Subgroups — #28404236

Order 14464.

1658 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#41×D₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
2	#20299267	C₂	1	2	yes	maximal, complemented
3	#20299255	C₂	1	2	yes	maximal, complemented
4	#20299242	C₂	1	6	yes	maximal, complemented
5	D₄×C₂²⋊D₁₁₃	C₂	1	6	yes	maximal, complemented
6	#20299176	C₂	1	3	yes	maximal, complemented
7	#20299165	C₂	1	3	yes	maximal, complemented
8	#20299105	C₂	1	3	yes	maximal, complemented
9	C₂×#16813593	C₂	1	3	yes	maximal, complemented
10	#20299081	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×H(5,𝔽₂)×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14268648	C₂²	1	1	yes	characteristic, complemented
13	C₂×#14268647	C₂²	1	3	yes	complemented
14	C₂×#14268646	C₂²	1	3	yes	complemented
15	#41×D₁₁₃	—	2	8	no	—
16	C₂×Q₈×D₁₁₃	C₂²	1	1	yes	characteristic, complemented
17	C₂²×C₂²⋊D₁₁₃	C₂²	1	6	yes	complemented
18	C₂×D₄×D₁₁₃	C₂²	1	3	yes	complemented
19	C₂²×C₄×D₁₁₃	C₂²	1	3	yes	complemented
20	#16813618	C₂²	1	6	yes	complemented
21	#16813609	C₂²	1	6	yes	complemented
22	#16813603	C₂²	1	6	yes	complemented
23	#16813601	C₂²	1	6	yes	complemented
24	#16813598	C₂²	1	1	yes	characteristic, complemented
25	#16813598	C₂²	1	3	yes	complemented
26	#16813596	C₂²	1	6	yes	complemented
27	#16813594	C₂²	1	12	yes	complemented
28	#16813593	C₂²	1	6	yes	complemented
29	#16813593	C₂²	1	3	yes	complemented
30	#16813593	C₂²	1	6	yes	complemented
31	#16813593	C₂²	1	6	yes	complemented
32	#16813593	C₂²	1	3	yes	complemented
33	#16813591	C₂²	1	12	yes	complemented
34	#16813590	C₂²	1	6	yes	complemented
35	Q₈×Dic₁₁₃	C₂²	1	2	yes	complemented
36	D₄×Dic₁₁₃	C₂²	1	6	yes	complemented
37	C₄×C₂²⋊D₁₁₃	C₂²	1	6	yes	complemented
38	C₄×C₂²⋊D₁₁₃	C₂²	1	6	yes	complemented
39	#16813581	C₂²	1	3	yes	complemented
40	C₂×C₄⋊Dic₁₁₃	C₂²	1	3	yes	complemented
41	C₂×C₂²⋊Dic₁₁₃	C₂²	1	6	yes	complemented
42	H(5,𝔽₂)×C₁₁₃	—	2	8	no	—
43	C₂×#41×C₁₁₃	C₂²	1	1	yes	characteristic
44	C₂×#41×C₁₁₃	C₂²	1	2	yes	complemented
45	C₂×#41×C₁₁₃	C₂²	1	3	yes	complemented
46	C₂²×D₄×C₁₁₃	C₂²	1	3	yes	complemented
47	C₂²×D₄×C₁₁₃	C₂²	1	6	yes	complemented
48	C₂³×D₁₁₃	C₂³	1	3	yes	—
49	#14268648	—	2	4	no	—
50	#14268647	—	2	12	no	—
51	#14268646	—	2	12	no	—
52	C₂×Dic₂₂₆	C₂³	1	3	yes	—
53	Q₈×D₁₁₃	—	2	4	no	—
54	C₂×D₄₅₂	C₂³	1	3	yes	—
55	C₂×C₂²⋊D₁₁₃	—	4	12	no	—
56	C₂×C₂²⋊D₁₁₃	C₂³	1	6	yes	—
57	C₂×C₂²⋊D₁₁₃	C₂³	1	6	yes	—
58	C₂×C₂²⋊D₁₁₃	C₂³	1	6	yes	—
59	C₂×C₂²⋊D₁₁₃	C₂³	1	12	yes	—
60	D₄×D₁₁₃	—	2	12	no	—
61	C₂²×Dic₁₁₃	C₂³	1	3	yes	—
62	C₂²×Dic₁₁₃	C₂³	1	6	yes	—
63	C₂×C₄×D₁₁₃	—	2	12	no	—
64	C₂×C₄×D₁₁₃	C₂³	1	3	yes	—
65	C₂×C₄×D₁₁₃	C₂³	1	3	yes	—
66	C₂×C₄×D₁₁₃	C₂³	1	1	yes	characteristic
67	C₂×C₄×D₁₁₃	C₂³	1	3	yes	—
68	C₄⋊Dic₁₁₃	C₂³	1	3	yes	—
69	C₄⋊Dic₁₁₃	C₂³	1	6	yes	—
70	C₄⋊Dic₁₁₃	C₂³	1	1	yes	characteristic
71	#14268626	C₂³	1	2	yes	—
72	#14268626	C₂³	1	6	yes	—
73	C₂²⋊Dic₁₁₃	C₂³	1	6	yes	—
74	C₂²⋊Dic₁₁₃	C₂³	1	12	yes	—
75	C₂²⋊Dic₁₁₃	C₂³	1	6	yes	—
76	C₂²⋊Dic₁₁₃	C₂³	1	6	yes	—
77	#14268624	C₂³	1	2	yes	—
78	#14268624	C₂³	1	6	yes	—
79	C₄×Dic₁₁₃	C₂³	1	6	yes	—
80	C₂⁴×C₁₁₃	C₂³	1	6	yes	—
81	#41×C₁₁₃	—	2	8	no	—
82	#41×C₁₁₃	—	2	12	no	—
83	#41×C₁₁₃	D₄	1	8	yes	complemented
84	C₂×Q₈×C₁₁₃	C₂³	1	1	yes	characteristic
85	C₂×Q₈×C₁₁₃	C₂³	1	1	yes	characteristic
86	C₂×D₄×C₁₁₃	—	2	24	no	—
87	C₂×D₄×C₁₁₃	—	2	12	no	—
88	C₂×D₄×C₁₁₃	C₂³	1	6	yes	—
89	C₂×D₄×C₁₁₃	C₂³	1	3	yes	—
90	C₂×D₄×C₁₁₃	C₂³	1	6	yes	—
91	C₂×D₄×C₁₁₃	C₂³	1	3	yes	—
92	C₂²×C₄×C₁₁₃	C₂³	1	6	yes	—
93	C₂²×C₄×C₁₁₃	C₂³	1	3	yes	—
94	C₂³×C₁₁₃	—	2	12	no	—
95	C₂³×C₁₁₃	—	4	6	no	—
96	C₂³×C₁₁₃	—	4	6	no	—
97	C₂³×C₁₁₃	C₂⁴	1	6	yes	—
98	C₂³×C₁₁₃	C₂⁴	1	3	yes	—
99	C₂²×D₁₁₃	—	2	6	no	—
100	C₂²×D₁₁₃	—	2	6	no	—
101	C₂²×D₁₁₃	—	4	3	no	—
102	C₂²×D₁₁₃	C₂⁴	1	3	yes	—
103	C₂²×D₁₁₃	C₂⁴	1	1	yes	characteristic
104	Q₈×C₁₁₃	—	2	2	no	—
105	Q₈×C₁₁₃	C₂×D₄	1	4	yes	complemented
106	D₄×C₁₁₃	—	2	12	no	—
107	D₄×C₁₁₃	—	2	12	no	—
108	D₄×C₁₁₃	—	2	6	no	—
109	D₄×C₁₁₃	C₂×D₄	1	12	yes	complemented
110	C₂×C₄×C₁₁₃	—	2	12	no	—
111	C₂×C₄×C₁₁₃	C₂×D₄	1	12	yes	complemented
112	C₂×C₄×C₁₁₃	C₂⁴	1	2	yes	—
113	C₂×C₄×C₁₁₃	C₂⁴	1	3	yes	—
114	C₂×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
115	C₂²⋊D₁₁₃	—	2	12	no	—
116	C₂²⋊D₁₁₃	—	2	12	no	—
117	C₂²⋊D₁₁₃	—	4	12	no	—
118	C₂²⋊D₁₁₃	—	4	6	no	—
119	C₂×Dic₁₁₃	—	2	6	no	—
120	C₂×Dic₁₁₃	—	4	6	no	—
121	C₂×Dic₁₁₃	C₂⁴	1	2	yes	—
122	C₂×Dic₁₁₃	C₂⁴	1	6	yes	—
123	C₂×Dic₁₁₃	C₂⁴	1	1	yes	characteristic
124	C₂×Dic₁₁₃	C₂⁴	1	3	yes	—
125	D₄₅₂	—	2	6	no	—
126	C₄×D₁₁₃	—	2	2	no	—
127	C₄×D₁₁₃	—	2	6	no	—
128	C₄×D₁₁₃	—	2	6	no	—
129	C₄×D₁₁₃	—	2	6	no	—
130	Dic₂₂₆	—	2	6	no	—
131	C₂²×C₁₁₃	—	2	3	no	—
132	C₂²×C₁₁₃	—	2	3	no	—
133	C₂²×C₁₁₃	—	2	6	no	—
134	C₂²×C₁₁₃	—	2	6	no	—
135	C₂²×C₁₁₃	—	2	6	no	—
136	C₂²×C₁₁₃	—	8	12	no	—
137	C₂²×C₁₁₃	C₂²×D₄	1	6	yes	—
138	C₂²×C₁₁₃	C₂⁵	1	1	yes	characteristic
139	C₂×D₁₁₃	—	2	1	no	—
140	C₂×D₁₁₃	—	2	1	no	—
141	C₂×D₁₁₃	—	2	3	no	—
142	C₂×D₁₁₃	—	2	3	no	—
143	C₂×D₁₁₃	—	2	3	no	—
144	C₂×D₁₁₃	—	4	12	no	—
145	C₂×D₁₁₃	H(5,𝔽₂)	1	2	yes	complemented
146	C₄×C₁₁₃	—	2	2	no	—
147	C₄×C₁₁₃	C₂²×D₄	1	6	yes	—
148	C₄×C₁₁₃	C₂²×D₄	1	2	yes	—
149	Dic₁₁₃	—	2	3	no	—
150	Dic₁₁₃	—	2	1	no	—
151	Dic₁₁₃	—	2	6	no	—
152	Dic₁₁₃	—	2	2	no	—
153	C₂×C₁₁₃	—	2	6	no	—
154	C₂×C₁₁₃	—	4	6	no	—
155	C₂×C₁₁₃	C₂³×D₄	1	1	yes	characteristic
156	C₂×C₁₁₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
157	C₂×C₁₁₃	#585	1	1	yes	characteristic
158	D₁₁₃	—	2	2	no	—
159	D₁₁₃	—	4	3	no	—
160	#3488	—	113	1	no	maximal, Sylow
161	C₁₁₃	#3488	1	1	yes	characteristic, complemented, Sylow
162	C₂×H(5,𝔽₂)	D₁₁₃	1	1	yes	characteristic, complemented
163	C₂²×#41	—	113	1	no	—
164	#550	—	113	2	no	—
165	#548	—	113	2	no	—
166	#546	—	113	6	no	—
167	D₄²	—	113	6	no	—
168	#537	—	113	3	no	—
169	#535	—	113	3	no	—
170	#525	—	113	3	no	—
171	C₂×#121	—	113	3	no	—
172	#520	—	113	1	no	—
173	H(5,𝔽₂)	—	2	8	no	—
174	C₂×#41	—	113	1	no	—
175	C₂×#41	—	113	3	no	—
176	C₂×#41	—	113	3	no	—
177	C₂×#41	—	226	8	no	—
178	C₂×#41	C₂×D₁₁₃	1	1	yes	characteristic
179	C₂×#41	C₂×D₁₁₃	1	3	yes	complemented
180	C₂×#41	C₂×D₁₁₃	1	2	yes	complemented
181	C₂²×Q₈	—	113	1	no	—
182	C₂²×D₄	—	113	6	no	—
183	C₂²×D₄	—	113	3	no	—
184	C₂²×D₄	C₂×D₁₁₃	1	3	yes	complemented
185	C₂²×D₄	C₂×D₁₁₃	1	6	yes	complemented
186	C₂³×C₄	—	113	3	no	—
187	#127	—	113	6	no	—
188	#124	—	113	6	no	—
189	#123	—	113	6	no	—
190	C₂²⋊Q₈	—	113	1	no	—
191	C₂²⋊Q₈	—	113	3	no	—
192	C₂²⋊Q₈	—	113	6	no	—
193	#121	—	113	3	no	—
194	#121	—	113	6	no	—
195	#121	—	113	6	no	—
196	#121	—	113	6	no	—
197	#121	—	113	3	no	—
198	#121	—	113	12	no	—
199	#121	—	113	6	no	—
200	#120	—	113	12	no	—
201	#120	—	113	6	no	—
202	C₄×Q₈	—	113	2	no	—
203	C₄×D₄	—	113	6	no	—
204	C₄×D₄	—	113	6	no	—
205	C₄×D₄	—	113	6	no	—
206	#117	—	113	3	no	—
207	C₂×C₄⋊C₄	—	113	3	no	—
208	C₂×C₂²⋊C₄	—	113	6	no	—
209	C₂⁴	—	113	3	no	—
210	C₂⁴	C₂²×D₁₁₃	1	6	yes	—
211	#41	—	2	8	no	—
212	#41	—	2	12	no	—
213	#41	—	226	4	no	—
214	#41	—	226	12	no	—
215	#41	—	226	12	no	—
216	#41	C₂²⋊D₁₁₃	1	8	yes	complemented
217	C₂×Q₈	—	113	3	no	—
218	C₂×Q₈	—	226	4	no	—
219	C₂×Q₈	C₂²×D₁₁₃	1	1	yes	characteristic
220	C₂×Q₈	C₂²×D₁₁₃	1	1	yes	characteristic
221	C₂×D₄	—	2	24	no	—
222	C₂×D₄	—	2	12	no	—
223	C₂×D₄	—	113	6	no	—
224	C₂×D₄	—	113	12	no	—
225	C₂×D₄	—	113	6	no	—
226	C₂×D₄	—	113	6	no	—
227	C₂×D₄	—	113	3	no	—
228	C₂×D₄	—	452	12	no	—
229	C₂×D₄	—	226	12	no	—
230	C₂×D₄	C₂²×D₁₁₃	1	6	yes	—
231	C₂×D₄	C₂²×D₁₁₃	1	3	yes	—
232	C₂×D₄	C₂²×D₁₁₃	1	6	yes	—
233	C₂×D₄	C₂²×D₁₁₃	1	3	yes	—
234	C₂²×C₄	—	113	6	no	—
235	C₂²×C₄	—	113	3	no	—
236	C₂²×C₄	—	113	3	no	—
237	C₂²×C₄	—	113	3	no	—
238	C₂²×C₄	—	113	1	no	—
239	C₂²×C₄	—	113	3	no	—
240	C₂²×C₄	—	226	12	no	—
241	C₂²×C₄	C₂²×D₁₁₃	1	6	yes	—
242	C₂²×C₄	C₂²×D₁₁₃	1	3	yes	—
243	C₄⋊C₄	—	113	3	no	—
244	C₄⋊C₄	—	113	6	no	—
245	C₄⋊C₄	—	113	1	no	—
246	C₄⋊C₄	—	113	2	no	—
247	C₄⋊C₄	—	113	6	no	—
248	C₂²⋊C₄	—	113	6	no	—
249	C₂²⋊C₄	—	113	12	no	—
250	C₂²⋊C₄	—	113	6	no	—
251	C₂²⋊C₄	—	113	6	no	—
252	C₂²⋊C₄	—	113	2	no	—
253	C₂²⋊C₄	—	113	6	no	—
254	C₄²	—	113	6	no	—
255	C₂³	—	2	12	no	—
256	C₂³	—	4	6	no	—
257	C₂³	—	4	6	no	—
258	C₂³	—	113	1	no	—
259	C₂³	—	113	3	no	—
260	C₂³	—	452	3	no	—
261	C₂³	—	226	6	no	—
262	C₂³	—	226	6	no	—
263	C₂³	C₂³×D₁₁₃	1	6	yes	—
264	C₂³	C₂³×D₁₁₃	1	3	yes	—
265	Q₈	—	2	2	no	—
266	Q₈	—	226	6	no	—
267	Q₈	C₂×C₂²⋊D₁₁₃	1	4	yes	complemented
268	D₄	—	2	12	no	—
269	D₄	—	2	12	no	—
270	D₄	—	2	6	no	—
271	D₄	—	452	12	no	—
272	D₄	—	452	6	no	—
273	D₄	—	226	12	no	—
274	D₄	—	226	12	no	—
275	D₄	—	226	6	no	—
276	D₄	C₂×C₂²⋊D₁₁₃	1	12	yes	complemented
277	C₂×C₄	—	2	12	no	—
278	C₂×C₄	—	113	2	no	—
279	C₂×C₄	—	113	6	no	—
280	C₂×C₄	—	113	1	no	—
281	C₂×C₄	—	113	3	no	—
282	C₂×C₄	—	452	6	no	—
283	C₂×C₄	—	226	6	no	—
284	C₂×C₄	—	226	2	no	—
285	C₂×C₄	—	226	6	no	—
286	C₂×C₄	—	226	6	no	—
287	C₂×C₄	—	226	6	no	—
288	C₂×C₄	C₂×C₂²⋊D₁₁₃	1	12	yes	complemented
289	C₂×C₄	C₂³×D₁₁₃	1	2	yes	—
290	C₂×C₄	C₂³×D₁₁₃	1	3	yes	—
291	C₂×C₄	C₂³×D₁₁₃	1	1	yes	characteristic
292	C₂²	—	2	3	no	—
293	C₂²	—	2	3	no	—
294	C₂²	—	2	6	no	—
295	C₂²	—	2	6	no	—
296	C₂²	—	2	6	no	—
297	C₂²	—	8	12	no	—
298	C₂²	—	113	2	no	—
299	C₂²	—	452	12	no	—
300	C₂²	—	226	3	no	—
301	C₂²	—	226	3	no	—
302	C₂²	—	226	3	no	—
303	C₂²	—	226	1	no	—
304	C₂²	—	226	1	no	—
305	C₂²	C₂²×C₂²⋊D₁₁₃	1	6	yes	—
306	C₂²	C₂⁴×D₁₁₃	1	1	yes	characteristic
307	C₄	—	2	2	no	—
308	C₄	—	226	1	no	—
309	C₄	—	226	3	no	—
310	C₄	—	226	6	no	—
311	C₄	—	226	2	no	—
312	C₄	C₂²×C₂²⋊D₁₁₃	1	6	yes	—
313	C₄	C₂²×C₂²⋊D₁₁₃	1	2	yes	—
314	C₂	—	2	6	no	—
315	C₂	—	4	6	no	—
316	C₂	—	452	3	no	—
317	C₂	—	226	2	no	—
318	C₂	C₂³×C₂²⋊D₁₁₃	1	1	yes	characteristic
319	C₂	H(5,𝔽₂)×D₁₁₃	1	1	yes	characteristic
320	C₂	#20299433	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₁₃
commutator, socle, cosocle	C₂²×C₁₁₃	C₂⁵
Fitting	C₂×H(5,𝔽₂)×C₁₁₃	C₂
hypercenter	C₂×H(5,𝔽₂)	D₁₁₃
hypocenter	C₁₁₃	#3488
LCS	G ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ C₂×H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
1. C₂×#41×D₁₁₃	C₂	semidirect
2. #20299267	C₂	semidirect
3. #20299255	C₂	semidirect
4. #20299242	C₂	semidirect
5. D₄×C₂²⋊D₁₁₃	C₂	semidirect
6. #20299176	C₂	semidirect
7. #20299165	C₂	semidirect
8. #20299105	C₂	semidirect
9. C₂×#16813593	C₂	semidirect
10. #20299081	C₂	semidirect
11. C₂×H(5,𝔽₂)×C₁₁₃	C₂	semidirect
12. C₂×#14268648	C₂²	semidirect
13. C₂×#14268647	C₂²	semidirect
14. C₂×#14268646	C₂²	semidirect
16. C₂×Q₈×D₁₁₃	C₂²	semidirect
17. C₂²×C₂²⋊D₁₁₃	C₂²	semidirect
18. C₂×D₄×D₁₁₃	C₂²	semidirect
19. C₂²×C₄×D₁₁₃	C₂²	semidirect
20. #16813618	C₂²	semidirect
21. #16813609	C₂²	semidirect
22. #16813603	C₂²	semidirect
23. #16813601	C₂²	semidirect
24. #16813598	C₂²	semidirect
25. #16813598	C₂²	semidirect
26. #16813596	C₂²	semidirect
27. #16813594	C₂²	semidirect
28. #16813593	C₂²	semidirect
29. #16813593	C₂²	semidirect
30. #16813593	C₂²	semidirect
31. #16813593	C₂²	semidirect
32. #16813593	C₂²	semidirect
33. #16813591	C₂²	semidirect
34. #16813590	C₂²	semidirect
35. Q₈×Dic₁₁₃	C₂²	semidirect
36. D₄×Dic₁₁₃	C₂²	semidirect
37. C₄×C₂²⋊D₁₁₃	C₂²	semidirect
38. C₄×C₂²⋊D₁₁₃	C₂²	semidirect
39. #16813581	C₂²	semidirect
40. C₂×C₄⋊Dic₁₁₃	C₂²	semidirect
41. C₂×C₂²⋊Dic₁₁₃	C₂²	semidirect
44. C₂×#41×C₁₁₃	C₂²	semidirect
45. C₂×#41×C₁₁₃	C₂²	semidirect
46. C₂²×D₄×C₁₁₃	C₂²	semidirect
47. C₂²×D₄×C₁₁₃	C₂²	semidirect
83. #41×C₁₁₃	D₄	semidirect
105. Q₈×C₁₁₃	C₂×D₄	semidirect
109. D₄×C₁₁₃	C₂×D₄	semidirect
111. C₂×C₄×C₁₁₃	C₂×D₄	semidirect
145. C₂×D₁₁₃	H(5,𝔽₂)	semidirect
161. C₁₁₃	#3488	semidirect
162. C₂×H(5,𝔽₂)	D₁₁₃	semidirect
179. C₂×#41	C₂×D₁₁₃	semidirect
180. C₂×#41	C₂×D₁₁₃	semidirect
184. C₂²×D₄	C₂×D₁₁₃	semidirect
185. C₂²×D₄	C₂×D₁₁₃	semidirect
216. #41	C₂²⋊D₁₁₃	semidirect
267. Q₈	C₂×C₂²⋊D₁₁₃	semidirect
276. D₄	C₂×C₂²⋊D₁₁₃	semidirect
288. C₂×C₄	C₂×C₂²⋊D₁₁₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#41×D₁₁₃	2
#20299267	2
#20299255	2
#20299242	2
D₄×C₂²⋊D₁₁₃	2
#20299176	2
#20299165	2
#20299105	2
C₂×#16813593	2
#20299081	2
C₂×H(5,𝔽₂)×C₁₁₃	2
#3488	113