Subgroups — C₂×#13068754 #14753999

Order 2816.

870 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10622537	C₂	1	2	yes	maximal
2	#13068754	C₂	1	8	yes	maximal, direct product factor
3	C₂×#10622190	C₂	1	4	yes	maximal
4	C₂²×C₄×C₈×C₁₁	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₈×C₁₁	C₂²	1	2	yes	—
6	C₂²×C₄²×C₁₁	C₂²	1	1	yes	characteristic
7	C₂²×#4318	C₂²	1	4	yes	—
8	C₂×C₄×C₈×C₁₁	C₄	1	2	yes	—
9	C₂×C₄×C₈×C₁₁	C₄	1	4	yes	—
10	C₂×C₄×C₈×C₁₁	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₁₁	C₂²	1	4	yes	—
12	#10622537	—	2	4	no	—
13	#10622537	C₂²	1	8	yes	—
14	C₂×C₄⋊(C₁₁⋊C₈)	—	2	4	no	—
15	#10622190	C₂²	1	16	yes	—
16	C₂³×C₄×C₁₁	Q₈	1	2	yes	—
17	C₂³×C₄×C₁₁	C₂³	1	1	yes	characteristic
18	C₂²×C₈×C₁₁	—	2	8	no	—
19	C₂²×C₈×C₁₁	C₂×C₄	1	4	yes	—
20	C₂²×C₈×C₁₁	C₂×C₄	1	4	yes	—
21	C₂²×C₈×C₁₁	C₂³	1	4	yes	—
22	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₁₁	C₂×C₄	1	2	yes	—
24	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₁₁	D₄	1	2	yes	—
26	C₂×C₄²×C₁₁	D₄	1	2	yes	—
27	C₂×C₄²×C₁₁	Q₈	1	2	yes	—
28	C₂×C₄²×C₁₁	C₂³	1	2	yes	—
29	C₂×#4318	—	2	4	no	—
30	C₂×#4318	—	2	16	no	—
31	C₂×#4318	C₂³	1	8	yes	—
32	C₂²×C₁₁⋊C₈	—	2	4	no	—
33	C₄×C₈×C₁₁	C₂×C₄	1	8	yes	—
34	C₄×C₈×C₁₁	C₂×C₄	1	8	yes	—
35	C₄⋊(C₁₁⋊C₈)	—	2	16	no	—
36	C₂×#1837	—	11	1	no	maximal, Sylow
37	C₂⁴×C₁₁	C₂×Q₈	1	1	yes	characteristic
38	C₂²×C₄×C₁₁	—	2	8	no	—
39	C₂²×C₄×C₁₁	C₄⋊C₄	1	2	yes	—
40	C₂²×C₄×C₁₁	C₄⋊C₄	1	4	yes	—
41	C₂²×C₄×C₁₁	C₄⋊C₄	1	2	yes	—
42	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₁	C₂×Q₈	1	4	yes	—
47	C₂²×C₄×C₁₁	C₂×Q₈	1	1	yes	characteristic
48	C₂²×C₄×C₁₁	#41	1	8	yes	—
49	C₂²×C₄×C₁₁	C₂⁴	1	1	yes	characteristic
50	C₂×C₈×C₁₁	—	2	8	no	—
51	C₂×C₈×C₁₁	—	2	8	no	—
52	C₂×C₈×C₁₁	—	2	8	no	—
53	C₂×C₈×C₁₁	C₂²×C₄	1	8	yes	—
54	C₄²×C₁₁	C₄⋊C₄	1	2	yes	—
55	C₄²×C₁₁	C₄⋊C₄	1	4	yes	—
56	C₄²×C₁₁	C₄⋊C₄	1	2	yes	—
57	C₄²×C₁₁	C₂²×C₄	1	2	yes	—
58	C₄²×C₁₁	C₂²×C₄	1	2	yes	—
59	C₄²×C₁₁	C₂×D₄	1	4	yes	—
60	#4318	—	2	16	no	—
61	#4318	—	4	8	no	—
62	C₂×C₁₁⋊C₈	—	2	8	no	—
63	C₂×C₁₁⋊C₈	—	4	8	no	—
64	C₂×#423	—	11	2	no	—
65	C₂²×C₄×C₈	D₁₁	1	1	yes	characteristic
66	#1837	—	11	8	no	—
67	C₂×#341	—	11	4	no	—
68	C₂³×C₁₁	—	2	4	no	—
69	C₂³×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂³×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂³×C₁₁	#125	1	4	yes	—
72	C₂³×C₁₁	C₂²×Q₈	1	1	yes	characteristic
73	C₂×C₄×C₁₁	—	2	4	no	—
74	C₂×C₄×C₁₁	—	2	8	no	—
75	C₂×C₄×C₁₁	—	2	8	no	—
76	C₂×C₄×C₁₁	#108	1	8	yes	—
77	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	4	yes	—
78	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	4	yes	—
81	C₂×C₄×C₁₁	#117	1	4	yes	—
82	C₂×C₄×C₁₁	#117	1	4	yes	—
83	C₂×C₄×C₁₁	#124	1	4	yes	—
84	C₂×C₄×C₁₁	#124	1	4	yes	—
85	C₂×C₄×C₁₁	#125	1	4	yes	—
86	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
87	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
88	C₂×C₄×C₁₁	C₂×#41	1	4	yes	—
89	C₈×C₁₁	—	2	8	no	—
90	C₁₁⋊C₈	—	4	8	no	—
91	C₂²×M₁₆	—	11	4	no	—
92	C₂³×C₈	C₂×D₁₁	1	2	yes	—
93	C₂²×C₄²	C₂×D₁₁	1	1	yes	characteristic
94	#423	—	11	8	no	—
95	#423	—	22	4	no	—
96	C₂×C₄⋊C₈	—	22	4	no	—
97	C₂×C₄×C₈	Dic₁₁	1	2	yes	—
98	C₂×C₄×C₈	Dic₁₁	1	4	yes	—
99	C₂×C₄×C₈	Dic₁₁	1	2	yes	—
100	C₂×C₄×C₈	C₂×D₁₁	1	4	yes	—
101	#341	—	11	16	no	—
102	C₂²×C₁₁	—	2	8	no	—
103	C₂²×C₁₁	—	2	2	no	—
104	C₂²×C₁₁	—	2	2	no	—
105	C₂²×C₁₁	#382	1	2	yes	—
106	C₂²×C₁₁	#382	1	2	yes	—
107	C₂²×C₁₁	C₂×#108	1	4	yes	—
108	C₂²×C₁₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
109	C₂²×C₁₁	C₂×#125	1	2	yes	—
110	C₄×C₁₁	—	2	4	no	—
111	C₄×C₁₁	#382	1	2	yes	—
112	C₄×C₁₁	#382	1	2	yes	—
113	C₄×C₁₁	C₂×#108	1	8	yes	—
114	C₄×C₁₁	C₂×#117	1	2	yes	—
115	C₄×C₁₁	C₂×#124	1	2	yes	—
116	C₂³×C₄	Dic₂₂	1	2	yes	—
117	C₂³×C₄	C₂²×D₁₁	1	1	yes	characteristic
118	C₂×M₁₆	—	11	8	no	—
119	C₂×M₁₆	—	22	4	no	—
120	C₂×M₁₆	—	22	16	no	—
121	C₂²×C₈	—	2	8	no	—
122	C₂²×C₈	—	22	4	no	—
123	C₂²×C₈	C₂×Dic₁₁	1	4	yes	—
124	C₂²×C₈	C₂×Dic₁₁	1	4	yes	—
125	C₂²×C₈	C₂²×D₁₁	1	4	yes	—
126	C₂×C₄²	Dic₂₂	1	2	yes	—
127	C₂×C₄²	D₄₄	1	2	yes	—
128	C₂×C₄²	D₄₄	1	2	yes	—
129	C₂×C₄²	C₂×Dic₁₁	1	1	yes	characteristic
130	C₂×C₄²	C₂×Dic₁₁	1	2	yes	—
131	C₂×C₄²	C₂×Dic₁₁	1	1	yes	characteristic
132	C₂×C₄²	C₂²×D₁₁	1	2	yes	—
133	C₄⋊C₈	—	22	16	no	—
134	C₄×C₈	C₂×Dic₁₁	1	8	yes	—
135	C₄×C₈	C₂×Dic₁₁	1	8	yes	—
136	C₂×C₁₁	—	2	2	no	—
137	C₂×C₁₁	—	2	2	no	—
138	C₂×C₁₁	#1837	1	4	yes	complemented
139	C₂×C₁₁	C₂×#382	1	1	yes	characteristic
140	C₂×C₁₁	C₂²×#108	1	2	yes	—
141	C₂⁴	C₂×Dic₂₂	1	1	yes	characteristic
142	C₂²×C₄	—	2	8	no	—
143	C₂²×C₄	C₄⋊Dic₁₁	1	2	yes	—
144	C₂²×C₄	C₄⋊Dic₁₁	1	4	yes	—
145	C₂²×C₄	C₄⋊Dic₁₁	1	2	yes	—
146	C₂²×C₄	C₂×Dic₂₂	1	4	yes	—
147	C₂²×C₄	C₂×Dic₂₂	1	1	yes	characteristic
148	C₂²×C₄	C₂×D₄₄	1	1	yes	characteristic
149	C₂²×C₄	C₂×D₄₄	1	1	yes	characteristic
150	C₂²×C₄	#4339	1	8	yes	—
151	C₂²×C₄	C₂²×Dic₁₁	1	1	yes	characteristic
152	C₂²×C₄	C₂²×Dic₁₁	1	1	yes	characteristic
153	C₂²×C₄	C₂³×D₁₁	1	1	yes	characteristic
154	M₁₆	—	22	16	no	—
155	M₁₆	—	44	8	no	—
156	C₂×C₈	—	2	8	no	—
157	C₂×C₈	—	2	8	no	—
158	C₂×C₈	—	2	8	no	—
159	C₂×C₈	—	22	8	no	—
160	C₂×C₈	—	44	8	no	—
161	C₂×C₈	C₂²×Dic₁₁	1	8	yes	—
162	C₄²	C₄⋊Dic₁₁	1	2	yes	—
163	C₄²	C₄⋊Dic₁₁	1	4	yes	—
164	C₄²	C₄⋊Dic₁₁	1	2	yes	—
165	C₄²	C₂×D₄₄	1	4	yes	—
166	C₄²	C₂²×Dic₁₁	1	2	yes	—
167	C₄²	C₂²×Dic₁₁	1	2	yes	—
168	C₁₁	C₂×#1837	1	1	yes	characteristic, complemented, Sylow
169	C₂³	—	2	4	no	—
170	C₂³	#66923	1	4	yes	—
171	C₂³	C₂×C₄⋊Dic₁₁	1	1	yes	characteristic
172	C₂³	C₂×C₄⋊Dic₁₁	1	1	yes	characteristic
173	C₂³	C₂²×Dic₂₂	1	1	yes	characteristic
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	—	2	8	no	—
176	C₂×C₄	—	2	8	no	—
177	C₂×C₄	#66883	1	8	yes	—
178	C₂×C₄	#66923	1	4	yes	—
179	C₂×C₄	#66928	1	4	yes	—
180	C₂×C₄	#66928	1	4	yes	—
181	C₂×C₄	C₂×C₄⋊Dic₁₁	1	4	yes	—
182	C₂×C₄	C₂×C₄⋊Dic₁₁	1	4	yes	—
183	C₂×C₄	C₂×C₄⋊Dic₁₁	1	1	yes	characteristic
184	C₂×C₄	C₂×C₄⋊Dic₁₁	1	1	yes	characteristic
185	C₂×C₄	#66979	1	4	yes	—
186	C₂×C₄	#66979	1	4	yes	—
187	C₂×C₄	C₂²×D₄₄	1	1	yes	characteristic
188	C₂×C₄	C₂×#4339	1	4	yes	—
189	C₂×C₄	C₂³×Dic₁₁	1	1	yes	characteristic
190	C₈	—	2	8	no	—
191	C₈	—	44	8	no	—
192	C₂²	—	2	8	no	—
193	C₂²	—	2	2	no	—
194	C₂²	—	2	2	no	—
195	C₂²	#10622549	1	2	yes	—
196	C₂²	#10622549	1	2	yes	—
197	C₂²	C₂×#66883	1	4	yes	—
198	C₂²	C₂×#66923	1	2	yes	—
199	C₂²	C₂²×C₄⋊Dic₁₁	1	1	yes	characteristic
200	C₄	—	2	4	no	—
201	C₄	#10622549	1	2	yes	—
202	C₄	#10622549	1	2	yes	—
203	C₄	C₂×#66883	1	8	yes	—
204	C₄	C₂×#66928	1	2	yes	—
205	C₄	C₂×#66979	1	2	yes	—
206	C₂	—	2	2	no	—
207	C₂	—	2	2	no	—
208	C₂	#13068754	1	4	yes	direct product factor
209	C₂	C₂×#10622549	1	1	yes	characteristic
210	C₂	C₂²×#66883	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₄₄
commutator	C₂×C₄×C₁₁	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₁
Fitting	C₂²×C₄×C₈×C₁₁	C₂
socle	C₂³×C₁₁	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₁	C₂×Q₈
cosocle	C₂²×C₄×C₁₁	C₂⁴
hypercenter	C₂²×C₄×C₈	D₁₁
hypocenter	C₁₁	C₂×#1837
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #13068754	C₂	direct (N × H)
138. C₂×C₁₁	#1837	semidirect
168. C₁₁	C₂×#1837	semidirect
208. C₂	#13068754	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×#10622537	2
#13068754	2
C₂×#10622190	2
C₂²×C₄×C₈×C₁₁	2
C₂×#1837	11

Subgroups — C2×#13068754 #14753999

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13068754 #14753999