Subgroups — #14757327

Order 2816.

874 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13077206	C₂	1	2	yes	maximal
2	C₂×#10622719	C₂	1	1	yes	maximal, characteristic
3	C₂×#10622714	C₂	1	1	yes	maximal, characteristic
4	#13069402	C₂	1	4	yes	maximal, complemented
5	#13069397	C₂	1	4	yes	maximal, complemented
6	#13068354	C₂	1	2	yes	maximal
7	C₂²×C₁₁×#108	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₁₁×M₁₆	C₂²	1	2	yes	—
9	C₂³×C₈×C₁₁	C₂²	1	1	yes	characteristic
10	C₂×#66979	C₂²	1	1	yes	characteristic
11	C₂×#66979	C₂²	1	2	yes	—
12	C₂²×#4318	C₂²	1	1	yes	characteristic
13	C₂×C₁₁×#108	C₄	1	8	yes	complemented
14	C₂×C₁₁×#108	C₂²	1	2	yes	—
15	C₂×C₁₁×#108	C₂²	1	2	yes	—
16	#10622734	—	2	4	no	—
17	#10622734	C₂²	1	8	yes	—
18	#10622719	—	2	1	no	—
19	#10622719	—	2	1	no	—
20	#10622719	—	2	1	no	—
21	#10622719	—	2	1	no	—
22	#10622719	—	2	4	no	—
23	C₂×#66882	C₂²	1	2	yes	—
24	C₂×#66881	C₂²	1	2	yes	—
25	#10622714	—	2	1	no	—
26	#10622714	—	2	1	no	—
27	#10622714	C₂²	1	2	yes	—
28	#10622714	C₂²	1	2	yes	—
29	C₂×#66879	—	2	1	no	—
30	C₂×C₈×Dic₁₁	—	2	1	no	—
31	#10622192	C₂²	1	8	yes	—
32	C₂³×C₄×C₁₁	C₂³	1	1	yes	characteristic
33	C₂×C₁₁×M₁₆	—	2	2	no	—
34	C₂×C₁₁×M₁₆	—	2	4	no	—
35	C₂×C₁₁×M₁₆	C₂×C₄	1	8	yes	—
36	C₂×C₁₁×M₁₆	C₂³	1	4	yes	—
37	C₂²×C₈×C₁₁	—	2	2	no	—
38	C₂²×C₈×C₁₁	C₂×C₄	1	4	yes	—
39	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₁₁	Q₈	1	4	yes	—
44	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
45	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
46	C₂×C₂²⋊Dic₁₁	—	2	1	no	—
47	C₂×C₂²⋊Dic₁₁	—	2	2	no	—
48	#66979	—	2	1	no	—
49	#66979	—	2	1	no	—
50	#66979	—	2	2	no	—
51	#66979	—	2	2	no	—
52	#66979	—	2	2	no	—
53	#66979	—	2	2	no	—
54	#66979	D₄	1	2	yes	—
55	#66979	C₂³	1	2	yes	—
56	C₂×C₄⋊Dic₁₁	D₄	1	2	yes	—
57	C₂×C₄⋊Dic₁₁	C₂³	1	4	yes	—
58	C₂×C₄×Dic₁₁	—	2	1	no	—
59	C₂×C₄×Dic₁₁	—	2	2	no	—
60	C₂×#4318	—	2	1	no	—
61	C₂×#4318	—	2	2	no	—
62	C₂×#4318	Q₈	1	4	yes	—
63	C₂×#4318	C₂³	1	2	yes	—
64	C₂²×C₁₁⋊C₈	—	2	1	no	—
65	C₁₁×#108	C₂×C₄	1	8	yes	complemented
66	C₁₁×#108	C₂×C₄	1	8	yes	complemented
67	#66882	—	2	2	no	—
68	#66882	—	2	2	no	—
69	#66882	—	2	8	no	—
70	#66881	—	2	2	no	—
71	#66881	—	2	2	no	—
72	#66881	—	2	8	no	—
73	#66879	—	2	2	no	—
74	#66879	—	2	2	no	—
75	C₈×Dic₁₁	—	2	2	no	—
76	C₈×Dic₁₁	—	2	2	no	—
77	#21181	—	11	1	no	maximal, Sylow
78	C₂⁴×C₁₁	#41	1	1	yes	characteristic
79	C₂²×C₄×C₁₁	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₁₁	C₂×Q₈	1	2	yes	—
85	C₂²×C₄×C₁₁	C₂×Q₈	1	2	yes	—
86	C₂²×C₄×C₁₁	#41	1	2	yes	—
87	C₂²×C₄×C₁₁	#41	1	1	yes	characteristic
88	C₂²×C₄×C₁₁	C₂⁴	1	1	yes	characteristic
89	C₂²×Dic₁₁	—	2	2	no	—
90	C₂²×Dic₁₁	—	2	4	no	—
91	C₁₁×M₁₆	—	2	8	no	—
92	C₁₁×M₁₆	—	2	4	no	—
93	C₁₁×M₁₆	C₂²×C₄	1	8	yes	—
94	C₂×C₈×C₁₁	—	2	8	no	—
95	C₂×C₈×C₁₁	—	2	4	no	—
96	C₂×C₈×C₁₁	C₄⋊C₄	1	4	yes	—
97	C₂×C₈×C₁₁	C₄⋊C₄	1	4	yes	—
98	C₂×C₈×C₁₁	C₄⋊C₄	1	4	yes	—
99	C₂×C₈×C₁₁	C₄⋊C₄	1	4	yes	—
100	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
101	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
102	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₁₁	C₂×Q₈	1	2	yes	—
107	C₂×C₈×C₁₁	C₂×Q₈	1	2	yes	—
108	C₂²⋊Dic₁₁	—	2	2	no	—
109	C₂²⋊Dic₁₁	—	2	2	no	—
110	C₂²⋊Dic₁₁	—	4	2	no	—
111	C₂²⋊Dic₁₁	—	4	2	no	—
112	C₄⋊Dic₁₁	—	2	2	no	—
113	C₄⋊Dic₁₁	—	2	4	no	—
114	C₄⋊Dic₁₁	—	2	4	no	—
115	C₄⋊Dic₁₁	C₂×D₄	1	4	yes	—
116	C₄×Dic₁₁	—	2	2	no	—
117	C₄×Dic₁₁	—	2	2	no	—
118	C₄×Dic₁₁	—	4	2	no	—
119	C₄×Dic₁₁	—	4	2	no	—
120	#4318	—	2	4	no	—
121	#4318	—	2	2	no	—
122	#4318	C₂×Q₈	1	4	yes	—
123	C₂×C₁₁⋊C₈	—	2	4	no	—
124	C₂×C₁₁⋊C₈	—	2	2	no	—
125	C₂²×#108	D₁₁	1	1	yes	characteristic
126	#2911	—	11	2	no	—
127	C₂×#427	—	11	1	no	—
128	C₂×#405	—	11	1	no	—
129	#1952	—	11	4	no	—
130	#1951	—	11	4	no	—
131	#1742	—	11	2	no	—
132	C₂³×C₁₁	—	4	2	no	—
133	C₂³×C₁₁	C₄×Q₈	1	2	yes	—
134	C₂³×C₁₁	#121	1	1	yes	characteristic
135	C₂³×C₁₁	#121	1	1	yes	characteristic
136	C₂³×C₁₁	#125	1	2	yes	—
137	C₂³×C₁₁	C₂×#41	1	1	yes	characteristic
138	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	2	yes	—
139	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	2	yes	—
140	C₂×C₄×C₁₁	C₄×D₄	1	2	yes	—
141	C₂×C₄×C₁₁	C₄×D₄	1	2	yes	—
142	C₂×C₄×C₁₁	C₄×Q₈	1	2	yes	—
143	C₂×C₄×C₁₁	#121	1	1	yes	characteristic
144	C₂×C₄×C₁₁	#121	1	1	yes	characteristic
145	C₂×C₄×C₁₁	C₂²⋊Q₈	1	2	yes	—
146	C₂×C₄×C₁₁	C₂²⋊Q₈	1	2	yes	—
147	C₂×C₄×C₁₁	#125	1	2	yes	—
148	C₂×C₄×C₁₁	C₄⋊Q₈	1	2	yes	—
149	C₂×C₄×C₁₁	C₄⋊Q₈	1	2	yes	—
150	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
151	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
152	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
153	C₂×C₄×C₁₁	C₂²×Q₈	1	1	yes	characteristic
154	C₂×C₄×C₁₁	C₂²×Q₈	1	1	yes	characteristic
155	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
156	C₂×Dic₁₁	—	2	4	no	—
157	C₂×Dic₁₁	—	4	4	no	—
158	C₂×Dic₁₁	—	4	4	no	—
159	C₂×Dic₁₁	—	8	4	no	—
160	C₈×C₁₁	—	2	8	no	—
161	C₈×C₁₁	C₂×C₄⋊C₄	1	2	yes	—
162	C₈×C₁₁	C₂×C₄⋊C₄	1	2	yes	—
163	C₈×C₁₁	C₂×C₄⋊C₄	1	2	yes	—
164	C₈×C₁₁	C₂×C₄⋊C₄	1	2	yes	—
165	C₁₁⋊C₈	—	2	4	no	—
166	C₂²×M₁₆	—	11	1	no	—
167	C₂²×M₁₆	C₂×D₁₁	1	2	yes	—
168	C₂³×C₈	C₂×D₁₁	1	1	yes	characteristic
169	C₂×#117	—	11	1	no	—
170	C₂×#117	—	11	2	no	—
171	C₂×#108	Dic₁₁	1	8	yes	complemented
172	C₂×#108	C₂×D₁₁	1	2	yes	—
173	C₂×#108	C₂×D₁₁	1	2	yes	—
174	#428	—	11	8	no	—
175	#428	—	22	4	no	—
176	#427	—	22	4	no	—
177	#427	—	22	1	no	—
178	#427	—	22	1	no	—
179	#427	—	22	1	no	—
180	#427	—	22	1	no	—
181	C₂×#107	—	11	2	no	—
182	C₂×#106	—	11	2	no	—
183	#405	—	11	2	no	—
184	#405	—	11	2	no	—
185	#405	—	22	1	no	—
186	#405	—	22	1	no	—
187	C₂×#97	—	22	1	no	—
188	C₂×C₄×C₈	—	22	1	no	—
189	#343	—	11	8	no	—
190	C₂²×C₁₁	—	2	2	no	—
191	C₂²×C₁₁	—	2	2	no	—
192	C₂²×C₁₁	—	2	2	no	—
193	C₂²×C₁₁	—	4	2	no	—
194	C₂²×C₁₁	—	4	2	no	—
195	C₂²×C₁₁	#389	1	2	yes	—
196	C₂²×C₁₁	#389	1	2	yes	—
197	C₂²×C₁₁	C₂×C₄×Q₈	1	1	yes	characteristic
198	C₂²×C₁₁	C₂×#121	1	1	yes	characteristic
199	C₂²×C₁₁	C₂×#125	1	1	yes	characteristic
200	C₄×C₁₁	#389	1	2	yes	—
201	C₄×C₁₁	#389	1	2	yes	—
202	C₄×C₁₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
203	C₄×C₁₁	C₂×C₄×D₄	1	1	yes	characteristic
204	C₄×C₁₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
205	C₄×C₁₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
206	Dic₁₁	—	4	4	no	—
207	Dic₁₁	—	8	4	no	—
208	C₂³×C₄	C₂²×D₁₁	1	1	yes	characteristic
209	C₂×M₁₆	—	2	2	no	—
210	C₂×M₁₆	—	2	4	no	—
211	C₂×M₁₆	—	11	4	no	—
212	C₂×M₁₆	—	11	2	no	—
213	C₂×M₁₆	—	22	1	no	—
214	C₂×M₁₆	—	22	2	no	—
215	C₂×M₁₆	C₂×Dic₁₁	1	8	yes	—
216	C₂×M₁₆	C₂²×D₁₁	1	4	yes	—
217	C₂²×C₈	—	2	2	no	—
218	C₂²×C₈	—	22	1	no	—
219	C₂²×C₈	Dic₂₂	1	4	yes	—
220	C₂²×C₈	D₄₄	1	1	yes	characteristic
221	C₂²×C₈	D₄₄	1	1	yes	characteristic
222	C₂²×C₈	D₄₄	1	1	yes	characteristic
223	C₂²×C₈	D₄₄	1	1	yes	characteristic
224	C₂²×C₈	C₂×Dic₁₁	1	4	yes	—
225	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
226	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
227	#117	—	11	2	no	—
228	#117	—	11	2	no	—
229	#117	—	22	1	no	—
230	#117	—	22	1	no	—
231	#117	—	22	2	no	—
232	#117	—	22	2	no	—
233	#117	—	22	2	no	—
234	#117	—	22	2	no	—
235	C₂×C₄⋊C₄	—	11	2	no	—
236	C₂×C₄⋊C₄	—	11	4	no	—
237	C₂×C₂²⋊C₄	—	22	1	no	—
238	C₂×C₂²⋊C₄	—	22	2	no	—
239	C₂×C₄²	—	22	1	no	—
240	C₂×C₄²	—	22	2	no	—
241	#108	C₂×Dic₁₁	1	8	yes	complemented
242	#108	C₂×Dic₁₁	1	8	yes	complemented
243	#107	—	22	8	no	—
244	#107	—	22	2	no	—
245	#107	—	22	2	no	—
246	#106	—	22	8	no	—
247	#106	—	22	2	no	—
248	#106	—	22	2	no	—
249	#97	—	22	2	no	—
250	#97	—	22	2	no	—
251	C₄×C₈	—	22	2	no	—
252	C₄×C₈	—	22	2	no	—
253	C₂×C₁₁	—	2	1	no	—
254	C₂×C₁₁	—	2	1	no	—
255	C₂×C₁₁	—	2	2	no	—
256	C₂×C₁₁	—	2	2	no	—
257	C₂×C₁₁	C₂×#389	1	1	yes	characteristic
258	C₂×C₁₁	#3067	1	2	yes	—
259	C₂⁴	#4343	1	1	yes	characteristic
260	C₂²×C₄	—	22	2	no	—
261	C₂²×C₄	—	22	4	no	—
262	C₂²×C₄	C₂×Dic₂₂	1	2	yes	—
263	C₂²×C₄	C₂×D₄₄	1	1	yes	characteristic
264	C₂²×C₄	C₂×D₄₄	1	1	yes	characteristic
265	C₂²×C₄	D₄×D₁₁	1	1	yes	characteristic
266	C₂²×C₄	D₄×D₁₁	1	1	yes	characteristic
267	C₂²×C₄	#4341	1	2	yes	—
268	C₂²×C₄	Q₈×D₁₁	1	2	yes	—
269	C₂²×C₄	#4343	1	1	yes	characteristic
270	C₂²×C₄	C₂²×Dic₁₁	1	2	yes	—
271	C₂²×C₄	C₂³×D₁₁	1	1	yes	characteristic
272	M₁₆	—	2	8	no	—
273	M₁₆	—	2	4	no	—
274	M₁₆	—	11	4	no	—
275	M₁₆	—	22	4	no	—
276	M₁₆	—	22	2	no	—
277	M₁₆	C₂²×Dic₁₁	1	8	yes	—
278	C₂×C₈	—	2	8	no	—
279	C₂×C₈	—	2	4	no	—
280	C₂×C₈	—	22	4	no	—
281	C₂×C₈	—	22	2	no	—
282	C₂×C₈	C₄⋊Dic₁₁	1	4	yes	—
283	C₂×C₈	C₄⋊Dic₁₁	1	4	yes	—
284	C₂×C₈	C₄⋊Dic₁₁	1	4	yes	—
285	C₂×C₈	C₄⋊Dic₁₁	1	4	yes	—
286	C₂×C₈	C₂×Dic₂₂	1	2	yes	—
287	C₂×C₈	C₂×Dic₂₂	1	2	yes	—
288	C₂×C₈	C₂×D₄₄	1	1	yes	characteristic
289	C₂×C₈	C₂×D₄₄	1	1	yes	characteristic
290	C₂×C₈	C₂×D₄₄	1	1	yes	characteristic
291	C₂×C₈	C₂×D₄₄	1	1	yes	characteristic
292	C₂×C₈	C₂²×Dic₁₁	1	2	yes	—
293	C₂×C₈	C₂²×Dic₁₁	1	2	yes	—
294	C₄⋊C₄	—	11	4	no	—
295	C₄⋊C₄	—	22	2	no	—
296	C₄⋊C₄	—	22	4	no	—
297	C₄⋊C₄	—	22	4	no	—
298	C₂²⋊C₄	—	22	2	no	—
299	C₂²⋊C₄	—	22	2	no	—
300	C₂²⋊C₄	—	44	2	no	—
301	C₂²⋊C₄	—	44	2	no	—
302	C₄²	—	22	2	no	—
303	C₄²	—	22	2	no	—
304	C₄²	—	44	2	no	—
305	C₄²	—	44	2	no	—
306	C₁₁	#21181	1	1	yes	characteristic, complemented, Sylow
307	C₂³	—	4	2	no	—
308	C₂³	#66943	1	2	yes	—
309	C₂³	#66948	1	1	yes	characteristic
310	C₂³	#66948	1	1	yes	characteristic
311	C₂³	Q₈×Dic₁₁	1	2	yes	—
312	C₂³	C₂×#4343	1	1	yes	characteristic
313	C₂×C₄	—	22	4	no	—
314	C₂×C₄	—	44	4	no	—
315	C₂×C₄	—	44	4	no	—
316	C₂×C₄	—	88	4	no	—
317	C₂×C₄	#66941	1	2	yes	—
318	C₂×C₄	#66941	1	2	yes	—
319	C₂×C₄	#66943	1	2	yes	—
320	C₂×C₄	#66948	1	1	yes	characteristic
321	C₂×C₄	#66948	1	1	yes	characteristic
322	C₂×C₄	#66950	1	2	yes	—
323	C₂×C₄	#66950	1	2	yes	—
324	C₂×C₄	C₂×C₄⋊Dic₁₁	1	2	yes	—
325	C₂×C₄	C₂×C₄⋊Dic₁₁	1	2	yes	—
326	C₂×C₄	D₄×Dic₁₁	1	2	yes	—
327	C₂×C₄	D₄×Dic₁₁	1	2	yes	—
328	C₂×C₄	Q₈×Dic₁₁	1	2	yes	—
329	C₂×C₄	C₂²×Dic₂₂	1	1	yes	characteristic
330	C₂×C₄	C₂²×D₄₄	1	1	yes	characteristic
331	C₂×C₄	C₂×D₄×D₁₁	1	1	yes	characteristic
332	C₂×C₄	C₂×#4341	1	1	yes	characteristic
333	C₂×C₄	C₂×Q₈×D₁₁	1	1	yes	characteristic
334	C₂×C₄	C₂³×Dic₁₁	1	1	yes	characteristic
335	C₈	—	2	8	no	—
336	C₈	—	22	4	no	—
337	C₈	C₂×C₄⋊Dic₁₁	1	2	yes	—
338	C₈	C₂×C₄⋊Dic₁₁	1	2	yes	—
339	C₈	C₂×C₄⋊Dic₁₁	1	2	yes	—
340	C₈	C₂×C₄⋊Dic₁₁	1	2	yes	—
341	C₂²	—	2	2	no	—
342	C₂²	—	2	2	no	—
343	C₂²	—	2	2	no	—
344	C₂²	—	4	2	no	—
345	C₂²	—	4	2	no	—
346	C₂²	#10622598	1	2	yes	—
347	C₂²	#10622598	1	2	yes	—
348	C₂²	C₂×#66943	1	1	yes	characteristic
349	C₂²	C₂×#66948	1	1	yes	characteristic
350	C₂²	C₂×Q₈×Dic₁₁	1	1	yes	characteristic
351	C₄	—	44	4	no	—
352	C₄	—	88	4	no	—
353	C₄	#10622598	1	2	yes	—
354	C₄	#10622598	1	2	yes	—
355	C₄	C₂×#66941	1	1	yes	characteristic
356	C₄	C₂×#66950	1	1	yes	characteristic
357	C₄	C₂²×C₄⋊Dic₁₁	1	1	yes	characteristic
358	C₄	C₂×D₄×Dic₁₁	1	1	yes	characteristic
359	C₂	—	2	1	no	—
360	C₂	—	2	1	no	—
361	C₂	—	2	2	no	—
362	C₂	—	2	2	no	—
363	C₂	C₂×#10622598	1	1	yes	characteristic
364	C₂	#13078337	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#66948
commutator	C₂×C₄×C₁₁	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₁
Fitting	C₂²×C₁₁×#108	C₂
socle	C₂²×C₁₁	C₂×#121
prime radical	C₂⁴×C₁₁	#41
cosocle	C₂²×C₄×C₁₁	C₂⁴
hypercenter	C₂²×#108	D₁₁
hypocenter	C₁₁	#21181
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#108

Complements

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

N	H ≅ G/N	type
4. #13069402	C₂	semidirect
5. #13069397	C₂	semidirect
13. C₂×C₁₁×#108	C₄	semidirect
65. C₁₁×#108	C₂×C₄	semidirect
66. C₁₁×#108	C₂×C₄	semidirect
171. C₂×#108	Dic₁₁	semidirect
241. #108	C₂×Dic₁₁	semidirect
242. #108	C₂×Dic₁₁	semidirect
306. C₁₁	#21181	semidirect

Maximal subgroups

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

subgroup	index
#13077206	2
C₂×#10622719	2
C₂×#10622714	2
#13069402	2
#13069397	2
#13068354	2
C₂²×C₁₁×#108	2
#21181	11