Subgroups — C₂²×#11719625 #16809652

Order 3328.

11118 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×#11719625	C₂	1	48	yes	maximal, direct product factor
2	C₂²×#88240	C₂	1	8	yes	maximal, complemented
3	C₂²×D₁₃⋊M₁₆	C₂	1	6	yes	maximal, complemented
4	C₂³×#6131	C₂	1	1	yes	maximal, characteristic
5	C₂³×Q₈×C₁₃	C₄	1	1	yes	characteristic
6	C₂²×#6131	C₄	1	4	yes	—
7	C₂²×#6131	C₄	1	24	yes	—
8	C₂²×#6131	C₂²	1	4	yes	—
9	C₂²×#6131	C₂²	1	24	yes	—
10	C₂³×D₅₂	C₄	1	3	yes	—
11	C₂³×C₄×D₁₃	C₂²	1	3	yes	—
12	C₂²×#6122	C₂²	1	4	yes	—
13	#11719625	C₂²	1	256	yes	direct product factor
14	C₂×#88240	C₂²	1	192	yes	complemented
15	C₂×D₁₃⋊M₁₆	C₂²	1	144	yes	complemented
16	C₂²×#6118	C₂²	1	12	yes	complemented
17	C₂²×D₁₃⋊C₈	C₂²	1	12	yes	complemented
18	C₂⁴×D₁₃	C₂×C₄	1	3	yes	—
19	C₂²×Q₈×C₁₃	C₂×C₄	1	4	yes	—
20	C₂²×Q₈×C₁₃	C₂×C₄	1	24	yes	—
21	C₂³×C₄×C₁₃	C₂×C₄	1	3	yes	—
22	C₂³×Dic₁₃	C₂³	1	1	yes	characteristic
23	C₂×#6131	C₂×C₄	1	48	yes	—
24	C₂×#6131	C₂×C₄	1	96	yes	—
25	C₂×#6131	C₂×C₄	1	192	yes	—
26	C₂×#6131	C₂³	1	48	yes	—
27	C₂×#6131	C₂³	1	64	yes	—
28	C₂²×D₅₂	C₂×C₄	1	12	yes	—
29	C₂²×D₅₂	C₂×C₄	1	72	yes	—
30	C₂²×C₄×D₁₃	C₂×C₄	1	6	yes	—
31	C₂²×C₄×D₁₃	C₂×C₄	1	36	yes	—
32	C₂²×C₄×D₁₃	C₂³	1	6	yes	—
33	C₂²×C₄×D₁₃	C₂³	1	36	yes	—
34	C₂×#6122	C₂³	1	48	yes	—
35	C₂²×#1057	C₂³	1	8	yes	—
36	#88240	C₂³	1	512	yes	complemented
37	D₁₃⋊M₁₆	C₂³	1	384	yes	complemented
38	C₂×#6118	C₂³	1	144	yes	complemented
39	C₂×D₁₃⋊C₈	C₂³	1	144	yes	complemented
40	C₂²×#568	—	13	1	no	maximal, Sylow
41	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
42	C₂³×D₁₃	—	4	12	no	—
43	C₂³×D₁₃	C₂²×C₄	1	6	yes	—
44	C₂³×D₁₃	C₂²×C₄	1	36	yes	—
45	C₂×Q₈×C₁₃	C₂²×C₄	1	48	yes	—
46	C₂×Q₈×C₁₃	C₂²×C₄	1	64	yes	—
47	C₂²×C₄×C₁₃	C₂²×C₄	1	36	yes	—
48	C₂²×C₄×C₁₃	C₂²×C₄	1	6	yes	—
49	C₂²×Dic₁₃	C₂²×C₄	1	6	yes	—
50	C₂²×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
51	C₂²×Dic₁₃	C₂⁴	1	1	yes	characteristic
52	C₂²×Dic₁₃	C₂⁴	1	6	yes	—
53	#6131	C₂²×C₄	1	192	yes	—
54	#6131	C₂²×C₄	1	256	yes	—
55	#6131	C₂⁴	1	64	yes	—
56	C₂×D₅₂	C₂²×C₄	1	144	yes	—
57	C₂×D₅₂	C₂²×C₄	1	192	yes	—
58	C₂×C₄×D₁₃	C₂²×C₄	1	36	yes	—
59	C₂×C₄×D₁₃	C₂²×C₄	1	72	yes	—
60	C₂×C₄×D₁₃	C₂²×C₄	1	144	yes	—
61	C₂×C₄×D₁₃	C₂⁴	1	36	yes	—
62	C₂×C₄×D₁₃	C₂⁴	1	48	yes	—
63	#6122	C₂⁴	1	64	yes	—
64	C₂×#1057	C₂⁴	1	48	yes	—
65	#6118	C₂⁴	1	192	yes	complemented
66	D₁₃⋊C₈	C₂⁴	1	192	yes	complemented
67	C₂³×#41	—	13	1	no	—
68	C₂×#568	—	13	48	no	—
69	C₂²×#131	—	13	8	no	—
70	C₂³×M₁₆	—	13	6	no	—
71	C₂³×C₁₃	—	2	4	no	—
72	C₂³×C₁₃	C₂³×C₄	1	6	yes	—
73	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
74	C₂²×D₁₃	—	2	24	no	—
75	C₂²×D₁₃	—	4	72	no	—
76	C₂²×D₁₃	C₂³×C₄	1	36	yes	—
77	C₂²×D₁₃	C₂³×C₄	1	48	yes	—
78	Q₈×C₁₃	C₂³×C₄	1	64	yes	—
79	C₂×C₄×C₁₃	C₂³×C₄	1	48	yes	—
80	C₂×C₄×C₁₃	C₂³×C₄	1	36	yes	—
81	C₂×Dic₁₃	C₂³×C₄	1	12	yes	—
82	C₂×Dic₁₃	C₂³×C₄	1	3	yes	—
83	C₂×Dic₁₃	C₂³×C₄	1	6	yes	—
84	C₂×Dic₁₃	C₂⁵	1	3	yes	—
85	C₂×Dic₁₃	C₂⁵	1	4	yes	—
86	D₅₂	C₂³×C₄	1	192	yes	—
87	C₄×D₁₃	C₂³×C₄	1	72	yes	—
88	C₄×D₁₃	C₂³×C₄	1	96	yes	—
89	C₄×D₁₃	C₂⁵	1	24	yes	—
90	#1057	C₂⁵	1	32	yes	—
91	C₂²×#41	—	13	24	no	—
92	C₂²×#41	—	13	4	no	—
93	C₂²×#41	—	13	4	no	—
94	C₂²×#41	—	13	24	no	—
95	C₂³×Q₈	#261	1	1	yes	characteristic
96	C₂³×D₄	—	13	3	no	—
97	C₂⁴×C₄	—	13	3	no	—
98	#568	—	13	256	no	—
99	C₂×#131	—	13	192	no	—
100	C₂²×M₁₆	—	13	12	no	—
101	C₂²×M₁₆	—	13	144	no	—
102	C₂²×M₁₆	—	13	4	no	—
103	C₂³×C₈	—	13	12	no	—
104	C₂²×C₁₃	—	2	12	no	—
105	C₂²×C₁₃	#568	1	4	yes	complemented
106	C₂²×C₁₃	C₂⁴×C₄	1	4	yes	—
107	C₂²×C₁₃	C₂⁴×C₄	1	3	yes	—
108	C₂×D₁₃	—	2	72	no	—
109	C₂×D₁₃	—	4	48	no	—
110	C₂×D₁₃	C₂⁴×C₄	1	24	yes	—
111	C₄×C₁₃	C₂⁴×C₄	1	24	yes	—
112	Dic₁₃	C₂⁴×C₄	1	3	yes	—
113	Dic₁₃	C₂⁴×C₄	1	4	yes	—
114	Dic₁₃	C₂⁶	1	1	yes	characteristic
115	C₂⁵	—	13	3	no	—
116	C₂×#41	—	13	48	no	—
117	C₂×#41	—	13	64	no	—
118	C₂×#41	—	13	48	no	—
119	C₂×#41	—	13	96	no	—
120	C₂×#41	—	13	192	no	—
121	C₂²×Q₈	C₂×#261	1	24	yes	—
122	C₂²×Q₈	C₂×#261	1	4	yes	—
123	C₂²×D₄	—	13	72	no	—
124	C₂²×D₄	—	13	12	no	—
125	C₂³×C₄	—	13	1	no	—
126	C₂³×C₄	—	13	6	no	—
127	C₂³×C₄	—	13	36	no	—
128	C₂³×C₄	—	13	36	no	—
129	C₂³×C₄	—	13	6	no	—
130	C₂³×C₄	C₂×#261	1	3	yes	—
131	#131	—	13	512	no	—
132	C₂×M₁₆	—	13	48	no	—
133	C₂×M₁₆	—	13	144	no	—
134	C₂×M₁₆	—	13	384	no	—
135	C₂²×C₈	—	13	8	no	—
136	C₂²×C₈	—	13	144	no	—
137	C₂×C₁₃	—	2	4	no	—
138	C₂×C₁₃	C₂×#568	1	6	yes	complemented
139	C₂×C₁₃	C₂⁵×C₄	1	1	yes	characteristic
140	D₁₃	—	2	24	no	—
141	C₂⁴	—	13	6	no	—
142	C₂⁴	—	13	36	no	—
143	C₂⁴	—	52	12	no	—
144	C₂⁴	C₂²×#261	1	1	yes	characteristic
145	#41	—	13	64	no	—
146	#41	—	13	192	no	—
147	#41	—	13	256	no	—
148	C₂×Q₈	C₂²×#261	1	48	yes	—
149	C₂×Q₈	C₂²×#261	1	64	yes	—
150	C₂×D₄	—	13	144	no	—
151	C₂×D₄	—	13	192	no	—
152	C₂²×C₄	—	13	6	no	—
153	C₂²×C₄	—	13	6	no	—
154	C₂²×C₄	—	13	1	no	—
155	C₂²×C₄	—	13	1	no	—
156	C₂²×C₄	—	13	36	no	—
157	C₂²×C₄	—	13	48	no	—
158	C₂²×C₄	—	13	36	no	—
159	C₂²×C₄	—	13	72	no	—
160	C₂²×C₄	—	13	144	no	—
161	C₂²×C₄	C₂²×#261	1	36	yes	—
162	C₂²×C₄	C₂²×#261	1	6	yes	—
163	M₁₆	—	13	64	no	—
164	M₁₆	—	13	192	no	—
165	C₂×C₈	—	13	48	no	—
166	C₂×C₈	—	13	192	no	—
167	C₁₃	C₂²×#568	1	1	yes	characteristic, complemented, Sylow
168	C₂³	—	2	4	no	—
169	C₂³	—	13	36	no	—
170	C₂³	—	13	48	no	—
171	C₂³	—	26	24	no	—
172	C₂³	—	52	72	no	—
173	C₂³	C₂³×#261	1	6	yes	—
174	C₂³	C₂³×#261	1	1	yes	characteristic
175	Q₈	C₂³×#261	1	64	yes	—
176	D₄	—	13	192	no	—
177	C₂×C₄	—	13	3	no	—
178	C₂×C₄	—	13	4	no	—
179	C₂×C₄	—	13	12	no	—
180	C₂×C₄	—	13	6	no	—
181	C₂×C₄	—	13	3	no	—
182	C₂×C₄	—	13	24	no	—
183	C₂×C₄	—	13	72	no	—
184	C₂×C₄	—	13	96	no	—
185	C₂×C₄	C₂³×#261	1	48	yes	—
186	C₂×C₄	C₂³×#261	1	36	yes	—
187	C₈	—	13	32	no	—
188	C₂²	—	2	12	no	—
189	C₂²	—	13	24	no	—
190	C₂²	—	26	72	no	—
191	C₂²	—	52	48	no	—
192	C₂²	#11719625	1	4	yes	direct product factor
193	C₂²	C₂⁴×#261	1	4	yes	—
194	C₂²	C₂⁴×#261	1	3	yes	—
195	C₄	—	13	3	no	—
196	C₄	—	13	4	no	—
197	C₄	—	13	1	no	—
198	C₄	C₂⁴×#261	1	24	yes	—
199	C₂	—	2	4	no	—
200	C₂	—	26	24	no	—
201	C₂	C₂×#11719625	1	6	yes	direct product factor
202	C₂	C₂⁵×#261	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×#261
commutator	C₂×C₁₃	C₂⁵×C₄
Frattini	C₂	C₂⁵×#261
Fitting	C₂³×Q₈×C₁₃	C₄
socle	C₂³×C₁₃	C₂³×C₄
prime radical	C₂³×#6131	C₂
cosocle	Dic₁₃	C₂⁶
hypercenter	C₂³×Q₈	#261
hypocenter	C₁₃	C₂²×#568
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂³×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×#11719625	C₂	direct (N × H)
2. C₂²×#88240	C₂	semidirect
3. C₂²×D₁₃⋊M₁₆	C₂	semidirect
13. #11719625	C₂²	direct (N × H)
14. C₂×#88240	C₂²	semidirect
15. C₂×D₁₃⋊M₁₆	C₂²	semidirect
16. C₂²×#6118	C₂²	semidirect
17. C₂²×D₁₃⋊C₈	C₂²	semidirect
36. #88240	C₂³	semidirect
37. D₁₃⋊M₁₆	C₂³	semidirect
38. C₂×#6118	C₂³	semidirect
39. C₂×D₁₃⋊C₈	C₂³	semidirect
65. #6118	C₂⁴	semidirect
66. D₁₃⋊C₈	C₂⁴	semidirect
105. C₂²×C₁₃	#568	semidirect
138. C₂×C₁₃	C₂×#568	semidirect
167. C₁₃	C₂²×#568	semidirect
192. C₂²	#11719625	direct (N × H)
201. C₂	C₂×#11719625	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×#11719625	2
C₂²×#88240	2
C₂²×D₁₃⋊M₁₆	2
C₂³×#6131	2
C₂²×#568	13

Subgroups — C22×#11719625 #16809652

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11719625 #16809652