Subgroups — C₂²×#11719096 #16809617

Order 3328.

2118 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11719096	C₂	1	24	yes	maximal, direct product factor
2	C₂²×#88098	C₂	1	2	yes	maximal, complemented
3	C₂²×#88106	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×D₁₃⋊C₁₆	C₂	1	2	yes	maximal, complemented
5	C₂³×#6092	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#88161	C₂	1	1	yes	maximal, characteristic
7	C₂²×#6127	C₄	1	1	yes	characteristic
8	C₂²×C₁₃×M₁₆	C₄	1	1	yes	characteristic
9	C₂³×#1055	C₂²	1	1	yes	characteristic
10	C₂×#88161	C₄	1	12	yes	—
11	C₂×#88161	C₂²	1	12	yes	—
12	C₂²×#6094	C₄	1	2	yes	—
13	C₂²×C₈×D₁₃	C₂²	1	2	yes	—
14	C₂×#88106	C₂²	1	12	yes	complemented
15	C₂²×#6092	C₂²	1	1	yes	characteristic
16	C₂²×#6092	C₂²	1	1	yes	characteristic
17	C₂²×#6092	C₂²	1	2	yes	—
18	C₂²×#6092	C₂²	1	12	yes	complemented
19	#11719096	C₂²	1	64	yes	direct product factor
20	C₂×#88098	C₂²	1	24	yes	complemented
21	C₂×D₁₃⋊C₁₆	C₂²	1	24	yes	complemented
22	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₂²⋊D₁₃	C₈	1	2	yes	—
24	C₂×#6127	C₈	1	12	yes	—
25	C₂×#6127	C₂×C₄	1	12	yes	—
26	C₂²×D₅₂	C₈	1	1	yes	characteristic
27	C₂²×C₄×D₁₃	C₂×C₄	1	2	yes	—
28	C₂²×Dic₂₆	C₈	1	1	yes	characteristic
29	C₂×C₁₃×M₁₆	C₂×C₄	1	12	yes	—
30	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
31	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
32	C₂²×#1055	C₂×C₄	1	6	yes	—
33	C₂²×#1055	C₂³	1	1	yes	characteristic
34	C₂²×#1055	C₂³	1	6	yes	—
35	#88161	C₂×C₄	1	48	yes	—
36	#88161	C₂³	1	16	yes	—
37	C₂×#6094	C₂×C₄	1	24	yes	—
38	C₂×C₈×D₁₃	C₂×C₄	1	12	yes	—
39	C₂×C₈×D₁₃	C₂³	1	12	yes	—
40	#88106	C₂³	1	16	yes	complemented
41	C₂×#6092	C₂³	1	6	yes	—
42	C₂×#6092	C₂³	1	12	yes	—
43	C₂×#6092	C₂³	1	6	yes	—
44	C₂×#6092	C₂³	1	16	yes	complemented
45	#88098	C₂³	1	32	yes	complemented
46	D₁₃⋊C₁₆	C₂³	1	32	yes	complemented
47	C₂²×#504	—	13	1	no	maximal, Sylow
48	C₂⁴×C₁₃	C₂×C₈	1	1	yes	characteristic
49	C₂³×D₁₃	C₂×C₈	1	2	yes	—
50	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
51	C₂²×C₄×C₁₃	C₂×C₈	1	6	yes	—
52	C₂²×C₄×C₁₃	C₂²×C₄	1	6	yes	—
53	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₂²⋊D₁₃	C₂×C₈	1	24	yes	—
55	C₂²×Dic₁₃	C₂×C₈	1	2	yes	—
56	#6127	C₂×C₈	1	48	yes	—
57	#6127	C₂²×C₄	1	16	yes	—
58	C₂×D₅₂	C₂×C₈	1	12	yes	—
59	C₂×C₄×D₁₃	C₂×C₈	1	12	yes	—
60	C₂×C₄×D₁₃	C₂²×C₄	1	12	yes	—
61	C₂×Dic₂₆	C₂×C₈	1	12	yes	—
62	C₁₃×M₁₆	C₂²×C₄	1	16	yes	—
63	C₂×C₈×C₁₃	C₂²×C₄	1	12	yes	—
64	C₂×#1055	C₂²×C₄	1	3	yes	—
65	C₂×#1055	C₂²×C₄	1	6	yes	—
66	C₂×#1055	C₂²×C₄	1	12	yes	—
67	C₂×#1055	C₂⁴	1	3	yes	—
68	C₂×#1055	C₂⁴	1	4	yes	—
69	#6094	C₂²×C₄	1	32	yes	—
70	C₈×D₁₃	C₂²×C₄	1	24	yes	—
71	C₈×D₁₃	C₂⁴	1	8	yes	—
72	#6092	C₂⁴	1	4	yes	—
73	#6092	C₂⁴	1	8	yes	—
74	#6092	C₂⁴	1	4	yes	—
75	C₂²×#131	—	13	1	no	—
76	C₂×#504	—	13	24	no	—
77	C₂²×M₃₂	—	13	1	no	—
78	C₂²×M₃₂	—	13	2	no	—
79	C₂³×C₁₆	—	13	1	no	—
80	C₂³×C₁₆	—	13	2	no	—
81	C₂³×C₁₃	—	2	4	no	—
82	C₂³×C₁₃	C₂²×C₈	1	6	yes	—
83	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
84	C₂²×D₁₃	—	2	8	no	—
85	C₂²×D₁₃	C₂²×C₈	1	12	yes	—
86	C₂×C₄×C₁₃	C₂²×C₈	1	6	yes	—
87	C₂×C₄×C₁₃	C₂²×C₈	1	3	yes	—
88	C₂×C₄×C₁₃	C₂²×C₈	1	12	yes	—
89	C₂×C₄×C₁₃	C₂³×C₄	1	3	yes	—
90	C₂×C₄×C₁₃	C₂³×C₄	1	4	yes	—
91	C₂²⋊D₁₃	C₂²×C₈	1	32	yes	—
92	C₂×Dic₁₃	C₂²×C₈	1	12	yes	—
93	D₅₂	C₂²×C₈	1	16	yes	—
94	C₄×D₁₃	C₂²×C₈	1	24	yes	—
95	C₄×D₁₃	C₂³×C₄	1	8	yes	—
96	Dic₂₆	C₂²×C₈	1	16	yes	—
97	C₈×C₁₃	C₂³×C₄	1	8	yes	—
98	#1055	C₂³×C₄	1	3	yes	—
99	#1055	C₂³×C₄	1	4	yes	—
100	#1055	C₂⁵	1	1	yes	characteristic
101	C₂²×#41	—	13	1	no	—
102	C₂×#131	—	13	12	no	—
103	C₂×#131	—	13	12	no	—
104	C₂²×M₁₆	—	13	2	no	—
105	C₂²×M₁₆	#261	1	1	yes	characteristic
106	C₂³×C₈	—	13	1	no	—
107	C₂³×C₈	—	13	2	no	—
108	#504	—	13	64	no	—
109	C₂×M₃₂	—	13	24	no	—
110	C₂×M₃₂	—	13	12	no	—
111	C₂²×C₁₆	—	13	1	no	—
112	C₂²×C₁₆	—	13	1	no	—
113	C₂²×C₁₆	—	13	2	no	—
114	C₂²×C₁₆	—	13	24	no	—
115	C₂²×C₁₆	—	13	12	no	—
116	C₂²×C₁₃	—	2	12	no	—
117	C₂²×C₁₃	#504	1	4	yes	complemented
118	C₂²×C₁₃	C₂³×C₈	1	4	yes	—
119	C₂²×C₁₃	C₂³×C₈	1	3	yes	—
120	C₂×D₁₃	—	2	24	no	—
121	C₂×D₁₃	C₂³×C₈	1	8	yes	—
122	C₄×C₁₃	C₂³×C₈	1	4	yes	—
123	C₄×C₁₃	C₂³×C₈	1	3	yes	—
124	C₄×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
125	Dic₁₃	C₂³×C₈	1	8	yes	—
126	C₂×#41	—	13	12	no	—
127	C₂×#41	—	13	12	no	—
128	C₂²×Q₈	—	13	1	no	—
129	C₂²×D₄	—	13	2	no	—
130	C₂²×D₄	—	13	1	no	—
131	C₂³×C₄	—	13	2	no	—
132	C₂³×C₄	C₂×#261	1	1	yes	characteristic
133	#131	—	13	48	no	—
134	#131	—	13	16	no	—
135	C₂×M₁₆	—	13	24	no	—
136	C₂×M₁₆	C₂×#261	1	12	yes	—
137	C₂²×C₈	—	13	6	no	—
138	C₂²×C₈	—	13	6	no	—
139	C₂²×C₈	—	13	1	no	—
140	C₂²×C₈	—	13	1	no	—
141	C₂²×C₈	—	13	12	no	—
142	C₂²×C₈	—	13	12	no	—
143	C₂²×C₈	C₂×#261	1	2	yes	—
144	M₃₂	—	13	32	no	—
145	M₃₂	—	13	16	no	—
146	C₂×C₁₆	—	13	32	no	—
147	C₂×C₁₆	—	13	6	no	—
148	C₂×C₁₆	—	13	12	no	—
149	C₂×C₁₆	—	13	6	no	—
150	C₂×C₁₆	—	13	16	no	—
151	C₂×C₁₃	—	2	4	no	—
152	C₂×C₁₃	C₂×#504	1	6	yes	complemented
153	C₂×C₁₃	C₂⁴×C₈	1	1	yes	characteristic
154	D₁₃	—	2	8	no	—
155	C₂⁴	—	13	2	no	—
156	C₂⁴	D₁₃⋊C₈	1	1	yes	characteristic
157	#41	—	13	48	no	—
158	#41	—	13	16	no	—
159	C₂×Q₈	—	13	12	no	—
160	C₂×D₄	—	13	24	no	—
161	C₂×D₄	—	13	12	no	—
162	C₂²×C₄	—	13	2	no	—
163	C₂²×C₄	—	13	12	no	—
164	C₂²×C₄	—	13	12	no	—
165	C₂²×C₄	D₁₃⋊C₈	1	1	yes	characteristic
166	C₂²×C₄	D₁₃⋊C₈	1	6	yes	—
167	C₂²×C₄	C₂²×#261	1	6	yes	—
168	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
169	M₁₆	—	13	32	no	—
170	M₁₆	C₂²×#261	1	16	yes	—
171	C₂×C₈	—	13	3	no	—
172	C₂×C₈	—	13	4	no	—
173	C₂×C₈	—	13	3	no	—
174	C₂×C₈	—	13	6	no	—
175	C₂×C₈	—	13	12	no	—
176	C₂×C₈	—	13	8	no	—
177	C₂×C₈	—	13	24	no	—
178	C₂×C₈	C₂²×#261	1	12	yes	—
179	C₁₆	—	13	4	no	—
180	C₁₆	—	13	8	no	—
181	C₁₆	—	13	4	no	—
182	C₁₃	C₂²×#504	1	1	yes	characteristic, complemented, Sylow
183	C₂³	—	2	4	no	—
184	C₂³	—	13	12	no	—
185	C₂³	—	26	8	no	—
186	C₂³	C₂×D₁₃⋊C₈	1	6	yes	—
187	C₂³	C₂×D₁₃⋊C₈	1	1	yes	characteristic
188	Q₈	—	13	16	no	—
189	D₄	—	13	32	no	—
190	D₄	—	13	16	no	—
191	C₂×C₄	—	13	12	no	—
192	C₂×C₄	—	13	24	no	—
193	C₂×C₄	—	13	8	no	—
194	C₂×C₄	C₂×D₁₃⋊C₈	1	3	yes	—
195	C₂×C₄	C₂×D₁₃⋊C₈	1	6	yes	—
196	C₂×C₄	C₂×D₁₃⋊C₈	1	12	yes	—
197	C₂×C₄	C₂³×#261	1	3	yes	—
198	C₂×C₄	C₂³×#261	1	4	yes	—
199	C₈	—	13	1	no	—
200	C₈	—	13	3	no	—
201	C₈	—	13	4	no	—
202	C₈	C₂³×#261	1	8	yes	—
203	C₂²	—	2	12	no	—
204	C₂²	—	13	8	no	—
205	C₂²	—	26	24	no	—
206	C₂²	#11719096	1	4	yes	direct product factor
207	C₂²	C₂²×D₁₃⋊C₈	1	4	yes	—
208	C₂²	C₂²×D₁₃⋊C₈	1	3	yes	—
209	C₄	—	13	8	no	—
210	C₄	C₂²×D₁₃⋊C₈	1	3	yes	—
211	C₄	C₂²×D₁₃⋊C₈	1	4	yes	—
212	C₄	C₂⁴×#261	1	1	yes	characteristic
213	C₂	—	2	4	no	—
214	C₂	—	26	8	no	—
215	C₂	C₂×#11719096	1	6	yes	direct product factor
216	C₂	C₂³×D₁₃⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×#261
commutator	C₂×C₁₃	C₂⁴×C₈
Frattini	C₄	C₂⁴×#261
Fitting	C₂²×C₁₃×M₁₆	C₄
socle	C₂³×C₁₃	C₂²×C₈
prime radical	C₂²×#6127	C₄
cosocle	#1055	C₂⁵
hypercenter	C₂²×M₁₆	#261
hypocenter	C₁₃	C₂²×#504
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#11719096	C₂	direct (N × H)
2. C₂²×#88098	C₂	semidirect
3. C₂²×#88106	C₂	semidirect
4. C₂²×D₁₃⋊C₁₆	C₂	semidirect
5. C₂³×#6092	C₂	semidirect
14. C₂×#88106	C₂²	semidirect
18. C₂²×#6092	C₂²	semidirect
19. #11719096	C₂²	direct (N × H)
20. C₂×#88098	C₂²	semidirect
21. C₂×D₁₃⋊C₁₆	C₂²	semidirect
40. #88106	C₂³	semidirect
44. C₂×#6092	C₂³	semidirect
45. #88098	C₂³	semidirect
46. D₁₃⋊C₁₆	C₂³	semidirect
117. C₂²×C₁₃	#504	semidirect
152. C₂×C₁₃	C₂×#504	semidirect
182. C₁₃	C₂²×#504	semidirect
206. C₂²	#11719096	direct (N × H)
215. C₂	C₂×#11719096	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#11719096	2
C₂²×#88098	2
C₂²×#88106	2
C₂²×D₁₃⋊C₁₆	2
C₂³×#6092	2
C₂²×#88161	2
C₂²×#504	13

Subgroups — C22×#11719096 #16809617

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11719096 #16809617