Subgroups — C₂×#20299257 #28404040

Order 14464.

1620 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

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

	subgroup	G/H
center	C₂³	C₂³×D₁₁₃
commutator, cosocle	C₂²×C₁₁₃	C₂⁵
Frattini	C₂²	C₂⁴×D₁₁₃
Fitting	C₂×C₄×D₄×C₁₁₃	C₂
socle	C₂³×C₁₁₃	C₂⁴
hypercenter	C₂×C₄×D₄	D₁₁₃
hypocenter	C₁₁₃	C₂×#548
LCS	G ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#14268647	C₂	semidirect
2. C₂³×Dic₂₂₆	C₂	semidirect
3. #20299257	C₂	direct (N × H)
4. C₂×#16813610	C₂	semidirect
5. C₂×#16813608	C₂	semidirect
6. C₂×#16813601	C₂	semidirect
7. C₂×#16813600	C₂	semidirect
8. C₂×#16813596	C₂	semidirect
9. C₂×#16813594	C₂	semidirect
10. C₂×C₄×Dic₂₂₆	C₂	semidirect
11. C₂×#16813582	C₂	semidirect
12. C₂×C₄×D₄×C₁₁₃	C₂	semidirect
14. C₂×#14268647	C₂²	semidirect
19. C₂²×Dic₂₂₆	C₂²	semidirect
25. #16813610	C₂²	semidirect
26. #16813608	C₂²	semidirect
27. #16813601	C₂²	semidirect
28. #16813600	C₂²	semidirect
29. #16813596	C₂²	semidirect
30. #16813594	C₂²	semidirect
31. C₄×Dic₂₂₆	C₂²	semidirect
32. #16813582	C₂²	semidirect
44. C₄×D₄×C₁₁₃	C₂²	semidirect
54. C₂×Dic₂₂₆	D₄	semidirect
101. C₂³×C₁₁₃	#41	semidirect
135. Dic₂₂₆	C₂×D₄	semidirect
145. C₂²×C₁₁₃	C₂×#41	semidirect
162. C₂×C₁₁₃	#548	semidirect
168. C₁₁₃	C₂×#548	semidirect
180. C₂×C₄×D₄	D₁₁₃	semidirect
204. C₄×D₄	C₂×D₁₁₃	semidirect
273. C₂³	#14268647	semidirect
317. C₂²	C₂×#14268647	semidirect
331. C₂	#20299257	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#14268647	2
C₂³×Dic₂₂₆	2
#20299257	2
C₂×#16813610	2
C₂×#16813608	2
C₂×#16813601	2
C₂×#16813600	2
C₂×#16813596	2
C₂×#16813594	2
C₂×C₄×Dic₂₂₆	2
C₂×#16813582	2
C₂×C₄×D₄×C₁₁₃	2
C₂×#548	113

Subgroups — C2×#20299257 #28404040

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20299257 #28404040