Subgroups — #16950681

Order 4352.

314 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11762630	C₂	1	1	yes	maximal, characteristic
2	C₄×#10587486	C₂	1	1	yes	maximal, characteristic
3	#14526867	C₂	1	2	yes	maximal
4	#14526863	C₂	1	2	yes	maximal
5	C₂×C₁₇×#364	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587483	C₂²	1	2	yes	—
7	C₂×#10587486	C₄	1	1	yes	characteristic
8	C₂×#10587486	C₄	1	1	yes	characteristic
9	C₂×#10587486	C₂²	1	1	yes	characteristic
10	C₂×#10587486	C₂²	1	1	yes	characteristic
11	C₂×C₄×#3622	C₄	1	1	yes	characteristic
12	C₄²×Dic₁₇	C₄	1	1	yes	characteristic
13	#11762630	—	2	2	no	—
14	C₂×C₁₇×M₃₂	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
16	C₁₇×#364	—	2	2	no	—
17	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
19	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×Dic₁₇	—	2	2	no	—
21	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
23	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
24	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
25	C₁₇×M₃₂	—	2	2	no	—
26	C₁₇×M₃₂	—	2	2	no	—
27	C₂×C₁₆×C₁₇	—	2	2	no	—
28	C₄×C₈×C₁₇	D₄	1	2	yes	—
29	C₄×C₈×C₁₇	Q₈	1	2	yes	—
30	#10587486	—	2	2	no	—
31	#10587486	—	2	1	no	—
32	#10587486	—	2	1	no	—
33	#10587486	—	2	1	no	—
34	#10587486	—	2	1	no	—
35	#10587486	C₈	1	4	yes	—
36	#10587483	—	2	2	no	—
37	#10587483	—	2	2	no	—
38	C₂×#63214	—	2	2	no	—
39	C₄×#3622	—	2	2	no	—
40	C₂²×C₄×C₁₇	—	2	1	no	—
41	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
42	C₂²×Dic₁₇	—	2	1	no	—
43	C₂²×Dic₁₇	M₁₆	1	2	yes	—
44	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
45	C₂×C₈×C₁₇	C₂×C₈	1	2	yes	—
46	C₂×C₈×C₁₇	M₁₆	1	2	yes	—
47	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
50	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
51	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
52	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
53	C₄²×C₁₇	—	2	1	no	—
54	C₄²×C₁₇	C₄⋊C₄	1	2	yes	—
55	C₄×Dic₁₇	—	2	1	no	—
56	C₄×Dic₁₇	—	2	1	no	—
57	C₄×Dic₁₇	—	2	1	no	—
58	C₄×Dic₁₇	—	4	4	no	—
59	C₄×Dic₁₇	C₂×C₈	1	2	yes	—
60	C₂×#3622	—	2	1	no	—
61	C₂×#3622	—	2	1	no	—
62	C₂×#3622	—	2	1	no	—
63	C₂×#3622	—	2	1	no	—
64	C₂×#3622	C₂×C₈	1	2	yes	—
65	C₂×#3622	M₁₆	1	2	yes	—
66	C₁₆×C₁₇	—	4	2	no	—
67	#63214	—	4	2	no	—
68	#12882	—	17	1	no	maximal, Sylow
69	C₂³×C₁₇	C₂×M₁₆	1	1	yes	characteristic
70	C₂×C₄×C₁₇	—	2	4	no	—
71	C₂×C₄×C₁₇	—	2	2	no	—
72	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₁₇	#117	1	1	yes	characteristic
74	C₂×C₄×C₁₇	#117	1	1	yes	characteristic
75	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
76	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
77	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
78	C₂×Dic₁₇	—	2	2	no	—
79	C₂×Dic₁₇	—	2	2	no	—
80	C₂×Dic₁₇	—	4	2	no	—
81	C₂×Dic₁₇	—	4	2	no	—
82	C₈×C₁₇	C₄⋊C₈	1	4	yes	—
83	C₈×C₁₇	#124	1	1	yes	characteristic
84	C₈×C₁₇	#124	1	1	yes	characteristic
85	C₈×C₁₇	#125	1	1	yes	characteristic
86	C₈×C₁₇	#125	1	1	yes	characteristic
87	#3622	—	2	2	no	—
88	#3622	—	2	2	no	—
89	C₂×#364	—	17	1	no	—
90	C₂×#364	D₁₇	1	1	yes	characteristic
91	C₄×#97	—	17	1	no	—
92	#1377	—	17	2	no	—
93	#1375	—	17	2	no	—
94	C₂²×C₁₇	—	2	2	no	—
95	C₂²×C₁₇	#423	1	1	yes	characteristic
96	C₂²×C₁₇	#431	1	1	yes	characteristic
97	C₂²×C₁₇	#432	1	1	yes	characteristic
98	C₄×C₁₇	—	2	4	no	—
99	C₄×C₁₇	#382	1	1	yes	characteristic
100	C₄×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
101	C₄×C₁₇	#431	1	1	yes	characteristic
102	C₄×C₁₇	#432	1	1	yes	characteristic
103	Dic₁₇	—	4	2	no	—
104	Dic₁₇	—	4	2	no	—
105	C₂×M₃₂	—	17	2	no	—
106	C₂×M₃₂	C₂×D₁₇	1	2	yes	—
107	C₂×#97	—	17	1	no	—
108	C₂×#97	—	17	1	no	—
109	C₂×#97	—	17	1	no	—
110	C₂×#97	—	17	1	no	—
111	C₂×C₄×C₈	—	17	1	no	—
112	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
113	C₄³	—	17	1	no	—
114	#364	—	2	2	no	—
115	#364	—	34	2	no	—
116	C₂×C₁₇	—	2	1	no	—
117	C₂×C₁₇	—	2	1	no	—
118	C₂×C₁₇	#1831	1	1	yes	characteristic
119	C₂×C₁₇	#2165	1	2	yes	—
120	C₂²×C₈	—	17	1	no	—
121	C₂²×C₈	—	17	1	no	—
122	C₂²×C₈	C₄×D₁₇	1	1	yes	characteristic
123	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
124	C₂×C₄²	—	17	1	no	—
125	C₂×C₄²	—	17	1	no	—
126	C₂×C₄²	—	34	2	no	—
127	C₂×C₄²	C₄×D₁₇	1	1	yes	characteristic
128	M₃₂	—	2	2	no	—
129	M₃₂	—	2	2	no	—
130	M₃₂	—	34	2	no	—
131	M₃₂	—	34	2	no	—
132	C₂×C₁₆	—	2	2	no	—
133	C₂×C₁₆	—	34	2	no	—
134	#97	—	17	4	no	—
135	#97	—	34	1	no	—
136	#97	—	34	1	no	—
137	#97	—	34	1	no	—
138	#97	—	34	1	no	—
139	#97	—	34	2	no	—
140	C₄×C₈	—	34	2	no	—
141	C₄×C₈	Dic₃₄	1	2	yes	—
142	C₄×C₈	C₂²⋊D₁₇	1	2	yes	—
143	C₁₇	#12882	1	1	yes	characteristic, complemented, Sylow
144	C₂²×C₄	—	2	1	no	—
145	C₂²×C₄	—	17	2	no	—
146	C₂²×C₄	—	34	1	no	—
147	C₂²×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
148	C₂×C₈	—	17	2	no	—
149	C₂×C₈	—	17	2	no	—
150	C₂×C₈	—	34	1	no	—
151	C₂×C₈	—	34	1	no	—
152	C₂×C₈	—	34	1	no	—
153	C₂×C₈	—	34	1	no	—
154	C₂×C₈	C₈×D₁₇	1	2	yes	—
155	C₂×C₈	#63218	1	2	yes	—
156	C₂×C₈	#63225	1	2	yes	—
157	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
158	C₂×C₈	#63254	1	1	yes	characteristic
159	C₂×C₈	#63254	1	1	yes	characteristic
160	C₂×C₈	#63256	1	1	yes	characteristic
161	C₂×C₈	#63256	1	1	yes	characteristic
162	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
163	C₄²	—	2	1	no	—
164	C₄²	—	17	2	no	—
165	C₄²	—	34	1	no	—
166	C₄²	—	34	1	no	—
167	C₄²	—	34	1	no	—
168	C₄²	—	68	4	no	—
169	C₄²	#63225	1	2	yes	—
170	C₁₆	—	4	2	no	—
171	C₁₆	—	68	2	no	—
172	C₂³	M₁₆×D₁₇	1	1	yes	characteristic
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	34	2	no	—
176	C₂×C₄	—	34	2	no	—
177	C₂×C₄	—	68	2	no	—
178	C₂×C₄	—	68	2	no	—
179	C₂×C₄	#10587578	1	1	yes	characteristic
180	C₂×C₄	#10587578	1	1	yes	characteristic
181	C₂×C₄	C₂×C₈×D₁₇	1	1	yes	characteristic
182	C₂×C₄	C₂×#63218	1	1	yes	characteristic
183	C₂×C₄	M₁₆×D₁₇	1	1	yes	characteristic
184	C₂×C₄	C₂×#63225	1	1	yes	characteristic
185	C₈	—	34	2	no	—
186	C₈	—	34	2	no	—
187	C₈	#10587485	1	4	yes	—
188	C₈	#10587576	1	1	yes	characteristic
189	C₈	#10587576	1	1	yes	characteristic
190	C₈	#10587622	1	1	yes	characteristic
191	C₈	#10587633	1	1	yes	characteristic
192	C₂²	—	2	2	no	—
193	C₂²	#11762832	1	1	yes	characteristic
194	C₂²	#11762861	1	1	yes	characteristic
195	C₂²	#11762865	1	1	yes	characteristic
196	C₄	—	2	4	no	—
197	C₄	—	68	2	no	—
198	C₄	—	68	2	no	—
199	C₄	#11762674	1	1	yes	characteristic
200	C₄	C₂×#10587485	1	1	yes	characteristic
201	C₄	#11762861	1	1	yes	characteristic
202	C₄	#11762865	1	1	yes	characteristic
203	C₂	—	2	1	no	—
204	C₂	—	2	1	no	—
205	C₂	#14528900	1	1	yes	characteristic
206	C₂	#14530595	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#10587578
commutator	C₂×C₄×C₁₇	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₂×C₁₇×#364	C₂
socle	C₂²×C₁₇	#432
prime radical	C₂³×C₁₇	C₂×M₁₆
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	C₂×#364	D₁₇
hypocenter	C₁₇	#12882
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
143. C₁₇	#12882	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11762630	2
C₄×#10587486	2
#14526867	2
#14526863	2
C₂×C₁₇×#364	2
#12882	17