Subgroups — C₂×#13067531 #14723544

Order 2816.

806 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₈₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#10622301	C₂	1	2	yes	maximal, complemented
3	#13067531	C₂	1	8	yes	maximal, direct product factor
4	C₂×#10622095	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#10622097	C₂	1	2	yes	maximal, complemented
6	C₂×C₁₁×#322	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₈₈	C₄	1	1	yes	characteristic
8	C₂²×#4313	—	2	2	no	—
9	C₂×C₄×D₄₄	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
13	C₂×#66885	C₄	1	2	yes	—
14	C₂×(C₂×D₁₁)⋊C₈	—	2	2	no	—
15	C₂×#66882	C₄	1	1	yes	characteristic
16	C₂×C₄⋊(C₁₁⋊C₈)	C₂²	1	2	yes	—
17	C₄×D₈₈	C₄	1	4	yes	—
18	C₄×D₈₈	C₂²	1	4	yes	—
19	#10622301	—	2	4	no	—
20	#10622301	C₂²	1	8	yes	complemented
21	C₁₁×#322	C₂²	1	4	yes	complemented
22	#10622097	C₂²	1	8	yes	complemented
23	#10622095	C₂²	1	4	yes	complemented
24	C₂²×D₄₄	C₂×C₄	1	2	yes	—
25	C₂²×C₄×D₁₁	—	2	2	no	—
26	C₂²×C₈×C₁₁	—	2	1	no	—
27	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₁	D₄	1	2	yes	—
30	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
31	C₂×(C₂×D₁₁)⋊C₄	—	2	2	no	—
32	C₂×C₄⋊Dic₁₁	C₂×C₄	1	2	yes	—
33	C₂²×C₁₁⋊C₈	—	2	2	no	—
34	C₂×D₈₈	—	2	4	no	—
35	C₂×D₈₈	C₂×C₄	1	4	yes	—
36	C₂×#4313	—	2	8	no	—
37	C₂×#4313	—	4	8	no	—
38	C₄×D₄₄	—	2	2	no	—
39	C₄×D₄₄	—	2	2	no	—
40	C₄×D₄₄	C₂×C₄	1	4	yes	—
41	C₄×D₄₄	C₂³	1	4	yes	—
42	C₄×C₈×C₁₁	—	2	1	no	—
43	C₄×C₈×C₁₁	C₂×C₄	1	2	yes	—
44	C₄×C₈×C₁₁	D₄	1	2	yes	—
45	C₄×C₈×C₁₁	C₂³	1	2	yes	—
46	C₄×C₈×C₁₁	C₂³	1	2	yes	—
47	C₄×C₈×C₁₁	C₂³	1	2	yes	—
48	#66885	C₂×C₄	1	8	yes	—
49	(C₂×D₁₁)⋊C₈	—	2	8	no	—
50	#66882	C₂×C₄	1	4	yes	—
51	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
52	C₄⋊(C₁₁⋊C₈)	C₂³	1	4	yes	—
53	C₂×#1594	—	11	1	no	maximal, Sylow
54	C₂³×D₁₁	—	2	2	no	—
55	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₁	#41	1	1	yes	characteristic
58	C₂²×Dic₁₁	—	2	2	no	—
59	C₂×D₄₄	—	2	2	no	—
60	C₂×D₄₄	—	2	4	no	—
61	C₂×D₄₄	—	2	4	no	—
62	C₂×D₄₄	C₂²×C₄	1	4	yes	—
63	C₂×C₄×D₁₁	—	2	4	no	—
64	C₂×C₄×D₁₁	—	2	4	no	—
65	C₂×C₄×D₁₁	—	4	4	no	—
66	C₂×C₄×D₁₁	—	4	4	no	—
67	C₂×C₈×C₁₁	—	2	2	no	—
68	C₂×C₈×C₁₁	—	2	4	no	—
69	C₂×C₈×C₁₁	—	2	2	no	—
70	C₂×C₈×C₁₁	—	4	2	no	—
71	C₂×C₈×C₁₁	M₁₆	1	4	yes	complemented
72	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
73	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
74	C₂×C₈×C₁₁	C₂×D₄	1	4	yes	—
75	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
76	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₁₁	#41	1	1	yes	characteristic
78	C₄²×C₁₁	C₂⁴	1	1	yes	characteristic
79	(C₂×D₁₁)⋊C₄	—	2	4	no	—
80	(C₂×D₁₁)⋊C₄	—	2	4	no	—
81	C₄⋊Dic₁₁	—	2	2	no	—
82	C₄⋊Dic₁₁	C₂²×C₄	1	4	yes	—
83	C₂×C₁₁⋊C₈	—	2	4	no	—
84	C₂×C₁₁⋊C₈	—	4	4	no	—
85	D₈₈	—	2	8	no	—
86	#4313	—	4	16	no	—
87	C₂×C₄×D₈	—	11	1	no	—
88	C₂×#436	—	11	2	no	—
89	#1594	—	11	8	no	—
90	C₂×#334	—	11	1	no	—
91	C₂×D₄⋊C₈	—	11	2	no	—
92	C₂×#322	D₁₁	1	1	yes	characteristic, complemented
93	C₂³×C₁₁	C₄×D₄	1	1	yes	characteristic
94	C₂²×D₁₁	—	2	4	no	—
95	C₂²×D₁₁	—	4	4	no	—
96	C₂²×D₁₁	—	4	4	no	—
97	C₂²×D₁₁	—	4	4	no	—
98	C₂×C₄×C₁₁	—	2	2	no	—
99	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
100	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
101	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
102	C₂×C₄×C₁₁	C₂×M₁₆	1	2	yes	—
103	C₂×C₄×C₁₁	#131	1	2	yes	—
104	C₂×C₄×C₁₁	AGL(1,ℤ/8)	1	4	yes	—
105	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
106	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
107	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
108	C₂×Dic₁₁	—	2	4	no	—
109	C₂×Dic₁₁	—	4	4	no	—
110	D₄₄	—	2	4	no	—
111	D₄₄	—	2	4	no	—
112	D₄₄	—	4	4	no	—
113	C₄×D₁₁	—	4	4	no	—
114	C₄×D₁₁	—	4	4	no	—
115	C₄×D₁₁	—	4	8	no	—
116	C₈×C₁₁	—	2	2	no	—
117	C₈×C₁₁	—	2	4	no	—
118	C₈×C₁₁	—	4	2	no	—
119	C₈×C₁₁	C₂×M₁₆	1	4	yes	complemented
120	C₁₁⋊C₈	—	4	4	no	—
121	C₂²×D₈	—	11	1	no	—
122	C₂²×M₁₆	—	22	2	no	—
123	C₂×C₄×D₄	—	11	2	no	—
124	C₄×D₈	—	11	4	no	—
125	C₄×D₈	—	11	4	no	—
126	#436	—	11	8	no	—
127	#436	—	22	4	no	—
128	C₂×#107	—	11	1	no	—
129	C₂×C₄⋊C₈	—	11	2	no	—
130	C₂×#102	—	11	2	no	—
131	C₂×C₂²⋊C₈	—	22	2	no	—
132	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
133	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
134	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
135	#334	—	11	4	no	—
136	D₄⋊C₈	—	11	8	no	—
137	#322	C₂×D₁₁	1	4	yes	complemented
138	C₂²×C₁₁	#436	1	2	yes	—
139	C₂²×C₁₁	#442	1	2	yes	—
140	C₂²×C₁₁	#444	1	2	yes	—
141	C₂²×C₁₁	C₂×C₄×D₄	1	1	yes	characteristic
142	C₂×D₁₁	—	4	4	no	—
143	C₂×D₁₁	—	4	4	no	—
144	C₂×D₁₁	—	4	4	no	—
145	C₂×D₁₁	—	8	8	no	—
146	C₄×C₁₁	—	2	1	no	—
147	C₄×C₁₁	—	2	1	no	—
148	C₄×C₁₁	#436	1	2	yes	—
149	C₄×C₁₁	#442	1	2	yes	—
150	C₄×C₁₁	C₂²×M₁₆	1	1	yes	characteristic
151	C₄×C₁₁	C₂×#131	1	1	yes	characteristic
152	C₄×C₁₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
153	Dic₁₁	—	4	4	no	—
154	C₂²×D₄	—	11	2	no	—
155	C₂³×C₄	—	22	2	no	—
156	C₂×D₈	—	11	4	no	—
157	C₂×D₈	—	22	4	no	—
158	C₂×M₁₆	—	22	8	no	—
159	C₂×M₁₆	—	44	8	no	—
160	C₂²×C₈	—	2	1	no	—
161	C₂²×C₈	—	22	2	no	—
162	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
163	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
164	C₂²×C₈	D₄₄	1	2	yes	—
165	C₄×D₄	—	11	4	no	—
166	C₄×D₄	—	11	4	no	—
167	C₄×D₄	—	22	2	no	—
168	C₄×D₄	—	22	2	no	—
169	C₂×C₄⋊C₄	—	11	2	no	—
170	C₂×C₂²⋊C₄	—	22	2	no	—
171	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
172	#107	—	11	4	no	—
173	C₄⋊C₈	—	11	4	no	—
174	C₄⋊C₈	—	22	2	no	—
175	#102	—	11	8	no	—
176	C₂²⋊C₈	—	22	8	no	—
177	C₄×C₈	—	2	1	no	—
178	C₄×C₈	C₄×D₁₁	1	2	yes	—
179	C₄×C₈	D₄₄	1	2	yes	—
180	C₄×C₈	C₂²×D₁₁	1	2	yes	—
181	C₄×C₈	C₂²×D₁₁	1	2	yes	—
182	C₄×C₈	C₂²×D₁₁	1	2	yes	—
183	C₂×C₁₁	#1594	1	4	yes	complemented
184	C₂×C₁₁	C₂×#436	1	1	yes	characteristic
185	C₂×C₁₁	C₂×#442	1	1	yes	characteristic
186	C₂×C₁₁	C₂×#444	1	1	yes	characteristic
187	D₁₁	—	8	4	no	—
188	C₂⁴	—	22	2	no	—
189	C₂×D₄	—	11	4	no	—
190	C₂×D₄	—	22	2	no	—
191	C₂×D₄	—	22	4	no	—
192	C₂×D₄	—	22	4	no	—
193	C₂²×C₄	—	22	2	no	—
194	C₂²×C₄	—	22	4	no	—
195	C₂²×C₄	—	22	4	no	—
196	C₂²×C₄	—	44	4	no	—
197	C₂²×C₄	—	44	4	no	—
198	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
199	C₂²×C₄	C₂×D₄₄	1	1	yes	characteristic
200	C₂²×C₄	#4339	1	1	yes	characteristic
201	D₈	—	22	8	no	—
202	M₁₆	—	44	16	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	4	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	4	2	no	—
207	C₂×C₈	—	22	4	no	—
208	C₂×C₈	—	44	4	no	—
209	C₂×C₈	#4313	1	4	yes	complemented
210	C₂×C₈	C₂×C₄×D₁₁	1	2	yes	—
211	C₂×C₈	C₂×C₄×D₁₁	1	2	yes	—
212	C₂×C₈	C₂×D₄₄	1	4	yes	—
213	C₄⋊C₄	—	11	4	no	—
214	C₄⋊C₄	—	22	2	no	—
215	C₂²⋊C₄	—	22	4	no	—
216	C₂²⋊C₄	—	22	4	no	—
217	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
218	C₄²	C₂×D₄₄	1	1	yes	characteristic
219	C₄²	#4339	1	1	yes	characteristic
220	C₄²	C₂³×D₁₁	1	1	yes	characteristic
221	C₁₁	C₂×#1594	1	1	yes	characteristic, complemented, Sylow
222	C₂³	—	22	4	no	—
223	C₂³	—	44	4	no	—
224	C₂³	—	44	4	no	—
225	C₂³	—	44	4	no	—
226	C₂³	C₄×D₄₄	1	1	yes	characteristic
227	D₄	—	22	4	no	—
228	D₄	—	22	4	no	—
229	D₄	—	44	4	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	22	4	no	—
232	C₂×C₄	—	44	4	no	—
233	C₂×C₄	—	44	4	no	—
234	C₂×C₄	—	44	4	no	—
235	C₂×C₄	—	44	8	no	—
236	C₂×C₄	C₄×D₄₄	1	1	yes	characteristic
237	C₂×C₄	C₄×D₄₄	1	1	yes	characteristic
238	C₂×C₄	C₄×D₄₄	1	1	yes	characteristic
239	C₂×C₄	C₂×#4313	1	2	yes	—
240	C₂×C₄	#66954	1	2	yes	—
241	C₂×C₄	#66961	1	4	yes	—
242	C₂×C₄	C₂²×C₄×D₁₁	1	1	yes	characteristic
243	C₂×C₄	C₂²×D₄₄	1	1	yes	characteristic
244	C₂×C₄	C₂×#4339	1	1	yes	characteristic
245	C₈	—	2	2	no	—
246	C₈	—	2	4	no	—
247	C₈	—	4	2	no	—
248	C₈	—	44	4	no	—
249	C₈	C₂×#4313	1	4	yes	complemented
250	C₂²	—	44	4	no	—
251	C₂²	—	44	4	no	—
252	C₂²	—	44	4	no	—
253	C₂²	—	88	8	no	—
254	C₂²	#10622301	1	2	yes	—
255	C₂²	#10622324	1	2	yes	—
256	C₂²	#10622330	1	2	yes	—
257	C₂²	C₂×C₄×D₄₄	1	1	yes	characteristic
258	C₄	—	2	1	no	—
259	C₄	—	2	1	no	—
260	C₄	—	44	4	no	—
261	C₄	#10622301	1	2	yes	—
262	C₄	#10622324	1	2	yes	—
263	C₄	C₂²×#4313	1	1	yes	characteristic
264	C₄	C₂×#66954	1	1	yes	characteristic
265	C₄	C₂×#66961	1	2	yes	—
266	C₂	—	88	4	no	—
267	C₂	#13067531	1	4	yes	direct product factor
268	C₂	C₂×#10622301	1	1	yes	characteristic
269	C₂	C₂×#10622324	1	1	yes	characteristic
270	C₂	C₂×#10622330	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₄₄
commutator	C₂×C₄×C₁₁	C₂³×C₄
Frattini	C₄²	C₂³×D₁₁
Fitting	C₂×C₁₁×#322	C₂
socle	C₂³×C₁₁	C₄×D₄
prime radical	C₂²×D₈₈	C₄
cosocle	C₄²×C₁₁	C₂⁴
hypercenter	C₂×#322	D₁₁
hypocenter	C₁₁	C₂×#1594
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
2. C₂×#10622301	C₂	semidirect
3. #13067531	C₂	direct (N × H)
4. C₂×#10622095	C₂	semidirect
5. C₂×#10622097	C₂	semidirect
6. C₂×C₁₁×#322	C₂	semidirect
20. #10622301	C₂²	semidirect
21. C₁₁×#322	C₂²	semidirect
22. #10622097	C₂²	semidirect
23. #10622095	C₂²	semidirect
71. C₂×C₈×C₁₁	M₁₆	semidirect
92. C₂×#322	D₁₁	semidirect
119. C₈×C₁₁	C₂×M₁₆	semidirect
137. #322	C₂×D₁₁	semidirect
183. C₂×C₁₁	#1594	semidirect
209. C₂×C₈	#4313	semidirect
221. C₁₁	C₂×#1594	semidirect
249. C₈	C₂×#4313	semidirect
267. C₂	#13067531	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×C₄×D₈₈	2
C₂×#10622301	2
#13067531	2
C₂×#10622095	2
C₂×#10622097	2
C₂×C₁₁×#322	2
C₂×#1594	11

Subgroups — C2×#13067531 #14723544

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13067531 #14723544