Subgroups — C₂×#20299077 #28403767

Order 14464.

1640 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14268646	C₂	1	1	yes	maximal, characteristic
2	#20299077	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₄×Dic₂₂₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#16813587	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₄×Dic₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×C₂²⋊D₁₁₃	C₂	1	2	yes	maximal, complemented
7	C₂×#16813582	C₂	1	2	yes	maximal, complemented
8	C₂×#16813579	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#16813578	C₂	1	2	yes	maximal, complemented
10	C₂²×#14268626	C₂	1	2	yes	maximal, complemented
11	C₂×C₄⋊C₄×D₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₄×D₄×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#14268646	C₄	1	16	yes	complemented
14	C₂×#14268646	C₂²	1	8	yes	—
15	C₂²×Dic₂₂₆	C₂²	1	1	yes	characteristic
16	C₂²×C₂²⋊D₁₁₃	C₂²	1	2	yes	—
17	C₂³×Dic₁₁₃	C₂²	1	2	yes	—
18	C₂³×Dic₁₁₃	C₂²	1	2	yes	—
19	C₂²×C₄×D₁₁₃	C₂²	1	2	yes	—
20	C₂²×C₄×D₁₁₃	C₂²	1	1	yes	characteristic
21	C₄×Dic₂₂₆	C₂²	1	8	yes	complemented
22	#16813587	C₂²	1	8	yes	complemented
23	D₄×Dic₁₁₃	C₂²	1	8	yes	complemented
24	C₄×C₂²⋊D₁₁₃	C₂²	1	16	yes	complemented
25	#16813582	C₂²	1	16	yes	complemented
26	#16813579	C₂²	1	8	yes	complemented
27	#16813578	C₂²	1	16	yes	complemented
28	C₂×C₄⋊Dic₁₁₃	C₂²	1	1	yes	characteristic
29	C₂×#14268626	C₂²	1	2	yes	—
30	C₂×#14268626	C₂²	1	1	yes	characteristic
31	C₂×#14268626	C₂²	1	2	yes	—
32	C₂×#14268626	C₂²	1	2	yes	—
33	C₂×#14268626	C₂²	1	1	yes	characteristic
34	C₂×#14268626	C₂²	1	16	yes	complemented
35	C₄⋊C₄×D₁₁₃	C₂²	1	8	yes	complemented
36	C₂×C₂²⋊Dic₁₁₃	C₂²	1	2	yes	—
37	C₂×#14268624	C₂²	1	1	yes	characteristic
38	C₂×#14268624	C₂²	1	1	yes	characteristic
39	C₂×C₄×Dic₁₁₃	C₂²	1	2	yes	—
40	C₂×C₄×Dic₁₁₃	C₂²	1	2	yes	—
41	C₂×C₄×Dic₁₁₃	C₂²	1	1	yes	characteristic
42	C₂²×D₄×C₁₁₃	C₂²	1	1	yes	characteristic
43	C₂³×C₄×C₁₁₃	C₂²	1	2	yes	—
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	2	yes	—
47	C₂×C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
48	C₂³×D₁₁₃	C₂³	1	1	yes	characteristic
49	#14268646	C₂×C₄	1	64	yes	complemented
50	C₂×Dic₂₂₆	C₂×C₄	1	8	yes	complemented
51	C₂×Dic₂₂₆	C₂³	1	4	yes	—
52	C₂×C₂²⋊D₁₁₃	C₂×C₄	1	16	yes	complemented
53	C₂×C₂²⋊D₁₁₃	C₂³	1	8	yes	—
54	C₂²×Dic₁₁₃	—	4	4	no	—
55	C₂²×Dic₁₁₃	C₂×C₄	1	16	yes	complemented
56	C₂²×Dic₁₁₃	C₂³	1	1	yes	characteristic
57	C₂²×Dic₁₁₃	C₂³	1	1	yes	characteristic
58	C₂²×Dic₁₁₃	C₂³	1	2	yes	—
59	C₂²×Dic₁₁₃	C₂³	1	8	yes	—
60	C₂²×Dic₁₁₃	C₂³	1	2	yes	—
61	C₂²×Dic₁₁₃	C₂³	1	1	yes	characteristic
62	C₂²×Dic₁₁₃	C₂³	1	8	yes	—
63	C₂×C₄×D₁₁₃	—	4	4	no	—
64	C₂×C₄×D₁₁₃	C₂×C₄	1	8	yes	complemented
65	C₂×C₄×D₁₁₃	C₂³	1	8	yes	—
66	C₂×C₄×D₁₁₃	C₂³	1	4	yes	—
67	C₄⋊Dic₁₁₃	C₂³	1	4	yes	—
68	#14268626	C₂³	1	8	yes	—
69	#14268626	C₂³	1	4	yes	—
70	#14268626	C₂³	1	8	yes	—
71	#14268626	C₂³	1	8	yes	—
72	#14268626	C₂³	1	4	yes	—
73	C₂²⋊Dic₁₁₃	C₂³	1	8	yes	—
74	#14268624	C₂³	1	4	yes	—
75	#14268624	C₂³	1	4	yes	—
76	C₄×Dic₁₁₃	C₂³	1	8	yes	—
77	C₄×Dic₁₁₃	C₂³	1	8	yes	—
78	C₄×Dic₁₁₃	C₂³	1	4	yes	—
79	C₂⁴×C₁₁₃	C₂³	1	2	yes	—
80	C₂×D₄×C₁₁₃	C₂×C₄	1	8	yes	complemented
81	C₂×D₄×C₁₁₃	C₂³	1	4	yes	—
82	C₂²×C₄×C₁₁₃	—	4	4	no	—
83	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
84	C₂²×C₄×C₁₁₃	C₂³	1	2	yes	—
85	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
86	C₂²×C₄×C₁₁₃	C₂³	1	8	yes	—
87	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
88	C₄⋊C₄×C₁₁₃	C₂³	1	4	yes	—
89	C₂²⋊C₄×C₁₁₃	C₂³	1	8	yes	—
90	C₄²×C₁₁₃	C₂³	1	4	yes	—
91	C₂³×C₁₁₃	—	2	4	no	—
92	C₂³×C₁₁₃	—	2	4	no	—
93	C₂³×C₁₁₃	C₂²×C₄	1	8	yes	—
94	C₂³×C₁₁₃	C₂⁴	1	4	yes	—
95	C₂³×C₁₁₃	C₂⁴	1	1	yes	characteristic
96	C₂²×D₁₁₃	—	2	2	no	—
97	C₂²×D₁₁₃	—	2	2	no	—
98	C₂²×D₁₁₃	C₂²×C₄	1	4	yes	—
99	C₂²×D₁₁₃	C₂⁴	1	2	yes	—
100	D₄×C₁₁₃	C₂²×C₄	1	16	yes	complemented
101	C₂×C₄×C₁₁₃	—	2	2	no	—
102	C₂×C₄×C₁₁₃	—	2	4	no	—
103	C₂×C₄×C₁₁₃	—	2	2	no	—
104	C₂×C₄×C₁₁₃	—	4	8	no	—
105	C₂×C₄×C₁₁₃	C₂²×C₄	1	4	yes	—
106	C₂×C₄×C₁₁₃	C₂⁴	1	2	yes	—
107	C₂×C₄×C₁₁₃	C₂⁴	1	2	yes	—
108	C₂×C₄×C₁₁₃	C₂⁴	1	4	yes	—
109	C₂×C₄×C₁₁₃	C₂⁴	1	2	yes	—
110	C₂²⋊D₁₁₃	C₂²×C₄	1	32	yes	complemented
111	C₂×Dic₁₁₃	—	2	4	no	—
112	C₂×Dic₁₁₃	—	2	2	no	—
113	C₂×Dic₁₁₃	—	2	2	no	—
114	C₂×Dic₁₁₃	—	4	8	no	—
115	C₂×Dic₁₁₃	C₂²×C₄	1	4	yes	—
116	C₂×Dic₁₁₃	C₂²×C₄	1	8	yes	—
117	C₂×Dic₁₁₃	C₂²×C₄	1	32	yes	complemented
118	C₂×Dic₁₁₃	C₂⁴	1	2	yes	—
119	C₂×Dic₁₁₃	C₂⁴	1	2	yes	—
120	C₂×Dic₁₁₃	C₂⁴	1	4	yes	—
121	C₂×Dic₁₁₃	C₂⁴	1	4	yes	—
122	C₂×Dic₁₁₃	C₂⁴	1	2	yes	—
123	C₄×D₁₁₃	—	4	8	no	—
124	C₄×D₁₁₃	C₂²×C₄	1	16	yes	complemented
125	Dic₂₂₆	C₂²×C₄	1	16	yes	complemented
126	C₂²×C₁₁₃	—	2	16	no	—
127	C₂²×C₁₁₃	—	2	4	no	—
128	C₂²×C₁₁₃	—	2	4	no	—
129	C₂²×C₁₁₃	C₂³×C₄	1	8	yes	—
130	C₂²×C₁₁₃	C₂³×C₄	1	2	yes	—
131	C₂²×C₁₁₃	H(5,𝔽₂)	1	2	yes	—
132	C₂²×C₁₁₃	2₋¹⁺⁴	1	2	yes	—
133	C₂²×C₁₁₃	C₂⁵	1	1	yes	characteristic
134	C₂×D₁₁₃	—	2	8	no	—
135	C₂×D₁₁₃	—	2	2	no	—
136	C₂×D₁₁₃	—	2	2	no	—
137	C₂×D₁₁₃	C₂³×C₄	1	4	yes	—
138	C₄×C₁₁₃	—	2	4	no	—
139	C₄×C₁₁₃	—	2	2	no	—
140	C₄×C₁₁₃	—	2	2	no	—
141	C₄×C₁₁₃	C₂³×C₄	1	4	yes	—
142	Dic₁₁₃	—	2	4	no	—
143	Dic₁₁₃	—	2	2	no	—
144	Dic₁₁₃	—	2	2	no	—
145	Dic₁₁₃	C₂³×C₄	1	8	yes	—
146	Dic₁₁₃	C₂³×C₄	1	4	yes	—
147	C₂×C₁₁₃	—	2	8	no	—
148	C₂×C₁₁₃	#520	1	4	yes	complemented
149	C₂×C₁₁₃	C₂⁴×C₄	1	1	yes	characteristic
150	C₂×C₁₁₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
151	C₂×C₁₁₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
152	D₁₁₃	—	2	4	no	—
153	C₂×#520	—	113	1	no	maximal, Sylow
154	C₁₁₃	C₂×#520	1	1	yes	characteristic, complemented, Sylow
155	C₂²×#41	—	113	1	no	—
156	#520	—	113	16	no	—
157	C₂×C₄×Q₈	—	113	1	no	—
158	C₂×C₄×Q₈	—	113	1	no	—
159	C₂×C₄×D₄	—	113	1	no	—
160	C₂×C₄×D₄	—	113	2	no	—
161	C₂×C₄×D₄	—	113	2	no	—
162	C₂×C₄×D₄	D₁₁₃	1	1	yes	characteristic, complemented
163	C₂×#117	—	113	2	no	—
164	C₂×#117	—	113	1	no	—
165	C₂²×C₄⋊C₄	—	113	2	no	—
166	C₂²×C₄⋊C₄	—	113	1	no	—
167	C₂×#41	—	113	16	no	—
168	C₂×#41	—	113	8	no	—
169	C₂²×Q₈	—	113	1	no	—
170	C₂²×D₄	—	113	2	no	—
171	C₂²×D₄	C₂×D₁₁₃	1	1	yes	characteristic
172	C₂³×C₄	—	113	2	no	—
173	C₂³×C₄	—	113	1	no	—
174	C₂³×C₄	—	113	2	no	—
175	C₂³×C₄	—	113	2	no	—
176	C₂³×C₄	C₂×D₁₁₃	1	2	yes	—
177	C₄×Q₈	—	113	8	no	—
178	C₄×Q₈	—	113	8	no	—
179	C₄×D₄	—	113	8	no	—
180	C₄×D₄	—	113	16	no	—
181	C₄×D₄	—	113	16	no	—
182	C₄×D₄	C₂×D₁₁₃	1	8	yes	complemented
183	#117	—	113	16	no	—
184	#117	—	113	8	no	—
185	C₂×C₄⋊C₄	—	113	2	no	—
186	C₂×C₄⋊C₄	—	113	1	no	—
187	C₂×C₄⋊C₄	—	113	2	no	—
188	C₂×C₄⋊C₄	—	113	2	no	—
189	C₂×C₄⋊C₄	—	113	1	no	—
190	C₂×C₄⋊C₄	—	113	16	no	—
191	C₂×C₄⋊C₄	—	113	1	no	—
192	C₂×C₄⋊C₄	—	113	8	no	—
193	C₂×C₄⋊C₄	C₂×D₁₁₃	1	1	yes	characteristic
194	C₂×C₂²⋊C₄	—	113	2	no	—
195	C₂×C₂²⋊C₄	—	113	1	no	—
196	C₂×C₂²⋊C₄	—	113	1	no	—
197	C₂×C₂²⋊C₄	C₂×D₁₁₃	1	2	yes	—
198	C₂×C₄²	—	113	2	no	—
199	C₂×C₄²	—	113	2	no	—
200	C₂×C₄²	—	113	1	no	—
201	C₂×C₄²	C₂×D₁₁₃	1	1	yes	characteristic
202	C₂⁴	—	113	1	no	—
203	C₂⁴	C₂²×D₁₁₃	1	2	yes	—
204	#41	—	113	64	no	—
205	C₂×Q₈	—	113	8	no	—
206	C₂×Q₈	—	113	4	no	—
207	C₂×D₄	—	113	16	no	—
208	C₂×D₄	—	113	8	no	—
209	C₂×D₄	C₄×D₁₁₃	1	8	yes	complemented
210	C₂×D₄	C₂²×D₁₁₃	1	4	yes	—
211	C₂²×C₄	—	4	4	no	—
212	C₂²×C₄	—	113	16	no	—
213	C₂²×C₄	—	113	8	no	—
214	C₂²×C₄	—	113	2	no	—
215	C₂²×C₄	—	113	1	no	—
216	C₂²×C₄	—	113	8	no	—
217	C₂²×C₄	—	113	4	no	—
218	C₂²×C₄	—	113	8	no	—
219	C₂²×C₄	—	113	1	no	—
220	C₂²×C₄	—	113	2	no	—
221	C₂²×C₄	—	113	1	no	—
222	C₂²×C₄	—	113	8	no	—
223	C₂²×C₄	—	452	4	no	—
224	C₂²×C₄	—	452	4	no	—
225	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
226	C₂²×C₄	C₂²×D₁₁₃	1	2	yes	—
227	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
228	C₂²×C₄	C₂²×D₁₁₃	1	8	yes	—
229	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
230	C₄⋊C₄	—	113	8	no	—
231	C₄⋊C₄	—	113	4	no	—
232	C₄⋊C₄	—	113	8	no	—
233	C₄⋊C₄	—	113	8	no	—
234	C₄⋊C₄	—	113	4	no	—
235	C₄⋊C₄	—	113	4	no	—
236	C₄⋊C₄	C₂²×D₁₁₃	1	4	yes	—
237	C₂²⋊C₄	—	113	8	no	—
238	C₂²⋊C₄	—	113	4	no	—
239	C₂²⋊C₄	—	113	4	no	—
240	C₂²⋊C₄	C₂²×D₁₁₃	1	8	yes	—
241	C₄²	—	113	8	no	—
242	C₄²	—	113	8	no	—
243	C₄²	—	113	4	no	—
244	C₄²	C₂²×D₁₁₃	1	4	yes	—
245	C₂³	—	2	4	no	—
246	C₂³	—	2	4	no	—
247	C₂³	—	113	4	no	—
248	C₂³	—	113	2	no	—
249	C₂³	—	226	2	no	—
250	C₂³	—	226	2	no	—
251	C₂³	C₂×C₄×D₁₁₃	1	8	yes	—
252	C₂³	C₂³×D₁₁₃	1	4	yes	—
253	C₂³	C₂³×D₁₁₃	1	1	yes	characteristic
254	Q₈	—	113	16	no	—
255	D₄	—	113	32	no	—
256	D₄	C₂×C₄×D₁₁₃	1	16	yes	complemented
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	4	8	no	—
261	C₂×C₄	—	113	4	no	—
262	C₂×C₄	—	113	8	no	—
263	C₂×C₄	—	113	32	no	—
264	C₂×C₄	—	113	4	no	—
265	C₂×C₄	—	113	2	no	—
266	C₂×C₄	—	113	16	no	—
267	C₂×C₄	—	113	2	no	—
268	C₂×C₄	—	113	4	no	—
269	C₂×C₄	—	113	2	no	—
270	C₂×C₄	—	452	8	no	—
271	C₂×C₄	—	452	8	no	—
272	C₂×C₄	—	226	4	no	—
273	C₂×C₄	—	226	2	no	—
274	C₂×C₄	—	226	2	no	—
275	C₂×C₄	C₂×C₄×D₁₁₃	1	4	yes	—
276	C₂×C₄	C₂³×D₁₁₃	1	2	yes	—
277	C₂×C₄	C₂³×D₁₁₃	1	4	yes	—
278	C₂×C₄	C₂³×D₁₁₃	1	2	yes	—
279	C₂×C₄	C₂³×D₁₁₃	1	2	yes	—
280	C₂²	—	2	16	no	—
281	C₂²	—	2	4	no	—
282	C₂²	—	2	4	no	—
283	C₂²	—	113	4	no	—
284	C₂²	—	226	8	no	—
285	C₂²	—	226	2	no	—
286	C₂²	—	226	2	no	—
287	C₂²	C₂²×C₄×D₁₁₃	1	8	yes	—
288	C₂²	C₂²×C₄×D₁₁₃	1	2	yes	—
289	C₂²	#16813667	1	2	yes	—
290	C₂²	#16813669	1	2	yes	—
291	C₂²	C₂⁴×D₁₁₃	1	1	yes	characteristic
292	C₄	—	2	2	no	—
293	C₄	—	2	4	no	—
294	C₄	—	2	2	no	—
295	C₄	—	113	4	no	—
296	C₄	—	113	8	no	—
297	C₄	—	226	4	no	—
298	C₄	—	226	2	no	—
299	C₄	—	226	2	no	—
300	C₄	C₂²×C₄×D₁₁₃	1	4	yes	—
301	C₂	—	2	8	no	—
302	C₂	—	226	4	no	—
303	C₂	#20299077	1	4	yes	direct product factor
304	C₂	C₂³×C₄×D₁₁₃	1	1	yes	characteristic
305	C₂	C₂×#16813667	1	1	yes	characteristic
306	C₂	C₂×#16813669	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₁₁₃
commutator	C₂×C₁₁₃	C₂⁴×C₄
Frattini	C₂²	C₂⁴×D₁₁₃
Fitting	C₂×C₄×D₄×C₁₁₃	C₂
socle	C₂³×C₁₁₃	C₂⁴
prime radical	C₂²×#14268646	C₂
cosocle	C₂²×C₁₁₃	C₂⁵
hypercenter	C₂×C₄×D₄	D₁₁₃
hypocenter	C₁₁₃	C₂×#520
LCS	G ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
2. #20299077	C₂	direct (N × H)
3. C₂×C₄×Dic₂₂₆	C₂	semidirect
4. C₂×#16813587	C₂	semidirect
5. C₂×D₄×Dic₁₁₃	C₂	semidirect
6. C₂×C₄×C₂²⋊D₁₁₃	C₂	semidirect
7. C₂×#16813582	C₂	semidirect
8. C₂×#16813579	C₂	semidirect
9. C₂×#16813578	C₂	semidirect
10. C₂²×#14268626	C₂	semidirect
11. C₂×C₄⋊C₄×D₁₁₃	C₂	semidirect
12. C₂×C₄×D₄×C₁₁₃	C₂	semidirect
13. C₂×#14268646	C₄	semidirect
21. C₄×Dic₂₂₆	C₂²	semidirect
22. #16813587	C₂²	semidirect
23. D₄×Dic₁₁₃	C₂²	semidirect
24. C₄×C₂²⋊D₁₁₃	C₂²	semidirect
25. #16813582	C₂²	semidirect
26. #16813579	C₂²	semidirect
27. #16813578	C₂²	semidirect
34. C₂×#14268626	C₂²	semidirect
35. C₄⋊C₄×D₁₁₃	C₂²	semidirect
44. C₄×D₄×C₁₁₃	C₂²	semidirect
49. #14268646	C₂×C₄	semidirect
50. C₂×Dic₂₂₆	C₂×C₄	semidirect
52. C₂×C₂²⋊D₁₁₃	C₂×C₄	semidirect
55. C₂²×Dic₁₁₃	C₂×C₄	semidirect
64. C₂×C₄×D₁₁₃	C₂×C₄	semidirect
80. C₂×D₄×C₁₁₃	C₂×C₄	semidirect
100. D₄×C₁₁₃	C₂²×C₄	semidirect
110. C₂²⋊D₁₁₃	C₂²×C₄	semidirect
117. C₂×Dic₁₁₃	C₂²×C₄	semidirect
124. C₄×D₁₁₃	C₂²×C₄	semidirect
125. Dic₂₂₆	C₂²×C₄	semidirect
148. C₂×C₁₁₃	#520	semidirect
154. C₁₁₃	C₂×#520	semidirect
162. C₂×C₄×D₄	D₁₁₃	semidirect
182. C₄×D₄	C₂×D₁₁₃	semidirect
209. C₂×D₄	C₄×D₁₁₃	semidirect
256. D₄	C₂×C₄×D₁₁₃	semidirect
303. C₂	#20299077	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×#14268646	2
#20299077	2
C₂×C₄×Dic₂₂₆	2
C₂×#16813587	2
C₂×D₄×Dic₁₁₃	2
C₂×C₄×C₂²⋊D₁₁₃	2
C₂×#16813582	2
C₂×#16813579	2
C₂×#16813578	2
C₂²×#14268626	2
C₂×C₄⋊C₄×D₁₁₃	2
C₂×C₄×D₄×C₁₁₃	2
C₂×#520	113

Subgroups — C2×#20299077 #28403767

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20299077 #28403767