Subgroups — #28404423

Order 14464.

1140 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20299330	C₂	1	4	yes	maximal, complemented
2	#20299322	C₂	1	2	yes	maximal, complemented
3	#20299313	C₂	1	2	yes	maximal, complemented
4	#20299294	C₂	1	4	yes	maximal, complemented
5	#20299268	C₂	1	2	yes	maximal, complemented
6	#20299252	C₂	1	2	yes	maximal, complemented
7	#20299185	C₂	1	4	yes	maximal, complemented
8	#20299162	C₂	1	2	yes	maximal, complemented
9	#20299155	C₂	1	1	yes	maximal, characteristic, complemented
10	#20299108	C₂	1	4	yes	maximal, complemented
11	#20299081	C₂	1	2	yes	maximal, complemented
12	C₂×C₄×D₄₅₂	C₂	1	1	yes	maximal, characteristic, complemented
13	#532×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×#14268646	C₂²	1	4	yes	complemented
15	C₂²×D₄₅₂	C₂²	1	1	yes	characteristic
16	C₂²×C₄×D₁₁₃	C₂²	1	2	yes	—
17	#16813622	C₂²	1	1	yes	characteristic, complemented
18	#16813622	C₂²	1	1	yes	characteristic, complemented
19	#16813618	C₂²	1	4	yes	—
20	#16813617	C₂²	1	8	yes	complemented
21	#16813614	C₂²	1	2	yes	complemented
22	#16813611	C₂²	1	4	yes	—
23	#16813608	C₂²	1	8	yes	complemented
24	#16813606	C₂²	1	8	yes	complemented
25	#16813604	C₂²	1	8	yes	complemented
26	#16813602	C₂²	1	4	yes	complemented
27	#16813601	C₂²	1	2	yes	—
28	#16813601	C₂²	1	2	yes	—
29	#16813596	C₂²	1	4	yes	—
30	#16813596	C₂²	1	4	yes	—
31	#16813594	C₂²	1	8	yes	complemented
32	#16813593	C₂²	1	4	yes	complemented
33	#16813590	C₂²	1	8	yes	—
34	Q₈×Dic₁₁₃	C₂²	1	4	yes	complemented
35	C₄×Dic₂₂₆	C₂²	1	2	yes	complemented
36	C₄×Dic₂₂₆	C₂²	1	2	yes	complemented
37	C₄×D₄₅₂	C₂²	1	2	yes	—
38	C₄×D₄₅₂	C₂²	1	2	yes	—
39	C₄×D₄₅₂	C₂²	1	2	yes	—
40	C₄×D₄₅₂	C₂²	1	2	yes	—
41	D₄×Dic₁₁₃	C₂²	1	8	yes	—
42	D₄×Dic₁₁₃	C₂²	1	4	yes	complemented
43	#16813582	C₂²	1	8	yes	complemented
44	#16813581	C₂²	1	2	yes	—
45	#16813581	C₂²	1	4	yes	complemented
46	#16813580	C₂²	1	4	yes	complemented
47	C₂×C₄⋊Dic₁₁₃	C₂²	1	1	yes	characteristic
48	C₂×C₄⋊Dic₁₁₃	C₂²	1	4	yes	complemented
49	C₂×#14268624	C₂²	1	2	yes	—
50	C₂×#41×C₁₁₃	C₂²	1	2	yes	—
51	C₄⋊Q₈×C₁₁₃	C₂²	1	1	yes	characteristic, complemented
52	#127×C₁₁₃	C₂²	1	1	yes	characteristic, complemented
53	#124×C₁₁₃	C₂²	1	2	yes	complemented
54	#121×C₁₁₃	C₂²	1	2	yes	—
55	#121×C₁₁₃	C₂²	1	2	yes	—
56	C₄×D₄×C₁₁₃	C₂²	1	2	yes	complemented
57	C₄×D₄×C₁₁₃	C₂²	1	2	yes	complemented
58	C₂×C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
59	C₂³×D₁₁₃	C₂³	1	2	yes	—
60	#14268646	—	4	8	no	—
61	C₂×Dic₂₂₆	C₂³	1	2	yes	—
62	C₂×Dic₂₂₆	C₂³	1	2	yes	—
63	C₂×D₄₅₂	—	2	4	no	—
64	C₂×D₄₅₂	C₂³	1	2	yes	—
65	C₂×D₄₅₂	C₂³	1	2	yes	—
66	C₂×C₂²⋊D₁₁₃	C₂³	1	8	yes	—
67	C₂×C₂²⋊D₁₁₃	C₂³	1	8	yes	—
68	C₂²×Dic₁₁₃	C₂³	1	2	yes	—
69	C₂²×Dic₁₁₃	C₂³	1	8	yes	—
70	C₂×C₄×D₁₁₃	—	4	4	no	—
71	C₂×C₄×D₁₁₃	C₂³	1	4	yes	—
72	C₂×C₄×D₁₁₃	C₂³	1	4	yes	—
73	C₄⋊Dic₁₁₃	C₂³	1	2	yes	—
74	C₄⋊Dic₁₁₃	C₂³	1	2	yes	—
75	C₄⋊Dic₁₁₃	C₂³	1	2	yes	—
76	C₄⋊Dic₁₁₃	C₂³	1	8	yes	—
77	C₄⋊Dic₁₁₃	C₂³	1	2	yes	—
78	C₄⋊Dic₁₁₃	C₂³	1	2	yes	—
79	C₄⋊Dic₁₁₃	C₂³	1	2	yes	—
80	#14268626	C₂³	1	4	yes	—
81	#14268626	C₂³	1	4	yes	—
82	C₂²⋊Dic₁₁₃	C₂³	1	4	yes	—
83	C₂²⋊Dic₁₁₃	C₂³	1	8	yes	—
84	#14268624	C₂³	1	4	yes	—
85	#14268624	C₂³	1	4	yes	—
86	#14268624	C₂³	1	8	yes	—
87	#14268624	C₂³	1	8	yes	—
88	C₄×Dic₁₁₃	C₂³	1	8	yes	—
89	C₄×Dic₁₁₃	C₂³	1	4	yes	—
90	#41×C₁₁₃	—	2	8	no	—
91	C₂×Q₈×C₁₁₃	C₂³	1	2	yes	—
92	C₂×D₄×C₁₁₃	C₂³	1	4	yes	—
93	C₂×D₄×C₁₁₃	C₂³	1	2	yes	—
94	C₂×D₄×C₁₁₃	C₂³	1	2	yes	—
95	C₂×D₄×C₁₁₃	C₂³	1	2	yes	—
96	C₂²×C₄×C₁₁₃	C₂³	1	4	yes	—
97	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
98	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
99	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
100	C₄⋊C₄×C₁₁₃	C₂³	1	2	yes	—
101	C₄⋊C₄×C₁₁₃	C₂³	1	2	yes	—
102	C₂²⋊C₄×C₁₁₃	C₂³	1	4	yes	—
103	C₂²⋊C₄×C₁₁₃	C₂³	1	4	yes	—
104	C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
105	C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
106	C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
107	C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
108	C₂³×C₁₁₃	C₂⁴	1	4	yes	—
109	C₂³×C₁₁₃	C₂⁴	1	1	yes	characteristic
110	C₂²×D₁₁₃	—	2	4	no	—
111	C₂²×D₁₁₃	—	4	2	no	—
112	C₂²×D₁₁₃	—	4	2	no	—
113	C₂²×D₁₁₃	C₂⁴	1	4	yes	—
114	Q₈×C₁₁₃	—	2	4	no	—
115	D₄×C₁₁₃	—	2	8	no	—
116	D₄×C₁₁₃	—	2	4	no	—
117	D₄×C₁₁₃	—	2	4	no	—
118	D₄×C₁₁₃	—	4	2	no	—
119	C₂×C₄×C₁₁₃	—	2	8	no	—
120	C₂×C₄×C₁₁₃	—	2	2	no	—
121	C₂×C₄×C₁₁₃	—	4	1	no	—
122	C₂×C₄×C₁₁₃	#41	1	4	yes	complemented
123	C₂×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
124	C₂×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
125	C₂×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
126	C₂×C₄×C₁₁₃	C₂⁴	1	4	yes	—
127	C₂×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
128	C₂×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
129	C₂×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
130	C₂²⋊D₁₁₃	—	4	8	no	—
131	C₂²⋊D₁₁₃	—	4	8	no	—
132	C₂×Dic₁₁₃	—	4	2	no	—
133	C₂×Dic₁₁₃	—	4	8	no	—
134	C₂×Dic₁₁₃	C₂⁴	1	4	yes	—
135	C₂×Dic₁₁₃	C₂⁴	1	4	yes	—
136	C₂×Dic₁₁₃	C₂⁴	1	4	yes	—
137	D₄₅₂	—	2	4	no	—
138	D₄₅₂	—	2	4	no	—
139	C₄×D₁₁₃	—	2	8	no	—
140	C₄×D₁₁₃	—	4	4	no	—
141	Dic₂₂₆	—	2	4	no	—
142	Dic₂₂₆	—	4	2	no	—
143	C₂²×C₁₁₃	—	2	1	no	—
144	C₂²×C₁₁₃	—	2	1	no	—
145	C₂²×C₁₁₃	—	2	4	no	—
146	C₂²×C₁₁₃	—	2	4	no	—
147	C₂²×C₁₁₃	—	2	4	no	—
148	C₂²×C₁₁₃	C₂×#41	1	2	yes	—
149	C₂²×C₁₁₃	C₂⁵	1	1	yes	characteristic
150	C₂×D₁₁₃	—	2	4	no	—
151	C₂×D₁₁₃	—	2	4	no	—
152	C₂×D₁₁₃	—	2	4	no	—
153	C₂×D₁₁₃	—	4	8	no	—
154	C₄×C₁₁₃	—	2	4	no	—
155	C₄×C₁₁₃	—	2	1	no	—
156	C₄×C₁₁₃	—	2	1	no	—
157	C₄×C₁₁₃	—	2	1	no	—
158	C₄×C₁₁₃	C₂×#41	1	2	yes	—
159	C₄×C₁₁₃	C₂×#41	1	2	yes	—
160	C₄×C₁₁₃	H(5,𝔽₂)	1	2	yes	—
161	Dic₁₁₃	—	2	4	no	—
162	Dic₁₁₃	—	2	4	no	—
163	Dic₁₁₃	—	2	4	no	—
164	C₂×C₁₁₃	—	2	2	no	—
165	C₂×C₁₁₃	—	4	4	no	—
166	C₂×C₁₁₃	C₂²×#41	1	1	yes	characteristic
167	C₂×C₁₁₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
168	C₂×C₁₁₃	#585	1	1	yes	characteristic
169	D₁₁₃	—	4	4	no	—
170	#3506	—	113	1	no	maximal, Sylow
171	C₁₁₃	#3506	1	1	yes	characteristic, complemented, Sylow
172	#559	—	113	4	no	—
173	#556	—	113	2	no	—
174	#555	—	113	2	no	—
175	#553	—	113	4	no	—
176	#550	—	113	2	no	—
177	#547	—	113	2	no	—
178	#538	—	113	4	no	—
179	#535	—	113	2	no	—
180	#533	—	113	1	no	—
181	#532	D₁₁₃	1	1	yes	characteristic, complemented
182	#525	—	113	4	no	—
183	#520	—	113	2	no	—
184	C₂×C₄×D₄	—	113	1	no	—
185	C₂×#41	—	113	4	no	—
186	C₂×#41	C₂×D₁₁₃	1	2	yes	—
187	C₂²×D₄	—	113	1	no	—
188	C₂³×C₄	—	113	2	no	—
189	C₄⋊Q₈	—	113	1	no	—
190	C₄⋊Q₈	—	113	1	no	—
191	C₄⋊Q₈	C₂×D₁₁₃	1	1	yes	characteristic, complemented
192	#127	—	113	4	no	—
193	#127	C₂×D₁₁₃	1	1	yes	characteristic, complemented
194	#126	—	113	8	no	—
195	#125	—	113	2	no	—
196	#124	—	113	4	no	—
197	#124	—	113	8	no	—
198	#124	C₂×D₁₁₃	1	2	yes	complemented
199	#123	—	113	8	no	—
200	#123	—	113	8	no	—
201	C₂²⋊Q₈	—	113	2	no	—
202	C₂²⋊Q₈	—	113	2	no	—
203	C₂²⋊Q₈	—	113	4	no	—
204	#121	—	113	4	no	—
205	#121	—	113	8	no	—
206	#121	—	113	4	no	—
207	#121	—	113	4	no	—
208	#121	C₂×D₁₁₃	1	2	yes	—
209	#121	C₂×D₁₁₃	1	2	yes	—
210	#120	—	113	8	no	—
211	C₄×Q₈	—	113	4	no	—
212	C₄×Q₈	—	113	2	no	—
213	C₄×Q₈	—	113	2	no	—
214	C₄×D₄	—	113	8	no	—
215	C₄×D₄	—	113	4	no	—
216	C₄×D₄	—	113	8	no	—
217	C₄×D₄	—	113	2	no	—
218	C₄×D₄	—	113	2	no	—
219	C₄×D₄	—	113	2	no	—
220	C₄×D₄	—	113	2	no	—
221	C₄×D₄	C₂×D₁₁₃	1	2	yes	complemented
222	C₄×D₄	C₂×D₁₁₃	1	2	yes	complemented
223	#117	—	113	2	no	—
224	#117	—	113	4	no	—
225	#117	—	113	4	no	—
226	C₂×C₄⋊C₄	—	113	1	no	—
227	C₂×C₄⋊C₄	—	113	4	no	—
228	C₂×C₂²⋊C₄	—	113	2	no	—
229	C₂×C₄²	C₂×D₁₁₃	1	1	yes	characteristic
230	C₂⁴	—	113	2	no	—
231	#41	—	2	8	no	—
232	#41	—	452	8	no	—
233	C₂×Q₈	—	113	2	no	—
234	C₂×Q₈	—	113	2	no	—
235	C₂×Q₈	C₂²×D₁₁₃	1	2	yes	—
236	C₂×D₄	—	113	8	no	—
237	C₂×D₄	—	113	8	no	—
238	C₂×D₄	—	113	2	no	—
239	C₂×D₄	—	113	2	no	—
240	C₂×D₄	—	226	4	no	—
241	C₂×D₄	C₂²×D₁₁₃	1	4	yes	—
242	C₂×D₄	C₂²×D₁₁₃	1	2	yes	—
243	C₂×D₄	C₂²×D₁₁₃	1	2	yes	—
244	C₂×D₄	C₂²×D₁₁₃	1	2	yes	—
245	C₂²×C₄	—	113	2	no	—
246	C₂²×C₄	—	113	8	no	—
247	C₂²×C₄	—	113	4	no	—
248	C₂²×C₄	—	113	4	no	—
249	C₂²×C₄	—	452	4	no	—
250	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
251	C₂²×C₄	C₂²×D₁₁₃	1	4	yes	—
252	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
253	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
254	C₄⋊C₄	—	113	2	no	—
255	C₄⋊C₄	—	113	2	no	—
256	C₄⋊C₄	—	113	2	no	—
257	C₄⋊C₄	—	113	8	no	—
258	C₄⋊C₄	—	113	2	no	—
259	C₄⋊C₄	—	113	2	no	—
260	C₄⋊C₄	—	113	4	no	—
261	C₄⋊C₄	—	113	4	no	—
262	C₄⋊C₄	—	113	2	no	—
263	C₄⋊C₄	C₂²×D₁₁₃	1	2	yes	—
264	C₄⋊C₄	C₂²×D₁₁₃	1	2	yes	—
265	C₂²⋊C₄	—	113	4	no	—
266	C₂²⋊C₄	—	113	8	no	—
267	C₂²⋊C₄	—	113	4	no	—
268	C₂²⋊C₄	—	113	4	no	—
269	C₂²⋊C₄	—	113	8	no	—
270	C₂²⋊C₄	—	113	8	no	—
271	C₂²⋊C₄	C₂²×D₁₁₃	1	4	yes	—
272	C₂²⋊C₄	C₂²×D₁₁₃	1	4	yes	—
273	C₄²	—	113	8	no	—
274	C₄²	—	113	4	no	—
275	C₄²	C₂²×D₁₁₃	1	1	yes	characteristic
276	C₄²	C₂²×D₁₁₃	1	1	yes	characteristic
277	C₄²	C₂²×D₁₁₃	1	1	yes	characteristic
278	C₄²	C₂²×D₁₁₃	1	1	yes	characteristic
279	C₂³	—	113	4	no	—
280	C₂³	—	452	2	no	—
281	C₂³	—	452	2	no	—
282	C₂³	—	226	4	no	—
283	C₂³	C₂³×D₁₁₃	1	4	yes	—
284	C₂³	C₂³×D₁₁₃	1	1	yes	characteristic
285	Q₈	—	2	4	no	—
286	Q₈	—	452	2	no	—
287	Q₈	—	226	4	no	—
288	D₄	—	2	8	no	—
289	D₄	—	2	4	no	—
290	D₄	—	2	4	no	—
291	D₄	—	4	2	no	—
292	D₄	—	452	8	no	—
293	D₄	—	452	8	no	—
294	D₄	—	226	4	no	—
295	D₄	—	226	4	no	—
296	C₂×C₄	—	2	8	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	113	4	no	—
300	C₂×C₄	—	113	4	no	—
301	C₂×C₄	—	113	4	no	—
302	C₂×C₄	—	452	4	no	—
303	C₂×C₄	—	452	2	no	—
304	C₂×C₄	—	452	8	no	—
305	C₂×C₄	—	226	8	no	—
306	C₂×C₄	#14268646	1	4	yes	complemented
307	C₂×C₄	C₂³×D₁₁₃	1	4	yes	—
308	C₂×C₄	C₂³×D₁₁₃	1	1	yes	characteristic
309	C₂×C₄	C₂³×D₁₁₃	1	1	yes	characteristic
310	C₂×C₄	C₂³×D₁₁₃	1	1	yes	characteristic
311	C₂×C₄	C₂³×D₁₁₃	1	1	yes	characteristic
312	C₂×C₄	C₂³×D₁₁₃	1	1	yes	characteristic
313	C₂×C₄	C₂³×D₁₁₃	1	1	yes	characteristic
314	C₂²	—	2	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	4	no	—
317	C₂²	—	2	4	no	—
318	C₂²	—	2	4	no	—
319	C₂²	—	452	8	no	—
320	C₂²	—	226	4	no	—
321	C₂²	—	226	4	no	—
322	C₂²	—	226	4	no	—
323	C₂²	C₂×#14268646	1	2	yes	—
324	C₂²	C₂⁴×D₁₁₃	1	1	yes	characteristic
325	C₄	—	2	4	no	—
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	226	4	no	—
330	C₄	—	226	4	no	—
331	C₄	—	226	4	no	—
332	C₄	C₂×#14268646	1	2	yes	—
333	C₄	C₂×#14268646	1	2	yes	—
334	C₄	#16813668	1	2	yes	—
335	C₂	—	2	2	no	—
336	C₂	—	4	4	no	—
337	C₂	—	452	4	no	—
338	C₂	C₂²×#14268646	1	1	yes	characteristic
339	C₂	C₂×#16813668	1	1	yes	characteristic
340	C₂	#20299434	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₁₃
commutator, socle, cosocle	C₂²×C₁₁₃	C₂⁵
Fitting	#532×C₁₁₃	C₂
hypercenter	#532	D₁₁₃
hypocenter	C₁₁₃	#3506
LCS	G ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ #532

Complements

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

N	H ≅ G/N	type
1. #20299330	C₂	semidirect
2. #20299322	C₂	semidirect
3. #20299313	C₂	semidirect
4. #20299294	C₂	semidirect
5. #20299268	C₂	semidirect
6. #20299252	C₂	semidirect
7. #20299185	C₂	semidirect
8. #20299162	C₂	semidirect
9. #20299155	C₂	semidirect
10. #20299108	C₂	semidirect
11. #20299081	C₂	semidirect
12. C₂×C₄×D₄₅₂	C₂	semidirect
13. #532×C₁₁₃	C₂	semidirect
14. C₂×#14268646	C₂²	semidirect
17. #16813622	C₂²	semidirect
18. #16813622	C₂²	semidirect
20. #16813617	C₂²	semidirect
21. #16813614	C₂²	semidirect
23. #16813608	C₂²	semidirect
24. #16813606	C₂²	semidirect
25. #16813604	C₂²	semidirect
26. #16813602	C₂²	semidirect
31. #16813594	C₂²	semidirect
32. #16813593	C₂²	semidirect
34. Q₈×Dic₁₁₃	C₂²	semidirect
35. C₄×Dic₂₂₆	C₂²	semidirect
36. C₄×Dic₂₂₆	C₂²	semidirect
42. D₄×Dic₁₁₃	C₂²	semidirect
43. #16813582	C₂²	semidirect
45. #16813581	C₂²	semidirect
46. #16813580	C₂²	semidirect
48. C₂×C₄⋊Dic₁₁₃	C₂²	semidirect
51. C₄⋊Q₈×C₁₁₃	C₂²	semidirect
52. #127×C₁₁₃	C₂²	semidirect
53. #124×C₁₁₃	C₂²	semidirect
56. C₄×D₄×C₁₁₃	C₂²	semidirect
57. C₄×D₄×C₁₁₃	C₂²	semidirect
122. C₂×C₄×C₁₁₃	#41	semidirect
171. C₁₁₃	#3506	semidirect
181. #532	D₁₁₃	semidirect
191. C₄⋊Q₈	C₂×D₁₁₃	semidirect
193. #127	C₂×D₁₁₃	semidirect
198. #124	C₂×D₁₁₃	semidirect
221. C₄×D₄	C₂×D₁₁₃	semidirect
222. C₄×D₄	C₂×D₁₁₃	semidirect
306. C₂×C₄	#14268646	semidirect

Maximal subgroups

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

subgroup	index
#20299330	2
#20299322	2
#20299313	2
#20299294	2
#20299268	2
#20299252	2
#20299185	2
#20299162	2
#20299155	2
#20299108	2
#20299081	2
C₂×C₄×D₄₅₂	2
#532×C₁₁₃	2
#3506	113