Subgroups — #14910494

Order 2816.

674 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10622517	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×Dic₈₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#13080359	C₂	1	2	yes	maximal, complemented
4	#13070877	C₂	1	1	yes	maximal, characteristic, complemented
5	#13070707	C₂	1	1	yes	maximal, characteristic, complemented
6	#13070702	C₂	1	1	yes	maximal, characteristic, complemented
7	#13070679	C₂	1	2	yes	maximal, complemented
8	#13070558	C₂	1	2	yes	maximal, complemented
9	#13070535	C₂	1	2	yes	maximal, complemented
10	C₄×Dic₈₈	C₂	1	1	yes	maximal, characteristic, complemented
11	D₄×C₁₁×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Dic₄₄	C₂²	1	2	yes	—
13	C₂×#66957	C₂²	1	2	yes	—
14	#10623083	—	2	2	no	—
15	C₂²×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
16	C₂²×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
17	C₈×D₄×C₁₁	C₂²	1	1	yes	characteristic
18	#10622729	C₂²	1	2	yes	—
19	#10622728	C₂²	1	2	yes	—
20	C₂×Dic₈₈	—	2	4	no	—
21	C₂×Dic₈₈	C₂²	1	1	yes	characteristic
22	C₂×Dic₈₈	C₂²	1	2	yes	—
23	C₂×Dic₈₈	C₂²	1	1	yes	characteristic
24	#10622517	—	2	4	no	—
25	C₂×#66864	C₂²	1	2	yes	—
26	C₂×#66864	C₂²	1	2	yes	—
27	C₂×D₁₇₆	C₂²	1	1	yes	characteristic
28	#10622458	—	2	2	no	—
29	#10622351	—	2	2	no	—
30	#10622346	—	2	2	no	—
31	C₄×Dic₄₄	C₂²	1	2	yes	—
32	#10622309	C₂²	1	2	yes	—
33	C₁₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
34	C₁₁×C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
35	C₁₁×C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
36	C₄×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
37	#10622147	C₂²	1	2	yes	—
38	#10622142	C₂²	1	1	yes	characteristic
39	#10622141	C₂²	1	2	yes	—
40	#10622141	C₂²	1	2	yes	—
41	C₂×#4339	—	2	2	no	—
42	C₂²×Dic₂₂	—	2	2	no	—
43	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
45	C₄×D₄×C₁₁	D₄	1	1	yes	characteristic
46	#66992	—	4	2	no	—
47	#66983	—	2	2	no	—
48	#66977	—	2	2	no	—
49	C₂×Dic₄₄	—	2	4	no	—
50	C₂×Dic₄₄	—	2	2	no	—
51	C₂×Dic₄₄	—	4	2	no	—
52	C₂×Dic₄₄	C₂³	1	2	yes	—
53	C₂×Dic₄₄	C₂³	1	2	yes	—
54	#66957	—	2	4	no	—
55	#66957	—	4	2	no	—
56	C₂×D₈₈	C₂³	1	2	yes	—
57	C₂×#4314	—	2	2	no	—
58	C₂×#4314	—	2	2	no	—
59	#66949	—	4	2	no	—
60	#66938	—	4	2	no	—
61	(C₂²⋊D₁₁)⋊C₄	—	4	2	no	—
62	#66928	—	2	2	no	—
63	C₄×Dic₂₂	—	2	2	no	—
64	C₂×C₁₁×C₁₆	—	2	2	no	—
65	C₂×C₁₁×C₁₆	—	2	2	no	—
66	C₂×C₁₁×C₁₆	C₂³	1	1	yes	characteristic
67	C₂×C₁₁×C₁₆	C₂³	1	1	yes	characteristic
68	C₂×C₁₁×C₁₆	C₂³	1	1	yes	characteristic
69	C₂×C₁₁×C₁₆	C₂³	1	1	yes	characteristic
70	C₁₁×C₄⋊C₈	D₄	1	1	yes	characteristic
71	C₁₁×C₂²⋊C₈	D₄	1	1	yes	characteristic
72	C₁₁×C₂²⋊C₈	D₄	1	1	yes	characteristic
73	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
74	#66885	—	2	2	no	—
75	#66882	C₂³	1	2	yes	—
76	#66880	—	2	2	no	—
77	#66880	—	2	2	no	—
78	Dic₈₈	—	2	4	no	—
79	Dic₈₈	—	2	2	no	—
80	Dic₈₈	D₄	1	4	yes	complemented
81	#66864	—	2	4	no	—
82	#66864	—	4	2	no	—
83	D₁₇₆	—	2	2	no	—
84	#33851	—	11	1	no	maximal, Sylow
85	C₂×D₄×C₁₁	D₈	1	1	yes	characteristic
86	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
88	C₂×C₂²⋊D₁₁	—	4	2	no	—
89	C₂×C₂²⋊D₁₁	—	4	2	no	—
90	C₂²×Dic₁₁	—	4	2	no	—
91	#4339	—	4	4	no	—
92	#4339	—	8	2	no	—
93	C₂×D₄₄	—	2	2	no	—
94	C₂×C₄×D₁₁	—	4	2	no	—
95	C₂×Dic₂₂	—	2	2	no	—
96	C₂×Dic₂₂	—	2	2	no	—
97	C₂×Dic₂₂	—	4	4	no	—
98	C₂×Dic₂₂	—	4	2	no	—
99	C₂×Dic₂₂	—	8	2	no	—
100	C₂×C₈×C₁₁	—	2	1	no	—
101	C₂×C₈×C₁₁	—	2	1	no	—
102	C₂×C₈×C₁₁	—	2	1	no	—
103	C₂×C₈×C₁₁	—	2	1	no	—
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×D₄	1	1	yes	characteristic
107	C₂×C₈×C₁₁	C₂⁴	1	1	yes	characteristic
108	C₁₁×C₄⋊C₄	D₈	1	1	yes	characteristic
109	C₁₁×C₂²⋊C₄	D₈	1	1	yes	characteristic
110	C₁₁×C₂²⋊C₄	D₈	1	1	yes	characteristic
111	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
112	C₂²⋊Dic₁₁	—	4	2	no	—
113	(C₂×D₁₁)⋊C₄	—	4	2	no	—
114	(C₂×D₁₁)⋊C₄	—	4	2	no	—
115	(C₂×D₁₁)⋊C₄	—	4	2	no	—
116	C₄⋊Dic₁₁	—	2	2	no	—
117	Dic₁₁⋊C₄	—	4	2	no	—
118	Dic₁₁⋊C₄	—	4	2	no	—
119	C₄×Dic₁₁	—	4	2	no	—
120	Dic₄₄	—	2	2	no	—
121	Dic₄₄	—	2	2	no	—
122	Dic₄₄	—	2	2	no	—
123	Dic₄₄	—	4	4	no	—
124	Dic₄₄	C₂×D₄	1	4	yes	—
125	D₈₈	—	2	2	no	—
126	D₈₈	—	2	2	no	—
127	#4314	—	4	4	no	—
128	#4314	—	8	2	no	—
129	C₁₁×C₁₆	—	2	1	no	—
130	C₁₁×C₁₆	—	2	1	no	—
131	C₁₁×C₁₆	—	2	1	no	—
132	C₁₁×C₁₆	C₂×D₄	1	2	yes	—
133	C₂×#508	—	11	1	no	—
134	C₂²×Q₃₂	—	11	1	no	—
135	#3285	—	11	2	no	—
136	#2249	—	11	1	no	—
137	#2216	—	11	1	no	—
138	#2215	—	11	1	no	—
139	#2211	—	11	2	no	—
140	#2189	—	11	2	no	—
141	#2185	—	11	2	no	—
142	C₄×Q₃₂	—	11	1	no	—
143	D₄×C₁₆	D₁₁	1	1	yes	characteristic, complemented
144	C₂³×C₁₁	C₂×D₈	1	1	yes	characteristic
145	C₂³×C₁₁	C₂×D₈	1	1	yes	characteristic
146	C₂²×D₁₁	—	4	2	no	—
147	D₄×C₁₁	—	2	2	no	—
148	C₂×C₄×C₁₁	—	2	1	no	—
149	C₂×C₄×C₁₁	—	2	1	no	—
150	C₂×C₄×C₁₁	—	2	1	no	—
151	C₂×C₄×C₁₁	—	2	1	no	—
152	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
153	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
154	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
155	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
156	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
157	C₂²⋊D₁₁	—	8	4	no	—
158	C₂²⋊D₁₁	—	8	4	no	—
159	C₂×Dic₁₁	—	4	2	no	—
160	C₂×Dic₁₁	—	4	2	no	—
161	C₂×Dic₁₁	—	4	2	no	—
162	C₂×Dic₁₁	—	8	4	no	—
163	D₄₄	—	4	2	no	—
164	D₄₄	—	4	2	no	—
165	C₄×D₁₁	—	8	4	no	—
166	Dic₂₂	—	2	4	no	—
167	Dic₂₂	—	4	2	no	—
168	Dic₂₂	—	4	2	no	—
169	Dic₂₂	—	4	2	no	—
170	Dic₂₂	—	8	4	no	—
171	C₈×C₁₁	—	2	1	no	—
172	C₈×C₁₁	—	2	1	no	—
173	C₈×C₁₁	—	2	1	no	—
174	C₈×C₁₁	C₂²×D₄	1	1	yes	characteristic
175	C₈×C₁₁	H(5,𝔽₂)	1	1	yes	characteristic
176	C₂×#135	—	11	2	no	—
177	C₂²×Q₁₆	—	11	2	no	—
178	#548	—	22	2	no	—
179	#508	—	22	4	no	—
180	C₂×Q₃₂	—	11	1	no	—
181	C₂×Q₃₂	—	11	2	no	—
182	C₂×Q₃₂	—	11	1	no	—
183	C₂×Q₃₂	—	22	4	no	—
184	C₂×SD₃₂	—	11	2	no	—
185	C₂×SD₃₂	—	11	2	no	—
186	C₂×D₁₆	—	11	1	no	—
187	C₂²×C₁₆	C₂×D₁₁	1	1	yes	characteristic
188	C₂²×C₁₆	C₂×D₁₁	1	1	yes	characteristic
189	#495	—	11	2	no	—
190	#467	—	11	2	no	—
191	#466	—	11	2	no	—
192	#464	—	22	2	no	—
193	#452	—	22	2	no	—
194	#448	—	22	2	no	—
195	C₄×Q₁₆	—	11	2	no	—
196	C₈×D₄	C₂×D₁₁	1	1	yes	characteristic
197	#366	—	11	1	no	—
198	C₄⋊C₁₆	C₂×D₁₁	1	1	yes	characteristic
199	#358	—	11	2	no	—
200	#358	—	11	2	no	—
201	#357	—	11	2	no	—
202	C₂²⋊C₁₆	C₂×D₁₁	1	1	yes	characteristic
203	C₂²⋊C₁₆	C₂×D₁₁	1	1	yes	characteristic
204	C₄×C₁₆	C₂×D₁₁	1	1	yes	characteristic
205	C₂²×C₁₁	—	2	1	no	—
206	C₂²×C₁₁	—	2	1	no	—
207	C₂²×C₁₁	—	2	1	no	—
208	C₂²×C₁₁	—	2	1	no	—
209	C₂²×C₁₁	—	2	1	no	—
210	C₂²×C₁₁	#508	1	2	yes	complemented
211	C₂²×C₁₁	C₂²×D₈	1	1	yes	characteristic
212	C₂×D₁₁	—	8	2	no	—
213	C₂×D₁₁	—	8	2	no	—
214	C₂×D₁₁	—	8	2	no	—
215	C₄×C₁₁	—	2	1	no	—
216	C₄×C₁₁	—	2	1	no	—
217	C₄×C₁₁	—	2	1	no	—
218	C₄×C₁₁	—	2	1	no	—
219	C₄×C₁₁	D₄²	1	1	yes	characteristic
220	C₄×C₁₁	#576	1	1	yes	characteristic
221	Dic₁₁	—	4	4	no	—
222	Dic₁₁	—	8	2	no	—
223	Dic₁₁	—	8	2	no	—
224	C₂×#41	—	22	2	no	—
225	C₂²×Q₈	—	22	2	no	—
226	#135	—	22	4	no	—
227	#135	—	44	2	no	—
228	C₂×Q₁₆	—	11	2	no	—
229	C₂×Q₁₆	—	11	2	no	—
230	C₂×Q₁₆	—	22	4	no	—
231	C₂×Q₁₆	—	22	2	no	—
232	C₂×Q₁₆	—	44	2	no	—
233	C₂×SD₁₆	—	22	2	no	—
234	C₂×SD₁₆	—	22	2	no	—
235	C₂×D₈	—	11	2	no	—
236	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
237	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
238	#124	—	22	2	no	—
239	#124	—	44	2	no	—
240	C₂²⋊Q₈	—	22	2	no	—
241	C₂²⋊Q₈	—	44	2	no	—
242	#121	—	22	2	no	—
243	#121	—	44	2	no	—
244	C₄×Q₈	—	22	2	no	—
245	C₄×D₄	—	44	2	no	—
246	C₄×D₄	D₄₄	1	1	yes	characteristic
247	Q₃₂	—	11	4	no	—
248	Q₃₂	—	22	4	no	—
249	Q₃₂	—	22	2	no	—
250	SD₃₂	—	22	4	no	—
251	SD₃₂	—	44	2	no	—
252	D₁₆	—	22	2	no	—
253	C₂×C₁₆	—	2	2	no	—
254	C₂×C₁₆	—	2	2	no	—
255	C₂×C₁₆	C₂²×D₁₁	1	1	yes	characteristic
256	C₂×C₁₆	C₂²×D₁₁	1	1	yes	characteristic
257	C₂×C₁₆	C₂²×D₁₁	1	1	yes	characteristic
258	C₂×C₁₆	C₂²×D₁₁	1	1	yes	characteristic
259	#107	—	11	2	no	—
260	C₄⋊C₈	D₄₄	1	1	yes	characteristic
261	#103	—	22	2	no	—
262	#103	—	22	2	no	—
263	#102	—	22	2	no	—
264	C₂²⋊C₈	D₄₄	1	1	yes	characteristic
265	C₂²⋊C₈	D₄₄	1	1	yes	characteristic
266	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
267	C₂×C₁₁	—	2	2	no	—
268	C₂×C₁₁	—	4	1	no	—
269	C₂×C₁₁	D₄×D₈	1	1	yes	characteristic
270	C₂×C₁₁	C₂×#508	1	1	yes	characteristic
271	C₂×C₁₁	#3417	1	1	yes	characteristic
272	D₁₁	—	16	2	no	—
273	#41	—	44	4	no	—
274	#41	—	88	2	no	—
275	C₂×Q₈	—	22	2	no	—
276	C₂×Q₈	—	22	2	no	—
277	C₂×Q₈	—	44	4	no	—
278	C₂×Q₈	—	44	2	no	—
279	C₂×Q₈	—	88	2	no	—
280	C₂×D₄	—	22	2	no	—
281	C₂×D₄	—	44	2	no	—
282	C₂×D₄	—	44	2	no	—
283	C₂×D₄	D₈₈	1	1	yes	characteristic
284	C₂²×C₄	—	44	2	no	—
285	C₂²×C₄	—	44	2	no	—
286	C₂²×C₄	C₂×D₄₄	1	1	yes	characteristic
287	C₂²×C₄	C₂×D₄₄	1	1	yes	characteristic
288	Q₁₆	—	11	4	no	—
289	Q₁₆	—	22	2	no	—
290	Q₁₆	—	22	2	no	—
291	Q₁₆	—	22	2	no	—
292	Q₁₆	—	44	4	no	—
293	SD₁₆	—	44	4	no	—
294	SD₁₆	—	88	2	no	—
295	D₈	—	22	2	no	—
296	D₈	—	22	2	no	—
297	C₂×C₈	—	2	1	no	—
298	C₂×C₈	—	2	1	no	—
299	C₂×C₈	—	2	1	no	—
300	C₂×C₈	—	2	1	no	—
301	C₂×C₈	C₂×D₄₄	1	1	yes	characteristic
302	C₂×C₈	C₂×D₄₄	1	1	yes	characteristic
303	C₂×C₈	C₂×D₄₄	1	1	yes	characteristic
304	C₂×C₈	C₂³×D₁₁	1	1	yes	characteristic
305	C₄⋊C₄	—	22	2	no	—
306	C₄⋊C₄	—	44	2	no	—
307	C₄⋊C₄	—	44	2	no	—
308	C₄⋊C₄	D₈₈	1	1	yes	characteristic
309	C₂²⋊C₄	—	44	2	no	—
310	C₂²⋊C₄	—	44	2	no	—
311	C₂²⋊C₄	—	44	2	no	—
312	C₂²⋊C₄	—	44	2	no	—
313	C₂²⋊C₄	D₈₈	1	1	yes	characteristic
314	C₂²⋊C₄	D₈₈	1	1	yes	characteristic
315	C₄²	—	44	2	no	—
316	C₄²	C₂×D₄₄	1	1	yes	characteristic
317	C₁₆	—	2	1	no	—
318	C₁₆	—	2	1	no	—
319	C₁₆	—	2	1	no	—
320	C₁₆	D₄×D₁₁	1	2	yes	—
321	C₁₁	#33851	1	1	yes	characteristic, complemented, Sylow
322	C₂³	—	44	2	no	—
323	C₂³	C₂×D₈₈	1	1	yes	characteristic
324	C₂³	C₂×D₈₈	1	1	yes	characteristic
325	Q₈	—	22	4	no	—
326	Q₈	—	44	2	no	—
327	Q₈	—	44	2	no	—
328	Q₈	—	44	2	no	—
329	Q₈	—	88	4	no	—
330	D₄	—	2	2	no	—
331	D₄	—	44	2	no	—
332	D₄	—	44	2	no	—
333	D₄	—	88	4	no	—
334	D₄	—	88	4	no	—
335	C₂×C₄	—	2	1	no	—
336	C₂×C₄	—	2	1	no	—
337	C₂×C₄	—	2	1	no	—
338	C₂×C₄	—	2	1	no	—
339	C₂×C₄	—	44	2	no	—
340	C₂×C₄	—	44	2	no	—
341	C₂×C₄	—	44	2	no	—
342	C₂×C₄	—	88	4	no	—
343	C₂×C₄	—	88	4	no	—
344	C₂×C₄	C₂×D₈₈	1	1	yes	characteristic
345	C₂×C₄	C₂×D₈₈	1	1	yes	characteristic
346	C₂×C₄	C₂×D₈₈	1	1	yes	characteristic
347	C₂×C₄	C₂×D₈₈	1	1	yes	characteristic
348	C₂×C₄	C₂²×D₄₄	1	1	yes	characteristic
349	C₈	—	2	1	no	—
350	C₈	—	2	1	no	—
351	C₈	—	2	1	no	—
352	C₈	C₂×D₄×D₁₁	1	1	yes	characteristic
353	C₈	#67042	1	1	yes	characteristic
354	C₂²	—	2	1	no	—
355	C₂²	—	2	1	no	—
356	C₂²	—	2	1	no	—
357	C₂²	—	2	1	no	—
358	C₂²	—	2	1	no	—
359	C₂²	—	88	2	no	—
360	C₂²	—	88	2	no	—
361	C₂²	—	88	2	no	—
362	C₂²	#10622517	1	2	yes	complemented
363	C₂²	C₂²×D₈₈	1	1	yes	characteristic
364	C₄	—	2	1	no	—
365	C₄	—	2	1	no	—
366	C₄	—	2	1	no	—
367	C₄	—	2	1	no	—
368	C₄	—	44	4	no	—
369	C₄	—	88	2	no	—
370	C₄	—	88	2	no	—
371	C₄	D₄×D₄₄	1	1	yes	characteristic
372	C₄	#10623283	1	1	yes	characteristic
373	C₂	—	2	2	no	—
374	C₂	—	4	1	no	—
375	C₂	—	176	2	no	—
376	C₂	D₄×D₈₈	1	1	yes	characteristic
377	C₂	C₂×#10622517	1	1	yes	characteristic
378	C₂	#13081769	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₈₈
commutator, cosocle	C₂×C₈×C₁₁	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₁
Fitting	D₄×C₁₁×C₁₆	C₂
socle	C₂²×C₁₁	C₂²×D₈
hypercenter	D₄×C₁₆	D₁₁
hypocenter	C₁₁	#33851
LCS	G ▷ C₂×C₈×C₁₁ ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ D₄×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#10622517	C₂	semidirect
2. C₂²×Dic₈₈	C₂	semidirect
3. #13080359	C₂	semidirect
4. #13070877	C₂	semidirect
5. #13070707	C₂	semidirect
6. #13070702	C₂	semidirect
7. #13070679	C₂	semidirect
8. #13070558	C₂	semidirect
9. #13070535	C₂	semidirect
10. C₄×Dic₈₈	C₂	semidirect
11. D₄×C₁₁×C₁₆	C₂	semidirect
80. Dic₈₈	D₄	semidirect
143. D₄×C₁₆	D₁₁	semidirect
210. C₂²×C₁₁	#508	semidirect
321. C₁₁	#33851	semidirect
362. C₂²	#10622517	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10622517	2
C₂²×Dic₈₈	2
#13080359	2
#13070877	2
#13070707	2
#13070702	2
#13070679	2
#13070558	2
#13070535	2
C₄×Dic₈₈	2
D₄×C₁₁×C₁₆	2
#33851	11