Subgroups — #16925633

Order 4352.

334 conjugacy classes of subgroups.

The subgroup lattice has 188 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	#14527071	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	#11762531	—	2	4	no	—
12	#11762530	C₄	1	2	yes	—
13	#11762530	C₄	1	2	yes	—
14	C₂×C₁₇×#108	C₄	1	2	yes	—
15	C₄×M₁₆×C₁₇	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
17	C₁₇×#341	C₄	1	2	yes	—
18	C₂×M₁₆×C₁₇	C₂×C₄	1	4	yes	—
19	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
22	C₂²×#3622	—	2	1	no	—
23	C₂²×#3622	Q₈	1	2	yes	—
24	C₁₇×#108	—	2	4	no	—
25	C₁₇×#97	—	2	2	no	—
26	C₄×C₈×C₁₇	—	2	2	no	—
27	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
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	C₂²⋊#3622	—	2	2	no	—
32	C₄⋊#3622	C₂×C₄	1	4	yes	—
33	C₄×#3622	—	2	2	no	—
34	C₄×#3622	—	2	1	no	—
35	C₄×#3622	C₂×C₄	1	2	yes	—
36	C₄×#3622	C₂×C₄	1	2	yes	—
37	C₄×#3622	D₄	1	2	yes	—
38	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
40	C₂²×C₄×C₁₇	#41	1	1	yes	characteristic
41	M₁₆×C₁₇	—	2	4	no	—
42	M₁₆×C₁₇	—	2	4	no	—
43	C₂×C₈×C₁₇	—	2	2	no	—
44	C₂×C₈×C₁₇	—	2	4	no	—
45	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
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	M₁₆	1	4	yes	—
50	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
51	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
52	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
53	C₄²×C₁₇	#41	1	1	yes	characteristic
54	C₂×#3622	—	2	2	no	—
55	C₂×#3622	—	2	4	no	—
56	C₂×#3622	—	4	2	no	—
57	C₂×#3622	—	4	2	no	—
58	C₂×#3622	C₄⋊C₄	1	4	yes	—
59	#8295	—	17	1	no	maximal, Sylow
60	C₂³×C₁₇	C₄×Q₈	1	1	yes	characteristic
61	C₂×C₄×C₁₇	—	2	2	no	—
62	C₂×C₄×C₁₇	—	2	2	no	—
63	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
64	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₁₇	#117	1	1	yes	characteristic
66	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
67	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
68	C₂×C₄×C₁₇	C₄×Q₈	1	1	yes	characteristic
69	C₂×C₄×C₁₇	C₂×M₁₆	1	2	yes	—
70	C₂×C₄×C₁₇	#131	1	2	yes	—
71	C₈×C₁₇	—	2	1	no	—
72	C₈×C₁₇	—	2	1	no	—
73	C₈×C₁₇	—	4	4	no	—
74	C₈×C₁₇	#97	1	2	yes	—
75	C₈×C₁₇	#97	1	2	yes	—
76	#3622	—	2	4	no	—
77	#3622	—	4	2	no	—
78	#3622	—	4	4	no	—
79	C₄×#108	D₁₇	1	1	yes	characteristic
80	#1564	—	17	1	no	—
81	#1451	—	17	2	no	—
82	C₂×#322	—	17	1	no	—
83	#1278	—	17	2	no	—
84	C₂²×C₁₇	—	2	1	no	—
85	C₂²×C₁₇	—	2	1	no	—
86	C₂²×C₁₇	C₄×C₄⋊C₄	1	1	yes	characteristic
87	C₂²×C₁₇	#446	1	2	yes	—
88	C₄×C₁₇	—	2	1	no	—
89	C₄×C₁₇	—	2	1	no	—
90	C₄×C₁₇	C₂×#97	1	1	yes	characteristic
91	C₄×C₁₇	#405	1	1	yes	characteristic
92	C₄×C₁₇	#430	1	2	yes	—
93	C₄×C₁₇	#436	1	2	yes	—
94	C₄×C₁₇	#444	1	2	yes	—
95	#431	—	17	2	no	—
96	C₂×#108	Dic₁₇	1	2	yes	—
97	C₄×M₁₆	C₂×D₁₇	1	2	yes	—
98	C₂×C₄×C₈	—	17	1	no	—
99	C₂×C₄×C₈	—	17	1	no	—
100	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
101	#341	Dic₁₇	1	2	yes	—
102	#335	—	17	2	no	—
103	#334	—	17	2	no	—
104	#322	—	17	2	no	—
105	#322	—	17	2	no	—
106	#322	—	34	4	no	—
107	C₂×C₁₇	—	2	2	no	—
108	C₂×C₁₇	#1770	1	1	yes	characteristic
109	C₂×C₁₇	#1773	1	1	yes	characteristic
110	C₂×C₁₇	#1778	1	1	yes	characteristic
111	C₂×M₁₆	C₂×Dic₁₇	1	4	yes	—
112	C₂²×C₈	—	17	2	no	—
113	C₂²×C₈	—	34	1	no	—
114	C₂²×C₈	C₂×Dic₁₇	1	1	yes	characteristic
115	C₂²×C₈	C₂×Dic₁₇	1	1	yes	characteristic
116	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
117	#108	—	2	4	no	—
118	C₄⋊C₈	—	17	4	no	—
119	C₂²⋊C₈	—	34	2	no	—
120	#97	—	2	2	no	—
121	C₄×C₈	—	2	2	no	—
122	C₄×C₈	—	17	2	no	—
123	C₄×C₈	—	17	2	no	—
124	C₄×C₈	—	17	2	no	—
125	C₄×C₈	—	34	2	no	—
126	C₄×C₈	—	34	1	no	—
127	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
128	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
129	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
130	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
131	C₁₇	#8295	1	1	yes	characteristic, complemented, Sylow
132	C₂²×C₄	Q₈×D₁₇	1	1	yes	characteristic
133	C₂²×C₄	#63256	1	1	yes	characteristic
134	C₂²×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
135	M₁₆	—	2	4	no	—
136	M₁₆	—	2	4	no	—
137	C₂×C₈	—	2	2	no	—
138	C₂×C₈	—	2	4	no	—
139	C₂×C₈	—	17	4	no	—
140	C₂×C₈	—	34	2	no	—
141	C₂×C₈	—	34	4	no	—
142	C₂×C₈	—	68	2	no	—
143	C₂×C₈	—	68	2	no	—
144	C₂×C₈	#63223	1	4	yes	—
145	C₂×C₈	C₄×Dic₁₇	1	1	yes	characteristic
146	C₂×C₈	C₄×Dic₁₇	1	1	yes	characteristic
147	C₂×C₈	C₄×Dic₁₇	1	1	yes	characteristic
148	C₂×C₈	C₄×Dic₁₇	1	1	yes	characteristic
149	C₄²	C₂×C₄×D₁₇	1	1	yes	characteristic
150	C₄²	C₂×C₄×D₁₇	1	1	yes	characteristic
151	C₄²	D₄×D₁₇	1	1	yes	characteristic
152	C₄²	#63254	1	1	yes	characteristic
153	C₂³	Q₈×Dic₁₇	1	1	yes	characteristic
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	#10587573	1	1	yes	characteristic
157	C₂×C₄	C₄⋊C₄×D₁₇	1	1	yes	characteristic
158	C₂×C₄	#10587578	1	1	yes	characteristic
159	C₂×C₄	#10587579	1	1	yes	characteristic
160	C₂×C₄	C₂×#63223	1	2	yes	—
161	C₂×C₄	C₂×C₄×Dic₁₇	1	1	yes	characteristic
162	C₂×C₄	D₄×Dic₁₇	1	1	yes	characteristic
163	C₂×C₄	#10587634	1	2	yes	—
164	C₈	—	2	1	no	—
165	C₈	—	2	1	no	—
166	C₈	—	4	4	no	—
167	C₈	—	34	4	no	—
168	C₈	—	68	2	no	—
169	C₈	—	68	4	no	—
170	C₈	#10587474	1	2	yes	—
171	C₈	#10587474	1	2	yes	—
172	C₂²	—	2	1	no	—
173	C₂²	—	2	1	no	—
174	C₂²	C₄⋊C₄×Dic₁₇	1	1	yes	characteristic
175	C₂²	#11762940	1	2	yes	—
176	C₄	—	2	1	no	—
177	C₄	—	2	1	no	—
178	C₄	C₂×#10587474	1	1	yes	characteristic
179	C₄	#11762765	1	1	yes	characteristic
180	C₄	#11762856	1	2	yes	—
181	C₄	#11762887	1	2	yes	—
182	C₄	#11762930	1	2	yes	—
183	C₂	—	2	2	no	—
184	C₂	#14528653	1	1	yes	characteristic
185	C₂	#14528671	1	1	yes	characteristic
186	C₂	#14528692	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₁₇
commutator	C₂×C₄×C₁₇	C₂×C₄²
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	#8295
LCS	G ▷ C₂×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
131. C₁₇	#8295	semidirect

Maximal subgroups

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

subgroup	index
#14527532	2
#14527071	2
C₂×#11762530	2
#14526622	2
C₄×C₁₇×#108	2
#8295	17