Subgroups — C₂×#13070702 #14908191

Order 2816.

806 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₁₇₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#10622728	C₂	1	2	yes	maximal, complemented
3	#13070702	C₂	1	8	yes	maximal, direct product factor
4	C₂×#10622142	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#10622147	C₂	1	2	yes	maximal, complemented
6	C₂³×C₁₁×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×C₈×C₁₁	C₂²	1	1	yes	characteristic
8	C₂×#66983	—	2	2	no	—
9	C₂²×D₈₈	C₂²	1	2	yes	—
10	C₂²×C₁₁×C₁₆	—	2	4	no	—
11	C₂²×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
12	C₂²×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
13	C₂²×C₁₁×C₁₆	C₂²	1	4	yes	complemented
14	#10622728	—	2	4	no	—
15	#10622728	C₂²	1	8	yes	complemented
16	C₂×#66885	—	2	2	no	—
17	C₂×#66882	C₂²	1	2	yes	—
18	C₂×D₁₇₆	—	2	4	no	—
19	C₂×D₁₇₆	C₂²	1	4	yes	complemented
20	#10622147	C₂²	1	8	yes	complemented
21	#10622142	C₂²	1	4	yes	complemented
22	C₂³×C₄×C₁₁	D₄	1	1	yes	characteristic
23	C₂²×C₂²⋊D₁₁	—	4	2	no	—
24	C₂²×D₄₄	—	2	2	no	—
25	C₂²×C₈×C₁₁	—	2	2	no	—
26	C₂²×C₈×C₁₁	—	2	2	no	—
27	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₁	D₄	1	2	yes	—
29	C₂²×C₈×C₁₁	C₂³	1	2	yes	—
30	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
31	#66983	—	2	8	no	—
32	#66983	—	4	4	no	—
33	C₂×(C₂×D₁₁)⋊C₄	—	4	2	no	—
34	C₂×C₄⋊Dic₁₁	—	2	2	no	—
35	C₂×D₈₈	—	2	4	no	—
36	C₂×D₈₈	—	2	4	no	—
37	C₂×D₈₈	—	2	2	no	—
38	C₂×D₈₈	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	complemented
42	C₂×C₁₁×C₁₆	C₂³	1	2	yes	—
43	C₂×C₁₁×C₁₆	C₂³	1	2	yes	—
44	#66885	—	2	8	no	—
45	#66882	—	2	2	no	—
46	#66882	C₂³	1	4	yes	—
47	D₁₇₆	—	2	8	no	—
48	C₂×#2215	—	11	1	no	maximal, Sylow
49	C₂⁴×C₁₁	D₈	1	1	yes	characteristic
50	C₂³×D₁₁	—	4	2	no	—
51	C₂²×C₄×C₁₁	—	2	2	no	—
52	C₂²×C₄×C₁₁	—	2	2	no	—
53	C₂²×C₄×C₁₁	D₈	1	1	yes	characteristic
54	C₂²×C₄×C₁₁	D₈	1	2	yes	—
55	C₂²×C₄×C₁₁	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
57	C₂×C₂²⋊D₁₁	—	4	8	no	—
58	C₂×C₂²⋊D₁₁	—	8	8	no	—
59	C₂²×Dic₁₁	—	4	2	no	—
60	C₂×D₄₄	—	2	4	no	—
61	C₂×D₄₄	—	4	4	no	—
62	C₂×D₄₄	—	4	2	no	—
63	C₂×D₄₄	—	4	4	no	—
64	C₂×C₈×C₁₁	—	2	4	no	—
65	C₂×C₈×C₁₁	—	2	2	no	—
66	C₂×C₈×C₁₁	—	2	2	no	—
67	C₂×C₈×C₁₁	—	2	2	no	—
68	C₂×C₈×C₁₁	C₂×D₄	1	2	yes	—
69	C₂×C₈×C₁₁	C₂×D₄	1	2	yes	—
70	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₁₁	#41	1	2	yes	—
72	C₂×C₈×C₁₁	C₂⁴	1	1	yes	characteristic
73	(C₂×D₁₁)⋊C₄	—	4	8	no	—
74	C₄⋊Dic₁₁	—	2	4	no	—
75	C₄⋊Dic₁₁	—	4	2	no	—
76	D₈₈	—	2	4	no	—
77	D₈₈	—	2	4	no	—
78	D₈₈	—	4	4	no	—
79	C₁₁×C₁₆	—	2	2	no	—
80	C₁₁×C₁₆	C₂×D₄	1	4	yes	complemented
81	C₂²×D₁₆	—	11	1	no	—
82	C₂³×C₁₆	D₁₁	1	1	yes	characteristic, complemented
83	C₂×#466	—	11	2	no	—
84	#2215	—	11	8	no	—
85	C₂×#366	—	11	1	no	—
86	C₂×#357	—	11	2	no	—
87	C₂³×C₁₁	—	2	2	no	—
88	C₂³×C₁₁	—	2	2	no	—
89	C₂³×C₁₁	D₁₆	1	4	yes	complemented
90	C₂³×C₁₁	C₂×D₈	1	2	yes	—
91	C₂³×C₁₁	C₂×D₈	1	1	yes	characteristic
92	C₂²×D₁₁	—	4	4	no	—
93	C₂²×D₁₁	—	8	4	no	—
94	C₂²×D₁₁	—	8	4	no	—
95	C₂²×D₁₁	—	8	4	no	—
96	C₂×C₄×C₁₁	—	2	2	no	—
97	C₂×C₄×C₁₁	—	2	2	no	—
98	C₂×C₄×C₁₁	—	2	2	no	—
99	C₂×C₄×C₁₁	—	2	4	no	—
100	C₂×C₄×C₁₁	#121	1	2	yes	—
101	C₂×C₄×C₁₁	C₂×D₈	1	2	yes	—
102	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
103	C₂×C₄×C₁₁	#135	1	2	yes	—
104	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
105	C₂²⋊D₁₁	—	8	16	no	—
106	C₂×Dic₁₁	—	4	4	no	—
107	C₂×Dic₁₁	—	8	4	no	—
108	D₄₄	—	4	4	no	—
109	D₄₄	—	4	4	no	—
110	D₄₄	—	8	4	no	—
111	C₈×C₁₁	—	2	2	no	—
112	C₈×C₁₁	#121	1	2	yes	—
113	C₈×C₁₁	C₂²×D₄	1	1	yes	characteristic
114	C₈×C₁₁	C₂×#41	1	1	yes	characteristic
115	C₂²×D₈	—	11	2	no	—
116	C₂³×C₈	C₂×D₁₁	1	1	yes	characteristic
117	C₂×#121	—	22	2	no	—
118	C₂×D₁₆	—	11	4	no	—
119	C₂×D₁₆	—	22	4	no	—
120	C₂²×C₁₆	—	2	4	no	—
121	C₂²×C₁₆	C₂×D₁₁	1	1	yes	characteristic
122	C₂²×C₁₆	C₂×D₁₁	1	1	yes	characteristic
123	C₂²×C₁₆	C₂×D₁₁	1	4	yes	complemented
124	#466	—	11	8	no	—
125	#466	—	22	4	no	—
126	C₂×#107	—	11	2	no	—
127	C₂×#102	—	22	2	no	—
128	#366	—	11	4	no	—
129	#357	—	11	8	no	—
130	C₂²×C₁₁	—	2	2	no	—
131	C₂²×C₁₁	—	2	8	no	—
132	C₂²×C₁₁	—	2	2	no	—
133	C₂²×C₁₁	#466	1	2	yes	—
134	C₂²×C₁₁	C₂×D₁₆	1	2	yes	—
135	C₂²×C₁₁	C₂×D₁₆	1	4	yes	complemented
136	C₂²×C₁₁	#508	1	2	yes	—
137	C₂²×C₁₁	C₂²×D₈	1	1	yes	characteristic
138	C₂×D₁₁	—	8	4	no	—
139	C₂×D₁₁	—	8	4	no	—
140	C₂×D₁₁	—	8	4	no	—
141	C₂×D₁₁	—	16	8	no	—
142	C₄×C₁₁	—	2	2	no	—
143	C₄×C₁₁	#466	1	2	yes	—
144	C₄×C₁₁	C₂×#121	1	1	yes	characteristic
145	C₄×C₁₁	C₂×#135	1	1	yes	characteristic
146	Dic₁₁	—	8	4	no	—
147	C₂²×D₄	—	22	2	no	—
148	C₂²×D₄	—	44	2	no	—
149	C₂³×C₄	D₄₄	1	1	yes	characteristic
150	C₂×D₈	—	11	4	no	—
151	C₂×D₈	—	22	2	no	—
152	C₂×D₈	—	22	4	no	—
153	C₂×D₈	—	22	4	no	—
154	C₂²×C₈	—	2	2	no	—
155	C₂²×C₈	—	2	2	no	—
156	C₂²×C₈	D₄₄	1	1	yes	characteristic
157	C₂²×C₈	D₄₄	1	2	yes	—
158	C₂²×C₈	C₂²×D₁₁	1	2	yes	—
159	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
160	#121	—	22	8	no	—
161	#121	—	44	4	no	—
162	C₂×C₄⋊C₄	—	22	2	no	—
163	C₂×C₂²⋊C₄	—	44	2	no	—
164	D₁₆	—	22	8	no	—
165	C₂×C₁₆	—	2	2	no	—
166	C₂×C₁₆	—	2	8	no	—
167	C₂×C₁₆	C₂²⋊D₁₁	1	4	yes	complemented
168	C₂×C₁₆	C₂²×D₁₁	1	2	yes	—
169	C₂×C₁₆	C₂²×D₁₁	1	2	yes	—
170	#107	—	11	4	no	—
171	#107	—	22	2	no	—
172	#102	—	22	8	no	—
173	C₂×C₁₁	—	2	4	no	—
174	C₂×C₁₁	#2215	1	4	yes	complemented
175	C₂×C₁₁	C₂×#466	1	1	yes	characteristic
176	C₂×C₁₁	C₂²×D₁₆	1	1	yes	characteristic
177	C₂×C₁₁	C₂×#508	1	1	yes	characteristic
178	D₁₁	—	16	4	no	—
179	C₂⁴	—	44	2	no	—
180	C₂⁴	D₈₈	1	1	yes	characteristic
181	C₂×D₄	—	22	4	no	—
182	C₂×D₄	—	44	8	no	—
183	C₂×D₄	—	44	2	no	—
184	C₂×D₄	—	44	4	no	—
185	C₂×D₄	—	44	4	no	—
186	C₂×D₄	—	88	8	no	—
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	44	2	no	—
190	C₂²×C₄	D₈₈	1	2	yes	—
191	C₂²×C₄	D₈₈	1	1	yes	characteristic
192	C₂²×C₄	C₂×D₄₄	1	2	yes	—
193	C₂²×C₄	C₂×D₄₄	1	1	yes	characteristic
194	D₈	—	22	4	no	—
195	D₈	—	22	4	no	—
196	D₈	—	44	4	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	2	4	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	C₂×D₄₄	1	2	yes	—
202	C₂×C₈	C₂×D₄₄	1	1	yes	characteristic
203	C₂×C₈	#4339	1	2	yes	—
204	C₂×C₈	C₂×C₂²⋊D₁₁	1	2	yes	—
205	C₂×C₈	C₂³×D₁₁	1	1	yes	characteristic
206	C₄⋊C₄	—	22	4	no	—
207	C₄⋊C₄	—	44	2	no	—
208	C₂²⋊C₄	—	44	8	no	—
209	C₁₆	—	2	2	no	—
210	C₁₆	C₂×C₂²⋊D₁₁	1	4	yes	complemented
211	C₁₁	C₂×#2215	1	1	yes	characteristic, complemented, Sylow
212	C₂³	—	2	2	no	—
213	C₂³	—	2	2	no	—
214	C₂³	—	44	4	no	—
215	C₂³	—	88	4	no	—
216	C₂³	—	88	4	no	—
217	C₂³	—	88	4	no	—
218	C₂³	D₁₇₆	1	4	yes	complemented
219	C₂³	C₂×D₈₈	1	2	yes	—
220	C₂³	C₂×D₈₈	1	1	yes	characteristic
221	D₄	—	44	4	no	—
222	D₄	—	44	4	no	—
223	D₄	—	88	16	no	—
224	D₄	—	88	4	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	44	4	no	—
230	C₂×C₄	—	88	4	no	—
231	C₂×C₄	C₂×D₈₈	1	2	yes	—
232	C₂×C₄	C₂×D₈₈	1	1	yes	characteristic
233	C₂×C₄	#66957	1	2	yes	—
234	C₂×C₄	#66983	1	2	yes	—
235	C₂×C₄	C₂²×D₄₄	1	1	yes	characteristic
236	C₈	—	2	2	no	—
237	C₈	#66983	1	2	yes	—
238	C₈	C₂×#4339	1	1	yes	characteristic
239	C₈	C₂²×C₂²⋊D₁₁	1	1	yes	characteristic
240	C₂²	—	2	8	no	—
241	C₂²	—	2	2	no	—
242	C₂²	—	2	2	no	—
243	C₂²	—	88	4	no	—
244	C₂²	—	88	4	no	—
245	C₂²	—	88	4	no	—
246	C₂²	—	176	8	no	—
247	C₂²	C₂×D₁₇₆	1	2	yes	—
248	C₂²	C₂×D₁₇₆	1	4	yes	complemented
249	C₂²	#10622517	1	2	yes	—
250	C₂²	#10622728	1	2	yes	—
251	C₂²	C₂²×D₈₈	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	88	4	no	—
254	C₄	#10622728	1	2	yes	—
255	C₄	C₂×#66957	1	1	yes	characteristic
256	C₄	C₂×#66983	1	1	yes	characteristic
257	C₂	—	2	4	no	—
258	C₂	—	176	4	no	—
259	C₂	#13070702	1	4	yes	direct product factor
260	C₂	C₂×#10622728	1	1	yes	characteristic
261	C₂	C₂²×D₁₇₆	1	1	yes	characteristic
262	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
hypercenter	C₂³×C₁₆	D₁₁
hypocenter	C₁₁	C₂×#2215
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×D₁₇₆	C₂	semidirect
2. C₂×#10622728	C₂	semidirect
3. #13070702	C₂	direct (N × H)
4. C₂×#10622142	C₂	semidirect
5. C₂×#10622147	C₂	semidirect
6. C₂³×C₁₁×C₁₆	C₂	semidirect
13. C₂²×C₁₁×C₁₆	C₂²	semidirect
15. #10622728	C₂²	semidirect
19. C₂×D₁₇₆	C₂²	semidirect
20. #10622147	C₂²	semidirect
21. #10622142	C₂²	semidirect
41. C₂×C₁₁×C₁₆	D₄	semidirect
80. C₁₁×C₁₆	C₂×D₄	semidirect
82. C₂³×C₁₆	D₁₁	semidirect
89. C₂³×C₁₁	D₁₆	semidirect
123. C₂²×C₁₆	C₂×D₁₁	semidirect
135. C₂²×C₁₁	C₂×D₁₆	semidirect
167. C₂×C₁₆	C₂²⋊D₁₁	semidirect
174. C₂×C₁₁	#2215	semidirect
210. C₁₆	C₂×C₂²⋊D₁₁	semidirect
211. C₁₁	C₂×#2215	semidirect
218. C₂³	D₁₇₆	semidirect
248. C₂²	C₂×D₁₇₆	semidirect
259. C₂	#13070702	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×D₁₇₆	2
C₂×#10622728	2
#13070702	2
C₂×#10622142	2
C₂×#10622147	2
C₂³×C₁₁×C₁₆	2
C₂×#2215	11

Subgroups — C2×#13070702 #14908191

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13070702 #14908191