Subgroups — #28404485

Order 14464.

1104 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20299321	C₂	1	2	yes	maximal, complemented
2	#20299313	C₂	1	2	yes	maximal, complemented
3	#20299308	C₂	1	2	yes	maximal, complemented
4	#20299294	C₂	1	4	yes	maximal, complemented
5	#20299257	C₂	1	4	yes	maximal, complemented
6	#20299253	C₂	1	2	yes	maximal, complemented
7	#20299219	C₂	1	4	yes	maximal, complemented
8	#20299176	C₂	1	2	yes	maximal, complemented
9	#20299155	C₂	1	2	yes	maximal, complemented
10	#20299148	C₂	1	2	yes	maximal, complemented
11	#20299148	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#16813622	C₂	1	1	yes	maximal, characteristic, complemented
13	#20299078	C₂	1	2	yes	maximal, complemented
14	C₄×#41×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×#14268647	C₂²	1	2	yes	—
16	C₂×#14268647	C₂²	1	4	yes	—
17	C₂²×Dic₂₂₆	C₂²	1	2	yes	—
18	#16813622	C₂²	1	2	yes	—
19	#16813622	C₂²	1	2	yes	—
20	#16813622	C₂²	1	2	yes	—
21	#16813622	C₂²	1	2	yes	—
22	#16813622	C₂²	1	1	yes	characteristic
23	#16813619	C₂²	1	1	yes	characteristic
24	#16813617	C₂²	1	8	yes	—
25	#16813614	C₂²	1	4	yes	—
26	#16813610	C₂²	1	4	yes	—
27	#16813610	C₂²	1	2	yes	—
28	#16813608	C₂²	1	8	yes	—
29	#16813604	C₂²	1	8	yes	—
30	#16813601	C₂²	1	4	yes	—
31	#16813601	C₂²	1	4	yes	—
32	#16813600	C₂²	1	8	yes	—
33	#16813598	C₂²	1	4	yes	—
34	#16813596	C₂²	1	2	yes	—
35	#16813596	C₂²	1	4	yes	—
36	#16813596	C₂²	1	4	yes	—
37	#16813596	C₂²	1	2	yes	—
38	#16813594	C₂²	1	8	yes	—
39	#16813593	C₂²	1	4	yes	—
40	C₄×Dic₂₂₆	C₂²	1	4	yes	—
41	C₄×Dic₂₂₆	C₂²	1	4	yes	—
42	C₄×D₄₅₂	C₂²	1	4	yes	—
43	C₄×D₄₅₂	C₂²	1	2	yes	—
44	C₄×D₄₅₂	C₂²	1	2	yes	—
45	C₄×D₄₅₂	C₂²	1	4	yes	—
46	#16813582	C₂²	1	8	yes	—
47	#16813581	C₂²	1	4	yes	—
48	#16813580	C₂²	1	4	yes	—
49	C₂×C₄⋊Dic₁₁₃	C₂²	1	2	yes	—
50	C₂×C₄⋊Dic₁₁₃	C₂²	1	2	yes	—
51	C₄⋊C₄×D₁₁₃	C₂²	1	4	yes	—
52	C₂×#41×C₁₁₃	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₁₁₃	C₂²	1	1	yes	characteristic
54	C₄×Q₈×C₁₁₃	C₂²	1	1	yes	characteristic
55	C₄×D₄×C₁₁₃	C₂²	1	2	yes	—
56	C₄×D₄×C₁₁₃	C₂²	1	1	yes	characteristic
57	C₄×D₄×C₁₁₃	C₂²	1	1	yes	characteristic
58	C₄×D₄×C₁₁₃	C₂²	1	2	yes	—
59	#117×C₁₁₃	C₂²	1	1	yes	characteristic
60	#117×C₁₁₃	C₂²	1	2	yes	—
61	C₂×C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
62	C₂×C₄²×C₁₁₃	C₂²	1	2	yes	—
63	#14268647	—	2	8	no	—
64	#14268647	—	4	8	no	—
65	C₂×Dic₂₂₆	—	4	4	no	—
66	C₂×Dic₂₂₆	C₂³	1	2	yes	—
67	C₂×Dic₂₂₆	C₂³	1	4	yes	—
68	C₂×Dic₂₂₆	C₂³	1	4	yes	—
69	C₂×D₄₅₂	C₂³	1	2	yes	—
70	C₂×D₄₅₂	C₂³	1	2	yes	—
71	C₂×D₄₅₂	C₂³	1	2	yes	—
72	C₂×C₂²⋊D₁₁₃	C₂³	1	4	yes	—
73	C₂×C₂²⋊D₁₁₃	C₂³	1	8	yes	—
74	C₂²×Dic₁₁₃	C₂³	1	4	yes	—
75	C₂×C₄×D₁₁₃	C₂³	1	4	yes	—
76	C₂×C₄×D₁₁₃	C₂³	1	8	yes	—
77	C₄⋊Dic₁₁₃	C₂³	1	4	yes	—
78	C₄⋊Dic₁₁₃	C₂³	1	4	yes	—
79	C₄⋊Dic₁₁₃	C₂³	1	4	yes	—
80	C₄⋊Dic₁₁₃	C₂³	1	4	yes	—
81	C₄⋊Dic₁₁₃	C₂³	1	2	yes	—
82	C₄⋊Dic₁₁₃	C₂³	1	2	yes	—
83	#14268626	C₂³	1	8	yes	—
84	#14268626	C₂³	1	8	yes	—
85	C₂²⋊Dic₁₁₃	C₂³	1	8	yes	—
86	#14268624	C₂³	1	8	yes	—
87	#14268624	C₂³	1	8	yes	—
88	#14268624	C₂³	1	4	yes	—
89	#14268624	C₂³	1	4	yes	—
90	C₄×Dic₁₁₃	C₂³	1	8	yes	—
91	#41×C₁₁₃	—	2	4	no	—
92	C₂×Q₈×C₁₁₃	C₂³	1	1	yes	characteristic
93	C₂×D₄×C₁₁₃	C₂³	1	1	yes	characteristic
94	C₂×D₄×C₁₁₃	C₂³	1	2	yes	—
95	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
96	C₂²×C₄×C₁₁₃	C₂³	1	2	yes	—
97	C₂²×C₄×C₁₁₃	C₂³	1	2	yes	—
98	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
99	C₂²×C₄×C₁₁₃	C₂³	1	2	yes	—
100	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
101	C₄⋊C₄×C₁₁₃	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₁₁₃	C₂³	1	1	yes	characteristic
103	C₄⋊C₄×C₁₁₃	C₂³	1	2	yes	—
104	C₄⋊C₄×C₁₁₃	C₂³	1	2	yes	—
105	C₂²⋊C₄×C₁₁₃	C₂³	1	2	yes	—
106	C₂²⋊C₄×C₁₁₃	C₂³	1	2	yes	—
107	C₂²⋊C₄×C₁₁₃	C₂³	1	2	yes	—
108	C₄²×C₁₁₃	C₂³	1	2	yes	—
109	C₄²×C₁₁₃	C₂³	1	2	yes	—
110	C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
111	C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
112	C₄²×C₁₁₃	C₂³	1	2	yes	—
113	C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
114	C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
115	C₂³×C₁₁₃	C₂⁴	1	2	yes	—
116	C₂³×C₁₁₃	C₂⁴	1	1	yes	characteristic
117	C₂²×D₁₁₃	C₂⁴	1	4	yes	—
118	Q₈×C₁₁₃	—	2	2	no	—
119	D₄×C₁₁₃	—	2	4	no	—
120	D₄×C₁₁₃	—	2	2	no	—
121	C₂×C₄×C₁₁₃	—	2	4	no	—
122	C₂×C₄×C₁₁₃	—	2	2	no	—
123	C₂×C₄×C₁₁₃	—	2	4	no	—
124	C₂×C₄×C₁₁₃	—	2	2	no	—
125	C₂×C₄×C₁₁₃	—	2	4	no	—
126	C₂×C₄×C₁₁₃	#41	1	4	yes	complemented
127	C₂×C₄×C₁₁₃	C₂⁴	1	2	yes	—
128	C₂×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
129	C₂×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
130	C₂×C₄×C₁₁₃	C₂⁴	1	2	yes	—
131	C₂×C₄×C₁₁₃	C₂⁴	1	2	yes	—
132	C₂×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
133	C₂×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
134	C₂×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
135	C₂×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
136	C₂²⋊D₁₁₃	—	2	8	no	—
137	C₂²⋊D₁₁₃	—	4	8	no	—
138	C₂×Dic₁₁₃	—	4	4	no	—
139	C₂×Dic₁₁₃	C₂⁴	1	4	yes	—
140	C₂×Dic₁₁₃	C₂⁴	1	8	yes	—
141	D₄₅₂	—	2	4	no	—
142	D₄₅₂	—	2	4	no	—
143	D₄₅₂	—	2	4	no	—
144	C₄×D₁₁₃	—	2	8	no	—
145	C₄×D₁₁₃	—	4	8	no	—
146	Dic₂₂₆	—	2	4	no	—
147	Dic₂₂₆	—	2	8	no	—
148	Dic₂₂₆	—	4	4	no	—
149	C₂²×C₁₁₃	—	2	1	no	—
150	C₂²×C₁₁₃	—	2	1	no	—
151	C₂²×C₁₁₃	—	2	2	no	—
152	C₂²×C₁₁₃	—	2	2	no	—
153	C₂²×C₁₁₃	—	2	2	no	—
154	C₂²×C₁₁₃	C₂×#41	1	2	yes	—
155	C₂²×C₁₁₃	C₂⁵	1	1	yes	characteristic
156	C₂×D₁₁₃	—	2	4	no	—
157	C₂×D₁₁₃	—	2	4	no	—
158	C₂×D₁₁₃	—	2	4	no	—
159	C₄×C₁₁₃	—	2	2	no	—
160	C₄×C₁₁₃	—	2	1	no	—
161	C₄×C₁₁₃	—	2	2	no	—
162	C₄×C₁₁₃	—	2	2	no	—
163	C₄×C₁₁₃	—	2	1	no	—
164	C₄×C₁₁₃	C₂×#41	1	2	yes	—
165	C₄×C₁₁₃	C₂×#41	1	2	yes	—
166	C₄×C₁₁₃	H(5,𝔽₂)	1	2	yes	—
167	C₄×C₁₁₃	2₋¹⁺⁴	1	2	yes	—
168	Dic₁₁₃	—	2	4	no	—
169	Dic₁₁₃	—	2	8	no	—
170	C₂×C₁₁₃	—	2	2	no	—
171	C₂×C₁₁₃	—	4	2	no	—
172	C₂×C₁₁₃	C₂²×#41	1	1	yes	characteristic
173	C₂×C₁₁₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
174	C₂×C₁₁₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
175	D₁₁₃	—	4	4	no	—
176	#3511	—	113	1	no	maximal, Sylow
177	C₁₁₃	#3511	1	1	yes	characteristic, complemented, Sylow
178	#556	—	113	2	no	—
179	#555	—	113	2	no	—
180	#554	—	113	2	no	—
181	#553	—	113	4	no	—
182	#548	—	113	4	no	—
183	#547	—	113	2	no	—
184	#542	—	113	4	no	—
185	#537	—	113	2	no	—
186	#533	—	113	2	no	—
187	#532	—	113	1	no	—
188	#532	—	113	2	no	—
189	C₂×C₄⋊Q₈	—	113	1	no	—
190	#520	—	113	2	no	—
191	C₄×#41	D₁₁₃	1	1	yes	characteristic, complemented
192	C₂×#41	—	113	4	no	—
193	C₂×#41	—	113	2	no	—
194	C₂×#41	C₂×D₁₁₃	1	1	yes	characteristic
195	C₂²×Q₈	—	113	2	no	—
196	C₄⋊Q₈	—	113	2	no	—
197	C₄⋊Q₈	—	113	2	no	—
198	C₄⋊Q₈	—	113	2	no	—
199	C₄⋊Q₈	—	113	2	no	—
200	C₄⋊Q₈	—	113	1	no	—
201	#127	—	113	1	no	—
202	#126	—	113	8	no	—
203	#125	—	113	4	no	—
204	#124	—	113	8	no	—
205	#124	—	113	2	no	—
206	#124	—	113	4	no	—
207	#123	—	113	8	no	—
208	C₂²⋊Q₈	—	113	4	no	—
209	C₂²⋊Q₈	—	113	4	no	—
210	C₂²⋊Q₈	—	113	4	no	—
211	C₂²⋊Q₈	—	113	8	no	—
212	#121	—	113	4	no	—
213	#121	—	113	8	no	—
214	#121	—	113	4	no	—
215	#121	—	113	2	no	—
216	#121	—	113	2	no	—
217	#121	—	113	4	no	—
218	C₄×Q₈	—	113	4	no	—
219	C₄×Q₈	—	113	4	no	—
220	C₄×Q₈	C₂×D₁₁₃	1	1	yes	characteristic
221	C₄×Q₈	C₂×D₁₁₃	1	1	yes	characteristic
222	C₄×D₄	—	113	8	no	—
223	C₄×D₄	—	113	4	no	—
224	C₄×D₄	—	113	2	no	—
225	C₄×D₄	—	113	4	no	—
226	C₄×D₄	—	113	2	no	—
227	C₄×D₄	C₂×D₁₁₃	1	1	yes	characteristic
228	C₄×D₄	C₂×D₁₁₃	1	2	yes	—
229	C₄×D₄	C₂×D₁₁₃	1	2	yes	—
230	C₄×D₄	C₂×D₁₁₃	1	1	yes	characteristic
231	#117	—	113	4	no	—
232	#117	—	113	4	no	—
233	#117	C₂×D₁₁₃	1	1	yes	characteristic
234	#117	C₂×D₁₁₃	1	2	yes	—
235	C₂×C₄⋊C₄	—	113	2	no	—
236	C₂×C₄⋊C₄	—	113	2	no	—
237	C₂×C₄⋊C₄	—	113	4	no	—
238	C₂×C₄²	C₂×D₁₁₃	1	1	yes	characteristic
239	C₂×C₄²	C₂×D₁₁₃	1	2	yes	—
240	#41	—	2	4	no	—
241	#41	—	452	8	no	—
242	#41	—	226	8	no	—
243	C₂×Q₈	—	113	4	no	—
244	C₂×Q₈	—	113	2	no	—
245	C₂×Q₈	—	113	4	no	—
246	C₂×Q₈	—	452	4	no	—
247	C₂×Q₈	C₂²×D₁₁₃	1	1	yes	characteristic
248	C₂×D₄	—	113	8	no	—
249	C₂×D₄	—	113	4	no	—
250	C₂×D₄	—	113	2	no	—
251	C₂×D₄	—	113	2	no	—
252	C₂×D₄	—	113	2	no	—
253	C₂×D₄	C₂²×D₁₁₃	1	2	yes	—
254	C₂×D₄	C₂²×D₁₁₃	1	1	yes	characteristic
255	C₂²×C₄	—	113	4	no	—
256	C₂²×C₄	—	113	8	no	—
257	C₂²×C₄	—	113	4	no	—
258	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
259	C₂²×C₄	C₂²×D₁₁₃	1	2	yes	—
260	C₂²×C₄	C₂²×D₁₁₃	1	2	yes	—
261	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
262	C₂²×C₄	C₂²×D₁₁₃	1	2	yes	—
263	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
264	C₄⋊C₄	—	113	4	no	—
265	C₄⋊C₄	—	113	4	no	—
266	C₄⋊C₄	—	113	4	no	—
267	C₄⋊C₄	—	113	8	no	—
268	C₄⋊C₄	—	113	4	no	—
269	C₄⋊C₄	—	113	2	no	—
270	C₄⋊C₄	—	113	2	no	—
271	C₄⋊C₄	—	113	8	no	—
272	C₄⋊C₄	C₂²×D₁₁₃	1	1	yes	characteristic
273	C₄⋊C₄	C₂²×D₁₁₃	1	1	yes	characteristic
274	C₄⋊C₄	C₂²×D₁₁₃	1	2	yes	—
275	C₄⋊C₄	C₂²×D₁₁₃	1	2	yes	—
276	C₂²⋊C₄	—	113	8	no	—
277	C₂²⋊C₄	—	113	8	no	—
278	C₂²⋊C₄	—	113	8	no	—
279	C₂²⋊C₄	—	113	4	no	—
280	C₂²⋊C₄	—	113	4	no	—
281	C₂²⋊C₄	C₂²×D₁₁₃	1	2	yes	—
282	C₂²⋊C₄	C₂²×D₁₁₃	1	2	yes	—
283	C₂²⋊C₄	C₂²×D₁₁₃	1	2	yes	—
284	C₄²	—	113	8	no	—
285	C₄²	C₂²×D₁₁₃	1	1	yes	characteristic
286	C₄²	C₂²×D₁₁₃	1	2	yes	—
287	C₄²	C₂²×D₁₁₃	1	1	yes	characteristic
288	C₄²	C₂²×D₁₁₃	1	2	yes	—
289	C₄²	C₂²×D₁₁₃	1	1	yes	characteristic
290	C₄²	C₂²×D₁₁₃	1	1	yes	characteristic
291	C₄²	C₂²×D₁₁₃	1	2	yes	—
292	C₂³	—	113	4	no	—
293	C₂³	C₂³×D₁₁₃	1	2	yes	—
294	C₂³	C₂³×D₁₁₃	1	1	yes	characteristic
295	Q₈	—	2	2	no	—
296	Q₈	—	452	4	no	—
297	Q₈	—	226	4	no	—
298	Q₈	—	226	8	no	—
299	D₄	—	2	4	no	—
300	D₄	—	2	2	no	—
301	D₄	—	452	8	no	—
302	D₄	—	226	8	no	—
303	D₄	—	226	4	no	—
304	D₄	—	226	4	no	—
305	D₄	—	226	4	no	—
306	C₂×C₄	—	2	4	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	4	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	4	no	—
311	C₂×C₄	—	113	4	no	—
312	C₂×C₄	—	113	8	no	—
313	C₂×C₄	—	452	8	no	—
314	C₂×C₄	—	452	4	no	—
315	C₂×C₄	—	226	8	no	—
316	C₂×C₄	#14268647	1	4	yes	complemented
317	C₂×C₄	C₂³×D₁₁₃	1	2	yes	—
318	C₂×C₄	C₂³×D₁₁₃	1	1	yes	characteristic
319	C₂×C₄	C₂³×D₁₁₃	1	1	yes	characteristic
320	C₂×C₄	C₂³×D₁₁₃	1	2	yes	—
321	C₂×C₄	C₂³×D₁₁₃	1	2	yes	—
322	C₂×C₄	C₂³×D₁₁₃	1	1	yes	characteristic
323	C₂×C₄	C₂³×D₁₁₃	1	1	yes	characteristic
324	C₂×C₄	C₂³×D₁₁₃	1	1	yes	characteristic
325	C₂×C₄	C₂³×D₁₁₃	1	1	yes	characteristic
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	2	2	no	—
329	C₂²	—	2	2	no	—
330	C₂²	—	2	2	no	—
331	C₂²	—	226	4	no	—
332	C₂²	—	226	4	no	—
333	C₂²	—	226	4	no	—
334	C₂²	C₂×#14268647	1	2	yes	—
335	C₂²	C₂⁴×D₁₁₃	1	1	yes	characteristic
336	C₄	—	2	2	no	—
337	C₄	—	2	2	no	—
338	C₄	—	2	1	no	—
339	C₄	—	2	2	no	—
340	C₄	—	2	1	no	—
341	C₄	—	226	4	no	—
342	C₄	—	226	8	no	—
343	C₄	C₂×#14268647	1	2	yes	—
344	C₄	C₂×#14268647	1	2	yes	—
345	C₄	#16813668	1	2	yes	—
346	C₄	#16813669	1	2	yes	—
347	C₂	—	2	2	no	—
348	C₂	—	4	2	no	—
349	C₂	—	452	4	no	—
350	C₂	C₂²×#14268647	1	1	yes	characteristic
351	C₂	C₂×#16813668	1	1	yes	characteristic
352	C₂	C₂×#16813669	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₁₃
commutator, socle, cosocle	C₂²×C₁₁₃	C₂⁵
Fitting	C₄×#41×C₁₁₃	C₂
hypercenter	C₄×#41	D₁₁₃
hypocenter	C₁₁₃	#3511
LCS	G ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
1. #20299321	C₂	semidirect
2. #20299313	C₂	semidirect
3. #20299308	C₂	semidirect
4. #20299294	C₂	semidirect
5. #20299257	C₂	semidirect
6. #20299253	C₂	semidirect
7. #20299219	C₂	semidirect
8. #20299176	C₂	semidirect
9. #20299155	C₂	semidirect
10. #20299148	C₂	semidirect
11. #20299148	C₂	semidirect
12. C₂×#16813622	C₂	semidirect
13. #20299078	C₂	semidirect
14. C₄×#41×C₁₁₃	C₂	semidirect
126. C₂×C₄×C₁₁₃	#41	semidirect
177. C₁₁₃	#3511	semidirect
191. C₄×#41	D₁₁₃	semidirect
316. C₂×C₄	#14268647	semidirect

Maximal subgroups

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

subgroup	index
#20299321	2
#20299313	2
#20299308	2
#20299294	2
#20299257	2
#20299253	2
#20299219	2
#20299176	2
#20299155	2
#20299148	2
C₂×#16813622	2
#20299078	2
C₄×#41×C₁₁₃	2
#3511	113