Subgroups — C₂×#13067374 #14722774

Order 2816.

854 conjugacy classes of subgroups.

The subgroup lattice has 298 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	1	yes	maximal, characteristic
2	C₂²×C₄×C₁₁⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#13067374	C₂	1	8	yes	maximal, direct product factor
4	C₂×#10622117	C₂	1	4	yes	maximal, complemented
5	C₂×C₁₁×#423	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₁×M₁₆	—	2	1	no	—
7	C₂²×C₄²×C₁₁	C₂²	1	1	yes	characteristic
8	C₂²×#4318	—	2	1	no	—
9	C₂³×C₁₁⋊C₈	C₄	1	2	yes	—
10	C₁₁×#423	—	2	2	no	—
11	C₁₁×#423	C₂²	1	4	yes	—
12	C₂×C₁₁×C₄⋊C₈	—	2	1	no	—
13	C₂×C₁₁×C₄⋊C₈	C₂²	1	2	yes	—
14	#10622537	—	2	2	no	—
15	#10622537	C₂²	1	4	yes	—
16	C₂×C₄⋊(C₁₁⋊C₈)	—	2	1	no	—
17	C₂×C₄⋊(C₁₁⋊C₈)	C₂²	1	2	yes	—
18	C₂×C₄×C₁₁⋊C₈	C₄	1	1	yes	characteristic
19	C₂×C₄×C₁₁⋊C₈	C₄	1	4	yes	—
20	C₂×C₄×C₁₁⋊C₈	C₄	1	1	yes	characteristic
21	C₂×C₄×C₁₁⋊C₈	C₂²	1	2	yes	—
22	C₂×C₄×C₁₁⋊C₈	C₂²	1	4	yes	—
23	#10622117	C₂²	1	16	yes	complemented
24	C₂³×C₄×C₁₁	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₁₁	D₄	1	1	yes	characteristic
26	C₂³×C₄×C₁₁	Q₈	1	1	yes	characteristic
27	C₂×C₁₁×M₁₆	—	2	4	no	—
28	C₂×C₁₁×M₁₆	—	4	4	no	—
29	C₂²×C₈×C₁₁	—	2	2	no	—
30	C₂×C₄²×C₁₁	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₁₁	D₄	1	2	yes	—
34	C₂×C₄²×C₁₁	Q₈	1	1	yes	characteristic
35	C₂×C₄²×C₁₁	Q₈	1	2	yes	—
36	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
37	C₂×C₄²×C₁₁	C₂³	1	2	yes	—
38	C₂×#4318	—	2	4	no	—
39	C₂×#4318	—	4	4	no	—
40	C₂²×C₁₁⋊C₈	—	2	8	no	—
41	C₂²×C₁₁⋊C₈	—	2	2	no	—
42	C₂²×C₁₁⋊C₈	C₂×C₄	1	8	yes	—
43	C₂²×C₁₁⋊C₈	C₂×C₄	1	2	yes	—
44	C₂²×C₁₁⋊C₈	C₂×C₄	1	2	yes	—
45	C₁₁×C₄⋊C₈	—	2	4	no	—
46	C₁₁×C₄⋊C₈	—	2	2	no	—
47	C₁₁×C₄⋊C₈	C₂³	1	4	yes	—
48	C₄⋊(C₁₁⋊C₈)	—	2	4	no	—
49	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
50	C₄⋊(C₁₁⋊C₈)	C₂³	1	4	yes	—
51	C₄×C₁₁⋊C₈	C₂×C₄	1	4	yes	—
52	C₄×C₁₁⋊C₈	C₂×C₄	1	4	yes	—
53	C₄×C₁₁⋊C₈	C₂×C₄	1	4	yes	—
54	C₄×C₁₁⋊C₈	C₂³	1	4	yes	—
55	C₂×#1567	—	11	1	no	maximal, Sylow
56	C₂⁴×C₁₁	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₁	—	2	4	no	—
58	C₂²×C₄×C₁₁	—	2	4	no	—
59	C₂²×C₄×C₁₁	C₄⋊C₄	1	2	yes	—
60	C₂²×C₄×C₁₁	C₄⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₁	C₄⋊C₄	1	2	yes	—
62	C₂²×C₄×C₁₁	C₄⋊C₄	1	2	yes	—
63	C₂²×C₄×C₁₁	C₄⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₁	C₄⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₁	D₈	1	4	yes	—
66	C₂²×C₄×C₁₁	Q₁₆	1	4	yes	—
67	C₂²×C₄×C₁₁	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₁	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₁	C₂×Q₈	1	1	yes	characteristic
72	C₂²×C₄×C₁₁	C₂×Q₈	1	2	yes	—
73	C₁₁×M₁₆	—	4	8	no	—
74	C₂×C₈×C₁₁	—	2	4	no	—
75	C₂×C₈×C₁₁	—	4	4	no	—
76	C₄²×C₁₁	C₄⋊C₄	1	2	yes	—
77	C₄²×C₁₁	C₄⋊C₄	1	2	yes	—
78	C₄²×C₁₁	C₄⋊C₄	1	2	yes	—
79	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
80	C₄²×C₁₁	C₂²×C₄	1	2	yes	—
81	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₁₁	C₂×D₄	1	2	yes	—
83	C₄²×C₁₁	C₂×Q₈	1	2	yes	—
84	C₄²×C₁₁	C₂×Q₈	1	1	yes	characteristic
85	C₄²×C₁₁	C₂⁴	1	1	yes	characteristic
86	#4318	—	4	8	no	—
87	C₂×C₁₁⋊C₈	—	2	16	no	—
88	C₂×C₁₁⋊C₈	—	2	4	no	—
89	C₂×C₁₁⋊C₈	—	2	4	no	—
90	C₂×C₁₁⋊C₈	—	4	4	no	—
91	C₂×C₁₁⋊C₈	M₁₆	1	8	yes	complemented
92	C₂×C₁₁⋊C₈	C₂²×C₄	1	4	yes	—
93	C₂×C₁₁⋊C₈	C₂²×C₄	1	4	yes	—
94	C₂×#423	—	11	1	no	—
95	C₂×#423	D₁₁	1	1	yes	characteristic
96	C₂²×C₄×C₈	—	11	1	no	—
97	#1567	—	11	8	no	—
98	C₂×#335	—	11	4	no	—
99	C₂³×C₁₁	—	2	2	no	—
100	C₂³×C₁₁	—	2	2	no	—
101	C₂³×C₁₁	#107	1	4	yes	—
102	C₂³×C₁₁	C₂×C₄⋊C₄	1	2	yes	—
103	C₂³×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₁₁	—	2	2	no	—
105	C₂×C₄×C₁₁	—	2	2	no	—
106	C₂×C₄×C₁₁	—	2	4	no	—
107	C₂×C₄×C₁₁	—	2	4	no	—
108	C₂×C₄×C₁₁	—	2	4	no	—
109	C₂×C₄×C₁₁	—	2	4	no	—
110	C₂×C₄×C₁₁	#107	1	8	yes	—
111	C₂×C₄×C₁₁	#107	1	4	yes	—
112	C₂×C₄×C₁₁	#108	1	4	yes	—
113	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	2	yes	—
115	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	2	yes	—
117	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	2	yes	—
119	C₂×C₄×C₁₁	C₂×M₁₆	1	4	yes	—
120	C₂×C₄×C₁₁	C₂×D₈	1	2	yes	—
121	C₂×C₄×C₁₁	C₂×D₈	1	4	yes	—
122	C₂×C₄×C₁₁	C₂×Q₁₆	1	4	yes	—
123	C₂×C₄×C₁₁	C₂×Q₁₆	1	2	yes	—
124	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
125	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₁₁	C₂²×Q₈	1	1	yes	characteristic
127	C₈×C₁₁	—	4	4	no	—
128	C₁₁⋊C₈	—	2	4	no	—
129	C₁₁⋊C₈	—	4	4	no	—
130	C₁₁⋊C₈	C₂×M₁₆	1	8	yes	complemented
131	C₂²×M₁₆	—	2	1	no	—
132	C₂²×M₁₆	—	22	1	no	—
133	C₂³×C₈	—	11	2	no	—
134	C₂²×C₄²	C₂×D₁₁	1	1	yes	characteristic
135	#423	—	2	2	no	—
136	#423	—	11	4	no	—
137	#423	—	22	2	no	—
138	#423	C₂×D₁₁	1	4	yes	—
139	C₂×C₄⋊C₈	—	2	1	no	—
140	C₂×C₄⋊C₈	—	11	2	no	—
141	C₂×C₄⋊C₈	—	22	1	no	—
142	C₂×C₄⋊C₈	C₂×D₁₁	1	2	yes	—
143	C₂×C₄×C₈	—	11	2	no	—
144	C₂×C₄×C₈	—	11	4	no	—
145	C₂×C₄×C₈	—	11	1	no	—
146	C₂×C₄×C₈	—	11	4	no	—
147	C₂×C₄×C₈	—	11	1	no	—
148	#335	—	11	16	no	—
149	C₂²×C₁₁	—	2	8	no	—
150	C₂²×C₁₁	—	2	2	no	—
151	C₂²×C₁₁	—	2	2	no	—
152	C₂²×C₁₁	#423	1	2	yes	—
153	C₂²×C₁₁	C₂×#107	1	4	yes	—
154	C₂²×C₁₁	C₂×#107	1	2	yes	—
155	C₂²×C₁₁	C₂×#108	1	2	yes	—
156	C₂²×C₁₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
157	C₄×C₁₁	—	2	2	no	—
158	C₄×C₁₁	—	2	1	no	—
159	C₄×C₁₁	—	2	1	no	—
160	C₄×C₁₁	#423	1	2	yes	—
161	C₄×C₁₁	C₂×#107	1	2	yes	—
162	C₄×C₁₁	C₂×#107	1	4	yes	—
163	C₄×C₁₁	C₂×#108	1	4	yes	—
164	C₄×C₁₁	C₂²×M₁₆	1	2	yes	—
165	C₄×C₁₁	C₂²×D₈	1	1	yes	characteristic
166	C₄×C₁₁	C₂²×Q₁₆	1	1	yes	characteristic
167	C₂³×C₄	Dic₂₂	1	1	yes	characteristic
168	C₂³×C₄	C₄×D₁₁	1	1	yes	characteristic
169	C₂³×C₄	C₂²⋊D₁₁	1	1	yes	characteristic
170	C₂×M₁₆	—	2	4	no	—
171	C₂×M₁₆	—	4	4	no	—
172	C₂×M₁₆	—	22	4	no	—
173	C₂×M₁₆	—	44	4	no	—
174	C₂²×C₈	—	2	2	no	—
175	C₂²×C₈	—	11	2	no	—
176	C₂²×C₈	—	11	8	no	—
177	C₂²×C₈	—	11	2	no	—
178	C₂²×C₈	—	22	8	no	—
179	C₂²×C₈	—	22	2	no	—
180	C₂×C₄²	Dic₂₂	1	1	yes	characteristic
181	C₂×C₄²	Dic₂₂	1	2	yes	—
182	C₂×C₄²	C₄×D₁₁	1	2	yes	—
183	C₂×C₄²	C₄×D₁₁	1	1	yes	characteristic
184	C₂×C₄²	C₂²⋊D₁₁	1	2	yes	—
185	C₂×C₄²	C₂²⋊D₁₁	1	1	yes	characteristic
186	C₂×C₄²	C₂²×D₁₁	1	2	yes	—
187	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
188	C₄⋊C₈	—	2	4	no	—
189	C₄⋊C₈	—	2	2	no	—
190	C₄⋊C₈	—	11	4	no	—
191	C₄⋊C₈	—	22	2	no	—
192	C₄⋊C₈	—	22	4	no	—
193	C₄⋊C₈	C₂²×D₁₁	1	4	yes	—
194	C₄×C₈	—	11	4	no	—
195	C₄×C₈	—	11	4	no	—
196	C₄×C₈	—	11	4	no	—
197	C₄×C₈	—	11	4	no	—
198	C₂×C₁₁	—	2	4	no	—
199	C₂×C₁₁	#1567	1	4	yes	complemented
200	C₂×C₁₁	C₂×#423	1	1	yes	characteristic
201	C₂×C₁₁	C₂²×#107	1	1	yes	characteristic
202	C₂×C₁₁	C₂²×#108	1	1	yes	characteristic
203	C₂⁴	Dic₁₁⋊C₄	1	1	yes	characteristic
204	C₂²×C₄	—	2	4	no	—
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	Dic₁₁⋊C₄	1	1	yes	characteristic
207	C₂²×C₄	Dic₁₁⋊C₄	1	2	yes	—
208	C₂²×C₄	Dic₁₁⋊C₄	1	2	yes	—
209	C₂²×C₄	Dic₁₁⋊C₄	1	1	yes	characteristic
210	C₂²×C₄	Dic₁₁⋊C₄	1	2	yes	—
211	C₂²×C₄	Dic₁₁⋊C₄	1	1	yes	characteristic
212	C₂²×C₄	#4323	1	4	yes	—
213	C₂²×C₄	#4326	1	4	yes	—
214	C₂²×C₄	C₂×Dic₂₂	1	1	yes	characteristic
215	C₂²×C₄	C₂×Dic₂₂	1	2	yes	—
216	C₂²×C₄	C₂×C₄×D₁₁	1	2	yes	—
217	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
218	C₂²×C₄	C₂×C₂²⋊D₁₁	1	2	yes	—
219	C₂²×C₄	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
220	M₁₆	—	4	8	no	—
221	M₁₆	—	44	8	no	—
222	C₂×C₈	—	2	4	no	—
223	C₂×C₈	—	4	4	no	—
224	C₂×C₈	—	11	8	no	—
225	C₂×C₈	—	11	4	no	—
226	C₂×C₈	—	11	4	no	—
227	C₂×C₈	—	22	16	no	—
228	C₂×C₈	—	22	4	no	—
229	C₂×C₈	—	22	4	no	—
230	C₂×C₈	—	44	4	no	—
231	C₄²	Dic₁₁⋊C₄	1	2	yes	—
232	C₄²	Dic₁₁⋊C₄	1	2	yes	—
233	C₄²	Dic₁₁⋊C₄	1	2	yes	—
234	C₄²	C₂×Dic₂₂	1	2	yes	—
235	C₄²	C₂×Dic₂₂	1	1	yes	characteristic
236	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
237	C₄²	C₂×C₄×D₁₁	1	2	yes	—
238	C₄²	C₂×C₂²⋊D₁₁	1	2	yes	—
239	C₄²	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
240	C₄²	C₂³×D₁₁	1	1	yes	characteristic
241	C₁₁	C₂×#1567	1	1	yes	characteristic, complemented, Sylow
242	C₂³	—	2	2	no	—
243	C₂³	—	2	2	no	—
244	C₂³	#66871	1	4	yes	—
245	C₂³	C₂×Dic₁₁⋊C₄	1	2	yes	—
246	C₂³	C₂×Dic₁₁⋊C₄	1	1	yes	characteristic
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	#66871	1	8	yes	—
254	C₂×C₄	#66871	1	4	yes	—
255	C₂×C₄	#66886	1	4	yes	—
256	C₂×C₄	M₁₆×D₁₁	1	4	yes	—
257	C₂×C₄	C₂×Dic₁₁⋊C₄	1	2	yes	—
258	C₂×C₄	C₂×Dic₁₁⋊C₄	1	1	yes	characteristic
259	C₂×C₄	C₂×Dic₁₁⋊C₄	1	1	yes	characteristic
260	C₂×C₄	C₂×Dic₁₁⋊C₄	1	2	yes	—
261	C₂×C₄	C₂×Dic₁₁⋊C₄	1	2	yes	—
262	C₂×C₄	C₂×Dic₁₁⋊C₄	1	1	yes	characteristic
263	C₂×C₄	C₂×#4323	1	2	yes	—
264	C₂×C₄	C₂×#4323	1	4	yes	—
265	C₂×C₄	C₂×#4326	1	4	yes	—
266	C₂×C₄	C₂×#4326	1	2	yes	—
267	C₂×C₄	C₂²×Dic₂₂	1	1	yes	characteristic
268	C₂×C₄	C₂²×C₄×D₁₁	1	1	yes	characteristic
269	C₂×C₄	C₂²×C₂²⋊D₁₁	1	1	yes	characteristic
270	C₈	—	4	4	no	—
271	C₈	—	11	8	no	—
272	C₈	—	22	4	no	—
273	C₈	—	44	4	no	—
274	C₂²	—	2	8	no	—
275	C₂²	—	2	2	no	—
276	C₂²	—	2	2	no	—
277	C₂²	C₂×#66871	1	4	yes	—
278	C₂²	C₂×#66871	1	2	yes	—
279	C₂²	#10622731	1	2	yes	—
280	C₂²	C₂×#66886	1	2	yes	—
281	C₂²	C₂²×Dic₁₁⋊C₄	1	1	yes	characteristic
282	C₄	—	2	2	no	—
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	C₂×#66871	1	4	yes	—
286	C₄	C₂×#66871	1	2	yes	—
287	C₄	#10622731	1	2	yes	—
288	C₄	C₂×#66886	1	4	yes	—
289	C₄	C₂×M₁₆×D₁₁	1	2	yes	—
290	C₄	C₂²×#4323	1	1	yes	characteristic
291	C₄	C₂²×#4326	1	1	yes	characteristic
292	C₂	—	2	4	no	—
293	C₂	#13067374	1	4	yes	direct product factor
294	C₂	C₂×#10622731	1	1	yes	characteristic
295	C₂	C₂²×#66871	1	1	yes	characteristic
296	C₂	C₂²×#66886	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₁₁
commutator	C₂×C₄×C₁₁	C₂³×C₄
Frattini	C₄²	C₂³×D₁₁
Fitting	C₂×C₁₁×#423	C₂
socle	C₂³×C₁₁	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₁	C₄⋊C₄
cosocle	C₄²×C₁₁	C₂⁴
hypercenter	C₂×#423	D₁₁
hypocenter	C₁₁	C₂×#1567
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
3. #13067374	C₂	direct (N × H)
4. C₂×#10622117	C₂	semidirect
23. #10622117	C₂²	semidirect
91. C₂×C₁₁⋊C₈	M₁₆	semidirect
130. C₁₁⋊C₈	C₂×M₁₆	semidirect
199. C₂×C₁₁	#1567	semidirect
241. C₁₁	C₂×#1567	semidirect
293. C₂	#13067374	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
C₂²×C₄×C₁₁⋊C₈	2
#13067374	2
C₂×#10622117	2
C₂×C₁₁×#423	2
C₂×#1567	11

Subgroups — C2×#13067374 #14722774

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13067374 #14722774