Subgroups — C₂²×C₄×#4313 #15617158

Order 2816.

3950 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#4313	C₂	1	2	yes	maximal
2	C₂²×C₄²×D₁₁	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×#4313	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#66879	C₂	1	2	yes	maximal, complemented
5	C₂²×C₄×C₁₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×C₈×C₁₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×C₄×D₁₁	C₄	1	2	yes	—
8	C₂³×C₄×D₁₁	C₂²	1	1	yes	characteristic
9	C₂³×C₈×C₁₁	C₂²	1	2	yes	—
10	C₂²×C₄²×C₁₁	C₂²	1	1	yes	characteristic
11	C₂²×C₄×Dic₁₁	C₄	1	2	yes	—
12	C₂²×C₄×Dic₁₁	C₂²	1	1	yes	characteristic
13	C₂³×C₁₁⋊C₈	C₂²	1	2	yes	—
14	C₂²×#4313	C₄	1	32	yes	direct product factor
15	C₂²×#4313	C₂²	1	24	yes	—
16	C₂×C₄²×D₁₁	C₄	1	12	yes	—
17	C₂×C₄²×D₁₁	C₂²	1	12	yes	—
18	C₂×C₄×C₈×C₁₁	C₂²	1	12	yes	complemented
19	C₂×#66879	C₂²	1	24	yes	complemented
20	C₂×C₄×C₁₁⋊C₈	C₂²	1	12	yes	complemented
21	C₄×#4313	C₂²	1	64	yes	direct product factor
22	C₂⁴×D₁₁	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₄×C₁₁	C₂×C₄	1	2	yes	—
24	C₂³×C₄×C₁₁	C₂³	1	1	yes	characteristic
25	C₂³×Dic₁₁	C₂×C₄	1	2	yes	—
26	C₂³×Dic₁₁	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×D₁₁	—	2	16	no	—
28	C₂²×C₄×D₁₁	C₂×C₄	1	24	yes	—
29	C₂²×C₄×D₁₁	C₂×C₄	1	6	yes	—
30	C₂²×C₄×D₁₁	C₂×C₄	1	16	yes	—
31	C₂²×C₄×D₁₁	C₂³	1	6	yes	—
32	C₂²×C₈×C₁₁	C₂×C₄	1	16	yes	complemented
33	C₂²×C₈×C₁₁	C₂³	1	12	yes	—
34	C₂×C₄²×C₁₁	C₂×C₄	1	6	yes	—
35	C₂×C₄²×C₁₁	C₂³	1	6	yes	—
36	C₂×C₄×Dic₁₁	C₂×C₄	1	6	yes	—
37	C₂×C₄×Dic₁₁	C₂×C₄	1	24	yes	—
38	C₂×C₄×Dic₁₁	C₂³	1	6	yes	—
39	C₂²×C₁₁⋊C₈	C₂×C₄	1	16	yes	complemented
40	C₂²×C₁₁⋊C₈	C₂³	1	12	yes	—
41	C₂×#4313	C₂×C₄	1	192	yes	direct product factor
42	C₂×#4313	C₂³	1	32	yes	—
43	C₄²×D₁₁	C₂×C₄	1	48	yes	—
44	C₄²×D₁₁	C₂³	1	16	yes	—
45	C₄×C₈×C₁₁	C₂³	1	16	yes	complemented
46	#66879	C₂³	1	32	yes	complemented
47	C₄×C₁₁⋊C₈	C₂³	1	16	yes	complemented
48	C₂²×C₄×M₁₆	—	11	1	no	maximal, Sylow
49	C₂⁴×C₁₁	C₂²×C₄	1	1	yes	characteristic
50	C₂³×D₁₁	—	2	8	no	—
51	C₂³×D₁₁	C₄²	1	8	yes	—
52	C₂³×D₁₁	C₂²×C₄	1	6	yes	—
53	C₂²×C₄×C₁₁	M₁₆	1	16	yes	complemented
54	C₂²×C₄×C₁₁	C₂²×C₄	1	3	yes	—
55	C₂²×C₄×C₁₁	C₂²×C₄	1	8	yes	—
56	C₂²×C₄×C₁₁	C₂²×C₄	1	12	yes	—
57	C₂²×C₄×C₁₁	C₂⁴	1	3	yes	—
58	C₂²×Dic₁₁	—	2	8	no	—
59	C₂²×Dic₁₁	C₄²	1	8	yes	—
60	C₂²×Dic₁₁	C₂²×C₄	1	12	yes	—
61	C₂²×Dic₁₁	C₂²×C₄	1	6	yes	—
62	C₂×C₄×D₁₁	—	2	96	no	—
63	C₂×C₄×D₁₁	C₄²	1	48	yes	—
64	C₂×C₄×D₁₁	C₂²×C₄	1	48	yes	—
65	C₂×C₄×D₁₁	C₂²×C₄	1	32	yes	—
66	C₂×C₄×D₁₁	C₂²×C₄	1	12	yes	—
67	C₂×C₄×D₁₁	C₂⁴	1	4	yes	—
68	C₂×C₈×C₁₁	C₂²×C₄	1	48	yes	complemented
69	C₂×C₈×C₁₁	C₂⁴	1	8	yes	—
70	C₄²×C₁₁	C₂²×C₄	1	12	yes	—
71	C₄²×C₁₁	C₂⁴	1	4	yes	—
72	C₄×Dic₁₁	C₂²×C₄	1	12	yes	—
73	C₄×Dic₁₁	C₂²×C₄	1	32	yes	—
74	C₄×Dic₁₁	C₂⁴	1	4	yes	—
75	C₂×C₁₁⋊C₈	C₂²×C₄	1	48	yes	complemented
76	C₂×C₁₁⋊C₈	C₂⁴	1	8	yes	—
77	#4313	C₂²×C₄	1	128	yes	direct product factor
78	C₂³×M₁₆	—	11	2	no	—
79	C₂³×C₄²	—	11	1	no	—
80	C₂×C₄×M₁₆	—	11	24	no	—
81	C₂²×#97	—	11	2	no	—
82	C₂²×C₄×C₈	—	11	1	no	—
83	C₂²×C₄×C₈	D₁₁	1	1	yes	characteristic, complemented
84	C₂³×C₁₁	C₂×C₄²	1	4	yes	—
85	C₂³×C₁₁	C₂×M₁₆	1	8	yes	—
86	C₂³×C₁₁	C₂³×C₄	1	3	yes	—
87	C₂²×D₁₁	—	2	24	no	—
88	C₂²×D₁₁	—	2	32	no	—
89	C₂²×D₁₁	C₂×C₄²	1	24	yes	—
90	C₂²×D₁₁	C₂³×C₄	1	4	yes	—
91	C₂×C₄×C₁₁	C₂×C₄²	1	12	yes	—
92	C₂×C₄×C₁₁	C₂×M₁₆	1	48	yes	complemented
93	C₂×C₄×C₁₁	C₂³×C₄	1	8	yes	—
94	C₂×C₄×C₁₁	C₂³×C₄	1	12	yes	—
95	C₂×C₄×C₁₁	C₂³×C₄	1	3	yes	—
96	C₂×C₄×C₁₁	C₂⁵	1	1	yes	characteristic
97	C₂×Dic₁₁	—	2	24	no	—
98	C₂×Dic₁₁	C₂×C₄²	1	24	yes	—
99	C₂×Dic₁₁	C₂³×C₄	1	8	yes	—
100	C₂×Dic₁₁	C₂³×C₄	1	4	yes	—
101	C₄×D₁₁	—	2	64	no	—
102	C₄×D₁₁	C₂×C₄²	1	48	yes	—
103	C₄×D₁₁	C₂³×C₄	1	16	yes	—
104	C₈×C₁₁	C₂³×C₄	1	16	yes	complemented
105	C₁₁⋊C₈	C₂³×C₄	1	16	yes	complemented
106	C₂⁴×C₄	—	11	1	no	—
107	C₂⁴×C₄	—	11	2	no	—
108	C₂²×M₁₆	—	11	24	no	—
109	C₂²×M₁₆	—	11	32	no	—
110	C₂³×C₈	—	11	2	no	—
111	C₂³×C₈	C₂×D₁₁	1	2	yes	—
112	C₂²×C₄²	—	11	1	no	—
113	C₂²×C₄²	—	11	2	no	—
114	C₂²×C₄²	—	11	12	no	—
115	C₂²×C₄²	—	11	12	no	—
116	C₂²×C₄²	C₂×D₁₁	1	1	yes	characteristic
117	C₄×M₁₆	—	11	64	no	—
118	C₂×#97	—	11	24	no	—
119	C₂×C₄×C₈	—	11	12	no	—
120	C₂×C₄×C₈	C₂×D₁₁	1	12	yes	complemented
121	C₂²×C₁₁	C₄×M₁₆	1	16	yes	complemented
122	C₂²×C₁₁	C₂²×C₄²	1	6	yes	—
123	C₂²×C₁₁	C₂²×M₁₆	1	12	yes	—
124	C₂²×C₁₁	C₂⁴×C₄	1	1	yes	characteristic
125	C₂×D₁₁	—	2	48	no	—
126	C₂×D₁₁	—	2	8	no	—
127	C₂×D₁₁	C₂²×C₄²	1	8	yes	—
128	C₄×C₁₁	C₂²×C₄²	1	6	yes	—
129	C₄×C₁₁	C₂²×M₁₆	1	16	yes	complemented
130	C₄×C₁₁	C₂⁴×C₄	1	2	yes	—
131	Dic₁₁	—	2	8	no	—
132	Dic₁₁	C₂²×C₄²	1	8	yes	—
133	C₂⁵	—	11	1	no	—
134	C₂³×C₄	—	11	1	no	—
135	C₂³×C₄	—	11	16	no	—
136	C₂³×C₄	—	11	6	no	—
137	C₂³×C₄	—	11	6	no	—
138	C₂³×C₄	—	11	2	no	—
139	C₂³×C₄	—	11	24	no	—
140	C₂³×C₄	—	22	16	no	—
141	C₂³×C₄	C₄×D₁₁	1	2	yes	—
142	C₂³×C₄	C₂²×D₁₁	1	1	yes	characteristic
143	C₂×M₁₆	—	11	192	no	—
144	C₂×M₁₆	—	11	32	no	—
145	C₂²×C₈	—	11	16	no	—
146	C₂²×C₈	—	11	12	no	—
147	C₂²×C₈	C₄×D₁₁	1	16	yes	complemented
148	C₂²×C₈	C₂²×D₁₁	1	12	yes	—
149	C₂×C₄²	—	11	24	no	—
150	C₂×C₄²	—	11	6	no	—
151	C₂×C₄²	—	11	6	no	—
152	C₂×C₄²	—	11	48	no	—
153	C₂×C₄²	—	11	16	no	—
154	C₂×C₄²	C₄×D₁₁	1	6	yes	—
155	C₂×C₄²	C₂²×D₁₁	1	6	yes	—
156	#97	—	11	32	no	—
157	C₄×C₈	—	11	16	no	—
158	C₄×C₈	C₂²×D₁₁	1	16	yes	complemented
159	C₂×C₁₁	C₂×C₄×M₁₆	1	12	yes	complemented
160	C₂×C₁₁	C₂³×C₄²	1	1	yes	characteristic
161	C₂×C₁₁	C₂³×M₁₆	1	2	yes	—
162	D₁₁	—	2	8	no	—
163	C₂⁴	—	11	8	no	—
164	C₂⁴	—	11	6	no	—
165	C₂⁴	—	22	8	no	—
166	C₂⁴	C₂×C₄×D₁₁	1	1	yes	characteristic
167	C₂²×C₄	—	11	8	no	—
168	C₂²×C₄	—	11	6	no	—
169	C₂²×C₄	—	11	48	no	—
170	C₂²×C₄	—	11	12	no	—
171	C₂²×C₄	—	11	48	no	—
172	C₂²×C₄	—	11	4	no	—
173	C₂²×C₄	—	11	12	no	—
174	C₂²×C₄	—	11	32	no	—
175	C₂²×C₄	—	22	8	no	—
176	C₂²×C₄	—	22	96	no	—
177	C₂²×C₄	#4313	1	16	yes	direct product factor
178	C₂²×C₄	C₂×C₄×D₁₁	1	3	yes	—
179	C₂²×C₄	C₂×C₄×D₁₁	1	8	yes	—
180	C₂²×C₄	C₂×C₄×D₁₁	1	12	yes	—
181	C₂²×C₄	C₂³×D₁₁	1	3	yes	—
182	M₁₆	—	11	128	no	—
183	C₂×C₈	—	11	48	no	—
184	C₂×C₈	—	11	8	no	—
185	C₂×C₈	C₂×C₄×D₁₁	1	48	yes	complemented
186	C₂×C₈	C₂³×D₁₁	1	8	yes	—
187	C₄²	—	11	32	no	—
188	C₄²	—	11	12	no	—
189	C₄²	—	11	4	no	—
190	C₄²	C₂×C₄×D₁₁	1	12	yes	—
191	C₄²	C₂³×D₁₁	1	4	yes	—
192	C₁₁	C₂²×C₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
193	C₂³	—	11	24	no	—
194	C₂³	—	11	4	no	—
195	C₂³	—	22	24	no	—
196	C₂³	—	22	32	no	—
197	C₂³	C₄²×D₁₁	1	4	yes	—
198	C₂³	C₂×#4313	1	8	yes	—
199	C₂³	C₂²×C₄×D₁₁	1	3	yes	—
200	C₂×C₄	—	11	24	no	—
201	C₂×C₄	—	11	4	no	—
202	C₂×C₄	—	11	48	no	—
203	C₂×C₄	—	11	16	no	—
204	C₂×C₄	—	11	8	no	—
205	C₂×C₄	—	22	24	no	—
206	C₂×C₄	—	22	64	no	—
207	C₂×C₄	C₄²×D₁₁	1	12	yes	—
208	C₂×C₄	C₂×#4313	1	48	yes	direct product factor
209	C₂×C₄	C₂²×C₄×D₁₁	1	3	yes	—
210	C₂×C₄	C₂²×C₄×D₁₁	1	12	yes	—
211	C₂×C₄	C₂²×C₄×D₁₁	1	8	yes	—
212	C₂×C₄	C₂⁴×D₁₁	1	1	yes	characteristic
213	C₈	—	11	16	no	—
214	C₈	C₂²×C₄×D₁₁	1	16	yes	complemented
215	C₂²	—	11	8	no	—
216	C₂²	—	22	48	no	—
217	C₂²	—	22	8	no	—
218	C₂²	C₄×#4313	1	16	yes	direct product factor
219	C₂²	C₂×C₄²×D₁₁	1	6	yes	—
220	C₂²	C₂²×#4313	1	12	yes	—
221	C₂²	C₂³×C₄×D₁₁	1	1	yes	characteristic
222	C₄	—	11	8	no	—
223	C₄	—	22	8	no	—
224	C₄	C₂×C₄²×D₁₁	1	6	yes	—
225	C₄	C₂²×#4313	1	16	yes	direct product factor
226	C₄	C₂³×C₄×D₁₁	1	2	yes	—
227	C₂	—	22	8	no	—
228	C₂	C₂×C₄×#4313	1	12	yes	direct product factor
229	C₂	C₂²×C₄²×D₁₁	1	1	yes	characteristic
230	C₂	C₂³×#4313	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄²	C₂×D₁₁
commutator	C₂×C₁₁	C₂³×C₄²
Frattini	C₂×C₄	C₂⁴×D₁₁
Fitting	C₂²×C₄×C₈×C₁₁	C₂
socle	C₂⁴×C₁₁	C₂²×C₄
prime radical	C₂⁴×D₁₁	C₂×C₄
cosocle	C₂×C₄×C₁₁	C₂⁵
hypercenter	C₂²×C₄×C₈	D₁₁
hypocenter	C₁₁	C₂²×C₄×M₁₆
LCS	G ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. C₂×C₄×#4313	C₂	direct (N × H)
4. C₂²×#66879	C₂	semidirect
5. C₂²×C₄×C₁₁⋊C₈	C₂	semidirect
6. C₂²×C₄×C₈×C₁₁	C₂	semidirect
14. C₂²×#4313	C₄	direct (N × H)
18. C₂×C₄×C₈×C₁₁	C₂²	semidirect
19. C₂×#66879	C₂²	semidirect
20. C₂×C₄×C₁₁⋊C₈	C₂²	semidirect
21. C₄×#4313	C₂²	direct (N × H)
32. C₂²×C₈×C₁₁	C₂×C₄	semidirect
39. C₂²×C₁₁⋊C₈	C₂×C₄	semidirect
41. C₂×#4313	C₂×C₄	direct (N × H)
45. C₄×C₈×C₁₁	C₂³	semidirect
46. #66879	C₂³	semidirect
47. C₄×C₁₁⋊C₈	C₂³	semidirect
53. C₂²×C₄×C₁₁	M₁₆	semidirect
68. C₂×C₈×C₁₁	C₂²×C₄	semidirect
75. C₂×C₁₁⋊C₈	C₂²×C₄	semidirect
77. #4313	C₂²×C₄	direct (N × H)
83. C₂²×C₄×C₈	D₁₁	semidirect
92. C₂×C₄×C₁₁	C₂×M₁₆	semidirect
104. C₈×C₁₁	C₂³×C₄	semidirect
105. C₁₁⋊C₈	C₂³×C₄	semidirect
120. C₂×C₄×C₈	C₂×D₁₁	semidirect
121. C₂²×C₁₁	C₄×M₁₆	semidirect
129. C₄×C₁₁	C₂²×M₁₆	semidirect
147. C₂²×C₈	C₄×D₁₁	semidirect
158. C₄×C₈	C₂²×D₁₁	semidirect
159. C₂×C₁₁	C₂×C₄×M₁₆	semidirect
177. C₂²×C₄	#4313	direct (N × H)
185. C₂×C₈	C₂×C₄×D₁₁	semidirect
192. C₁₁	C₂²×C₄×M₁₆	semidirect
208. C₂×C₄	C₂×#4313	direct (N × H)
214. C₈	C₂²×C₄×D₁₁	semidirect
218. C₂²	C₄×#4313	direct (N × H)
225. C₄	C₂²×#4313	direct (N × H)
228. C₂	C₂×C₄×#4313	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#4313	2
C₂²×C₄²×D₁₁	2
C₂×C₄×#4313	2
C₂²×#66879	2
C₂²×C₄×C₁₁⋊C₈	2
C₂²×C₄×C₈×C₁₁	2
C₂²×C₄×M₁₆	11

Subgroups — C22×C4×#4313 #15617158

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄×#4313 #15617158