Subgroups — #28404386

Order 14464.

1060 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×Dic₂₂₆	C₂	1	2	yes	maximal, complemented
2	#20299326	C₂	1	2	yes	maximal, complemented
3	#20299308	C₂	1	2	yes	maximal, complemented
4	#20299306	C₂	1	4	yes	maximal, complemented
5	Q₈×D₄₅₂	C₂	1	2	yes	maximal, complemented
6	#20299261	C₂	1	4	yes	maximal, complemented
7	#20299228	C₂	1	4	yes	maximal, complemented
8	#20299226	C₂	1	2	yes	maximal, complemented
9	#20299153	C₂	1	2	yes	maximal, complemented
10	C₂×#16813622	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄⋊Q₈×D₁₁₃	C₂	1	2	yes	maximal, complemented
12	#20299078	C₂	1	2	yes	maximal
13	C₄×#14268647	C₂	1	1	yes	maximal, characteristic
14	#533×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×#14268647	C₂²	1	1	yes	characteristic
16	C₂²×Dic₂₂₆	C₂²	1	2	yes	—
17	C₂×Q₈×D₁₁₃	C₂²	1	4	yes	—
18	#16813622	C₂²	1	2	yes	—
19	#16813622	C₂²	1	4	yes	—
20	#16813622	C₂²	1	2	yes	—
21	#16813622	C₂²	1	2	yes	—
22	#16813622	C₂²	1	2	yes	—
23	#16813621	C₂²	1	4	yes	—
24	#16813621	C₂²	1	8	yes	—
25	#16813621	C₂²	1	4	yes	—
26	#16813620	C₂²	1	4	yes	—
27	#16813614	C₂²	1	4	yes	—
28	#16813613	C₂²	1	4	yes	—
29	#16813612	C₂²	1	8	yes	—
30	#16813602	C₂²	1	8	yes	—
31	#16813602	C₂²	1	4	yes	—
32	#16813600	C₂²	1	8	yes	—
33	#16813599	C₂²	1	8	yes	—
34	#16813598	C₂²	1	4	yes	—
35	Q₈×Dic₁₁₃	C₂²	1	4	yes	—
36	C₄×Dic₂₂₆	C₂²	1	1	yes	characteristic
37	C₄×Dic₂₂₆	C₂²	1	1	yes	characteristic
38	C₄×Dic₂₂₆	C₂²	1	4	yes	—
39	#16813587	C₂²	1	4	yes	—
40	C₄×D₄₅₂	C₂²	1	1	yes	characteristic
41	C₄×D₄₅₂	C₂²	1	4	yes	—
42	C₄×D₄₅₂	C₂²	1	1	yes	characteristic
43	C₄×C₂²⋊D₁₁₃	C₂²	1	2	yes	—
44	C₄×C₂²⋊D₁₁₃	C₂²	1	2	yes	—
45	#16813582	C₂²	1	8	yes	—
46	#16813581	C₂²	1	1	yes	characteristic
47	#16813580	C₂²	1	4	yes	—
48	#16813579	C₂²	1	2	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	C₄⋊C₄×D₁₁₃	C₂²	1	4	yes	—
53	C₄²×D₁₁₃	C₂²	1	2	yes	—
54	C₄⋊Q₈×C₁₁₃	C₂²	1	1	yes	characteristic
55	C₄⋊Q₈×C₁₁₃	C₂²	1	1	yes	characteristic
56	#125×C₁₁₃	C₂²	1	2	yes	—
57	C₂²⋊Q₈×C₁₁₃	C₂²	1	2	yes	—
58	C₂²⋊Q₈×C₁₁₃	C₂²	1	2	yes	—
59	C₄×Q₈×C₁₁₃	C₂²	1	2	yes	—
60	C₄×Q₈×C₁₁₃	C₂²	1	2	yes	—
61	#117×C₁₁₃	C₂²	1	2	yes	—
62	C₂×C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
63	#14268647	Q₈	1	8	yes	complemented
64	C₂×Dic₂₂₆	—	4	4	no	—
65	C₂×Dic₂₂₆	C₂³	1	4	yes	—
66	C₂×Dic₂₂₆	C₂³	1	8	yes	—
67	C₂×Dic₂₂₆	C₂³	1	4	yes	—
68	C₂×Dic₂₂₆	C₂³	1	1	yes	characteristic
69	Q₈×D₁₁₃	—	4	8	no	—
70	C₂×D₄₅₂	C₂³	1	1	yes	characteristic
71	C₂×C₂²⋊D₁₁₃	C₂³	1	2	yes	—
72	C₂²×Dic₁₁₃	C₂³	1	4	yes	—
73	C₂×C₄×D₁₁₃	C₂³	1	2	yes	—
74	C₂×C₄×D₁₁₃	C₂³	1	2	yes	—
75	C₂×C₄×D₁₁₃	C₂³	1	8	yes	—
76	C₂×C₄×D₁₁₃	C₂³	1	2	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	1	yes	characteristic
82	C₄⋊Dic₁₁₃	C₂³	1	1	yes	characteristic
83	C₄⋊Dic₁₁₃	C₂³	1	8	yes	—
84	#14268626	C₂³	1	8	yes	—
85	#14268626	C₂³	1	8	yes	—
86	#14268626	C₂³	1	2	yes	—
87	#14268626	C₂³	1	2	yes	—
88	#14268626	C₂³	1	8	yes	—
89	C₂²⋊Dic₁₁₃	C₂³	1	2	yes	—
90	#14268624	C₂³	1	2	yes	—
91	#14268624	C₂³	1	2	yes	—
92	#14268624	C₂³	1	8	yes	—
93	C₄×Dic₁₁₃	C₂³	1	2	yes	—
94	C₄×Dic₁₁₃	C₂³	1	2	yes	—
95	C₄×Dic₁₁₃	C₂³	1	2	yes	—
96	C₄×Dic₁₁₃	C₂³	1	8	yes	—
97	C₂×Q₈×C₁₁₃	C₂³	1	2	yes	—
98	C₂×Q₈×C₁₁₃	C₂³	1	2	yes	—
99	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
100	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
101	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₁₁₃	C₂³	1	2	yes	—
103	C₄⋊C₄×C₁₁₃	C₂³	1	2	yes	—
104	C₄⋊C₄×C₁₁₃	C₂³	1	2	yes	—
105	C₄⋊C₄×C₁₁₃	C₂³	1	2	yes	—
106	C₄⋊C₄×C₁₁₃	C₂³	1	4	yes	—
107	C₄⋊C₄×C₁₁₃	C₂³	1	4	yes	—
108	C₂²⋊C₄×C₁₁₃	C₂³	1	4	yes	—
109	C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
110	C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
111	C₄²×C₁₁₃	C₂³	1	4	yes	—
112	C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
113	C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
114	C₂³×C₁₁₃	C₂⁴	1	1	yes	characteristic
115	C₂²×D₁₁₃	C₂⁴	1	2	yes	—
116	Q₈×C₁₁₃	—	2	4	no	—
117	Q₈×C₁₁₃	—	2	4	no	—
118	C₂×C₄×C₁₁₃	—	2	2	no	—
119	C₂×C₄×C₁₁₃	—	2	2	no	—
120	C₂×C₄×C₁₁₃	C₂×Q₈	1	4	yes	complemented
121	C₂×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
122	C₂×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
123	C₂×C₄×C₁₁₃	C₂⁴	1	4	yes	—
124	C₂×C₄×C₁₁₃	C₂⁴	1	4	yes	—
125	C₂×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
126	C₂×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
127	C₂×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
128	C₂×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
129	C₂²⋊D₁₁₃	C₂×Q₈	1	8	yes	complemented
130	C₂×Dic₁₁₃	—	4	4	no	—
131	C₂×Dic₁₁₃	C₂⁴	1	2	yes	—
132	C₂×Dic₁₁₃	C₂⁴	1	2	yes	—
133	C₂×Dic₁₁₃	C₂⁴	1	8	yes	—
134	C₂×Dic₁₁₃	C₂⁴	1	2	yes	—
135	D₄₅₂	C₂×Q₈	1	4	yes	complemented
136	C₄×D₁₁₃	—	2	4	no	—
137	C₄×D₁₁₃	—	2	4	no	—
138	C₄×D₁₁₃	—	4	8	no	—
139	C₄×D₁₁₃	C₂×Q₈	1	8	yes	complemented
140	Dic₂₂₆	—	2	8	no	—
141	Dic₂₂₆	—	4	8	no	—
142	Dic₂₂₆	—	4	4	no	—
143	Dic₂₂₆	C₂×Q₈	1	4	yes	complemented
144	C₂²×C₁₁₃	—	2	1	no	—
145	C₂²×C₁₁₃	—	2	1	no	—
146	C₂²×C₁₁₃	C₂²×Q₈	1	2	yes	—
147	C₂²×C₁₁₃	C₂⁵	1	1	yes	characteristic
148	C₂×D₁₁₃	—	2	2	no	—
149	C₂×D₁₁₃	—	2	2	no	—
150	C₂×D₁₁₃	C₂²×Q₈	1	4	yes	—
151	C₄×C₁₁₃	—	2	1	no	—
152	C₄×C₁₁₃	—	2	4	no	—
153	C₄×C₁₁₃	—	2	1	no	—
154	C₄×C₁₁₃	—	2	4	no	—
155	C₄×C₁₁₃	C₂²×Q₈	1	2	yes	—
156	C₄×C₁₁₃	C₂²×Q₈	1	2	yes	—
157	C₄×C₁₁₃	H(5,𝔽₂)	1	2	yes	—
158	C₄×C₁₁₃	2₋¹⁺⁴	1	2	yes	—
159	Dic₁₁₃	—	2	8	no	—
160	Dic₁₁₃	—	2	2	no	—
161	Dic₁₁₃	—	2	2	no	—
162	Dic₁₁₃	C₂²×Q₈	1	4	yes	—
163	C₂×C₁₁₃	—	2	2	no	—
164	C₂×C₁₁₃	C₂³×Q₈	1	1	yes	characteristic
165	C₂×C₁₁₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
166	C₂×C₁₁₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
167	D₁₁₃	—	2	4	no	—
168	#3503	—	113	1	no	maximal, Sylow
169	C₁₁₃	#3503	1	1	yes	characteristic, complemented, Sylow
170	Q₈²	—	113	2	no	—
171	#557	—	113	2	no	—
172	#554	—	113	4	no	—
173	#554	—	113	2	no	—
174	D₄×Q₈	—	113	4	no	—
175	D₄×Q₈	—	113	2	no	—
176	#544	—	113	4	no	—
177	#544	—	113	2	no	—
178	#533	—	113	2	no	—
179	#533	D₁₁₃	1	1	yes	characteristic, complemented
180	C₂×C₄⋊Q₈	—	113	2	no	—
181	C₂×C₄⋊Q₈	—	113	1	no	—
182	#520	—	113	2	no	—
183	C₄×#41	—	113	1	no	—
184	C₂×#41	—	113	1	no	—
185	C₂²×Q₈	—	113	4	no	—
186	C₂²×Q₈	—	113	2	no	—
187	C₄⋊Q₈	—	113	4	no	—
188	C₄⋊Q₈	—	113	4	no	—
189	C₄⋊Q₈	—	113	4	no	—
190	C₄⋊Q₈	—	113	8	no	—
191	C₄⋊Q₈	—	113	2	no	—
192	C₄⋊Q₈	—	113	2	no	—
193	C₄⋊Q₈	—	113	2	no	—
194	C₄⋊Q₈	—	113	2	no	—
195	C₄⋊Q₈	—	113	4	no	—
196	C₄⋊Q₈	C₂×D₁₁₃	1	1	yes	characteristic
197	C₄⋊Q₈	C₂×D₁₁₃	1	1	yes	characteristic
198	#125	—	113	4	no	—
199	#125	—	113	8	no	—
200	#125	—	113	4	no	—
201	#125	C₂×D₁₁₃	1	2	yes	—
202	C₂²⋊Q₈	—	113	4	no	—
203	C₂²⋊Q₈	—	113	4	no	—
204	C₂²⋊Q₈	—	113	8	no	—
205	C₂²⋊Q₈	—	113	8	no	—
206	C₂²⋊Q₈	—	113	8	no	—
207	C₂²⋊Q₈	C₂×D₁₁₃	1	2	yes	—
208	C₂²⋊Q₈	C₂×D₁₁₃	1	2	yes	—
209	C₄×Q₈	—	113	4	no	—
210	C₄×Q₈	—	113	4	no	—
211	C₄×Q₈	—	113	1	no	—
212	C₄×Q₈	—	113	1	no	—
213	C₄×Q₈	—	113	4	no	—
214	C₄×Q₈	C₂×D₁₁₃	1	2	yes	—
215	C₄×Q₈	C₂×D₁₁₃	1	2	yes	—
216	C₄×D₄	—	113	2	no	—
217	C₄×D₄	—	113	2	no	—
218	C₄×D₄	—	113	8	no	—
219	C₄×D₄	—	113	1	no	—
220	C₄×D₄	—	113	1	no	—
221	C₄×D₄	—	113	4	no	—
222	#117	—	113	1	no	—
223	#117	—	113	4	no	—
224	#117	—	113	2	no	—
225	#117	C₂×D₁₁₃	1	2	yes	—
226	C₂×C₄⋊C₄	—	113	2	no	—
227	C₂×C₄⋊C₄	—	113	2	no	—
228	C₂×C₄⋊C₄	—	113	4	no	—
229	C₂×C₄⋊C₄	—	113	4	no	—
230	C₂×C₄²	—	113	2	no	—
231	C₂×C₄²	C₂×D₁₁₃	1	1	yes	characteristic
232	#41	—	113	8	no	—
233	C₂×Q₈	—	113	1	no	—
234	C₂×Q₈	—	113	8	no	—
235	C₂×Q₈	—	113	4	no	—
236	C₂×Q₈	—	113	4	no	—
237	C₂×Q₈	—	452	4	no	—
238	C₂×Q₈	—	452	8	no	—
239	C₂×Q₈	C₂²×D₁₁₃	1	2	yes	—
240	C₂×Q₈	C₂²×D₁₁₃	1	2	yes	—
241	C₂×D₄	—	113	2	no	—
242	C₂×D₄	—	113	1	no	—
243	C₂²×C₄	—	113	2	no	—
244	C₂²×C₄	—	113	4	no	—
245	C₂²×C₄	—	113	8	no	—
246	C₂²×C₄	—	113	2	no	—
247	C₂²×C₄	—	113	2	no	—
248	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
249	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
250	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
251	C₄⋊C₄	—	113	4	no	—
252	C₄⋊C₄	—	113	4	no	—
253	C₄⋊C₄	—	113	8	no	—
254	C₄⋊C₄	—	113	4	no	—
255	C₄⋊C₄	—	113	8	no	—
256	C₄⋊C₄	—	113	4	no	—
257	C₄⋊C₄	—	113	2	no	—
258	C₄⋊C₄	—	113	1	no	—
259	C₄⋊C₄	—	113	2	no	—
260	C₄⋊C₄	—	113	1	no	—
261	C₄⋊C₄	—	113	8	no	—
262	C₄⋊C₄	—	113	8	no	—
263	C₄⋊C₄	C₂²×D₁₁₃	1	2	yes	—
264	C₄⋊C₄	C₂²×D₁₁₃	1	2	yes	—
265	C₄⋊C₄	C₂²×D₁₁₃	1	2	yes	—
266	C₄⋊C₄	C₂²×D₁₁₃	1	2	yes	—
267	C₄⋊C₄	C₂²×D₁₁₃	1	4	yes	—
268	C₄⋊C₄	C₂²×D₁₁₃	1	4	yes	—
269	C₂²⋊C₄	—	113	2	no	—
270	C₂²⋊C₄	—	113	2	no	—
271	C₂²⋊C₄	—	113	2	no	—
272	C₂²⋊C₄	—	113	8	no	—
273	C₂²⋊C₄	C₂²×D₁₁₃	1	4	yes	—
274	C₄²	—	113	2	no	—
275	C₄²	—	113	2	no	—
276	C₄²	—	113	2	no	—
277	C₄²	—	113	8	no	—
278	C₄²	C₂²×D₁₁₃	1	1	yes	characteristic
279	C₄²	C₂²×D₁₁₃	1	1	yes	characteristic
280	C₄²	C₂²×D₁₁₃	1	1	yes	characteristic
281	C₄²	C₂²×D₁₁₃	1	1	yes	characteristic
282	C₄²	C₂²×D₁₁₃	1	4	yes	—
283	C₂³	—	113	2	no	—
284	C₂³	C₂³×D₁₁₃	1	1	yes	characteristic
285	Q₈	—	2	4	no	—
286	Q₈	—	2	4	no	—
287	Q₈	—	113	4	no	—
288	Q₈	—	452	4	no	—
289	Q₈	—	452	8	no	—
290	Q₈	—	226	8	no	—
291	D₄	—	113	8	no	—
292	D₄	—	113	4	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	113	2	no	—
296	C₂×C₄	—	113	8	no	—
297	C₂×C₄	—	113	2	no	—
298	C₂×C₄	—	113	2	no	—
299	C₂×C₄	—	113	8	no	—
300	C₂×C₄	—	452	8	no	—
301	C₂×C₄	—	452	4	no	—
302	C₂×C₄	—	226	4	no	—
303	C₂×C₄	—	226	4	no	—
304	C₂×C₄	Q₈×D₁₁₃	1	4	yes	complemented
305	C₂×C₄	C₂³×D₁₁₃	1	4	yes	—
306	C₂×C₄	C₂³×D₁₁₃	1	1	yes	characteristic
307	C₂×C₄	C₂³×D₁₁₃	1	1	yes	characteristic
308	C₂×C₄	C₂³×D₁₁₃	1	4	yes	—
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	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	113	4	no	—
316	C₂²	—	226	2	no	—
317	C₂²	—	226	2	no	—
318	C₂²	C₂×Q₈×D₁₁₃	1	2	yes	—
319	C₂²	C₂⁴×D₁₁₃	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	2	4	no	—
322	C₄	—	2	1	no	—
323	C₄	—	2	4	no	—
324	C₄	—	113	4	no	—
325	C₄	—	226	8	no	—
326	C₄	—	226	2	no	—
327	C₄	—	226	2	no	—
328	C₄	C₂×Q₈×D₁₁₃	1	2	yes	—
329	C₄	C₂×Q₈×D₁₁₃	1	2	yes	—
330	C₄	#16813668	1	2	yes	—
331	C₄	#16813669	1	2	yes	—
332	C₂	—	2	2	no	—
333	C₂	—	226	4	no	—
334	C₂	C₂²×Q₈×D₁₁₃	1	1	yes	characteristic
335	C₂	C₂×#16813668	1	1	yes	characteristic
336	C₂	C₂×#16813669	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₁₃
commutator, socle, cosocle	C₂²×C₁₁₃	C₂⁵
Fitting	#533×C₁₁₃	C₂
prime radical	C₂×#14268647	C₂²
hypercenter	#533	D₁₁₃
hypocenter	C₁₁₃	#3503
LCS	G ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ #533

Complements

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

N	H ≅ G/N	type
1. Q₈×Dic₂₂₆	C₂	semidirect
2. #20299326	C₂	semidirect
3. #20299308	C₂	semidirect
4. #20299306	C₂	semidirect
5. Q₈×D₄₅₂	C₂	semidirect
6. #20299261	C₂	semidirect
7. #20299228	C₂	semidirect
8. #20299226	C₂	semidirect
9. #20299153	C₂	semidirect
10. C₂×#16813622	C₂	semidirect
11. C₄⋊Q₈×D₁₁₃	C₂	semidirect
14. #533×C₁₁₃	C₂	semidirect
63. #14268647	Q₈	semidirect
120. C₂×C₄×C₁₁₃	C₂×Q₈	semidirect
129. C₂²⋊D₁₁₃	C₂×Q₈	semidirect
135. D₄₅₂	C₂×Q₈	semidirect
139. C₄×D₁₁₃	C₂×Q₈	semidirect
143. Dic₂₂₆	C₂×Q₈	semidirect
169. C₁₁₃	#3503	semidirect
179. #533	D₁₁₃	semidirect
304. C₂×C₄	Q₈×D₁₁₃	semidirect

Maximal subgroups

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

subgroup	index
Q₈×Dic₂₂₆	2
#20299326	2
#20299308	2
#20299306	2
Q₈×D₄₅₂	2
#20299261	2
#20299228	2
#20299226	2
#20299153	2
C₂×#16813622	2
C₄⋊Q₈×D₁₁₃	2
#20299078	2
C₄×#14268647	2
#533×C₁₁₃	2
#3503	113