Subgroups — #16952350

Order 4352.

284 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	#14530550	C₂	1	2	yes	maximal
2	C₂×#11762630	C₂	1	1	yes	maximal, characteristic
3	C₄×#10587483	C₂	1	1	yes	maximal, characteristic
4	#14526870	C₂	1	2	yes	maximal
5	C₁₇×#1772	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587483	C₂²	1	2	yes	—
7	C₂×#10587483	C₂²	1	1	yes	characteristic
8	C₂×#10587483	C₂²	1	1	yes	characteristic
9	#11762630	—	2	1	no	—
10	#11762630	—	2	1	no	—
11	#11762630	—	2	4	no	—
12	#11762585	—	2	1	no	—
13	C₄×#63214	—	2	1	no	—
14	C₂×C₁₇×#108	C₄	1	1	yes	characteristic
15	C₁₇×#427	C₄	1	1	yes	characteristic
16	C₁₇×#405	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
18	C₁₇×#339	C₄	1	2	yes	—
19	C₂×M₁₆×C₁₇	C₂×C₄	1	2	yes	—
20	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
22	C₁₇×#117	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
24	C₁₇×#108	C₈	1	2	yes	—
25	C₁₇×#108	C₈	1	2	yes	—
26	C₁₇×#107	—	2	1	no	—
27	C₁₇×#106	—	2	1	no	—
28	C₁₇×#97	—	2	2	no	—
29	C₄×C₈×C₁₇	—	2	2	no	—
30	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
31	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
32	C₄×C₈×C₁₇	Q₈	1	1	yes	characteristic
33	C₄×C₈×C₁₇	Q₈	1	1	yes	characteristic
34	#10587483	—	2	2	no	—
35	#10587483	—	2	2	no	—
36	#10587483	—	2	1	no	—
37	#10587483	—	2	1	no	—
38	#10587483	D₄	1	2	yes	—
39	#10587483	Q₈	1	2	yes	—
40	C₂×#63214	—	2	1	no	—
41	C₂×#63214	—	2	1	no	—
42	C₂×#63214	—	2	2	no	—
43	C₂²×C₄×C₁₇	—	2	1	no	—
44	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
45	M₁₆×C₁₇	—	2	2	no	—
46	M₁₆×C₁₇	C₂×C₈	1	4	yes	—
47	C₂×C₈×C₁₇	—	2	2	no	—
48	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
51	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
52	C₂×C₈×C₁₇	M₁₆	1	1	yes	characteristic
53	C₂×C₈×C₁₇	M₁₆	1	1	yes	characteristic
54	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
55	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
59	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
60	C₄⋊C₄×C₁₇	—	2	2	no	—
61	C₂²⋊C₄×C₁₇	—	2	2	no	—
62	C₄²×C₁₇	—	2	1	no	—
63	C₄²×C₁₇	—	2	2	no	—
64	C₄²×C₁₇	C₄⋊C₄	1	1	yes	characteristic
65	C₄²×C₁₇	C₄⋊C₄	1	1	yes	characteristic
66	#63214	—	2	2	no	—
67	#63214	—	2	2	no	—
68	#63214	—	4	2	no	—
69	#13164	—	17	1	no	maximal, Sylow
70	C₂³×C₁₇	#131	1	1	yes	characteristic
71	C₂×C₄×C₁₇	—	2	1	no	—
72	C₂×C₄×C₁₇	—	2	1	no	—
73	C₂×C₄×C₁₇	—	4	2	no	—
74	C₂×C₄×C₁₇	—	4	2	no	—
75	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
77	C₂×C₄×C₁₇	C₄×Q₈	1	1	yes	characteristic
78	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
79	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
80	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
81	C₈×C₁₇	—	2	4	no	—
82	C₈×C₁₇	C₄⋊C₈	1	1	yes	characteristic
83	C₈×C₁₇	C₄⋊C₈	1	1	yes	characteristic
84	C₈×C₁₇	C₄⋊C₈	1	1	yes	characteristic
85	C₈×C₁₇	C₄⋊C₈	1	1	yes	characteristic
86	C₈×C₁₇	#121	1	1	yes	characteristic
87	C₈×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
88	C₈×C₁₇	#125	1	1	yes	characteristic
89	C₈×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
90	#2153	—	17	2	no	—
91	C₂×#364	—	17	1	no	—
92	C₄×M₃₂	—	17	1	no	—
93	#1772	D₁₇	1	1	yes	characteristic
94	#1378	—	17	2	no	—
95	C₂²×C₁₇	—	4	1	no	—
96	C₂²×C₁₇	#424	1	1	yes	characteristic
97	C₂²×C₁₇	#436	1	1	yes	characteristic
98	C₂²×C₁₇	C₈×Q₈	1	1	yes	characteristic
99	C₄×C₁₇	—	2	2	no	—
100	C₄×C₁₇	—	4	1	no	—
101	C₄×C₁₇	—	8	2	no	—
102	C₄×C₁₇	#389	1	1	yes	characteristic
103	C₄×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
104	C₄×C₁₇	C₈×D₄	1	1	yes	characteristic
105	C₄×C₁₇	#446	1	1	yes	characteristic
106	C₂×M₃₂	—	17	1	no	—
107	C₂×M₃₂	—	17	1	no	—
108	C₂×M₃₂	—	17	2	no	—
109	C₂×#108	Dic₁₇	1	1	yes	characteristic
110	#427	Dic₁₇	1	1	yes	characteristic
111	#405	C₂×D₁₇	1	2	yes	—
112	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
113	#364	—	34	4	no	—
114	#364	—	34	1	no	—
115	#364	—	34	1	no	—
116	#346	—	34	1	no	—
117	C₄×C₁₆	—	34	1	no	—
118	#339	Dic₁₇	1	2	yes	—
119	C₂×C₁₇	—	2	1	no	—
120	C₂×C₁₇	—	2	1	no	—
121	C₂×C₁₇	—	2	1	no	—
122	C₂×C₁₇	#1939	1	1	yes	characteristic
123	C₂×M₁₆	C₂×Dic₁₇	1	2	yes	—
124	C₂²×C₈	C₂×Dic₁₇	1	1	yes	characteristic
125	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
126	#117	C₂×Dic₁₇	1	2	yes	—
127	C₂×C₄²	C₂×Dic₁₇	1	1	yes	characteristic
128	M₃₂	—	17	2	no	—
129	M₃₂	—	17	2	no	—
130	M₃₂	—	34	1	no	—
131	M₃₂	—	34	1	no	—
132	M₃₂	—	34	2	no	—
133	M₃₂	—	34	2	no	—
134	C₂×C₁₆	—	34	1	no	—
135	C₂×C₁₆	—	34	1	no	—
136	C₂×C₁₆	—	34	2	no	—
137	#108	#3622	1	2	yes	—
138	#108	#3622	1	2	yes	—
139	#107	—	2	1	no	—
140	#106	—	2	1	no	—
141	#97	—	2	2	no	—
142	C₄×C₈	—	2	2	no	—
143	C₄×C₈	Dic₃₄	1	1	yes	characteristic
144	C₄×C₈	Dic₃₄	1	1	yes	characteristic
145	C₄×C₈	D₆₈	1	1	yes	characteristic
146	C₄×C₈	D₆₈	1	1	yes	characteristic
147	C₁₇	#13164	1	1	yes	characteristic, complemented, Sylow
148	C₂²×C₄	—	2	1	no	—
149	C₂²×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
150	M₁₆	—	2	2	no	—
151	M₁₆	C₂×#3622	1	4	yes	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	C₂×#3622	1	1	yes	characteristic
154	C₂×C₈	C₂×#3622	1	1	yes	characteristic
155	C₂×C₈	#63223	1	1	yes	characteristic
156	C₂×C₈	#63223	1	1	yes	characteristic
157	C₂×C₈	C₄⋊Dic₁₇	1	1	yes	characteristic
158	C₂×C₈	C₄⋊Dic₁₇	1	1	yes	characteristic
159	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
160	C₂×C₈	C₂×D₆₈	1	1	yes	characteristic
161	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
162	C₂×C₈	#63254	1	1	yes	characteristic
163	C₂×C₈	Q₈×D₁₇	1	1	yes	characteristic
164	C₂×C₈	#63256	1	1	yes	characteristic
165	C₄⋊C₄	—	2	2	no	—
166	C₂²⋊C₄	—	2	2	no	—
167	C₄²	—	2	1	no	—
168	C₄²	—	2	2	no	—
169	C₄²	C₄⋊Dic₁₇	1	1	yes	characteristic
170	C₄²	C₄⋊Dic₁₇	1	1	yes	characteristic
171	C₁₆	—	34	2	no	—
172	C₁₆	—	34	2	no	—
173	C₁₆	—	68	2	no	—
174	C₂³	#10587634	1	1	yes	characteristic
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	C₂²×#3622	1	1	yes	characteristic
180	C₂×C₄	C₂×#63223	1	1	yes	characteristic
181	C₂×C₄	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
182	C₂×C₄	D₄×Dic₁₇	1	1	yes	characteristic
183	C₂×C₄	Q₈×Dic₁₇	1	1	yes	characteristic
184	C₂×C₄	#10587634	1	1	yes	characteristic
185	C₈	—	2	4	no	—
186	C₈	C₄⋊#3622	1	1	yes	characteristic
187	C₈	C₄⋊#3622	1	1	yes	characteristic
188	C₈	C₄⋊#3622	1	1	yes	characteristic
189	C₈	C₄⋊#3622	1	1	yes	characteristic
190	C₈	#10587574	1	1	yes	characteristic
191	C₈	#10587576	1	1	yes	characteristic
192	C₈	#10587581	1	1	yes	characteristic
193	C₈	#10587583	1	1	yes	characteristic
194	C₂²	—	4	1	no	—
195	C₂²	#11762836	1	1	yes	characteristic
196	C₂²	#11762887	1	1	yes	characteristic
197	C₂²	Q₈×#3622	1	1	yes	characteristic
198	C₄	—	2	2	no	—
199	C₄	—	4	1	no	—
200	C₄	—	8	2	no	—
201	C₄	#11762708	1	1	yes	characteristic
202	C₄	C₂×C₄⋊#3622	1	1	yes	characteristic
203	C₄	D₄×#3622	1	1	yes	characteristic
204	C₄	#11762940	1	1	yes	characteristic
205	C₂	—	2	1	no	—
206	C₂	—	2	1	no	—
207	C₂	—	2	1	no	—
208	C₂	#14529503	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#10587581
commutator	C₂×C₄×C₁₇	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₁₇×#1772	C₂
socle	C₂×C₁₇	#1939
prime radical	C₂³×C₁₇	#131
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	#1772	D₁₇
hypocenter	C₁₇	#13164
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #1772

Complements

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

N	H ≅ G/N	type
147. C₁₇	#13164	semidirect

Maximal subgroups

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

subgroup	index
#14530550	2
C₂×#11762630	2
C₄×#10587483	2
#14526870	2
C₁₇×#1772	2
#13164	17