Subgroups — C₂×#13082243 #15660878

Order 2816.

8418 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13082243	C₂	1	32	yes	maximal, direct product factor
2	C₂×#10623265	C₂	1	9	yes	maximal, complemented
3	C₂×#10623108	C₂	1	6	yes	maximal, complemented
4	C₂×#10623256	C₂	1	9	yes	maximal, complemented
5	C₂×C₄×#4343	C₂	1	6	yes	maximal, complemented
6	C₂×Q₈²×C₁₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#4343	C₂²	1	6	yes	—
8	C₂²×#4343	C₂²	1	6	yes	—
9	Q₈²×C₁₁	C₂²	1	16	yes	complemented
10	C₂×C₁₁×C₄⋊Q₈	C₂²	1	9	yes	—
11	C₂×C₄×Q₈×C₁₁	C₂²	1	6	yes	—
12	C₂×#67000	C₂²	1	18	yes	—
13	C₂×Q₈×Dic₁₁	C₂²	1	2	yes	—
14	#10623265	C₂²	1	144	yes	complemented
15	#10623256	C₂²	1	144	yes	complemented
16	#10623108	C₂²	1	96	yes	complemented
17	C₄×#4343	C₂²	1	96	yes	complemented
18	C₂×#66948	C₂²	1	36	yes	—
19	C₂×#66946	C₂²	1	18	yes	—
20	C₂×#66945	C₂²	1	18	yes	—
21	C₂×#66927	C₂²	1	9	yes	—
22	C₂×C₄×D₄₄	C₂²	1	18	yes	—
23	C₂×C₄²×D₁₁	C₂²	1	9	yes	—
24	C₂²×Q₈×C₁₁	C₂³	1	6	yes	—
25	C₂²×Q₈×C₁₁	C₂³	1	2	yes	—
26	C₂×#4343	—	2	48	no	—
27	C₂×#4343	—	4	24	no	—
28	C₂×#4343	C₂³	1	48	yes	—
29	C₂×#4343	C₂³	1	48	yes	—
30	C₂²×D₄₄	C₂³	1	18	yes	—
31	C₂²×D₄₄	C₂³	1	18	yes	—
32	C₂²×C₄×D₁₁	C₂³	1	18	yes	—
33	C₂²×C₄×D₁₁	C₂³	1	9	yes	—
34	C₁₁×C₄⋊Q₈	C₂³	1	72	yes	—
35	C₄×Q₈×C₁₁	C₂³	1	48	yes	—
36	C₂×C₁₁×C₄⋊C₄	C₂³	1	18	yes	—
37	C₂×C₄²×C₁₁	C₂³	1	9	yes	—
38	#67000	C₂³	1	144	yes	—
39	Q₈×Dic₁₁	C₂³	1	16	yes	—
40	C₂×(C₂×D₁₁)⋊C₄	C₂³	1	36	yes	—
41	C₂×C₄⋊Dic₁₁	C₂³	1	6	yes	—
42	C₂×C₄×Dic₁₁	C₂³	1	6	yes	—
43	C₂×C₄×Dic₁₁	C₂³	1	9	yes	—
44	#66948	C₂³	1	288	yes	—
45	#66946	C₂³	1	144	yes	—
46	#66945	C₂³	1	144	yes	—
47	#66927	C₂³	1	72	yes	—
48	C₄×D₄₄	C₂³	1	144	yes	—
49	C₄²×D₁₁	C₂³	1	72	yes	—
50	C₂×#3483	—	11	1	no	maximal, Sylow
51	C₂³×D₁₁	C₂⁴	1	9	yes	—
52	C₂×Q₈×C₁₁	—	4	12	no	—
53	C₂×Q₈×C₁₁	#41	1	16	yes	complemented
54	C₂×Q₈×C₁₁	C₂⁴	1	24	yes	—
55	C₂×Q₈×C₁₁	C₂⁴	1	8	yes	—
56	C₂²×C₄×C₁₁	C₂⁴	1	9	yes	—
57	C₂²×C₄×C₁₁	C₂⁴	1	6	yes	—
58	C₂²×Dic₁₁	C₂⁴	1	6	yes	—
59	C₂²×Dic₁₁	C₂⁴	1	1	yes	characteristic
60	#4343	—	2	192	no	—
61	#4343	—	4	96	no	—
62	C₂×D₄₄	—	2	72	no	—
63	C₂×D₄₄	—	4	36	no	—
64	C₂×D₄₄	C₂⁴	1	72	yes	—
65	C₂×D₄₄	C₂⁴	1	72	yes	—
66	C₂×C₄×D₁₁	—	2	36	no	—
67	C₂×C₄×D₁₁	—	2	72	no	—
68	C₂×C₄×D₁₁	C₂⁴	1	72	yes	—
69	C₂×C₄×D₁₁	C₂⁴	1	36	yes	—
70	C₁₁×C₄⋊C₄	C₂⁴	1	72	yes	—
71	C₄²×C₁₁	C₂⁴	1	36	yes	—
72	(C₂×D₁₁)⋊C₄	C₂⁴	1	144	yes	—
73	C₄⋊Dic₁₁	C₂⁴	1	24	yes	—
74	C₄×Dic₁₁	C₂⁴	1	24	yes	—
75	C₄×Dic₁₁	C₂⁴	1	36	yes	—
76	#3483	—	11	32	no	—
77	C₂×Q₈²	D₁₁	1	1	yes	characteristic, complemented
78	C₂×#555	—	11	9	no	—
79	C₂×#550	—	11	6	no	—
80	C₂×#532	—	11	9	no	—
81	C₂×C₄×#41	—	11	6	no	—
82	C₂³×C₁₁	C₂⁵	1	1	yes	characteristic
83	C₂²×D₁₁	—	2	36	no	—
84	C₂²×D₁₁	—	2	18	no	—
85	C₂²×D₁₁	C₂⁵	1	18	yes	—
86	Q₈×C₁₁	—	4	24	no	—
87	Q₈×C₁₁	C₂×#41	1	32	yes	complemented
88	C₂×C₄×C₁₁	—	2	18	no	—
89	C₂×C₄×C₁₁	C₂×#41	1	24	yes	—
90	C₂×C₄×C₁₁	C₂⁵	1	18	yes	—
91	C₂×C₄×C₁₁	C₂⁵	1	12	yes	—
92	C₂×Dic₁₁	—	2	12	no	—
93	C₂×Dic₁₁	H(5,𝔽₂)	1	4	yes	—
94	C₂×Dic₁₁	C₂⁵	1	12	yes	—
95	C₂×Dic₁₁	C₂⁵	1	2	yes	—
96	D₄₄	—	2	144	no	—
97	D₄₄	—	4	72	no	—
98	C₄×D₁₁	—	2	144	no	—
99	C₄×D₁₁	—	2	72	no	—
100	C₂²×#41	—	11	6	no	—
101	C₂²×#41	—	11	6	no	—
102	Q₈²	C₂×D₁₁	1	16	yes	complemented
103	#555	—	11	144	no	—
104	#550	—	11	96	no	—
105	#532	—	11	144	no	—
106	C₂×C₄⋊Q₈	C₂×D₁₁	1	9	yes	—
107	C₂×#127	—	11	9	no	—
108	C₂×#124	—	11	18	no	—
109	C₂×#121	—	11	36	no	—
110	C₄×#41	—	11	96	no	—
111	C₂×C₄×Q₈	—	11	2	no	—
112	C₂×C₄×Q₈	C₂×D₁₁	1	6	yes	—
113	C₂×C₄×D₄	—	11	18	no	—
114	C₂×C₄×D₄	—	11	18	no	—
115	C₂×#117	—	11	18	no	—
116	C₂²×C₄²	—	11	9	no	—
117	C₂²×C₁₁	C₂²×#41	1	4	yes	—
118	C₂²×C₁₁	C₂×H(5,𝔽₂)	1	2	yes	—
119	C₂²×C₁₁	C₂⁶	1	1	yes	characteristic
120	C₂×D₁₁	—	2	36	no	—
121	C₂×D₁₁	—	2	18	no	—
122	C₂×D₁₁	—	4	36	no	—
123	C₄×C₁₁	—	2	18	no	—
124	C₄×C₁₁	C₂²×#41	1	24	yes	—
125	Dic₁₁	—	2	12	no	—
126	Dic₁₁	C₂×H(5,𝔽₂)	1	4	yes	—
127	C₂×#41	—	11	48	no	—
128	C₂×#41	—	11	48	no	—
129	C₂×#41	—	22	48	no	—
130	C₂×#41	—	44	24	no	—
131	C₂²×Q₈	C₂²×D₁₁	1	6	yes	—
132	C₂²×Q₈	C₂²×D₁₁	1	2	yes	—
133	C₂²×D₄	—	11	18	no	—
134	C₂²×D₄	—	11	18	no	—
135	C₂³×C₄	—	11	18	no	—
136	C₂³×C₄	—	11	9	no	—
137	C₄⋊Q₈	C₂²×D₁₁	1	72	yes	—
138	#127	—	11	72	no	—
139	#124	—	11	144	no	—
140	#121	—	11	288	no	—
141	C₄×Q₈	—	11	16	no	—
142	C₄×Q₈	C₂²×D₁₁	1	48	yes	—
143	C₄×D₄	—	11	144	no	—
144	C₄×D₄	—	11	144	no	—
145	#117	—	11	144	no	—
146	C₂×C₄⋊C₄	—	11	6	no	—
147	C₂×C₄⋊C₄	C₂²×D₁₁	1	18	yes	—
148	C₂×C₂²⋊C₄	—	11	36	no	—
149	C₂×C₄²	—	11	9	no	—
150	C₂×C₄²	—	11	6	no	—
151	C₂×C₄²	—	11	72	no	—
152	C₂×C₄²	C₂²×D₁₁	1	9	yes	—
153	C₂×C₁₁	#3483	1	4	yes	complemented
154	C₂×C₁₁	C₂³×#41	1	2	yes	—
155	C₂×C₁₁	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
156	D₁₁	—	4	18	no	—
157	C₂⁴	—	11	9	no	—
158	#41	—	22	192	no	—
159	#41	—	44	96	no	—
160	C₂×Q₈	—	4	12	no	—
161	C₂×Q₈	#4343	1	16	yes	complemented
162	C₂×Q₈	C₂³×D₁₁	1	24	yes	—
163	C₂×Q₈	C₂³×D₁₁	1	8	yes	—
164	C₂×D₄	—	11	72	no	—
165	C₂×D₄	—	11	72	no	—
166	C₂×D₄	—	22	72	no	—
167	C₂×D₄	—	44	36	no	—
168	C₂²×C₄	—	11	1	no	—
169	C₂²×C₄	—	11	6	no	—
170	C₂²×C₄	—	11	72	no	—
171	C₂²×C₄	—	11	36	no	—
172	C₂²×C₄	—	22	36	no	—
173	C₂²×C₄	—	22	72	no	—
174	C₂²×C₄	C₂³×D₁₁	1	9	yes	—
175	C₂²×C₄	C₂³×D₁₁	1	6	yes	—
176	C₄⋊C₄	—	11	24	no	—
177	C₄⋊C₄	C₂³×D₁₁	1	72	yes	—
178	C₂²⋊C₄	—	11	144	no	—
179	C₄²	—	11	24	no	—
180	C₄²	—	11	36	no	—
181	C₄²	C₂³×D₁₁	1	36	yes	—
182	C₁₁	C₂×#3483	1	1	yes	characteristic, complemented, Sylow
183	C₂³	—	11	18	no	—
184	C₂³	—	22	36	no	—
185	C₂³	—	22	18	no	—
186	C₂³	C₂⁴×D₁₁	1	1	yes	characteristic
187	Q₈	—	4	24	no	—
188	Q₈	C₂×#4343	1	32	yes	complemented
189	D₄	—	22	144	no	—
190	D₄	—	44	72	no	—
191	C₂×C₄	—	2	18	no	—
192	C₂×C₄	—	11	4	no	—
193	C₂×C₄	—	11	2	no	—
194	C₂×C₄	—	11	12	no	—
195	C₂×C₄	—	22	72	no	—
196	C₂×C₄	—	22	12	no	—
197	C₂×C₄	—	22	144	no	—
198	C₂×C₄	C₂×#4343	1	24	yes	—
199	C₂×C₄	C₂⁴×D₁₁	1	18	yes	—
200	C₂×C₄	C₂⁴×D₁₁	1	12	yes	—
201	C₂²	—	22	36	no	—
202	C₂²	—	22	18	no	—
203	C₂²	—	44	36	no	—
204	C₂²	C₂²×#4343	1	4	yes	—
205	C₂²	D₁₁×H(5,𝔽₂)	1	2	yes	—
206	C₂²	C₂⁵×D₁₁	1	1	yes	characteristic
207	C₄	—	2	18	no	—
208	C₄	—	11	4	no	—
209	C₄	—	22	12	no	—
210	C₄	C₂²×#4343	1	24	yes	—
211	C₂	—	44	18	no	—
212	C₂	#13082243	1	4	yes	direct product factor
213	C₂	C₂³×#4343	1	2	yes	—
214	C₂	C₂×D₁₁×H(5,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂⁴×D₁₁
commutator, cosocle	C₂²×C₁₁	C₂⁶
Frattini	C₂²	C₂⁵×D₁₁
Fitting	C₂×Q₈²×C₁₁	C₂
socle	C₂³×C₁₁	C₂⁵
hypercenter	C₂×Q₈²	D₁₁
hypocenter	C₁₁	C₂×#3483
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂³ ◁ C₂×Q₈²

Complements

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

N	H ≅ G/N	type
1. #13082243	C₂	direct (N × H)
2. C₂×#10623265	C₂	semidirect
3. C₂×#10623108	C₂	semidirect
4. C₂×#10623256	C₂	semidirect
5. C₂×C₄×#4343	C₂	semidirect
6. C₂×Q₈²×C₁₁	C₂	semidirect
9. Q₈²×C₁₁	C₂²	semidirect
14. #10623265	C₂²	semidirect
15. #10623256	C₂²	semidirect
16. #10623108	C₂²	semidirect
17. C₄×#4343	C₂²	semidirect
53. C₂×Q₈×C₁₁	#41	semidirect
77. C₂×Q₈²	D₁₁	semidirect
87. Q₈×C₁₁	C₂×#41	semidirect
102. Q₈²	C₂×D₁₁	semidirect
153. C₂×C₁₁	#3483	semidirect
161. C₂×Q₈	#4343	semidirect
182. C₁₁	C₂×#3483	semidirect
188. Q₈	C₂×#4343	semidirect
212. C₂	#13082243	direct (N × H)

Maximal subgroups

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

subgroup	index
#13082243	2
C₂×#10623265	2
C₂×#10623108	2
C₂×#10623256	2
C₂×C₄×#4343	2
C₂×Q₈²×C₁₁	2
C₂×#3483	11

Subgroups — C2×#13082243 #15660878

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13082243 #15660878