Subgroups — #16807061

Order 3328.

590 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131201	C₂	1	8	yes	maximal, complemented
2	#13131168	C₂	1	2	yes	maximal, complemented
3	#13131063	C₂	1	2	yes	maximal
4	C₄×#88106	C₂	1	2	yes	maximal, complemented
5	C₂×#11718658	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×#117	C₄	1	1	yes	characteristic
7	C₂²×#6099	C₄	1	1	yes	characteristic
8	C₂³×#1055	C₂²	1	1	yes	characteristic
9	C₂×#88106	C₂²	1	4	yes	—
10	#11718658	C₄	1	4	yes	—
11	#11718658	C₂²	1	4	yes	—
12	C₂×#88043	C₄	1	1	yes	characteristic
13	C₂×#88043	C₄	1	1	yes	characteristic
14	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
15	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
16	C₂²⋊#6092	C₂²	1	8	yes	—
17	#11718231	C₂²	1	4	yes	complemented
18	C₄⋊#6092	C₂²	1	8	yes	complemented
19	C₄×#6092	C₂²	1	4	yes	complemented
20	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₁₃×#117	C₈	1	4	yes	—
22	C₁₃×#117	C₂×C₄	1	4	yes	—
23	C₂×C₁₃×C₄⋊C₄	C₈	1	2	yes	—
24	C₂×C₁₃×C₂²⋊C₄	C₈	1	2	yes	—
25	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂×#6099	—	2	4	no	—
28	C₂×#6099	C₂×C₄	1	4	yes	—
29	C₂²×#1055	—	2	4	no	—
30	C₂²×#1055	C₂×C₄	1	2	yes	—
31	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
32	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
33	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
34	C₂²×#1055	C₂³	1	2	yes	—
35	C₂²×#1055	C₂³	1	1	yes	characteristic
36	#88106	—	2	8	no	—
37	C₂×#6092	C₂³	1	8	yes	—
38	#88043	C₂×C₄	1	4	yes	—
39	#88043	C₂×C₄	1	4	yes	—
40	C₄×#1055	C₂×C₄	1	2	yes	—
41	C₄×#1055	C₂×C₄	1	2	yes	—
42	C₄×#1055	C₂³	1	2	yes	—
43	C₄×#1055	C₂³	1	2	yes	—
44	#33419	—	13	1	no	maximal, Sylow
45	C₂⁴×C₁₃	C₂×C₈	1	1	yes	characteristic
46	C₂²×C₄×C₁₃	—	2	2	no	—
47	C₂²×C₄×C₁₃	—	2	2	no	—
48	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
49	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
50	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
51	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
52	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
53	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
54	C₁₃×C₄⋊C₄	C₂×C₈	1	8	yes	—
55	C₁₃×C₂²⋊C₄	C₂×C₈	1	8	yes	—
56	C₄²×C₁₃	C₂×C₈	1	2	yes	—
57	C₄²×C₁₃	C₂×C₈	1	2	yes	—
58	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
59	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
60	#6099	—	2	8	no	—
61	C₂×#1055	—	2	2	no	—
62	C₂×#1055	—	2	2	no	—
63	C₂×#1055	—	2	4	no	—
64	C₂×#1055	—	2	4	no	—
65	C₂×#1055	M₁₆	1	4	yes	—
66	C₂×#1055	C₂²×C₄	1	2	yes	—
67	C₂×#1055	C₂²×C₄	1	2	yes	—
68	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
69	C₂×#1055	C₂²×C₄	1	2	yes	—
70	C₂×#1055	#41	1	4	yes	—
71	C₂×#1055	C₂⁴	1	1	yes	characteristic
72	#6092	—	2	8	no	—
73	C₂×#405	—	13	1	no	—
74	#2164	—	13	8	no	—
75	#2150	—	13	2	no	—
76	#2112	—	13	2	no	—
77	C₄×M₃₂	—	13	2	no	—
78	C₂³×C₁₃	—	2	2	no	—
79	C₂³×C₁₃	—	4	2	no	—
80	C₂³×C₁₃	C₂²×C₈	1	2	yes	—
81	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
82	C₂×C₄×C₁₃	—	2	2	no	—
83	C₂×C₄×C₁₃	—	2	2	no	—
84	C₂×C₄×C₁₃	—	2	2	no	—
85	C₂×C₄×C₁₃	—	2	4	no	—
86	C₂×C₄×C₁₃	—	4	2	no	—
87	C₂×C₄×C₁₃	—	4	2	no	—
88	C₂×C₄×C₁₃	#117	1	2	yes	—
89	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
90	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
91	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
92	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
93	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
94	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
95	#1055	—	2	2	no	—
96	#1055	—	2	2	no	—
97	#1055	#117	1	2	yes	—
98	#1055	C₂×M₁₆	1	4	yes	—
99	#1055	C₂×#41	1	2	yes	—
100	C₂²×M₁₆	—	13	1	no	—
101	C₂³×C₈	—	13	1	no	—
102	C₂×#117	#261	1	1	yes	characteristic
103	C₂×M₃₂	—	13	4	no	—
104	#405	—	13	4	no	—
105	#405	—	13	4	no	—
106	C₂×#97	—	13	1	no	—
107	C₂×#97	—	13	1	no	—
108	C₂×C₄×C₈	—	13	1	no	—
109	C₂×C₄×C₈	—	13	1	no	—
110	C₄⋊C₁₆	—	13	8	no	—
111	C₂²⋊C₁₆	—	13	8	no	—
112	#346	—	13	4	no	—
113	C₄×C₁₆	—	13	4	no	—
114	C₂²×C₁₃	—	2	2	no	—
115	C₂²×C₁₃	—	2	4	no	—
116	C₂²×C₁₃	—	2	2	no	—
117	C₂²×C₁₃	—	4	4	no	—
118	C₂²×C₁₃	#431	1	2	yes	—
119	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
120	C₄×C₁₃	—	2	4	no	—
121	C₄×C₁₃	—	2	4	no	—
122	C₄×C₁₃	—	2	2	no	—
123	C₄×C₁₃	#431	1	2	yes	—
124	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
125	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
126	C₂³×C₄	C₂×#261	1	1	yes	characteristic
127	C₂×M₁₆	—	13	4	no	—
128	C₂×M₁₆	—	26	4	no	—
129	C₂²×C₈	—	13	1	no	—
130	C₂²×C₈	—	13	1	no	—
131	C₂²×C₈	—	13	2	no	—
132	C₂²×C₈	—	13	1	no	—
133	C₂²×C₈	—	13	2	no	—
134	C₂²×C₈	—	13	1	no	—
135	C₂²×C₈	—	26	4	no	—
136	#117	#1057	1	4	yes	—
137	#117	C₂×#261	1	4	yes	—
138	C₂×C₄⋊C₄	#1057	1	2	yes	—
139	C₂×C₂²⋊C₄	#1057	1	2	yes	—
140	C₂×C₄²	C₂×#261	1	1	yes	characteristic
141	C₂×C₄²	C₂×#261	1	1	yes	characteristic
142	M₃₂	—	26	8	no	—
143	C₂×C₁₆	—	13	8	no	—
144	#97	—	13	4	no	—
145	#97	—	13	4	no	—
146	C₄×C₈	—	13	2	no	—
147	C₄×C₈	—	13	2	no	—
148	C₄×C₈	—	13	2	no	—
149	C₄×C₈	—	13	2	no	—
150	C₂×C₁₃	—	2	2	no	—
151	C₂×C₁₃	—	4	2	no	—
152	C₂×C₁₃	C₂×#431	1	1	yes	characteristic
153	C₂×C₁₃	#3407	1	2	yes	—
154	C₂⁴	C₂×#1057	1	1	yes	characteristic
155	C₂²×C₄	—	2	2	no	—
156	C₂²×C₄	—	2	2	no	—
157	C₂²×C₄	C₂×#1057	1	1	yes	characteristic
158	C₂²×C₄	C₂×#1057	1	2	yes	—
159	C₂²×C₄	C₂×#1057	1	2	yes	—
160	C₂²×C₄	C₂×#1057	1	2	yes	—
161	C₂²×C₄	C₂²×#261	1	2	yes	—
162	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
163	M₁₆	—	26	8	no	—
164	C₂×C₈	—	13	4	no	—
165	C₂×C₈	—	13	4	no	—
166	C₂×C₈	—	13	1	no	—
167	C₂×C₈	—	13	1	no	—
168	C₂×C₈	—	13	2	no	—
169	C₂×C₈	—	13	2	no	—
170	C₂×C₈	—	13	2	no	—
171	C₂×C₈	—	26	4	no	—
172	C₂×C₈	—	26	4	no	—
173	C₂×C₈	—	26	2	no	—
174	C₂×C₈	—	26	2	no	—
175	C₄⋊C₄	C₂×#1057	1	8	yes	—
176	C₂²⋊C₄	C₂×#1057	1	8	yes	—
177	C₄²	C₂×#1057	1	2	yes	—
178	C₄²	C₂×#1057	1	2	yes	—
179	C₄²	C₂²×#261	1	2	yes	—
180	C₄²	C₂²×#261	1	2	yes	—
181	C₁₆	—	26	8	no	—
182	C₁₃	#33419	1	1	yes	characteristic, complemented, Sylow
183	C₂³	—	2	2	no	—
184	C₂³	—	4	2	no	—
185	C₂³	C₂²×#1057	1	2	yes	—
186	C₂³	C₂²×#1057	1	1	yes	characteristic
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	4	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	D₁₃⋊M₁₆	1	2	yes	—
194	C₂×C₄	#88237	1	2	yes	—
195	C₂×C₄	C₂²×#1057	1	4	yes	—
196	C₂×C₄	C₂²×#1057	1	4	yes	—
197	C₂×C₄	C₂²×#1057	1	2	yes	—
198	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
199	C₂×C₄	C₂³×#261	1	1	yes	characteristic
200	C₈	—	13	4	no	—
201	C₈	—	13	2	no	—
202	C₈	—	13	2	no	—
203	C₈	—	26	2	no	—
204	C₈	—	26	2	no	—
205	C₂²	—	2	2	no	—
206	C₂²	—	2	4	no	—
207	C₂²	—	2	2	no	—
208	C₂²	—	4	4	no	—
209	C₂²	#11719119	1	2	yes	—
210	C₂²	C₂³×#1057	1	1	yes	characteristic
211	C₄	—	2	4	no	—
212	C₄	—	2	4	no	—
213	C₄	—	2	2	no	—
214	C₄	#11719119	1	2	yes	—
215	C₄	C₂×D₁₃⋊M₁₆	1	1	yes	characteristic
216	C₄	C₂×#88237	1	1	yes	characteristic
217	C₂	—	2	2	no	—
218	C₂	—	4	2	no	—
219	C₂	C₂×#11719119	1	1	yes	characteristic
220	C₂	#13131803	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#261
commutator, socle	C₂²×C₁₃	C₂³×C₈
Fitting	C₂×C₁₃×#117	C₄
prime radical	C₂⁴×C₁₃	C₂×C₈
cosocle	C₂×#1055	C₂⁴
hypercenter	C₂×#117	#261
hypocenter	C₁₃	#33419
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
1. #13131201	C₂	semidirect
2. #13131168	C₂	semidirect
4. C₄×#88106	C₂	semidirect
17. #11718231	C₂²	semidirect
18. C₄⋊#6092	C₂²	semidirect
19. C₄×#6092	C₂²	semidirect
182. C₁₃	#33419	semidirect

Maximal subgroups

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

subgroup	index
#13131201	2
#13131168	2
#13131063	2
C₄×#88106	2
C₂×#11718658	2
#33419	13