#63231	1	2	yes	—
195	C₂×C₄	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
196	C₈	—	68	2	no	—
197	C₈	—	68	2	no	—
198	C₈	#10587506	1	8	yes	complemented
199	C₂²	#11762712	1	1	yes	characteristic
200	C₂²	#11763183	1	1	yes	characteristic
201	C₂²	#11763188	1	1	yes	characteristic
202	C₂²	#11763197	1	2	yes	—
203	C₂²	#11763202	1	2	yes	—
204	C₄	—	8	4	no	—
205	C₄	—	272	2	no	—
206	C₄	C₂×#10587506	1	2	yes	—
207	C₄	#11762916	1	2	yes	—
208	C₄	#11762921	1	2	yes	—
209	C₄	#11763059	1	2	yes	—
210	C₄	#11763064	1	2	yes	—
211	C₄	#11763159	1	2	yes	—
212	C₂	#14528407	1	2	yes	—
213	C₂	#14528462	1	2	yes	—
214	C₂	#14529587	1	1	yes	characteristic
215	C₂	#14529656	1	1	yes	characteristic
216	C₂	#14529661	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#10587626
commutator	C₄²×C₁₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₂×C₁₇×#500	C₂
socle, prime radical	C₂³×C₁₇	#127
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	C₂×#500	D₁₇
hypocenter	C₁₇	#9836
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
15. C₁₇×#500	C₄	semidirect
88. C₈×C₁₇	#103	semidirect
111. #500	Dic₁₇	semidirect
149. C₁₇	#9836	semidirect
198. C₈	#10587506	semidirect

Maximal subgroups

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

subgroup	index
#14529746	2
#14529587	2
#14528941	2
C₂×#11762530	2
C₂×C₁₇×#500	2
#9836	17