Subgroups — C₂×#13070707 #14908197

Order 2816.

702 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₈₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#10622729	C₂	1	2	yes	maximal
3	#13070707	C₂	1	8	yes	maximal, direct product factor
4	C₂×#10622142	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#10622141	C₂	1	2	yes	maximal, complemented
6	C₂³×C₁₁×C₁₆	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₈×C₁₁	C₂²	1	1	yes	characteristic
8	C₂×#66977	—	2	2	no	—
9	C₂²×Dic₄₄	C₂²	1	2	yes	—
10	C₂²×C₁₁×C₁₆	—	2	4	no	—
11	C₂²×C₁₁×C₁₆	C₂²	1	4	yes	—
12	C₂²×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
13	C₂²×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
14	#10622729	—	2	4	no	—
15	#10622729	C₂²	1	8	yes	—
16	C₂×#66882	C₂²	1	2	yes	—
17	C₂×#66880	—	2	2	no	—
18	C₂×Dic₈₈	—	2	4	no	—
19	C₂×Dic₈₈	C₂²	1	4	yes	complemented
20	#10622142	C₂²	1	4	yes	complemented
21	#10622141	C₂²	1	8	yes	complemented
22	C₂³×C₄×C₁₁	D₄	1	1	yes	characteristic
23	C₂²×Dic₂₂	—	2	2	no	—
24	C₂²×C₈×C₁₁	—	2	2	no	—
25	C₂²×C₈×C₁₁	—	2	2	no	—
26	C₂²×C₈×C₁₁	D₄	1	2	yes	—
27	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₁	C₂³	1	2	yes	—
29	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
30	C₂×C₂²⋊Dic₁₁	—	4	2	no	—
31	C₂×C₄⋊Dic₁₁	—	2	2	no	—
32	#66977	—	2	8	no	—
33	#66977	—	4	4	no	—
34	C₂×Dic₁₁⋊C₄	—	4	2	no	—
35	C₂×Dic₄₄	—	2	4	no	—
36	C₂×Dic₄₄	—	2	4	no	—
37	C₂×Dic₄₄	—	2	2	no	—
38	C₂×Dic₄₄	C₂³	1	4	yes	—
39	C₂×C₁₁×C₁₆	—	2	2	no	—
40	C₂×C₁₁×C₁₆	—	2	8	no	—
41	C₂×C₁₁×C₁₆	D₄	1	4	yes	—
42	C₂×C₁₁×C₁₆	C₂³	1	2	yes	—
43	C₂×C₁₁×C₁₆	C₂³	1	2	yes	—
44	#66882	—	2	2	no	—
45	#66882	C₂³	1	4	yes	—
46	#66880	—	2	8	no	—
47	Dic₈₈	—	2	8	no	—
48	C₂×#2216	—	11	1	no	maximal, Sylow
49	C₂⁴×C₁₁	D₈	1	1	yes	characteristic
50	C₂²×C₄×C₁₁	—	2	2	no	—
51	C₂²×C₄×C₁₁	—	2	2	no	—
52	C₂²×C₄×C₁₁	D₈	1	1	yes	characteristic
53	C₂²×C₄×C₁₁	D₈	1	2	yes	—
54	C₂²×C₄×C₁₁	C₂×D₄	1	2	yes	—
55	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
56	C₂²×Dic₁₁	—	4	2	no	—
57	C₂²×Dic₁₁	—	4	2	no	—
58	C₂×Dic₂₂	—	2	4	no	—
59	C₂×Dic₂₂	—	4	4	no	—
60	C₂×Dic₂₂	—	4	2	no	—
61	C₂×Dic₂₂	—	4	4	no	—
62	C₂×C₈×C₁₁	—	2	4	no	—
63	C₂×C₈×C₁₁	—	2	2	no	—
64	C₂×C₈×C₁₁	—	2	2	no	—
65	C₂×C₈×C₁₁	—	2	2	no	—
66	C₂×C₈×C₁₁	C₂×D₄	1	2	yes	—
67	C₂×C₈×C₁₁	C₂×D₄	1	2	yes	—
68	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₁₁	#41	1	2	yes	—
70	C₂×C₈×C₁₁	C₂⁴	1	1	yes	characteristic
71	C₂²⋊Dic₁₁	—	4	8	no	—
72	C₄⋊Dic₁₁	—	2	4	no	—
73	C₄⋊Dic₁₁	—	4	2	no	—
74	Dic₁₁⋊C₄	—	4	8	no	—
75	Dic₄₄	—	2	4	no	—
76	Dic₄₄	—	2	4	no	—
77	Dic₄₄	—	4	4	no	—
78	C₁₁×C₁₆	—	2	2	no	—
79	C₁₁×C₁₆	C₂×D₄	1	4	yes	—
80	C₂²×Q₃₂	—	11	1	no	—
81	C₂³×C₁₆	D₁₁	1	1	yes	characteristic
82	C₂×#467	—	11	2	no	—
83	#2216	—	11	8	no	—
84	C₂×#366	—	11	1	no	—
85	C₂×#358	—	11	2	no	—
86	C₂³×C₁₁	—	2	2	no	—
87	C₂³×C₁₁	—	2	2	no	—
88	C₂³×C₁₁	Q₃₂	1	4	yes	complemented
89	C₂³×C₁₁	C₂×D₈	1	2	yes	—
90	C₂³×C₁₁	C₂×D₈	1	1	yes	characteristic
91	C₂×C₄×C₁₁	—	2	2	no	—
92	C₂×C₄×C₁₁	—	2	2	no	—
93	C₂×C₄×C₁₁	—	2	2	no	—
94	C₂×C₄×C₁₁	—	2	4	no	—
95	C₂×C₄×C₁₁	#121	1	2	yes	—
96	C₂×C₄×C₁₁	C₂×D₈	1	2	yes	—
97	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
98	C₂×C₄×C₁₁	#135	1	2	yes	—
99	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
100	C₂×Dic₁₁	—	4	4	no	—
101	C₂×Dic₁₁	—	4	4	no	—
102	C₂×Dic₁₁	—	8	4	no	—
103	C₂×Dic₁₁	—	8	4	no	—
104	Dic₂₂	—	4	4	no	—
105	Dic₂₂	—	4	4	no	—
106	Dic₂₂	—	8	4	no	—
107	C₈×C₁₁	—	2	2	no	—
108	C₈×C₁₁	#121	1	2	yes	—
109	C₈×C₁₁	C₂²×D₄	1	1	yes	characteristic
110	C₈×C₁₁	C₂×#41	1	1	yes	characteristic
111	C₂²×Q₁₆	—	11	2	no	—
112	C₂³×C₈	C₂×D₁₁	1	1	yes	characteristic
113	C₂×C₂²⋊Q₈	—	22	2	no	—
114	C₂×Q₃₂	—	11	4	no	—
115	C₂×Q₃₂	—	22	4	no	—
116	C₂²×C₁₆	—	2	4	no	—
117	C₂²×C₁₆	C₂×D₁₁	1	1	yes	characteristic
118	C₂²×C₁₆	C₂×D₁₁	1	4	yes	—
119	C₂²×C₁₆	C₂×D₁₁	1	1	yes	characteristic
120	#467	—	11	8	no	—
121	#467	—	22	4	no	—
122	C₂×#107	—	11	2	no	—
123	C₂×#103	—	22	2	no	—
124	#366	—	11	4	no	—
125	#358	—	11	8	no	—
126	C₂²×C₁₁	—	2	2	no	—
127	C₂²×C₁₁	—	2	8	no	—
128	C₂²×C₁₁	—	2	2	no	—
129	C₂²×C₁₁	#466	1	2	yes	—
130	C₂²×C₁₁	C₂×Q₃₂	1	2	yes	—
131	C₂²×C₁₁	C₂×Q₃₂	1	4	yes	complemented
132	C₂²×C₁₁	#508	1	2	yes	—
133	C₂²×C₁₁	C₂²×D₈	1	1	yes	characteristic
134	C₄×C₁₁	—	2	2	no	—
135	C₄×C₁₁	#466	1	2	yes	—
136	C₄×C₁₁	C₂×#121	1	1	yes	characteristic
137	C₄×C₁₁	C₂×#135	1	1	yes	characteristic
138	Dic₁₁	—	8	4	no	—
139	Dic₁₁	—	8	4	no	—
140	C₂²×Q₈	—	22	2	no	—
141	C₂³×C₄	D₄₄	1	1	yes	characteristic
142	C₂×Q₁₆	—	11	4	no	—
143	C₂×Q₁₆	—	22	2	no	—
144	C₂×Q₁₆	—	22	4	no	—
145	C₂×Q₁₆	—	22	4	no	—
146	C₂²×C₈	—	2	2	no	—
147	C₂²×C₈	—	2	2	no	—
148	C₂²×C₈	D₄₄	1	1	yes	characteristic
149	C₂²×C₈	D₄₄	1	2	yes	—
150	C₂²×C₈	C₂²×D₁₁	1	2	yes	—
151	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
152	C₂²⋊Q₈	—	22	8	no	—
153	C₂²⋊Q₈	—	44	4	no	—
154	C₂×C₄⋊C₄	—	22	2	no	—
155	C₂×C₄⋊C₄	—	44	2	no	—
156	C₂×C₂²⋊C₄	—	44	2	no	—
157	Q₃₂	—	22	8	no	—
158	C₂×C₁₆	—	2	2	no	—
159	C₂×C₁₆	—	2	8	no	—
160	C₂×C₁₆	C₂²⋊D₁₁	1	4	yes	—
161	C₂×C₁₆	C₂²×D₁₁	1	2	yes	—
162	C₂×C₁₆	C₂²×D₁₁	1	2	yes	—
163	#107	—	11	4	no	—
164	#107	—	22	2	no	—
165	#103	—	22	8	no	—
166	C₂×C₁₁	—	2	4	no	—
167	C₂×C₁₁	#2216	1	4	yes	complemented
168	C₂×C₁₁	C₂×#466	1	1	yes	characteristic
169	C₂×C₁₁	C₂²×Q₃₂	1	1	yes	characteristic
170	C₂×C₁₁	C₂×#508	1	1	yes	characteristic
171	C₂⁴	D₈₈	1	1	yes	characteristic
172	C₂×Q₈	—	22	4	no	—
173	C₂×Q₈	—	44	2	no	—
174	C₂×Q₈	—	44	4	no	—
175	C₂×Q₈	—	44	4	no	—
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	—	2	2	no	—
178	C₂²×C₄	—	44	2	no	—
179	C₂²×C₄	—	44	2	no	—
180	C₂²×C₄	D₈₈	1	2	yes	—
181	C₂²×C₄	D₈₈	1	1	yes	characteristic
182	C₂²×C₄	C₂×D₄₄	1	2	yes	—
183	C₂²×C₄	C₂×D₄₄	1	1	yes	characteristic
184	Q₁₆	—	22	4	no	—
185	Q₁₆	—	22	4	no	—
186	Q₁₆	—	44	4	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	2	4	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	C₂×D₄₄	1	2	yes	—
192	C₂×C₈	C₂×D₄₄	1	1	yes	characteristic
193	C₂×C₈	#4339	1	2	yes	—
194	C₂×C₈	C₂×C₂²⋊D₁₁	1	2	yes	—
195	C₂×C₈	C₂³×D₁₁	1	1	yes	characteristic
196	C₄⋊C₄	—	22	4	no	—
197	C₄⋊C₄	—	44	2	no	—
198	C₄⋊C₄	—	44	8	no	—
199	C₂²⋊C₄	—	44	8	no	—
200	C₁₆	—	2	2	no	—
201	C₁₆	C₂×C₂²⋊D₁₁	1	4	yes	—
202	C₁₁	C₂×#2216	1	1	yes	characteristic, complemented, Sylow
203	C₂³	—	2	2	no	—
204	C₂³	—	2	2	no	—
205	C₂³	Dic₈₈	1	4	yes	complemented
206	C₂³	C₂×D₈₈	1	2	yes	—
207	C₂³	C₂×D₈₈	1	1	yes	characteristic
208	Q₈	—	44	4	no	—
209	Q₈	—	44	4	no	—
210	Q₈	—	88	4	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	4	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	44	4	no	—
216	C₂×C₄	—	44	4	no	—
217	C₂×C₄	—	88	4	no	—
218	C₂×C₄	—	88	4	no	—
219	C₂×C₄	C₂×D₈₈	1	2	yes	—
220	C₂×C₄	C₂×D₈₈	1	1	yes	characteristic
221	C₂×C₄	#66957	1	2	yes	—
222	C₂×C₄	#66983	1	2	yes	—
223	C₂×C₄	C₂²×D₄₄	1	1	yes	characteristic
224	C₈	—	2	2	no	—
225	C₈	#66983	1	2	yes	—
226	C₈	C₂×#4339	1	1	yes	characteristic
227	C₈	C₂²×C₂²⋊D₁₁	1	1	yes	characteristic
228	C₂²	—	2	8	no	—
229	C₂²	—	2	2	no	—
230	C₂²	—	2	2	no	—
231	C₂²	#10622517	1	2	yes	—
232	C₂²	C₂×Dic₈₈	1	2	yes	—
233	C₂²	C₂×Dic₈₈	1	4	yes	complemented
234	C₂²	#10622728	1	2	yes	—
235	C₂²	C₂²×D₈₈	1	1	yes	characteristic
236	C₄	—	2	2	no	—
237	C₄	—	88	4	no	—
238	C₄	—	88	4	no	—
239	C₄	#10622728	1	2	yes	—
240	C₄	C₂×#66957	1	1	yes	characteristic
241	C₄	C₂×#66983	1	1	yes	characteristic
242	C₂	—	2	4	no	—
243	C₂	#13070707	1	4	yes	direct product factor
244	C₂	C₂×#10622728	1	1	yes	characteristic
245	C₂	C₂²×Dic₈₈	1	1	yes	characteristic
246	C₂	C₂×#10622517	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₈₈
commutator, cosocle	C₂×C₈×C₁₁	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₁
Fitting	C₂³×C₁₁×C₁₆	C₂
socle	C₂³×C₁₁	C₂×D₈
prime radical	C₂⁴×C₁₁	D₈
hypercenter	C₂³×C₁₆	D₁₁
hypocenter	C₁₁	C₂×#2216
LCS	G ▷ C₂×C₈×C₁₁ ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂²×Dic₈₈	C₂	semidirect
3. #13070707	C₂	direct (N × H)
4. C₂×#10622142	C₂	semidirect
5. C₂×#10622141	C₂	semidirect
19. C₂×Dic₈₈	C₂²	semidirect
20. #10622142	C₂²	semidirect
21. #10622141	C₂²	semidirect
88. C₂³×C₁₁	Q₃₂	semidirect
131. C₂²×C₁₁	C₂×Q₃₂	semidirect
167. C₂×C₁₁	#2216	semidirect
202. C₁₁	C₂×#2216	semidirect
205. C₂³	Dic₈₈	semidirect
233. C₂²	C₂×Dic₈₈	semidirect
243. C₂	#13070707	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×Dic₈₈	2
C₂×#10622729	2
#13070707	2
C₂×#10622142	2
C₂×#10622141	2
C₂³×C₁₁×C₁₆	2
C₂×#2216	11

Subgroups — C2×#13070707 #14908197

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13070707 #14908197