Subgroups — #16925583

Order 4352.

398 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#10587489	C₂	1	1	yes	maximal, characteristic
2	#14528651	C₂	1	2	yes	maximal
3	C₂×#11762560	C₂	1	1	yes	maximal, characteristic
4	#14526624	C₂	1	2	yes	maximal
5	C₂×C₁₇×#322	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587489	C₄	1	2	yes	—
7	C₂×C₄⋊#3622	C₂²	1	2	yes	—
8	C₂×#10587486	—	2	2	no	—
9	C₄×C₄⋊Dic₁₇	C₂²	1	2	yes	—
10	#11762572	C₄	1	2	yes	—
11	#11762563	—	2	2	no	—
12	#11762560	C₄	1	4	yes	complemented
13	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
16	C₁₇×#322	C₄	1	4	yes	complemented
17	C₂²×C₈×C₁₇	—	2	1	no	—
18	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₇	D₄	1	2	yes	—
21	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
22	C₂×C₄⋊Dic₁₇	C₂×C₄	1	4	yes	—
23	C₂×C₄×Dic₁₇	—	2	2	no	—
24	C₂²×#3622	—	2	2	no	—
25	C₄×C₈×C₁₇	—	2	1	no	—
26	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
27	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
28	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
29	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
30	C₄×C₈×C₁₇	Q₈	1	2	yes	—
31	#10587502	—	2	2	no	—
32	#10587489	—	2	4	no	—
33	#10587486	—	4	4	no	—
34	C₄⋊#3622	—	2	2	no	—
35	C₄⋊#3622	C₂×C₄	1	4	yes	—
36	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₇	#41	1	1	yes	characteristic
39	C₂²×Dic₁₇	—	2	4	no	—
40	C₂×C₈×C₁₇	—	2	2	no	—
41	C₂×C₈×C₁₇	—	2	2	no	—
42	C₂×C₈×C₁₇	—	2	2	no	—
43	C₂×C₈×C₁₇	—	2	2	no	—
44	C₂×C₈×C₁₇	—	4	2	no	—
45	C₂×C₈×C₁₇	C₄²	1	2	yes	—
46	C₂×C₈×C₁₇	C₄²	1	2	yes	—
47	C₂×C₈×C₁₇	C₄⋊C₄	1	4	yes	—
48	C₂×C₈×C₁₇	M₁₆	1	4	yes	—
49	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
50	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
51	C₄²×C₁₇	C₂×Q₈	1	1	yes	characteristic
52	C₄²×C₁₇	#41	1	1	yes	characteristic
53	C₄⋊Dic₁₇	—	2	4	no	—
54	C₄⋊Dic₁₇	—	2	4	no	—
55	C₄×Dic₁₇	—	4	2	no	—
56	C₄×Dic₁₇	—	4	2	no	—
57	C₂×#3622	—	2	4	no	—
58	C₂×#3622	—	4	2	no	—
59	C₂×#3622	—	4	2	no	—
60	#8287	—	17	1	no	maximal, Sylow
61	C₂³×C₁₇	C₄×D₄	1	1	yes	characteristic
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×Q₈	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	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	2	yes	—
72	C₂×C₄×C₁₇	#137	1	2	yes	—
73	C₂×Dic₁₇	—	4	4	no	—
74	C₂×Dic₁₇	—	4	4	no	—
75	C₂×Dic₁₇	—	4	4	no	—
76	C₈×C₁₇	—	2	2	no	—
77	C₈×C₁₇	—	2	4	no	—
78	C₈×C₁₇	—	4	2	no	—
79	C₈×C₁₇	#97	1	4	yes	complemented
80	#3622	—	4	4	no	—
81	C₄×#107	—	17	1	no	—
82	#1770	—	17	2	no	—
83	C₂×#334	—	17	1	no	—
84	C₂×#322	D₁₇	1	1	yes	characteristic
85	#1279	—	17	2	no	—
86	C₂²×C₁₇	C₄×C₄⋊C₄	1	1	yes	characteristic
87	C₂²×C₁₇	#436	1	2	yes	—
88	C₂²×C₁₇	#441	1	1	yes	characteristic
89	C₂²×C₁₇	#442	1	1	yes	characteristic
90	C₂²×C₁₇	#444	1	2	yes	—
91	C₄×C₁₇	—	2	1	no	—
92	C₄×C₁₇	—	2	1	no	—
93	C₄×C₁₇	C₂×#97	1	1	yes	characteristic
94	C₄×C₁₇	#405	1	1	yes	characteristic
95	C₄×C₁₇	#428	1	2	yes	—
96	C₄×C₁₇	#440	1	2	yes	—
97	C₄×C₁₇	#446	1	2	yes	—
98	Dic₁₇	—	8	4	no	—
99	C₂×#107	—	17	2	no	—
100	C₂×C₄⋊C₈	—	17	2	no	—
101	C₂×#97	—	34	2	no	—
102	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
103	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
104	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
105	C₄×C₄⋊C₄	—	17	2	no	—
106	#340	—	17	2	no	—
107	#336	—	34	2	no	—
108	#334	—	17	4	no	—
109	#322	Dic₁₇	1	4	yes	complemented
110	C₂×C₁₇	#1593	1	2	yes	—
111	C₂×C₁₇	#1594	1	2	yes	—
112	C₂×C₁₇	#1770	1	1	yes	characteristic
113	C₂×C₁₇	#1773	1	1	yes	characteristic
114	C₂×C₁₇	#1776	1	1	yes	characteristic
115	C₂²×C₈	—	2	1	no	—
116	C₂²×C₈	—	34	2	no	—
117	C₂²×C₈	C₄×D₁₇	1	1	yes	characteristic
118	C₂²×C₈	C₄×D₁₇	1	1	yes	characteristic
119	C₂²×C₈	D₆₈	1	2	yes	—
120	C₂×C₄⋊C₄	—	17	4	no	—
121	C₂×C₄²	—	34	2	no	—
122	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
123	#107	—	34	4	no	—
124	C₄⋊C₈	—	17	4	no	—
125	C₄⋊C₈	—	34	2	no	—
126	#97	—	68	4	no	—
127	C₄×C₈	—	2	1	no	—
128	C₄×C₈	Dic₃₄	1	2	yes	—
129	C₄×C₈	C₄×D₁₇	1	2	yes	—
130	C₄×C₈	C₂×Dic₁₇	1	2	yes	—
131	C₄×C₈	C₂×Dic₁₇	1	2	yes	—
132	C₄×C₈	C₂×Dic₁₇	1	2	yes	—
133	#95	—	34	2	no	—
134	C₁₇	#8287	1	1	yes	characteristic, complemented, Sylow
135	C₂²×C₄	—	34	4	no	—
136	C₂²×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
137	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
138	C₂²×C₄	#63252	1	1	yes	characteristic
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	4	2	no	—
144	C₂×C₈	—	34	4	no	—
145	C₂×C₈	—	68	2	no	—
146	C₂×C₈	—	68	2	no	—
147	C₂×C₈	#63218	1	4	yes	—
148	C₂×C₈	C₄×Dic₁₇	1	2	yes	—
149	C₂×C₈	C₄×Dic₁₇	1	2	yes	—
150	C₂×C₈	C₄⋊Dic₁₇	1	4	yes	—
151	C₄⋊C₄	—	34	4	no	—
152	C₄⋊C₄	—	34	4	no	—
153	C₄²	—	68	2	no	—
154	C₄²	—	68	2	no	—
155	C₄²	C₂×Dic₃₄	1	1	yes	characteristic
156	C₄²	C₂×C₄×D₁₇	1	1	yes	characteristic
157	C₄²	#63252	1	1	yes	characteristic
158	C₄²	C₂²×Dic₁₇	1	1	yes	characteristic
159	C₂³	C₄×D₆₈	1	1	yes	characteristic
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	68	4	no	—
162	C₂×C₄	—	68	4	no	—
163	C₂×C₄	—	68	4	no	—
164	C₂×C₄	C₄×Dic₃₄	1	1	yes	characteristic
165	C₂×C₄	C₄×Dic₃₄	1	1	yes	characteristic
166	C₂×C₄	C₄×D₆₈	1	1	yes	characteristic
167	C₂×C₄	C₂×#63218	1	2	yes	—
168	C₂×C₄	#10587587	1	2	yes	—
169	C₂×C₄	#10587594	1	2	yes	—
170	C₂×C₄	#10587595	1	2	yes	—
171	C₂×C₄	C₂×C₄×Dic₁₇	1	1	yes	characteristic
172	C₂×C₄	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
173	C₂×C₄	#10587612	1	1	yes	characteristic
174	C₈	—	2	2	no	—
175	C₈	—	2	4	no	—
176	C₈	—	4	2	no	—
177	C₈	—	68	4	no	—
178	C₈	#10587486	1	4	yes	complemented
179	C₂²	C₄×C₄⋊Dic₁₇	1	1	yes	characteristic
180	C₂²	#11762888	1	2	yes	—
181	C₂²	#11762917	1	1	yes	characteristic
182	C₂²	#11762922	1	1	yes	characteristic
183	C₂²	#11762932	1	2	yes	—
184	C₄	—	2	1	no	—
185	C₄	—	2	1	no	—
186	C₄	—	136	4	no	—
187	C₄	C₂×#10587486	1	1	yes	characteristic
188	C₄	#11762764	1	1	yes	characteristic
189	C₄	#11762851	1	2	yes	—
190	C₄	#11762911	1	2	yes	—
191	C₄	#11762939	1	2	yes	—
192	C₂	#14527694	1	2	yes	—
193	C₂	#14527699	1	2	yes	—
194	C₂	#14528651	1	1	yes	characteristic
195	C₂	#14528668	1	1	yes	characteristic
196	C₂	#14528681	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₆₈
commutator	C₂×C₄×C₁₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₂×C₁₇×#322	C₂
socle, prime radical	C₂³×C₁₇	C₄×D₄
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	C₂×#322	D₁₇
hypocenter	C₁₇	#8287
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
12. #11762560	C₄	semidirect
16. C₁₇×#322	C₄	semidirect
79. C₈×C₁₇	#97	semidirect
109. #322	Dic₁₇	semidirect
134. C₁₇	#8287	semidirect
178. C₈	#10587486	semidirect

Maximal subgroups

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

subgroup	index
C₄×#10587489	2
#14528651	2
C₂×#11762560	2
#14526624	2
C₂×C₁₇×#322	2
#8287	17