Subgroups — #14533819

Order 2176.

452 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10587556	C₂	1	2	yes	maximal
2	#11762743	C₂	1	4	yes	maximal
3	#11762707	C₂	1	2	yes	maximal
4	#11762705	C₂	1	2	yes	maximal
5	#11762679	C₂	1	1	yes	maximal, characteristic
6	#11762678	C₂	1	2	yes	maximal
7	C₄×C₄⋊Dic₁₇	C₂	1	1	yes	maximal, characteristic
8	C₄×C₄⋊C₄×C₁₇	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄⋊C₄×C₁₇	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊C₄×C₁₇	C₂²	1	1	yes	characteristic
11	C₂×C₄²×C₁₇	C₂²	1	2	yes	—
12	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Dic₁₇	C₂²	1	2	yes	—
14	C₂×C₄⋊Dic₁₇	C₂²	1	2	yes	—
15	C₂×C₄⋊Dic₁₇	C₂²	1	2	yes	—
16	C₂×C₄⋊Dic₁₇	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Dic₁₇	C₂²	1	1	yes	characteristic
18	C₂×#63225	C₂²	1	4	yes	—
19	C₂×#63225	C₂²	1	4	yes	—
20	C₂×#63225	C₂²	1	4	yes	—
21	C₂×C₄×Dic₁₇	C₂²	1	2	yes	—
22	#10587556	—	2	8	no	—
23	C₁₇×#95	C₂²	1	2	yes	—
24	#10587502	C₂²	1	4	yes	—
25	#10587502	C₂²	1	2	yes	—
26	C₂²×C₄×C₁₇	C₂³	1	2	yes	—
27	C₂²×C₄×C₁₇	C₂³	1	2	yes	—
28	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
30	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
31	C₂²×Dic₁₇	C₂³	1	4	yes	—
32	C₂²×Dic₁₇	C₂³	1	2	yes	—
33	C₂²×Dic₁₇	C₂³	1	2	yes	—
34	C₄⋊C₄×C₁₇	—	2	2	no	—
35	C₄⋊C₄×C₁₇	—	2	2	no	—
36	C₄²×C₁₇	—	2	4	no	—
37	C₄²×C₁₇	Q₈	1	4	yes	—
38	C₄⋊Dic₁₇	—	2	4	no	—
39	C₄⋊Dic₁₇	—	2	4	no	—
40	C₄⋊Dic₁₇	—	4	2	no	—
41	C₄⋊Dic₁₇	D₄	1	4	yes	—
42	C₄⋊Dic₁₇	Q₈	1	4	yes	—
43	#63225	—	2	8	no	—
44	#63225	—	2	8	no	—
45	#63225	—	4	4	no	—
46	C₄×Dic₁₇	—	2	4	no	—
47	C₂³×C₁₇	C₂⁴	1	1	yes	characteristic
48	C₂×C₄×C₁₇	—	2	1	no	—
49	C₂×C₄×C₁₇	—	2	1	no	—
50	C₂×C₄×C₁₇	—	2	2	no	—
51	C₂×C₄×C₁₇	—	2	2	no	—
52	C₂×C₄×C₁₇	—	2	2	no	—
53	C₂×C₄×C₁₇	—	2	2	no	—
54	C₂×C₄×C₁₇	—	2	2	no	—
55	C₂×C₄×C₁₇	—	2	2	no	—
56	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
57	C₂×C₄×C₁₇	C₂×Q₈	1	2	yes	—
58	C₂×C₄×C₁₇	C₂×Q₈	1	2	yes	—
59	C₂×C₄×C₁₇	C₂×Q₈	1	2	yes	—
60	C₂×C₄×C₁₇	C₂×Q₈	1	2	yes	—
61	C₂×C₄×C₁₇	#41	1	2	yes	—
62	C₂×C₄×C₁₇	#41	1	2	yes	—
63	C₂×Dic₁₇	—	2	2	no	—
64	C₂×Dic₁₇	—	2	2	no	—
65	C₂×Dic₁₇	—	2	2	no	—
66	C₂×Dic₁₇	—	2	2	no	—
67	C₂×Dic₁₇	—	2	4	no	—
68	C₂×Dic₁₇	—	2	4	no	—
69	C₂×Dic₁₇	—	2	4	no	—
70	C₂×Dic₁₇	C₂×D₄	1	4	yes	—
71	C₂×Dic₁₇	C₂×Q₈	1	4	yes	—
72	#2552	—	17	1	no	maximal, Sylow
73	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
74	C₂²×C₁₇	C₂²×Q₈	1	1	yes	characteristic
75	C₂²×C₁₇	C₂²×Q₈	1	1	yes	characteristic
76	C₂²×C₁₇	C₂×#41	1	1	yes	characteristic
77	C₂²×C₁₇	C₂×#41	1	1	yes	characteristic
78	C₂²×C₁₇	H(5,𝔽₂)	1	1	yes	characteristic
79	C₂²×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
80	C₄×C₁₇	—	2	2	no	—
81	C₄×C₁₇	—	4	2	no	—
82	C₄×C₁₇	—	4	2	no	—
83	C₄×C₁₇	#125	1	4	yes	complemented
84	C₄×C₁₇	C₄⋊Q₈	1	4	yes	—
85	Dic₁₇	—	2	4	no	—
86	Dic₁₇	—	2	4	no	—
87	Dic₁₇	—	4	4	no	—
88	C₂×#125	—	17	2	no	—
89	#398	—	17	4	no	—
90	#389	—	17	2	no	—
91	#389	—	17	2	no	—
92	#384	—	17	2	no	—
93	#384	—	17	1	no	—
94	C₄×C₄⋊C₄	—	17	1	no	—
95	C₄×C₄⋊C₄	D₁₇	1	1	yes	characteristic
96	C₂×C₁₇	C₂×#125	1	1	yes	characteristic
97	C₂×C₁₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
98	C₂×C₁₇	#544	1	1	yes	characteristic
99	C₂×C₁₇	#547	1	1	yes	characteristic
100	C₂×C₁₇	#550	1	1	yes	characteristic
101	C₂×C₁₇	#554	1	1	yes	characteristic
102	C₂×C₁₇	#557	1	1	yes	characteristic
103	#125	—	34	8	no	—
104	C₂×C₄⋊C₄	—	17	1	no	—
105	C₂×C₄⋊C₄	—	17	1	no	—
106	C₂×C₄⋊C₄	—	17	2	no	—
107	C₂×C₄⋊C₄	—	17	4	no	—
108	C₂×C₄⋊C₄	—	17	4	no	—
109	C₂×C₄⋊C₄	—	17	2	no	—
110	C₂×C₄⋊C₄	—	17	4	no	—
111	C₂×C₄⋊C₄	—	17	2	no	—
112	C₂×C₄⋊C₄	C₂×D₁₇	1	1	yes	characteristic
113	C₂×C₄⋊C₄	C₂×D₁₇	1	1	yes	characteristic
114	C₂×C₄²	—	17	2	no	—
115	C₂×C₄²	C₂×D₁₇	1	2	yes	—
116	C₂×C₄²	C₂×D₁₇	1	1	yes	characteristic
117	#95	—	17	2	no	—
118	#95	—	17	4	no	—
119	#95	C₂×D₁₇	1	2	yes	—
120	C₁₇	#2552	1	1	yes	characteristic, complemented, Sylow
121	C₂²×C₄	—	17	2	no	—
122	C₂²×C₄	—	17	2	no	—
123	C₂²×C₄	—	17	4	no	—
124	C₂²×C₄	C₂²×D₁₇	1	2	yes	—
125	C₂²×C₄	C₂²×D₁₇	1	2	yes	—
126	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
127	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
128	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
129	C₄⋊C₄	—	2	2	no	—
130	C₄⋊C₄	—	2	2	no	—
131	C₄⋊C₄	—	17	4	no	—
132	C₄⋊C₄	—	17	4	no	—
133	C₄⋊C₄	—	34	8	no	—
134	C₄⋊C₄	—	34	8	no	—
135	C₄⋊C₄	—	34	4	no	—
136	C₄⋊C₄	—	34	4	no	—
137	C₄⋊C₄	—	68	4	no	—
138	C₄⋊C₄	—	68	2	no	—
139	C₄²	—	2	4	no	—
140	C₄²	—	34	4	no	—
141	C₄²	Dic₃₄	1	4	yes	—
142	C₂³	C₂³×D₁₇	1	1	yes	characteristic
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	17	4	no	—
152	C₂×C₄	—	17	4	no	—
153	C₂×C₄	—	34	2	no	—
154	C₂×C₄	—	34	2	no	—
155	C₂×C₄	—	34	2	no	—
156	C₂×C₄	—	34	2	no	—
157	C₂×C₄	—	34	4	no	—
158	C₂×C₄	—	34	4	no	—
159	C₂×C₄	—	34	4	no	—
160	C₂×C₄	C₂×Dic₃₄	1	2	yes	—
161	C₂×C₄	C₂×Dic₃₄	1	2	yes	—
162	C₂×C₄	C₂×Dic₃₄	1	2	yes	—
163	C₂×C₄	#63252	1	2	yes	—
164	C₂×C₄	#63252	1	2	yes	—
165	C₂×C₄	D₄×D₁₇	1	2	yes	—
166	C₂×C₄	Q₈×D₁₇	1	2	yes	—
167	C₂²	C₂²×Dic₃₄	1	1	yes	characteristic
168	C₂²	C₂×#63252	1	1	yes	characteristic
169	C₂²	C₂×#63252	1	1	yes	characteristic
170	C₂²	C₂×D₄×D₁₇	1	1	yes	characteristic
171	C₂²	C₂×Q₈×D₁₇	1	1	yes	characteristic
172	C₂²	#10587696	1	1	yes	characteristic
173	C₂²	#10587697	1	1	yes	characteristic
174	C₄	—	2	2	no	—
175	C₄	—	4	2	no	—
176	C₄	—	4	2	no	—
177	C₄	—	34	4	no	—
178	C₄	—	34	4	no	—
179	C₄	—	68	4	no	—
180	C₄	#10587556	1	4	yes	complemented
181	C₄	#10587574	1	4	yes	—
182	C₂	C₂×#10587556	1	1	yes	characteristic
183	C₂	C₂×#10587574	1	1	yes	characteristic
184	C₂	#11763437	1	1	yes	characteristic
185	C₂	#11763464	1	1	yes	characteristic
186	C₂	#11763480	1	1	yes	characteristic
187	C₂	#11763519	1	1	yes	characteristic
188	C₂	#11763537	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₇
commutator, socle, prime radical, cosocle	C₂³×C₁₇	C₂⁴
Fitting	C₄×C₄⋊C₄×C₁₇	C₂
hypercenter	C₄×C₄⋊C₄	D₁₇
hypocenter	C₁₇	#2552
LCS	G ▷ C₂³×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
83. C₄×C₁₇	#125	semidirect
120. C₁₇	#2552	semidirect
180. C₄	#10587556	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10587556	2
#11762743	2
#11762707	2
#11762705	2
#11762679	2
#11762678	2
C₄×C₄⋊Dic₁₇	2
C₄×C₄⋊C₄×C₁₇	2
#2552	17