Subgroups — #16934336

Order 4352.

494 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14528901	C₂	1	1	yes	maximal, characteristic
2	C₄²×#3622	C₂	1	1	yes	maximal, characteristic
3	#14526625	C₂	1	4	yes	maximal
4	C₄×C₁₇×#125	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#3622	C₂²	1	2	yes	—
6	C₂×C₄×#3622	—	2	2	no	—
7	C₂×C₄×#3622	C₂²	1	2	yes	—
8	#11762564	—	2	2	no	—
9	C₂×C₁₇×#125	C₄	1	1	yes	characteristic
10	C₁₇×#389	C₄	1	2	yes	—
11	C₁₇×#387	—	2	2	no	—
12	C₁₇×#382	C₄	1	1	yes	characteristic
13	C₄×C₄⋊C₄×C₁₇	—	2	2	no	—
14	C₄×C₄⋊C₄×C₁₇	C₂²	1	2	yes	—
15	C₄³×C₁₇	C₂²	1	1	yes	characteristic
16	C₁₇×#125	—	2	2	no	—
17	C₁₇×#125	C₈	1	4	yes	complemented
18	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
19	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
20	C₂×C₄⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
21	C₂×C₄⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
22	C₂×C₄²×C₁₇	—	2	2	no	—
23	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₁₇	D₄	1	2	yes	—
26	C₂×C₄²×C₁₇	D₄	1	2	yes	—
27	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
28	C₂²×#3622	—	2	2	no	—
29	C₂²×#3622	—	2	2	no	—
30	C₁₇×#95	—	2	2	no	—
31	C₁₇×#95	—	2	2	no	—
32	C₁₇×#95	—	2	2	no	—
33	C₄⋊#3622	—	2	4	no	—
34	C₄×#3622	—	2	8	no	—
35	C₄×#3622	—	2	2	no	—
36	C₄×#3622	Q₈	1	4	yes	—
37	C₂²×C₄×C₁₇	—	2	2	no	—
38	C₂²×C₄×C₁₇	—	2	2	no	—
39	C₂²×C₄×C₁₇	C₂²⋊C₄	1	2	yes	—
40	C₂²×C₄×C₁₇	C₂²⋊C₄	1	2	yes	—
41	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
44	C₄⋊C₄×C₁₇	—	2	8	no	—
45	C₄⋊C₄×C₁₇	—	2	2	no	—
46	C₄⋊C₄×C₁₇	—	2	4	no	—
47	C₄⋊C₄×C₁₇	—	4	4	no	—
48	C₄⋊C₄×C₁₇	C₂×C₈	1	4	yes	—
49	C₄²×C₁₇	—	2	4	no	—
50	C₄²×C₁₇	—	2	2	no	—
51	C₄²×C₁₇	—	2	4	no	—
52	C₄²×C₁₇	—	4	4	no	—
53	C₄²×C₁₇	C₂×C₈	1	2	yes	—
54	C₄²×C₁₇	M₁₆	1	2	yes	—
55	C₄²×C₁₇	C₂×Q₈	1	2	yes	—
56	C₄²×C₁₇	#41	1	2	yes	—
57	C₂×#3622	—	2	8	no	—
58	C₂×#3622	—	2	4	no	—
59	C₂×#3622	—	4	2	no	—
60	C₂×#3622	—	4	4	no	—
61	#9871	—	17	1	no	maximal, Sylow
62	C₂³×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₁₇	—	2	2	no	—
64	C₂×C₄×C₁₇	—	2	2	no	—
65	C₂×C₄×C₁₇	—	2	2	no	—
66	C₂×C₄×C₁₇	—	2	2	no	—
67	C₂×C₄×C₁₇	—	2	2	no	—
68	C₂×C₄×C₁₇	—	2	4	no	—
69	C₂×C₄×C₁₇	—	4	2	no	—
70	C₂×C₄×C₁₇	—	4	2	no	—
71	C₂×C₄×C₁₇	—	4	2	no	—
72	C₂×C₄×C₁₇	—	4	2	no	—
73	C₂×C₄×C₁₇	—	4	2	no	—
74	C₂×C₄×C₁₇	C₂²⋊C₈	1	4	yes	—
75	C₂×C₄×C₁₇	#104	1	4	yes	—
76	C₂×C₄×C₁₇	C₄×Q₈	1	2	yes	—
77	C₂×C₄×C₁₇	C₂²⋊Q₈	1	2	yes	—
78	C₂×C₄×C₁₇	#124	1	1	yes	characteristic
79	C₂×C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
80	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
81	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
82	C₂×C₄×C₁₇	#131	1	2	yes	—
83	C₂×C₄×C₁₇	#135	1	4	yes	—
84	#3622	—	2	8	no	—
85	#3622	—	8	2	no	—
86	C₄×#125	D₁₇	1	1	yes	characteristic
87	#1831	—	17	1	no	—
88	C₄²×C₈	—	17	1	no	—
89	#1279	—	17	4	no	—
90	C₂²×C₁₇	#391	1	1	yes	characteristic
91	C₂²×C₁₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
92	C₂²×C₁₇	#408	1	1	yes	characteristic
93	C₂²×C₁₇	#416	1	2	yes	—
94	C₂²×C₁₇	C₂×#104	1	2	yes	—
95	C₄×C₁₇	—	2	4	no	—
96	C₄×C₁₇	—	2	2	no	—
97	C₄×C₁₇	—	2	4	no	—
98	C₄×C₁₇	—	4	4	no	—
99	C₄×C₁₇	—	8	2	no	—
100	C₄×C₁₇	C₈×Q₈	1	2	yes	—
101	C₄×C₁₇	#446	1	2	yes	—
102	C₄×C₁₇	#488	1	2	yes	—
103	C₄×C₁₇	#499	1	2	yes	—
104	C₂×#125	Dic₁₇	1	1	yes	characteristic
105	C₂×C₄⋊C₈	—	17	2	no	—
106	C₂×C₄×C₈	—	17	2	no	—
107	C₂×C₄×C₈	—	34	2	no	—
108	#389	Dic₁₇	1	2	yes	—
109	#387	—	2	2	no	—
110	#382	Dic₁₇	1	1	yes	characteristic
111	C₄×C₄⋊C₄	—	2	2	no	—
112	C₄×C₄⋊C₄	C₂×D₁₇	1	2	yes	—
113	C₄³	C₂×D₁₇	1	1	yes	characteristic
114	#336	—	34	2	no	—
115	C₂×C₁₇	#1476	1	2	yes	—
116	C₂×C₁₇	#1494	1	2	yes	—
117	C₂×C₁₇	#1984	1	1	yes	characteristic
118	C₂×C₁₇	#1991	1	1	yes	characteristic
119	C₂×C₁₇	#1994	1	1	yes	characteristic
120	C₂²×C₈	—	34	2	no	—
121	C₂²×C₈	—	34	2	no	—
122	#125	—	2	2	no	—
123	#125	#3622	1	4	yes	complemented
124	C₂×C₄⋊C₄	—	2	2	no	—
125	C₂×C₄⋊C₄	—	2	2	no	—
126	C₂×C₄⋊C₄	C₂×Dic₁₇	1	2	yes	—
127	C₂×C₄⋊C₄	C₂×Dic₁₇	1	2	yes	—
128	C₂×C₄²	—	2	2	no	—
129	C₂×C₄²	C₂×Dic₁₇	1	1	yes	characteristic
130	C₂×C₄²	C₂×Dic₁₇	1	1	yes	characteristic
131	C₂×C₄²	C₂²⋊D₁₇	1	2	yes	—
132	C₂×C₄²	C₂²⋊D₁₇	1	2	yes	—
133	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
134	C₄⋊C₈	—	34	4	no	—
135	C₄×C₈	—	17	4	no	—
136	C₄×C₈	—	34	8	no	—
137	C₄×C₈	—	34	2	no	—
138	#95	—	2	2	no	—
139	#95	—	2	2	no	—
140	#95	—	2	2	no	—
141	C₁₇	#9871	1	1	yes	characteristic, complemented, Sylow
142	C₂²×C₄	—	2	2	no	—
143	C₂²×C₄	—	2	2	no	—
144	C₂²×C₄	C₂²⋊Dic₁₇	1	2	yes	—
145	C₂²×C₄	C₂²⋊Dic₁₇	1	2	yes	—
146	C₂²×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
147	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
148	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
149	C₂×C₈	—	34	8	no	—
150	C₂×C₈	—	34	4	no	—
151	C₂×C₈	—	68	2	no	—
152	C₂×C₈	—	68	4	no	—
153	C₄⋊C₄	—	2	8	no	—
154	C₄⋊C₄	—	2	2	no	—
155	C₄⋊C₄	—	2	4	no	—
156	C₄⋊C₄	—	4	4	no	—
157	C₄⋊C₄	C₂×#3622	1	4	yes	—
158	C₄²	—	2	4	no	—
159	C₄²	—	2	2	no	—
160	C₄²	—	2	4	no	—
161	C₄²	—	4	4	no	—
162	C₄²	C₂×#3622	1	2	yes	—
163	C₄²	#63223	1	2	yes	—
164	C₄²	Q₈×D₁₇	1	2	yes	—
165	C₄²	#63256	1	2	yes	—
166	C₂³	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	4	2	no	—
174	C₂×C₄	—	4	2	no	—
175	C₂×C₄	—	4	2	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	C₂²⋊#3622	1	4	yes	—
179	C₂×C₄	#10587508	1	4	yes	—
180	C₂×C₄	C₂²×#3622	1	1	yes	characteristic
181	C₂×C₄	C₂×#63223	1	1	yes	characteristic
182	C₂×C₄	#10587630	1	1	yes	characteristic
183	C₂×C₄	Q₈×Dic₁₇	1	2	yes	—
184	C₂×C₄	#10587632	1	2	yes	—
185	C₂×C₄	#10587633	1	1	yes	characteristic
186	C₂×C₄	#10587634	1	2	yes	—
187	C₂×C₄	#10587636	1	4	yes	—
188	C₈	—	34	8	no	—
189	C₈	—	136	2	no	—
190	C₂²	#11762716	1	1	yes	characteristic
191	C₂²	C₂×C₂²⋊#3622	1	1	yes	characteristic
192	C₂²	#11762776	1	1	yes	characteristic
193	C₂²	#11762805	1	2	yes	—
194	C₂²	C₂×#10587508	1	2	yes	—
195	C₄	—	2	4	no	—
196	C₄	—	2	4	no	—
197	C₄	—	2	2	no	—
198	C₄	—	4	4	no	—
199	C₄	—	8	2	no	—
200	C₄	Q₈×#3622	1	2	yes	—
201	C₄	#11762940	1	2	yes	—
202	C₄	#11763166	1	2	yes	—
203	C₄	#11763208	1	2	yes	—
204	C₂	#14527166	1	2	yes	—
205	C₂	#14527244	1	2	yes	—
206	C₂	#14529721	1	1	yes	characteristic
207	C₂	#14529750	1	1	yes	characteristic
208	C₂	#14529762	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₁₇
commutator	C₂×C₄×C₁₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₄×C₁₇×#125	C₂
socle, prime radical	C₂³×C₁₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	C₄×#125	D₁₇
hypocenter	C₁₇	#9871
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×#125

Complements

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

N	H ≅ G/N	type
17. C₁₇×#125	C₈	semidirect
123. #125	#3622	semidirect
141. C₁₇	#9871	semidirect

Maximal subgroups

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

subgroup	index
#14528901	2
C₄²×#3622	2
#14526625	2
C₄×C₁₇×#125	2
#9871	17