Subgroups — #16933697

Order 4352.

394 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14539298	C₂	1	1	yes	maximal, characteristic
2	#14527347	C₂	1	4	yes	maximal
3	#14527067	C₂	1	1	yes	maximal, characteristic
4	C₁₇×#1846	C₂	1	1	yes	maximal, characteristic
5	#11763366	C₂²	1	2	yes	—
6	#11763214	C₄	1	2	yes	complemented
7	#11763211	—	2	1	no	—
8	#11763211	—	2	1	no	—
9	#11763207	C₄	1	2	yes	complemented
10	#11762862	C₂²	1	2	yes	—
11	#11762847	—	2	1	no	—
12	#11762847	—	2	1	no	—
13	#11762545	—	2	8	no	—
14	#11762530	—	2	1	no	—
15	C₈×#3622	—	2	1	no	—
16	C₂×C₁₇×#107	—	2	1	no	—
17	C₂×C₁₇×#106	—	2	1	no	—
18	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
19	C₁₇×#382	C₂²	1	2	yes	—
20	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
22	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
23	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
24	#10587612	—	2	2	no	—
25	#10587610	—	2	4	no	—
26	#10587556	—	2	2	no	—
27	#10587555	C₂×C₄	1	4	yes	—
28	C₄×Dic₃₄	—	2	4	no	—
29	C₁₇×#107	—	4	2	no	—
30	C₁₇×#106	—	4	2	no	—
31	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
34	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
35	C₁₇×#95	—	2	4	no	—
36	C₂²⋊#3622	D₄	1	4	yes	—
37	#10587489	—	2	2	no	—
38	#10587489	—	2	2	no	—
39	#10587488	—	2	2	no	—
40	#10587488	—	2	2	no	—
41	C₄⋊#3622	—	2	2	no	—
42	C₄×#3622	—	2	2	no	—
43	C₂²×C₄×C₁₇	—	4	2	no	—
44	C₂²×C₄×C₁₇	C₂×D₄	1	2	yes	—
45	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
46	C₂×Dic₃₄	—	2	4	no	—
47	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₈×C₁₇	SD₁₆	1	2	yes	—
52	C₂×C₈×C₁₇	SD₁₆	1	2	yes	—
53	C₂×C₈×C₁₇	Q₁₆	1	2	yes	—
54	C₂×C₈×C₁₇	Q₁₆	1	2	yes	—
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	2	yes	—
60	C₂²⋊Dic₁₇	—	4	2	no	—
61	C₄⋊Dic₁₇	—	2	4	no	—
62	C₄⋊Dic₁₇	—	2	4	no	—
63	#63225	—	4	4	no	—
64	C₄×Dic₁₇	—	4	2	no	—
65	C₂×#3622	—	2	4	no	—
66	#9755	—	17	1	no	maximal, Sylow
67	C₂³×C₁₇	#127	1	1	yes	characteristic
68	C₂×C₄×C₁₇	—	4	2	no	—
69	C₂×C₄×C₁₇	—	8	4	no	—
70	C₂×C₄×C₁₇	—	8	2	no	—
71	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₁₇	C₄×D₄	1	2	yes	—
73	C₂×C₄×C₁₇	#121	1	2	yes	—
74	C₂×C₄×C₁₇	#124	1	1	yes	characteristic
75	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
76	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
77	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
78	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
79	C₂×Dic₁₇	—	4	4	no	—
80	Dic₃₄	—	8	4	no	—
81	C₈×C₁₇	#103	1	2	yes	—
82	C₈×C₁₇	#103	1	2	yes	—
83	C₈×C₁₇	#103	1	2	yes	—
84	C₈×C₁₇	#103	1	2	yes	—
85	#3622	—	4	4	no	—
86	#3160	—	17	1	no	—
87	#1846	D₁₇	1	1	yes	characteristic
88	#1518	—	17	4	no	—
89	#1450	—	17	1	no	—
90	C₂²×C₁₇	—	2	2	no	—
91	C₂²×C₁₇	#390	1	1	yes	characteristic
92	C₂²×C₁₇	#492	1	1	yes	characteristic
93	C₂²×C₁₇	#494	1	1	yes	characteristic
94	C₄×C₁₇	—	2	2	no	—
95	C₄×C₁₇	—	2	2	no	—
96	C₄×C₁₇	—	16	2	no	—
97	C₄×C₁₇	C₂×#103	1	1	yes	characteristic
98	C₄×C₁₇	C₂×#103	1	1	yes	characteristic
99	C₄×C₁₇	#487	1	1	yes	characteristic
100	C₄×C₁₇	#487	1	1	yes	characteristic
101	Dic₁₇	—	8	4	no	—
102	#533	—	17	2	no	—
103	#500	—	17	2	no	—
104	#499	—	34	1	no	—
105	#499	—	34	1	no	—
106	#498	—	17	2	no	—
107	#431	—	17	2	no	—
108	#427	—	34	1	no	—
109	#427	—	34	1	no	—
110	C₂×#107	—	2	1	no	—
111	C₂×#106	—	2	1	no	—
112	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
113	#382	C₂×D₁₇	1	2	yes	—
114	#328	—	34	8	no	—
115	#322	—	34	1	no	—
116	C₈²	—	34	1	no	—
117	C₂×C₁₇	—	2	1	no	—
118	C₂×C₁₇	—	2	1	no	—
119	C₂×C₁₇	#1956	1	1	yes	characteristic
120	C₂×C₁₇	#1976	1	2	yes	—
121	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
122	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
123	C₄⋊Q₈	—	17	4	no	—
124	#125	—	34	2	no	—
125	C₂²⋊Q₈	—	34	4	no	—
126	C₄×Q₈	—	34	4	no	—
127	#117	—	34	2	no	—
128	C₂×C₄⋊C₄	—	2	2	no	—
129	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
130	#107	—	4	2	no	—
131	#107	—	34	2	no	—
132	#107	—	34	2	no	—
133	#106	—	4	2	no	—
134	#106	—	34	2	no	—
135	#106	—	34	2	no	—
136	C₄⋊C₈	—	34	2	no	—
137	C₂²⋊C₈	—	17	4	no	—
138	C₄×C₈	—	34	2	no	—
139	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
140	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
141	C₄×C₈	D₆₈	1	1	yes	characteristic
142	C₄×C₈	D₆₈	1	1	yes	characteristic
143	#95	—	2	4	no	—
144	C₁₇	#9755	1	1	yes	characteristic, complemented, Sylow
145	C₂×Q₈	—	34	4	no	—
146	C₂²×C₄	—	4	2	no	—
147	C₂²×C₄	D₄×D₁₇	1	2	yes	—
148	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
149	C₂×C₈	—	34	4	no	—
150	C₂×C₈	#63227	1	1	yes	characteristic
151	C₂×C₈	#63227	1	1	yes	characteristic
152	C₂×C₈	#63227	1	1	yes	characteristic
153	C₂×C₈	#63227	1	1	yes	characteristic
154	C₂×C₈	#63229	1	2	yes	—
155	C₂×C₈	#63229	1	2	yes	—
156	C₂×C₈	#63231	1	2	yes	—
157	C₂×C₈	#63231	1	2	yes	—
158	C₄⋊C₄	—	4	2	no	—
159	C₄⋊C₄	—	4	2	no	—
160	C₄⋊C₄	—	34	4	no	—
161	C₄⋊C₄	—	34	4	no	—
162	C₄⋊C₄	—	68	4	no	—
163	C₂²⋊C₄	—	68	2	no	—
164	C₄²	—	68	2	no	—
165	C₄²	C₂×C₄×D₁₇	1	1	yes	characteristic
166	C₄²	C₂×D₆₈	1	1	yes	characteristic
167	C₄²	#63256	1	2	yes	—
168	C₂³	#10587626	1	1	yes	characteristic
169	Q₈	—	136	4	no	—
170	C₂×C₄	—	4	2	no	—
171	C₂×C₄	—	8	4	no	—
172	C₂×C₄	—	8	2	no	—
173	C₂×C₄	—	68	4	no	—
174	C₂×C₄	#10587579	1	2	yes	—
175	C₂×C₄	#10587581	1	2	yes	—
176	C₂×C₄	C₂×#63227	1	1	yes	characteristic
177	C₂×C₄	C₂×#63229	1	1	yes	characteristic
178	C₂×C₄	C₂×#63229	1	1	yes	characteristic
179	C₂×C₄	C₂×#63231	1	1	yes	characteristic
180	C₂×C₄	C₂×#63231	1	1	yes	characteristic
181	C₂×C₄	#10587633	1	1	yes	characteristic
182	C₈	—	68	4	no	—
183	C₈	#10587481	1	2	yes	—
184	C₈	#10587481	1	2	yes	—
185	C₈	#10587481	1	2	yes	—
186	C₈	#10587481	1	2	yes	—
187	C₂²	—	2	2	no	—
188	C₂²	#11762710	1	1	yes	characteristic
189	C₂²	#11763181	1	1	yes	characteristic
190	C₂²	#11763188	1	1	yes	characteristic
191	C₄	—	2	2	no	—
192	C₄	—	2	2	no	—
193	C₄	—	16	2	no	—
194	C₄	—	136	4	no	—
195	C₄	C₂×#10587481	1	1	yes	characteristic
196	C₄	C₂×#10587481	1	1	yes	characteristic
197	C₄	#11763161	1	1	yes	characteristic
198	C₄	#11763161	1	1	yes	characteristic
199	C₂	—	2	1	no	—
200	C₂	—	2	1	no	—
201	C₂	#14529585	1	1	yes	characteristic
202	C₂	#14529680	1	2	yes	—

	subgroup	G/H
center	C₂²	#11762710
commutator	C₄²×C₁₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₁₇×#1846	C₂
socle	C₂²×C₁₇	#390
prime radical	C₂³×C₁₇	#127
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	#1846	D₁₇
hypocenter	C₁₇	#9755
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1846

Complements

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

N	H ≅ G/N	type
6. #11763214	C₄	semidirect
9. #11763207	C₄	semidirect
144. C₁₇	#9755	semidirect

Maximal subgroups

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

subgroup	index
#14539298	2
#14527347	2
#14527067	2
C₁₇×#1846	2
#9755	17