Subgroups — #16925628

Order 4352.

358 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14527532	C₂	1	1	yes	maximal, characteristic
2	M₁₆×#3622	C₂	1	2	yes	maximal
3	C₂×#11762530	C₂	1	1	yes	maximal, characteristic
4	#14526622	C₂	1	2	yes	maximal
5	C₄×C₁₇×#108	C₂	1	1	yes	maximal, characteristic
6	#11762862	C₂²	1	2	yes	—
7	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
8	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
9	#11762562	C₄	1	2	yes	—
10	#11762560	C₄	1	2	yes	—
11	#11762530	—	2	2	no	—
12	#11762530	C₄	1	2	yes	—
13	#11762530	C₄	1	2	yes	—
14	C₈×#3622	—	2	2	no	—
15	C₂×C₁₇×#108	C₄	1	2	yes	—
16	C₄×M₁₆×C₁₇	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
18	C₁₇×#341	C₄	1	2	yes	—
19	C₂×M₁₆×C₁₇	C₂×C₄	1	4	yes	—
20	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
23	C₂²×#3622	—	2	1	no	—
24	C₂²×#3622	Q₈	1	2	yes	—
25	C₁₇×#108	C₈	1	8	yes	complemented
26	C₁₇×#97	—	2	2	no	—
27	C₄×C₈×C₁₇	—	2	2	no	—
28	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
32	C₂²⋊#3622	—	2	2	no	—
33	C₄⋊#3622	C₂×C₄	1	4	yes	—
34	C₄×#3622	—	2	2	no	—
35	C₄×#3622	—	2	1	no	—
36	C₄×#3622	C₂×C₄	1	2	yes	—
37	C₄×#3622	C₂×C₄	1	2	yes	—
38	C₄×#3622	D₄	1	2	yes	—
39	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
41	C₂²×C₄×C₁₇	#41	1	1	yes	characteristic
42	M₁₆×C₁₇	—	2	4	no	—
43	M₁₆×C₁₇	C₂×C₈	1	8	yes	—
44	C₂×C₈×C₁₇	—	2	4	no	—
45	C₂×C₈×C₁₇	—	2	2	no	—
46	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
47	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
48	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
49	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
50	C₂×C₈×C₁₇	C₂×C₈	1	4	yes	—
51	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
52	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
53	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
54	C₄²×C₁₇	#41	1	1	yes	characteristic
55	C₂×#3622	—	2	2	no	—
56	C₂×#3622	—	2	4	no	—
57	C₂×#3622	—	4	2	no	—
58	C₂×#3622	—	4	2	no	—
59	C₂×#3622	C₄⋊C₄	1	4	yes	—
60	#8294	—	17	1	no	maximal, Sylow
61	C₂³×C₁₇	C₄×Q₈	1	1	yes	characteristic
62	C₂×C₄×C₁₇	—	2	2	no	—
63	C₂×C₄×C₁₇	—	2	2	no	—
64	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
65	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₁₇	#117	1	1	yes	characteristic
67	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
68	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
69	C₂×C₄×C₁₇	C₄×Q₈	1	1	yes	characteristic
70	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
71	C₂×C₄×C₁₇	#131	1	2	yes	—
72	C₈×C₁₇	—	2	1	no	—
73	C₈×C₁₇	—	2	1	no	—
74	C₈×C₁₇	—	2	8	no	—
75	C₈×C₁₇	C₄×C₈	1	2	yes	—
76	C₈×C₁₇	C₄×C₈	1	2	yes	—
77	#3622	—	2	4	no	—
78	#3622	—	4	2	no	—
79	#3622	—	4	4	no	—
80	C₄×#108	D₁₇	1	1	yes	characteristic
81	#1564	—	17	1	no	—
82	C₈×M₁₆	—	17	2	no	—
83	C₂×#322	—	17	1	no	—
84	#1278	—	17	2	no	—
85	C₂²×C₁₇	—	2	1	no	—
86	C₂²×C₁₇	—	2	1	no	—
87	C₂²×C₁₇	C₄×C₄⋊C₄	1	1	yes	characteristic
88	C₂²×C₁₇	C₈×Q₈	1	2	yes	—
89	C₄×C₁₇	—	2	1	no	—
90	C₄×C₁₇	—	2	1	no	—
91	C₄×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
92	C₄×C₁₇	#405	1	1	yes	characteristic
93	C₄×C₁₇	#430	1	2	yes	—
94	C₄×C₁₇	C₈×D₄	1	2	yes	—
95	C₄×C₁₇	#444	1	2	yes	—
96	#431	—	17	2	no	—
97	C₂×#108	Dic₁₇	1	2	yes	—
98	C₄×M₁₆	C₂×D₁₇	1	2	yes	—
99	C₂×C₄×C₈	—	17	1	no	—
100	C₂×C₄×C₈	—	17	1	no	—
101	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
102	#341	Dic₁₇	1	2	yes	—
103	#335	—	17	2	no	—
104	#334	—	17	2	no	—
105	#322	—	17	2	no	—
106	#322	—	17	2	no	—
107	#322	—	34	2	no	—
108	C₈²	—	34	2	no	—
109	C₂×C₁₇	—	2	2	no	—
110	C₂×C₁₇	C₈×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₁₇	#1773	1	1	yes	characteristic
112	C₂×C₁₇	#1778	1	1	yes	characteristic
113	C₂×M₁₆	C₂×Dic₁₇	1	4	yes	—
114	C₂²×C₈	—	17	2	no	—
115	C₂²×C₈	—	34	1	no	—
116	C₂²×C₈	C₂×Dic₁₇	1	1	yes	characteristic
117	C₂²×C₈	C₂×Dic₁₇	1	1	yes	characteristic
118	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
119	#108	#3622	1	8	yes	complemented
120	C₄⋊C₈	—	17	4	no	—
121	C₂²⋊C₈	—	34	2	no	—
122	#97	—	2	2	no	—
123	C₄×C₈	—	2	2	no	—
124	C₄×C₈	—	17	2	no	—
125	C₄×C₈	—	17	2	no	—
126	C₄×C₈	—	17	2	no	—
127	C₄×C₈	—	34	2	no	—
128	C₄×C₈	—	34	1	no	—
129	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
130	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
131	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
132	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
133	C₁₇	#8294	1	1	yes	characteristic, complemented, Sylow
134	C₂²×C₄	Q₈×D₁₇	1	1	yes	characteristic
135	C₂²×C₄	#63256	1	1	yes	characteristic
136	C₂²×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
137	M₁₆	—	2	4	no	—
138	M₁₆	C₂×#3622	1	8	yes	—
139	C₂×C₈	—	2	4	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	17	4	no	—
142	C₂×C₈	—	34	2	no	—
143	C₂×C₈	—	34	4	no	—
144	C₂×C₈	—	68	2	no	—
145	C₂×C₈	—	68	2	no	—
146	C₂×C₈	C₂×#3622	1	4	yes	—
147	C₂×C₈	C₄×Dic₁₇	1	1	yes	characteristic
148	C₂×C₈	C₄×Dic₁₇	1	1	yes	characteristic
149	C₂×C₈	C₄×Dic₁₇	1	1	yes	characteristic
150	C₂×C₈	C₄×Dic₁₇	1	1	yes	characteristic
151	C₄²	C₂×C₄×D₁₇	1	1	yes	characteristic
152	C₄²	C₂×C₄×D₁₇	1	1	yes	characteristic
153	C₄²	D₄×D₁₇	1	1	yes	characteristic
154	C₄²	#63254	1	1	yes	characteristic
155	C₂³	Q₈×Dic₁₇	1	1	yes	characteristic
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	#10587573	1	1	yes	characteristic
159	C₂×C₄	C₄⋊C₄×D₁₇	1	1	yes	characteristic
160	C₂×C₄	#10587578	1	1	yes	characteristic
161	C₂×C₄	#10587579	1	1	yes	characteristic
162	C₂×C₄	C₂²×#3622	1	2	yes	—
163	C₂×C₄	C₂×C₄×Dic₁₇	1	1	yes	characteristic
164	C₂×C₄	D₄×Dic₁₇	1	1	yes	characteristic
165	C₂×C₄	#10587634	1	2	yes	—
166	C₈	—	2	1	no	—
167	C₈	—	2	1	no	—
168	C₈	—	2	8	no	—
169	C₈	—	34	4	no	—
170	C₈	—	68	2	no	—
171	C₈	—	68	4	no	—
172	C₈	C₄×#3622	1	2	yes	—
173	C₈	C₄×#3622	1	2	yes	—
174	C₂²	—	2	1	no	—
175	C₂²	—	2	1	no	—
176	C₂²	C₄⋊C₄×Dic₁₇	1	1	yes	characteristic
177	C₂²	Q₈×#3622	1	2	yes	—
178	C₄	—	2	1	no	—
179	C₄	—	2	1	no	—
180	C₄	C₂×C₄×#3622	1	1	yes	characteristic
181	C₄	#11762765	1	1	yes	characteristic
182	C₄	#11762856	1	2	yes	—
183	C₄	D₄×#3622	1	2	yes	—
184	C₄	#11762930	1	2	yes	—
185	C₂	—	2	2	no	—
186	C₂	C₄⋊C₄×#3622	1	1	yes	characteristic
187	C₂	#14528671	1	1	yes	characteristic
188	C₂	#14528692	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₁₇
commutator	C₄×C₁₇	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₄×C₁₇×#108	C₂
socle	C₂²×C₁₇	C₄×C₄⋊C₄
prime radical	C₂³×C₁₇	C₄×Q₈
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	C₄×#108	D₁₇
hypocenter	C₁₇	#8294
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄×#108

Complements

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

N	H ≅ G/N	type
25. C₁₇×#108	C₈	semidirect
119. #108	#3622	semidirect
133. C₁₇	#8294	semidirect

Maximal subgroups

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

subgroup	index
#14527532	2
M₁₆×#3622	2
C₂×#11762530	2
#14526622	2
C₄×C₁₇×#108	2
#8294	17