Subgroups — #16952340

Order 4352.

290 conjugacy classes of subgroups.

The subgroup lattice has 186 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	#14530543	C₂	1	2	yes	maximal
3	C₂×#11762585	C₂	1	1	yes	maximal, characteristic
4	#14526873	C₂	1	2	yes	maximal
5	C₁₇×#1772	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587483	C₂²	1	2	yes	—
7	C₂²×#63214	C₂²	1	1	yes	characteristic
8	C₂²×#63214	C₂²	1	1	yes	characteristic
9	#11762630	—	2	2	no	—
10	C₄⋊#63214	—	2	4	no	—
11	#11762585	—	2	2	no	—
12	C₂×C₁₇×#108	C₄	1	1	yes	characteristic
13	C₁₇×#427	C₄	1	1	yes	characteristic
14	C₁₇×#405	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
16	C₁₇×#339	C₄	1	2	yes	—
17	C₂×M₁₆×C₁₇	C₂×C₄	1	2	yes	—
18	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
20	C₁₇×#117	C₂×C₄	1	2	yes	—
21	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
22	C₁₇×#108	—	2	2	no	—
23	C₁₇×#107	C₈	1	2	yes	—
24	C₁₇×#106	C₈	1	2	yes	—
25	C₁₇×#97	—	2	2	no	—
26	C₄×C₈×C₁₇	—	2	2	no	—
27	C₄×C₈×C₁₇	D₄	1	2	yes	—
28	C₄×C₈×C₁₇	Q₈	1	2	yes	—
29	#10587483	—	2	2	no	—
30	#10587483	—	2	2	no	—
31	C₂×#63214	—	2	1	no	—
32	C₂×#63214	—	2	1	no	—
33	C₂×#63214	—	2	2	no	—
34	C₂×#63214	—	2	1	no	—
35	C₂×#63214	—	2	1	no	—
36	C₂×#63214	D₄	1	2	yes	—
37	C₂×#63214	Q₈	1	2	yes	—
38	C₂²×C₄×C₁₇	—	2	1	no	—
39	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
40	M₁₆×C₁₇	—	2	2	no	—
41	M₁₆×C₁₇	—	2	2	no	—
42	C₂×C₈×C₁₇	—	2	2	no	—
43	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
44	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
45	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
46	C₂×C₈×C₁₇	M₁₆	1	2	yes	—
47	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
52	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
53	C₄⋊C₄×C₁₇	C₂×C₈	1	4	yes	—
54	C₂²⋊C₄×C₁₇	—	2	2	no	—
55	C₄²×C₁₇	—	2	2	no	—
56	C₄²×C₁₇	—	2	1	no	—
57	C₄²×C₁₇	C₄⋊C₄	1	2	yes	—
58	#63214	—	2	2	no	—
59	#63214	—	2	2	no	—
60	#63214	—	4	2	no	—
61	#13162	—	17	1	no	maximal, Sylow
62	C₂³×C₁₇	#131	1	1	yes	characteristic
63	C₂×C₄×C₁₇	—	2	2	no	—
64	C₂×C₄×C₁₇	—	2	4	no	—
65	C₂×C₄×C₁₇	—	4	2	no	—
66	C₂×C₄×C₁₇	C₂×C₄⋊C₄	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	C₂²×C₈	1	1	yes	characteristic
70	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
71	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
72	C₈×C₁₇	—	4	2	no	—
73	C₈×C₁₇	C₄⋊C₈	1	2	yes	—
74	C₈×C₁₇	C₄⋊C₈	1	2	yes	—
75	C₈×C₁₇	#121	1	1	yes	characteristic
76	C₈×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
77	C₈×C₁₇	#125	1	1	yes	characteristic
78	C₈×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
79	C₂×#364	—	17	1	no	—
80	#2151	—	17	2	no	—
81	C₂×#346	—	17	1	no	—
82	#1772	D₁₇	1	1	yes	characteristic
83	#1379	—	17	2	no	—
84	C₂²×C₁₇	—	2	2	no	—
85	C₂²×C₁₇	#424	1	1	yes	characteristic
86	C₂²×C₁₇	C₈×D₄	1	1	yes	characteristic
87	C₂²×C₁₇	#446	1	1	yes	characteristic
88	C₄×C₁₇	—	4	2	no	—
89	C₄×C₁₇	—	4	4	no	—
90	C₄×C₁₇	#389	1	1	yes	characteristic
91	C₄×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
92	C₄×C₁₇	#436	1	1	yes	characteristic
93	C₄×C₁₇	C₈×Q₈	1	1	yes	characteristic
94	C₂×M₃₂	—	17	2	no	—
95	C₂²×C₁₆	—	17	1	no	—
96	C₂²×C₁₆	—	17	1	no	—
97	C₂×#108	Dic₁₇	1	1	yes	characteristic
98	#427	Dic₁₇	1	1	yes	characteristic
99	#405	C₂×D₁₇	1	2	yes	—
100	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
101	#364	—	34	2	no	—
102	C₄⋊C₁₆	—	34	4	no	—
103	#346	—	34	2	no	—
104	#339	Dic₁₇	1	2	yes	—
105	C₂×C₁₇	—	2	1	no	—
106	C₂×C₁₇	—	2	1	no	—
107	C₂×C₁₇	#1939	1	1	yes	characteristic
108	C₂×C₁₇	#2171	1	2	yes	—
109	C₂×M₁₆	C₂×Dic₁₇	1	2	yes	—
110	C₂²×C₈	C₂×Dic₁₇	1	1	yes	characteristic
111	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
112	#117	C₂×Dic₁₇	1	2	yes	—
113	C₂×C₄²	C₂×Dic₁₇	1	1	yes	characteristic
114	M₃₂	—	34	2	no	—
115	M₃₂	—	34	2	no	—
116	C₂×C₁₆	—	17	2	no	—
117	C₂×C₁₆	—	17	2	no	—
118	C₂×C₁₆	—	34	1	no	—
119	C₂×C₁₆	—	34	1	no	—
120	C₂×C₁₆	—	34	1	no	—
121	C₂×C₁₆	—	34	1	no	—
122	C₂×C₁₆	—	34	2	no	—
123	#108	—	2	2	no	—
124	#107	#3622	1	2	yes	—
125	#106	#3622	1	2	yes	—
126	#97	—	2	2	no	—
127	C₄×C₈	—	2	2	no	—
128	C₄×C₈	Dic₃₄	1	2	yes	—
129	C₄×C₈	D₆₈	1	2	yes	—
130	C₁₇	#13162	1	1	yes	characteristic, complemented, Sylow
131	C₂²×C₄	—	2	1	no	—
132	C₂²×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
133	M₁₆	—	2	2	no	—
134	M₁₆	—	2	2	no	—
135	C₂×C₈	—	2	2	no	—
136	C₂×C₈	C₂×#3622	1	1	yes	characteristic
137	C₂×C₈	C₂×#3622	1	1	yes	characteristic
138	C₂×C₈	#63223	1	2	yes	—
139	C₂×C₈	C₄⋊Dic₁₇	1	2	yes	—
140	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
141	C₂×C₈	C₂×D₆₈	1	1	yes	characteristic
142	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
143	C₂×C₈	#63254	1	1	yes	characteristic
144	C₂×C₈	Q₈×D₁₇	1	1	yes	characteristic
145	C₂×C₈	#63256	1	1	yes	characteristic
146	C₄⋊C₄	C₂×#3622	1	4	yes	—
147	C₂²⋊C₄	—	2	2	no	—
148	C₄²	—	2	2	no	—
149	C₄²	—	2	1	no	—
150	C₄²	C₄⋊Dic₁₇	1	2	yes	—
151	C₁₆	—	34	2	no	—
152	C₁₆	—	34	2	no	—
153	C₁₆	—	68	2	no	—
154	C₂³	#10587634	1	1	yes	characteristic
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	4	no	—
157	C₂×C₄	—	4	2	no	—
158	C₂×C₄	C₂²×#3622	1	1	yes	characteristic
159	C₂×C₄	C₂×#63223	1	1	yes	characteristic
160	C₂×C₄	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
161	C₂×C₄	D₄×Dic₁₇	1	1	yes	characteristic
162	C₂×C₄	Q₈×Dic₁₇	1	1	yes	characteristic
163	C₂×C₄	#10587634	1	1	yes	characteristic
164	C₈	—	4	2	no	—
165	C₈	C₄⋊#3622	1	2	yes	—
166	C₈	C₄⋊#3622	1	2	yes	—
167	C₈	#10587574	1	1	yes	characteristic
168	C₈	#10587576	1	1	yes	characteristic
169	C₈	#10587581	1	1	yes	characteristic
170	C₈	#10587583	1	1	yes	characteristic
171	C₂²	—	2	2	no	—
172	C₂²	#11762836	1	1	yes	characteristic
173	C₂²	D₄×#3622	1	1	yes	characteristic
174	C₂²	#11762940	1	1	yes	characteristic
175	C₄	—	4	2	no	—
176	C₄	—	4	4	no	—
177	C₄	#11762708	1	1	yes	characteristic
178	C₄	C₂×C₄⋊#3622	1	1	yes	characteristic
179	C₄	#11762887	1	1	yes	characteristic
180	C₄	Q₈×#3622	1	1	yes	characteristic
181	C₂	—	2	1	no	—
182	C₂	—	2	1	no	—
183	C₂	#14529503	1	1	yes	characteristic
184	C₂	#14530626	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	D₄×D₁₇
commutator	C₂×C₄×C₁₇	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₁₇×#1772	C₂
socle	C₂²×C₁₇	C₈×D₄
prime radical	C₂³×C₁₇	#131
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	#1772	D₁₇
hypocenter	C₁₇	#13162
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #1772

Complements

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

N	H ≅ G/N	type
130. C₁₇	#13162	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
#14530543	2
C₂×#11762585	2
#14526873	2
C₁₇×#1772	2
#13162	17