Subgroups — C₂×#20299304 #28404104

Order 14464.

1200 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20299304	C₂	1	16	yes	maximal, direct product factor
2	C₂×#16813621	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16813613	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#16813602	C₂	1	2	yes	maximal, complemented
5	C₂×#16813600	C₂	1	2	yes	maximal, complemented
6	C₂×#16813600	C₂	1	2	yes	maximal, complemented
7	C₂×C₄×Dic₂₂₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₄₅₂	C₂	1	1	yes	maximal, characteristic
9	C₂×#16813584	C₂	1	1	yes	maximal, characteristic
10	C₂×#16813584	C₂	1	1	yes	maximal, characteristic
11	C₂×C₄⋊C₄×D₁₁₃	C₂	1	2	yes	maximal, complemented
12	C₂×#125×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×Dic₂₂₆	C₂²	1	1	yes	characteristic
14	C₂²×Dic₂₂₆	C₂²	1	2	yes	—
15	C₂²×D₄₅₂	C₂²	1	1	yes	characteristic
16	C₂²×C₄×D₁₁₃	C₂²	1	2	yes	—
17	C₂²×C₄×D₁₁₃	C₂²	1	2	yes	—
18	C₂²×C₄×D₁₁₃	C₂²	1	2	yes	—
19	#16813621	C₂²	1	8	yes	complemented
20	#16813613	C₂²	1	8	yes	complemented
21	#16813602	C₂²	1	16	yes	complemented
22	#16813600	C₂²	1	16	yes	complemented
23	#16813600	C₂²	1	16	yes	complemented
24	C₄×Dic₂₂₆	C₂²	1	8	yes	complemented
25	C₄×D₄₅₂	C₂²	1	8	yes	—
26	#16813584	C₂²	1	8	yes	—
27	#16813584	C₂²	1	8	yes	—
28	C₂×C₄⋊Dic₁₁₃	C₂²	1	2	yes	—
29	C₂×C₄⋊Dic₁₁₃	C₂²	1	1	yes	characteristic
30	C₂×C₄⋊Dic₁₁₃	C₂²	1	1	yes	characteristic
31	C₂×#14268626	C₂²	1	1	yes	characteristic
32	C₂×#14268626	C₂²	1	2	yes	—
33	C₂×#14268626	C₂²	1	1	yes	characteristic
34	C₂×#14268626	C₂²	1	2	yes	—
35	C₄⋊C₄×D₁₁₃	C₂²	1	16	yes	complemented
36	C₂×#14268624	C₂²	1	2	yes	—
37	C₂×#14268624	C₂²	1	2	yes	—
38	C₂×#14268624	C₂²	1	2	yes	—
39	C₂×C₄×Dic₁₁₃	C₂²	1	1	yes	characteristic
40	C₂×C₄×Dic₁₁₃	C₂²	1	1	yes	characteristic
41	#125×C₁₁₃	C₂²	1	8	yes	complemented
42	C₂×C₄⋊C₄×C₁₁₃	C₂²	1	2	yes	—
43	C₂×C₄⋊C₄×C₁₁₃	C₂²	1	1	yes	characteristic
44	C₂×C₄⋊C₄×C₁₁₃	C₂²	1	1	yes	characteristic
45	C₂×C₄⋊C₄×C₁₁₃	C₂²	1	2	yes	—
46	C₂×C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
47	C₂³×D₁₁₃	C₂³	1	2	yes	—
48	C₂×Dic₂₂₆	—	2	4	no	—
49	C₂×Dic₂₂₆	—	4	4	no	—
50	C₂×Dic₂₂₆	C₂³	1	4	yes	—
51	C₂×Dic₂₂₆	C₂³	1	8	yes	—
52	C₂×D₄₅₂	Q₈	1	8	yes	complemented
53	C₂×D₄₅₂	C₂³	1	4	yes	—
54	C₂²×Dic₁₁₃	C₂³	1	1	yes	characteristic
55	C₂²×Dic₁₁₃	C₂³	1	1	yes	characteristic
56	C₂²×Dic₁₁₃	C₂³	1	1	yes	characteristic
57	C₂²×Dic₁₁₃	C₂³	1	2	yes	—
58	C₂²×Dic₁₁₃	C₂³	1	1	yes	characteristic
59	C₂×C₄×D₁₁₃	—	2	8	no	—
60	C₂×C₄×D₁₁₃	—	4	4	no	—
61	C₂×C₄×D₁₁₃	—	4	4	no	—
62	C₂×C₄×D₁₁₃	C₂³	1	8	yes	—
63	C₂×C₄×D₁₁₃	C₂³	1	8	yes	—
64	C₂×C₄×D₁₁₃	C₂³	1	8	yes	—
65	C₄⋊Dic₁₁₃	C₂³	1	8	yes	—
66	C₄⋊Dic₁₁₃	C₂³	1	4	yes	—
67	C₄⋊Dic₁₁₃	C₂³	1	4	yes	—
68	#14268626	C₂³	1	4	yes	—
69	#14268626	C₂³	1	8	yes	—
70	#14268626	C₂³	1	4	yes	—
71	#14268626	C₂³	1	8	yes	—
72	#14268624	C₂³	1	8	yes	—
73	#14268624	C₂³	1	8	yes	—
74	#14268624	C₂³	1	8	yes	—
75	C₄×Dic₁₁₃	C₂³	1	4	yes	—
76	C₄×Dic₁₁₃	C₂³	1	4	yes	—
77	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
78	C₂²×C₄×C₁₁₃	C₂³	1	2	yes	—
79	C₂²×C₄×C₁₁₃	C₂³	1	2	yes	—
80	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
81	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
82	C₄⋊C₄×C₁₁₃	C₂³	1	8	yes	—
83	C₄⋊C₄×C₁₁₃	C₂³	1	4	yes	—
84	C₄⋊C₄×C₁₁₃	C₂³	1	4	yes	—
85	C₄⋊C₄×C₁₁₃	C₂³	1	8	yes	—
86	C₄²×C₁₁₃	C₂³	1	4	yes	—
87	C₂³×C₁₁₃	C₂⁴	1	1	yes	characteristic
88	C₂²×D₁₁₃	—	2	4	no	—
89	C₂²×D₁₁₃	—	2	4	no	—
90	C₂²×D₁₁₃	C₂×Q₈	1	8	yes	—
91	C₂²×D₁₁₃	C₂⁴	1	4	yes	—
92	C₂×C₄×C₁₁₃	—	2	4	no	—
93	C₂×C₄×C₁₁₃	—	2	2	no	—
94	C₂×C₄×C₁₁₃	—	2	2	no	—
95	C₂×C₄×C₁₁₃	—	2	4	no	—
96	C₂×C₄×C₁₁₃	C₂×Q₈	1	4	yes	—
97	C₂×C₄×C₁₁₃	C₂⁴	1	2	yes	—
98	C₂×C₄×C₁₁₃	C₂⁴	1	2	yes	—
99	C₂×C₄×C₁₁₃	C₂⁴	1	4	yes	—
100	C₂×C₄×C₁₁₃	C₂⁴	1	4	yes	—
101	C₂×C₄×C₁₁₃	C₂⁴	1	2	yes	—
102	C₂×Dic₁₁₃	—	2	2	no	—
103	C₂×Dic₁₁₃	—	2	2	no	—
104	C₂×Dic₁₁₃	—	2	4	no	—
105	C₂×Dic₁₁₃	—	2	2	no	—
106	C₂×Dic₁₁₃	#41	1	4	yes	—
107	C₂×Dic₁₁₃	C₂⁴	1	2	yes	—
108	C₂×Dic₁₁₃	C₂⁴	1	2	yes	—
109	C₂×Dic₁₁₃	C₂⁴	1	2	yes	—
110	C₂×Dic₁₁₃	C₂⁴	1	4	yes	—
111	C₂×Dic₁₁₃	C₂⁴	1	2	yes	—
112	D₄₅₂	C₂×Q₈	1	16	yes	complemented
113	C₄×D₁₁₃	—	2	16	no	—
114	C₄×D₁₁₃	—	4	8	no	—
115	C₄×D₁₁₃	—	4	8	no	—
116	Dic₂₂₆	—	2	8	no	—
117	Dic₂₂₆	—	4	8	no	—
118	C₂²×C₁₁₃	C₂²×Q₈	1	2	yes	—
119	C₂²×C₁₁₃	C₂×#41	1	2	yes	—
120	C₂²×C₁₁₃	H(5,𝔽₂)	1	2	yes	—
121	C₂²×C₁₁₃	C₂⁵	1	1	yes	characteristic
122	C₂×D₁₁₃	—	2	16	no	—
123	C₂×D₁₁₃	—	2	4	no	—
124	C₂×D₁₁₃	—	2	4	no	—
125	C₂×D₁₁₃	C₂²×Q₈	1	8	yes	—
126	C₄×C₁₁₃	—	2	2	no	—
127	C₄×C₁₁₃	—	2	4	no	—
128	C₄×C₁₁₃	—	2	4	no	—
129	C₄×C₁₁₃	—	2	2	no	—
130	C₄×C₁₁₃	C₂²×Q₈	1	4	yes	—
131	Dic₁₁₃	—	2	2	no	—
132	Dic₁₁₃	—	2	2	no	—
133	Dic₁₁₃	—	2	4	no	—
134	Dic₁₁₃	—	2	2	no	—
135	Dic₁₁₃	C₂×#41	1	4	yes	—
136	C₂×C₁₁₃	#554	1	4	yes	complemented
137	C₂×C₁₁₃	C₂³×Q₈	1	1	yes	characteristic
138	C₂×C₁₁₃	C₂²×#41	1	1	yes	characteristic
139	C₂×C₁₁₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
140	D₁₁₃	—	2	8	no	—
141	C₂×#554	—	113	1	no	maximal, Sylow
142	C₁₁₃	C₂×#554	1	1	yes	characteristic, complemented, Sylow
143	#554	—	113	16	no	—
144	C₂×C₄⋊Q₈	—	113	1	no	—
145	C₂×#125	—	113	1	no	—
146	C₂×#125	D₁₁₃	1	1	yes	characteristic, complemented
147	C₂×C₂²⋊Q₈	—	113	2	no	—
148	C₂×C₂²⋊Q₈	—	113	2	no	—
149	C₂×C₂²⋊Q₈	—	113	2	no	—
150	C₂×C₄×Q₈	—	113	1	no	—
151	C₂×C₄×D₄	—	113	1	no	—
152	C₂×C₄×D₄	—	113	1	no	—
153	C₂×C₄×D₄	—	113	1	no	—
154	C₂²×C₄⋊C₄	—	113	2	no	—
155	C₂²×Q₈	—	113	2	no	—
156	C₂²×Q₈	—	113	1	no	—
157	C₂²×D₄	—	113	1	no	—
158	C₂³×C₄	—	113	2	no	—
159	C₂³×C₄	—	113	2	no	—
160	C₂³×C₄	—	113	2	no	—
161	C₄⋊Q₈	—	113	8	no	—
162	#125	—	113	8	no	—
163	#125	C₂×D₁₁₃	1	8	yes	complemented
164	C₂²⋊Q₈	—	113	16	no	—
165	C₂²⋊Q₈	—	113	16	no	—
166	C₂²⋊Q₈	—	113	16	no	—
167	C₄×Q₈	—	113	8	no	—
168	C₄×D₄	—	113	8	no	—
169	C₄×D₄	—	113	8	no	—
170	C₄×D₄	—	113	8	no	—
171	C₂×C₄⋊C₄	—	113	1	no	—
172	C₂×C₄⋊C₄	—	113	2	no	—
173	C₂×C₄⋊C₄	—	113	2	no	—
174	C₂×C₄⋊C₄	—	113	1	no	—
175	C₂×C₄⋊C₄	—	113	1	no	—
176	C₂×C₄⋊C₄	—	113	1	no	—
177	C₂×C₄⋊C₄	—	113	2	no	—
178	C₂×C₄⋊C₄	—	113	16	no	—
179	C₂×C₄⋊C₄	C₂×D₁₁₃	1	2	yes	—
180	C₂×C₄⋊C₄	C₂×D₁₁₃	1	1	yes	characteristic
181	C₂×C₄⋊C₄	C₂×D₁₁₃	1	1	yes	characteristic
182	C₂×C₄⋊C₄	C₂×D₁₁₃	1	2	yes	—
183	C₂×C₂²⋊C₄	—	113	2	no	—
184	C₂×C₂²⋊C₄	—	113	2	no	—
185	C₂×C₂²⋊C₄	—	113	2	no	—
186	C₂×C₄²	—	113	1	no	—
187	C₂×C₄²	—	113	1	no	—
188	C₂×C₄²	C₂×D₁₁₃	1	1	yes	characteristic
189	C₂⁴	—	113	2	no	—
190	C₂×Q₈	—	113	8	no	—
191	C₂×Q₈	—	113	4	no	—
192	C₂×Q₈	—	452	4	no	—
193	C₂×Q₈	—	226	4	no	—
194	C₂×D₄	—	113	8	no	—
195	C₂×D₄	—	113	4	no	—
196	C₂²×C₄	—	113	1	no	—
197	C₂²×C₄	—	113	1	no	—
198	C₂²×C₄	—	113	1	no	—
199	C₂²×C₄	—	113	1	no	—
200	C₂²×C₄	—	113	2	no	—
201	C₂²×C₄	—	113	8	no	—
202	C₂²×C₄	—	113	8	no	—
203	C₂²×C₄	—	113	8	no	—
204	C₂²×C₄	—	452	4	no	—
205	C₂²×C₄	—	452	4	no	—
206	C₂²×C₄	—	226	8	no	—
207	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
208	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
209	C₂²×C₄	C₂²×D₁₁₃	1	2	yes	—
210	C₂²×C₄	C₂²×D₁₁₃	1	2	yes	—
211	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
212	C₄⋊C₄	—	113	4	no	—
213	C₄⋊C₄	—	113	8	no	—
214	C₄⋊C₄	—	113	8	no	—
215	C₄⋊C₄	—	113	4	no	—
216	C₄⋊C₄	—	113	4	no	—
217	C₄⋊C₄	—	113	4	no	—
218	C₄⋊C₄	—	113	8	no	—
219	C₄⋊C₄	C₂²×D₁₁₃	1	8	yes	—
220	C₄⋊C₄	C₂²×D₁₁₃	1	4	yes	—
221	C₄⋊C₄	C₂²×D₁₁₃	1	4	yes	—
222	C₄⋊C₄	C₂²×D₁₁₃	1	8	yes	—
223	C₂²⋊C₄	—	113	8	no	—
224	C₂²⋊C₄	—	113	8	no	—
225	C₂²⋊C₄	—	113	8	no	—
226	C₄²	—	113	4	no	—
227	C₄²	—	113	4	no	—
228	C₄²	C₂²×D₁₁₃	1	4	yes	—
229	C₂³	—	113	8	no	—
230	C₂³	—	113	4	no	—
231	C₂³	—	226	4	no	—
232	C₂³	—	226	4	no	—
233	C₂³	C₂³×D₁₁₃	1	1	yes	characteristic
234	Q₈	—	452	8	no	—
235	Q₈	—	226	8	no	—
236	D₄	—	113	16	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	113	4	no	—
242	C₂×C₄	—	113	2	no	—
243	C₂×C₄	—	113	2	no	—
244	C₂×C₄	—	113	2	no	—
245	C₂×C₄	—	113	2	no	—
246	C₂×C₄	—	113	4	no	—
247	C₂×C₄	—	452	8	no	—
248	C₂×C₄	—	452	8	no	—
249	C₂×C₄	—	226	16	no	—
250	C₂×C₄	—	226	2	no	—
251	C₂×C₄	—	226	2	no	—
252	C₂×C₄	—	226	4	no	—
253	C₂×C₄	—	226	2	no	—
254	C₂×C₄	Q₈×D₁₁₃	1	4	yes	—
255	C₂×C₄	C₂³×D₁₁₃	1	2	yes	—
256	C₂×C₄	C₂³×D₁₁₃	1	2	yes	—
257	C₂×C₄	C₂³×D₁₁₃	1	4	yes	—
258	C₂×C₄	C₂³×D₁₁₃	1	4	yes	—
259	C₂×C₄	C₂³×D₁₁₃	1	2	yes	—
260	C₂²	—	113	8	no	—
261	C₂²	—	226	16	no	—
262	C₂²	—	226	4	no	—
263	C₂²	—	226	4	no	—
264	C₂²	C₂×Q₈×D₁₁₃	1	2	yes	—
265	C₂²	#41×D₁₁₃	1	2	yes	—
266	C₂²	#16813668	1	2	yes	—
267	C₂²	C₂⁴×D₁₁₃	1	1	yes	characteristic
268	C₄	—	2	4	no	—
269	C₄	—	2	2	no	—
270	C₄	—	2	2	no	—
271	C₄	—	2	4	no	—
272	C₄	—	113	4	no	—
273	C₄	—	226	2	no	—
274	C₄	—	226	2	no	—
275	C₄	—	226	4	no	—
276	C₄	—	226	2	no	—
277	C₄	C₂×Q₈×D₁₁₃	1	4	yes	—
278	C₂	—	226	8	no	—
279	C₂	#20299304	1	4	yes	direct product factor
280	C₂	C₂²×Q₈×D₁₁₃	1	1	yes	characteristic
281	C₂	C₂×#41×D₁₁₃	1	1	yes	characteristic
282	C₂	C₂×#16813668	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₁₁₃
commutator, cosocle	C₂²×C₁₁₃	C₂⁵
Frattini	C₂²	C₂⁴×D₁₁₃
Fitting	C₂×#125×C₁₁₃	C₂
socle	C₂³×C₁₁₃	C₂⁴
prime radical	C₂²×D₄₅₂	C₂²
hypercenter	C₂×#125	D₁₁₃
hypocenter	C₁₁₃	C₂×#554
LCS	G ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×#125

Complements

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

N	H ≅ G/N	type
1. #20299304	C₂	direct (N × H)
2. C₂×#16813621	C₂	semidirect
3. C₂×#16813613	C₂	semidirect
4. C₂×#16813602	C₂	semidirect
5. C₂×#16813600	C₂	semidirect
6. C₂×#16813600	C₂	semidirect
7. C₂×C₄×Dic₂₂₆	C₂	semidirect
11. C₂×C₄⋊C₄×D₁₁₃	C₂	semidirect
12. C₂×#125×C₁₁₃	C₂	semidirect
19. #16813621	C₂²	semidirect
20. #16813613	C₂²	semidirect
21. #16813602	C₂²	semidirect
22. #16813600	C₂²	semidirect
23. #16813600	C₂²	semidirect
24. C₄×Dic₂₂₆	C₂²	semidirect
35. C₄⋊C₄×D₁₁₃	C₂²	semidirect
41. #125×C₁₁₃	C₂²	semidirect
52. C₂×D₄₅₂	Q₈	semidirect
112. D₄₅₂	C₂×Q₈	semidirect
136. C₂×C₁₁₃	#554	semidirect
142. C₁₁₃	C₂×#554	semidirect
146. C₂×#125	D₁₁₃	semidirect
163. #125	C₂×D₁₁₃	semidirect
279. C₂	#20299304	direct (N × H)

Maximal subgroups

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

subgroup	index
#20299304	2
C₂×#16813621	2
C₂×#16813613	2
C₂×#16813602	2
C₂×#16813600	2
C₂×C₄×Dic₂₂₆	2
C₂×C₄×D₄₅₂	2
C₂×#16813584	2
C₂×C₄⋊C₄×D₁₁₃	2
C₂×#125×C₁₁₃	2
C₂×#554	113

Subgroups — C2×#20299304 #28404104

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20299304 #28404104