Subgroups — #14528764

Order 2176.

400 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#3622	C₂	1	2	yes	maximal, complemented
2	C₂×C₂²⋊#3622	C₂	1	2	yes	maximal
3	#11762564	C₂	1	2	yes	maximal, complemented
4	C₂²×C₄²×C₁₇	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₄×C₁₇	C₄	1	1	yes	characteristic
6	C₂³×C₄×C₁₇	C₄	1	1	yes	characteristic
7	C₂³×C₄×C₁₇	C₂²	1	1	yes	characteristic
8	C₂×C₄²×C₁₇	—	2	4	no	—
9	C₂×C₄²×C₁₇	C₄	1	1	yes	characteristic
10	C₂×C₄²×C₁₇	C₄	1	1	yes	characteristic
11	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
12	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
13	C₂²×#3622	C₂²	1	4	yes	—
14	C₂²⋊#3622	—	2	4	no	—
15	C₄⋊#3622	—	2	4	no	—
16	C₂⁴×C₁₇	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₄×C₁₇	—	2	2	no	—
18	C₂²×C₄×C₁₇	—	2	2	no	—
19	C₂²×C₄×C₁₇	—	2	2	no	—
20	C₂²×C₄×C₁₇	—	2	2	no	—
21	C₂²×C₄×C₁₇	—	2	2	no	—
22	C₂²×C₄×C₁₇	—	2	2	no	—
23	C₂²×C₄×C₁₇	—	2	2	no	—
24	C₂²×C₄×C₁₇	C₈	1	4	yes	complemented
25	C₂²×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₁₇	D₄	1	2	yes	—
31	C₂²×C₄×C₁₇	Q₈	1	2	yes	—
32	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
33	C₄²×C₁₇	—	2	2	no	—
34	C₄²×C₁₇	—	2	2	no	—
35	C₄²×C₁₇	—	2	2	no	—
36	C₄²×C₁₇	D₄	1	4	yes	—
37	C₂×#3622	—	2	4	no	—
38	C₂×#3622	—	2	4	no	—
39	C₂×#3622	—	2	4	no	—
40	C₂³×C₁₇	—	2	1	no	—
41	C₂³×C₁₇	—	2	1	no	—
42	C₂³×C₁₇	—	2	1	no	—
43	C₂³×C₁₇	—	2	1	no	—
44	C₂³×C₁₇	C₄⋊C₄	1	2	yes	—
45	C₂³×C₁₇	C₂×C₈	1	2	yes	—
46	C₂³×C₁₇	M₁₆	1	2	yes	—
47	C₂³×C₁₇	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₁₇	—	2	8	no	—
49	C₂×C₄×C₁₇	—	2	1	no	—
50	C₂×C₄×C₁₇	—	2	1	no	—
51	C₂×C₄×C₁₇	—	2	1	no	—
52	C₂×C₄×C₁₇	—	2	1	no	—
53	C₂×C₄×C₁₇	—	2	1	no	—
54	C₂×C₄×C₁₇	—	2	1	no	—
55	C₂×C₄×C₁₇	—	2	1	no	—
56	C₂×C₄×C₁₇	—	2	1	no	—
57	C₂×C₄×C₁₇	—	2	8	no	—
58	C₂×C₄×C₁₇	—	2	4	no	—
59	C₂×C₄×C₁₇	—	2	4	no	—
60	C₂×C₄×C₁₇	C₂²⋊C₄	1	2	yes	—
61	C₂×C₄×C₁₇	C₂²⋊C₄	1	2	yes	—
62	C₂×C₄×C₁₇	C₄⋊C₄	1	2	yes	—
63	C₂×C₄×C₁₇	C₂×C₈	1	2	yes	—
64	C₂×C₄×C₁₇	C₂×C₈	1	2	yes	—
65	C₂×C₄×C₁₇	M₁₆	1	2	yes	—
66	C₂×C₄×C₁₇	M₁₆	1	2	yes	—
67	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
68	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
69	C₂×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
70	C₂×C₄×C₁₇	#41	1	2	yes	—
71	#3622	—	4	4	no	—
72	#1797	—	17	1	no	maximal, Sylow
73	C₂²×C₁₇	—	2	2	no	—
74	C₂²×C₁₇	—	2	2	no	—
75	C₂²×C₁₇	—	2	2	no	—
76	C₂²×C₁₇	—	2	2	no	—
77	C₂²×C₁₇	—	2	2	no	—
78	C₂²×C₁₇	—	2	2	no	—
79	C₂²×C₁₇	C₄⋊C₈	1	4	yes	complemented
80	C₂²×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	C₂²×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂²×C₁₇	#117	1	1	yes	characteristic
83	C₂²×C₁₇	C₂²×C₈	1	1	yes	characteristic
84	C₂²×C₁₇	C₂×M₁₆	1	1	yes	characteristic
85	C₂²×C₁₇	C₂×M₁₆	1	1	yes	characteristic
86	C₂²×C₁₇	C₂×M₁₆	1	1	yes	characteristic
87	C₄×C₁₇	—	2	2	no	—
88	C₄×C₁₇	—	2	2	no	—
89	C₄×C₁₇	—	2	2	no	—
90	C₄×C₁₇	—	2	2	no	—
91	C₄×C₁₇	C₂²⋊C₈	1	4	yes	complemented
92	C₄×C₁₇	#121	1	2	yes	—
93	C₄×C₁₇	C₂²⋊Q₈	1	2	yes	—
94	C₂²×C₄²	D₁₇	1	1	yes	characteristic
95	C₂×C₄⋊C₈	—	17	2	no	—
96	C₂×C₂²⋊C₈	—	17	2	no	—
97	#336	—	17	2	no	—
98	C₂×C₁₇	—	2	4	no	—
99	C₂×C₁₇	#380	1	1	yes	characteristic
100	C₂×C₁₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
101	C₂×C₁₇	#407	1	1	yes	characteristic
102	C₂×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
103	C₂×C₁₇	#423	1	1	yes	characteristic
104	C₂×C₁₇	#431	1	1	yes	characteristic
105	C₂×C₁₇	#432	1	1	yes	characteristic
106	C₂³×C₄	Dic₁₇	1	1	yes	characteristic
107	C₂³×C₄	Dic₁₇	1	1	yes	characteristic
108	C₂³×C₄	C₂×D₁₇	1	1	yes	characteristic
109	C₂²×C₈	—	17	4	no	—
110	C₂×C₄²	—	2	4	no	—
111	C₂×C₄²	Dic₁₇	1	1	yes	characteristic
112	C₂×C₄²	Dic₁₇	1	1	yes	characteristic
113	C₂×C₄²	C₂×D₁₇	1	1	yes	characteristic
114	C₂×C₄²	C₂×D₁₇	1	1	yes	characteristic
115	C₄⋊C₈	—	34	4	no	—
116	C₂²⋊C₈	—	34	4	no	—
117	C₁₇	#1797	1	1	yes	characteristic, complemented, Sylow
118	C₂⁴	C₂×Dic₁₇	1	1	yes	characteristic
119	C₂²×C₄	—	2	2	no	—
120	C₂²×C₄	—	2	2	no	—
121	C₂²×C₄	—	2	2	no	—
122	C₂²×C₄	—	2	2	no	—
123	C₂²×C₄	—	2	2	no	—
124	C₂²×C₄	—	2	2	no	—
125	C₂²×C₄	—	2	2	no	—
126	C₂²×C₄	#3622	1	4	yes	complemented
127	C₂²×C₄	Dic₃₄	1	2	yes	—
128	C₂²×C₄	D₆₈	1	2	yes	—
129	C₂²×C₄	C₂×Dic₁₇	1	1	yes	characteristic
130	C₂²×C₄	C₂×Dic₁₇	1	1	yes	characteristic
131	C₂²×C₄	C₂×Dic₁₇	1	1	yes	characteristic
132	C₂²×C₄	C₂×Dic₁₇	1	1	yes	characteristic
133	C₂²×C₄	C₂×Dic₁₇	1	1	yes	characteristic
134	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
135	C₂×C₈	—	34	4	no	—
136	C₂×C₈	—	34	4	no	—
137	C₂×C₈	—	34	4	no	—
138	C₄²	—	2	2	no	—
139	C₄²	—	2	2	no	—
140	C₄²	—	2	2	no	—
141	C₄²	C₂²⋊D₁₇	1	4	yes	—
142	C₂³	—	2	1	no	—
143	C₂³	—	2	1	no	—
144	C₂³	—	2	1	no	—
145	C₂³	—	2	1	no	—
146	C₂³	C₂×#3622	1	2	yes	—
147	C₂³	#63223	1	2	yes	—
148	C₂³	C₄⋊Dic₁₇	1	2	yes	—
149	C₂³	C₂²×Dic₁₇	1	1	yes	characteristic
150	C₂×C₄	—	2	8	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	2	1	no	—
157	C₂×C₄	—	2	1	no	—
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	2	8	no	—
160	C₂×C₄	—	2	4	no	—
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	C₂×#3622	1	2	yes	—
163	C₂×C₄	C₂×#3622	1	2	yes	—
164	C₂×C₄	#63223	1	2	yes	—
165	C₂×C₄	#63223	1	2	yes	—
166	C₂×C₄	C₄⋊Dic₁₇	1	2	yes	—
167	C₂×C₄	C₂²⋊Dic₁₇	1	2	yes	—
168	C₂×C₄	C₂²⋊Dic₁₇	1	2	yes	—
169	C₂×C₄	C₂×Dic₃₄	1	1	yes	characteristic
170	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
171	C₂×C₄	#63252	1	2	yes	—
172	C₂×C₄	C₂×C₂²⋊D₁₇	1	2	yes	—
173	C₈	—	68	4	no	—
174	C₂²	—	2	2	no	—
175	C₂²	—	2	2	no	—
176	C₂²	—	2	2	no	—
177	C₂²	—	2	2	no	—
178	C₂²	—	2	2	no	—
179	C₂²	—	2	2	no	—
180	C₂²	C₄⋊#3622	1	4	yes	complemented
181	C₂²	C₂²×#3622	1	1	yes	characteristic
182	C₂²	C₂×#63223	1	1	yes	characteristic
183	C₂²	C₂×#63223	1	1	yes	characteristic
184	C₂²	C₂×#63223	1	1	yes	characteristic
185	C₂²	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
186	C₂²	#10587612	1	1	yes	characteristic
187	C₂²	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
188	C₄	—	2	2	no	—
189	C₄	—	2	2	no	—
190	C₄	—	2	2	no	—
191	C₄	—	2	2	no	—
192	C₄	C₂²⋊#3622	1	4	yes	complemented
193	C₄	#10587610	1	2	yes	—
194	C₄	#10587616	1	2	yes	—
195	C₂	—	2	4	no	—
196	C₂	#11762669	1	1	yes	characteristic
197	C₂	C₂×C₂²⋊#3622	1	1	yes	characteristic
198	C₂	#11762773	1	1	yes	characteristic
199	C₂	C₂×C₄⋊#3622	1	1	yes	characteristic
200	C₂	#11762833	1	1	yes	characteristic
201	C₂	#11762862	1	1	yes	characteristic
202	C₂	#11762866	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₇
commutator	C₂²×C₁₇	C₂²×C₈
Fitting	C₂²×C₄²×C₁₇	C₂
socle	C₂³×C₁₇	C₂²×C₄
prime radical	C₂⁴×C₁₇	C₂×C₄
cosocle	C₂²×C₄×C₁₇	C₂³
hypercenter	C₂²×C₄²	D₁₇
hypocenter	C₁₇	#1797
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
1. C₂×C₄⋊#3622	C₂	semidirect
3. #11762564	C₂	semidirect
24. C₂²×C₄×C₁₇	C₈	semidirect
79. C₂²×C₁₇	C₄⋊C₈	semidirect
91. C₄×C₁₇	C₂²⋊C₈	semidirect
117. C₁₇	#1797	semidirect
126. C₂²×C₄	#3622	semidirect
180. C₂²	C₄⋊#3622	semidirect
192. C₄	C₂²⋊#3622	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#3622	2
C₂×C₂²⋊#3622	2
#11762564	2
C₂²×C₄²×C₁₇	2
#1797	17