Subgroups — C₂×#20298614 #28399115

Order 14464.

772 conjugacy classes of subgroups.

The subgroup lattice has 304 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
2	C₂×#16813581	C₂	1	1	yes	maximal, characteristic, complemented
3	#20298614	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16813548	C₂	1	4	yes	maximal, complemented
5	C₂²×M₁₆×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14268647	—	2	1	no	—
7	C₂×#14268647	—	2	2	no	—
8	C₂×#14268647	—	2	1	no	—
9	C₂×#14268647	—	2	2	no	—
10	C₂×#14268647	C₄	1	2	yes	complemented
11	C₂×#14268647	C₄	1	4	yes	complemented
12	C₂×#14268647	C₂²	1	2	yes	—
13	C₂×#14268647	C₂²	1	4	yes	—
14	C₂²×Dic₂₂₆	C₄	1	1	yes	characteristic, complemented
15	C₂²×D₄₅₂	C₄	1	1	yes	characteristic, complemented
16	C₂²×C₂²⋊D₁₁₃	—	2	1	no	—
17	C₂²×C₄×D₁₁₃	—	2	1	no	—
18	#16813581	—	2	2	no	—
19	#16813581	C₂²	1	4	yes	complemented
20	C₂×C₄⋊Dic₁₁₃	—	2	1	no	—
21	C₂×C₂²⋊Dic₁₁₃	—	2	1	no	—
22	C₂×C₄×Dic₁₁₃	C₂²	1	2	yes	complemented
23	#16813548	C₂²	1	16	yes	complemented
24	C₂³×C₄×C₁₁₃	C₂²	1	1	yes	characteristic
25	C₂×M₁₆×C₁₁₃	—	2	1	no	—
26	C₂×M₁₆×C₁₁₃	—	2	2	no	—
27	C₂×M₁₆×C₁₁₃	C₂²	1	2	yes	complemented
28	C₂×M₁₆×C₁₁₃	C₂²	1	4	yes	complemented
29	C₂²×C₈×C₁₁₃	—	2	1	no	—
30	C₂³×D₁₁₃	—	2	1	no	—
31	#14268647	—	2	4	no	—
32	#14268647	—	2	4	no	—
33	#14268647	—	2	4	no	—
34	#14268647	—	2	2	no	—
35	#14268647	—	2	4	no	—
36	#14268647	—	2	2	no	—
37	#14268647	—	2	4	no	—
38	#14268647	C₂×C₄	1	4	yes	—
39	#14268647	C₂×C₄	1	8	yes	complemented
40	#14268647	C₂³	1	4	yes	—
41	C₂×Dic₂₂₆	—	2	2	no	—
42	C₂×Dic₂₂₆	—	2	1	no	—
43	C₂×Dic₂₂₆	C₂×C₄	1	2	yes	—
44	C₂×Dic₂₂₆	C₂×C₄	1	4	yes	complemented
45	C₂×D₄₅₂	—	2	2	no	—
46	C₂×D₄₅₂	—	2	1	no	—
47	C₂×D₄₅₂	C₂×C₄	1	2	yes	—
48	C₂×D₄₅₂	C₂×C₄	1	4	yes	complemented
49	C₂×C₂²⋊D₁₁₃	—	2	2	no	—
50	C₂×C₂²⋊D₁₁₃	—	2	4	no	—
51	C₂×C₂²⋊D₁₁₃	—	2	2	no	—
52	C₂×C₂²⋊D₁₁₃	—	2	4	no	—
53	C₂²×Dic₁₁₃	—	2	1	no	—
54	C₂²×Dic₁₁₃	—	2	2	no	—
55	C₂×C₄×D₁₁₃	—	2	4	no	—
56	C₂×C₄×D₁₁₃	—	2	2	no	—
57	C₂×C₄×D₁₁₃	—	2	4	no	—
58	C₂×C₄×D₁₁₃	—	2	2	no	—
59	C₄⋊Dic₁₁₃	—	2	4	no	—
60	C₂²⋊Dic₁₁₃	—	2	4	no	—
61	C₄×Dic₁₁₃	—	2	2	no	—
62	C₄×Dic₁₁₃	C₂³	1	4	yes	complemented
63	C₂⁴×C₁₁₃	D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₁₃	C₂×C₄	1	2	yes	—
67	C₂²×C₄×C₁₁₃	D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₁₃	D₄	1	2	yes	—
69	C₂²×C₄×C₁₁₃	D₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₁₃	D₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₁₃	D₄	1	2	yes	—
72	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
73	C₂²×C₄×C₁₁₃	C₂³	1	2	yes	—
74	M₁₆×C₁₁₃	—	2	4	no	—
75	M₁₆×C₁₁₃	—	2	2	no	—
76	M₁₆×C₁₁₃	C₂³	1	4	yes	complemented
77	C₂×C₈×C₁₁₃	—	2	4	no	—
78	C₂×C₈×C₁₁₃	—	2	2	no	—
79	C₂³×C₁₁₃	—	4	2	no	—
80	C₂³×C₁₁₃	C₂²⋊C₄	1	1	yes	characteristic
81	C₂³×C₁₁₃	C₂²⋊C₄	1	1	yes	characteristic
82	C₂³×C₁₁₃	C₂²⋊C₄	1	2	yes	—
83	C₂³×C₁₁₃	C₂×D₄	1	1	yes	characteristic
84	C₂³×C₁₁₃	C₂×D₄	1	2	yes	—
85	C₂²×D₁₁₃	—	2	2	no	—
86	C₂²×D₁₁₃	—	2	4	no	—
87	C₂²×D₁₁₃	—	4	4	no	—
88	C₂×C₄×C₁₁₃	C₂²⋊C₄	1	2	yes	—
89	C₂×C₄×C₁₁₃	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₁₁₃	C₂²⋊C₄	1	2	yes	—
91	C₂×C₄×C₁₁₃	C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₁₁₃	C₂²⋊C₄	1	2	yes	—
93	C₂×C₄×C₁₁₃	C₂²⋊C₄	1	2	yes	—
94	C₂×C₄×C₁₁₃	C₂²⋊C₄	1	2	yes	—
95	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
97	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
99	C₂×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
100	C₂×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
101	C₂×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
105	C₂×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
106	C₂²⋊D₁₁₃	—	2	4	no	—
107	C₂²⋊D₁₁₃	—	2	8	no	—
108	C₂²⋊D₁₁₃	—	2	4	no	—
109	C₂×Dic₁₁₃	—	2	2	no	—
110	C₂×Dic₁₁₃	—	2	4	no	—
111	C₂×Dic₁₁₃	—	2	4	no	—
112	C₂×Dic₁₁₃	—	4	4	no	—
113	D₄₅₂	—	2	2	no	—
114	D₄₅₂	—	2	4	no	—
115	D₄₅₂	C₂²×C₄	1	4	yes	—
116	C₄×D₁₁₃	—	2	4	no	—
117	C₄×D₁₁₃	—	2	8	no	—
118	C₄×D₁₁₃	—	2	4	no	—
119	Dic₂₂₆	—	2	2	no	—
120	Dic₂₂₆	—	2	4	no	—
121	Dic₂₂₆	C₂²×C₄	1	4	yes	—
122	C₈×C₁₁₃	—	2	4	no	—
123	C₂²×C₁₁₃	—	2	2	no	—
124	C₂²×C₁₁₃	—	2	2	no	—
125	C₂²×C₁₁₃	—	2	2	no	—
126	C₂²×C₁₁₃	—	4	2	no	—
127	C₂²×C₁₁₃	—	4	2	no	—
128	C₂²×C₁₁₃	C₂×C₂²⋊C₄	1	2	yes	—
129	C₂²×C₁₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₂²×C₁₁₃	C₂×C₂²⋊C₄	1	2	yes	—
131	C₂²×C₁₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₂²×C₁₁₃	C₂²×D₄	1	1	yes	characteristic
133	C₂×D₁₁₃	—	2	4	no	—
134	C₂×D₁₁₃	—	4	8	no	—
135	C₂×D₁₁₃	—	4	4	no	—
136	C₄×C₁₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
137	C₄×C₁₁₃	C₂×C₂²⋊C₄	1	2	yes	—
138	C₄×C₁₁₃	C₂×C₂²⋊C₄	1	2	yes	—
139	C₄×C₁₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
140	C₄×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
141	C₄×C₁₁₃	C₂²×D₄	1	1	yes	characteristic
142	Dic₁₁₃	—	2	4	no	—
143	Dic₁₁₃	—	4	4	no	—
144	C₂×C₁₁₃	—	2	1	no	—
145	C₂×C₁₁₃	—	2	2	no	—
146	C₂×C₁₁₃	—	2	1	no	—
147	C₂×C₁₁₃	—	2	2	no	—
148	C₂×C₁₁₃	#421	1	2	yes	complemented
149	C₂×C₁₁₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
150	D₁₁₃	—	4	4	no	—
151	C₂×#421	—	113	1	no	maximal, Sylow
152	C₁₁₃	C₂×#421	1	1	yes	characteristic, complemented, Sylow
153	C₂²×#41	—	113	1	no	—
154	C₂²×M₁₆	D₁₁₃	1	1	yes	characteristic, complemented
155	C₂×#117	—	113	1	no	—
156	#421	—	113	8	no	—
157	C₂×#104	—	113	4	no	—
158	C₂×#41	—	113	2	no	—
159	C₂×#41	—	113	4	no	—
160	C₂×#41	—	113	2	no	—
161	C₂×#41	—	113	4	no	—
162	C₂×#41	—	226	1	no	—
163	C₂×#41	—	226	2	no	—
164	C₂×#41	—	226	2	no	—
165	C₂×#41	—	226	1	no	—
166	C₂²×Q₈	—	113	1	no	—
167	C₂²×D₄	—	113	1	no	—
168	C₂²×D₄	—	226	1	no	—
169	C₂³×C₄	—	226	1	no	—
170	C₂³×C₄	C₂×D₁₁₃	1	1	yes	characteristic
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	2	1	no	—
173	C₂×M₁₆	C₂×D₁₁₃	1	2	yes	complemented
174	C₂×M₁₆	C₂×D₁₁₃	1	4	yes	complemented
175	C₂²×C₈	—	2	1	no	—
176	#117	—	113	4	no	—
177	#117	—	226	2	no	—
178	C₂×C₄⋊C₄	—	226	1	no	—
179	C₂×C₂²⋊C₄	—	226	1	no	—
180	C₂×C₄²	—	113	2	no	—
181	#104	—	113	16	no	—
182	C₂⁴	—	226	1	no	—
183	C₂⁴	D₄₅₂	1	1	yes	characteristic
184	#41	—	113	4	no	—
185	#41	—	113	8	no	—
186	#41	—	113	4	no	—
187	#41	—	226	4	no	—
188	#41	—	226	4	no	—
189	#41	—	226	4	no	—
190	#41	—	226	4	no	—
191	#41	—	226	2	no	—
192	#41	—	226	2	no	—
193	#41	—	226	4	no	—
194	C₂×Q₈	—	113	2	no	—
195	C₂×Q₈	—	113	4	no	—
196	C₂×Q₈	—	226	2	no	—
197	C₂×Q₈	—	226	1	no	—
198	C₂×D₄	—	113	2	no	—
199	C₂×D₄	—	113	4	no	—
200	C₂×D₄	—	226	2	no	—
201	C₂×D₄	—	226	4	no	—
202	C₂×D₄	—	226	2	no	—
203	C₂×D₄	—	226	4	no	—
204	C₂×D₄	—	226	2	no	—
205	C₂×D₄	—	226	1	no	—
206	C₂²×C₄	—	226	2	no	—
207	C₂²×C₄	—	226	1	no	—
208	C₂²×C₄	—	226	4	no	—
209	C₂²×C₄	—	226	2	no	—
210	C₂²×C₄	—	226	4	no	—
211	C₂²×C₄	—	226	2	no	—
212	C₂²×C₄	C₄×D₁₁₃	1	1	yes	characteristic
213	C₂²×C₄	C₄×D₁₁₃	1	1	yes	characteristic
214	C₂²×C₄	C₄×D₁₁₃	1	2	yes	—
215	C₂²×C₄	D₄₅₂	1	1	yes	characteristic
216	C₂²×C₄	D₄₅₂	1	2	yes	—
217	C₂²×C₄	C₂²⋊D₁₁₃	1	2	yes	—
218	C₂²×C₄	C₂²⋊D₁₁₃	1	1	yes	characteristic
219	C₂²×C₄	C₂²⋊D₁₁₃	1	1	yes	characteristic
220	C₂²×C₄	C₂²×D₁₁₃	1	2	yes	—
221	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
222	M₁₆	—	2	4	no	—
223	M₁₆	—	2	2	no	—
224	M₁₆	C₂²×D₁₁₃	1	4	yes	complemented
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	—	2	2	no	—
227	C₄⋊C₄	—	226	4	no	—
228	C₂²⋊C₄	—	226	4	no	—
229	C₄²	—	113	4	no	—
230	C₄²	—	226	2	no	—
231	C₂³	—	4	2	no	—
232	C₂³	—	452	4	no	—
233	C₂³	—	226	4	no	—
234	C₂³	—	226	2	no	—
235	C₂³	#14268624	1	1	yes	characteristic
236	C₂³	#14268624	1	1	yes	characteristic
237	C₂³	#14268624	1	2	yes	—
238	C₂³	C₂×D₄₅₂	1	1	yes	characteristic
239	C₂³	C₂×D₄₅₂	1	2	yes	—
240	Q₈	—	113	4	no	—
241	Q₈	—	226	2	no	—
242	Q₈	—	226	4	no	—
243	D₄	—	113	4	no	—
244	D₄	—	226	4	no	—
245	D₄	—	226	8	no	—
246	D₄	—	226	4	no	—
247	D₄	—	226	2	no	—
248	D₄	—	226	4	no	—
249	C₂×C₄	—	452	4	no	—
250	C₂×C₄	—	226	4	no	—
251	C₂×C₄	—	226	4	no	—
252	C₂×C₄	—	226	2	no	—
253	C₂×C₄	—	226	8	no	—
254	C₂×C₄	—	226	4	no	—
255	C₂×C₄	—	226	4	no	—
256	C₂×C₄	#14268624	1	1	yes	characteristic
257	C₂×C₄	#14268624	1	2	yes	—
258	C₂×C₄	#14268624	1	2	yes	—
259	C₂×C₄	#14268624	1	1	yes	characteristic
260	C₂×C₄	#14268624	1	2	yes	—
261	C₂×C₄	#14268624	1	2	yes	—
262	C₂×C₄	#14268624	1	2	yes	—
263	C₂×C₄	C₂×C₄×D₁₁₃	1	2	yes	—
264	C₂×C₄	C₂×C₄×D₁₁₃	1	1	yes	characteristic
265	C₂×C₄	C₂×C₄×D₁₁₃	1	2	yes	—
266	C₂×C₄	C₂×C₄×D₁₁₃	1	1	yes	characteristic
267	C₂×C₄	C₂×C₂²⋊D₁₁₃	1	2	yes	—
268	C₂×C₄	C₂×C₂²⋊D₁₁₃	1	2	yes	—
269	C₂×C₄	C₂×C₂²⋊D₁₁₃	1	1	yes	characteristic
270	C₂×C₄	C₂×C₂²⋊D₁₁₃	1	1	yes	characteristic
271	C₂×C₄	C₂×D₄₅₂	1	1	yes	characteristic
272	C₂×C₄	C₂×D₄₅₂	1	2	yes	—
273	C₂×C₄	C₂³×D₁₁₃	1	1	yes	characteristic
274	C₈	—	2	4	no	—
275	C₂²	—	2	2	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	4	2	no	—
279	C₂²	—	4	2	no	—
280	C₂²	—	452	8	no	—
281	C₂²	—	452	4	no	—
282	C₂²	—	226	4	no	—
283	C₂²	C₂×#14268624	1	2	yes	—
284	C₂²	C₂×#14268624	1	2	yes	—
285	C₂²	C₂×#14268624	1	1	yes	characteristic
286	C₂²	C₂×#14268624	1	1	yes	characteristic
287	C₂²	C₂²×D₄₅₂	1	1	yes	characteristic
288	C₄	—	452	4	no	—
289	C₄	—	226	4	no	—
290	C₄	C₂×#14268624	1	2	yes	—
291	C₄	C₂×#14268624	1	1	yes	characteristic
292	C₄	C₂×#14268624	1	2	yes	—
293	C₄	C₂×#14268624	1	1	yes	characteristic
294	C₄	C₂²×C₄×D₁₁₃	1	1	yes	characteristic
295	C₄	C₂²×C₂²⋊D₁₁₃	1	1	yes	characteristic
296	C₂	—	2	1	no	—
297	C₂	—	2	2	no	—
298	C₂	—	2	1	no	—
299	C₂	—	2	2	no	—
300	C₂	—	452	4	no	—
301	C₂	#20298614	1	2	yes	direct product factor
302	C₂	C₂²×#14268624	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₁₃
commutator	C₄×C₁₁₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₁₃
Fitting	C₂²×M₁₆×C₁₁₃	C₂
socle	C₂²×C₁₁₃	C₂×C₂²⋊C₄
prime radical	C₂²×#14268647	C₂
cosocle	C₂×C₄×C₁₁₃	C₂⁴
hypercenter	C₂²×M₁₆	D₁₁₃
hypocenter	C₁₁₃	C₂×#421
LCS	G ▷ C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×#16813581	C₂	semidirect
3. #20298614	C₂	direct (N × H)
4. C₂×#16813548	C₂	semidirect
5. C₂²×M₁₆×C₁₁₃	C₂	semidirect
10. C₂×#14268647	C₄	semidirect
11. C₂×#14268647	C₄	semidirect
14. C₂²×Dic₂₂₆	C₄	semidirect
15. C₂²×D₄₅₂	C₄	semidirect
19. #16813581	C₂²	semidirect
22. C₂×C₄×Dic₁₁₃	C₂²	semidirect
23. #16813548	C₂²	semidirect
27. C₂×M₁₆×C₁₁₃	C₂²	semidirect
28. C₂×M₁₆×C₁₁₃	C₂²	semidirect
39. #14268647	C₂×C₄	semidirect
44. C₂×Dic₂₂₆	C₂×C₄	semidirect
48. C₂×D₄₅₂	C₂×C₄	semidirect
62. C₄×Dic₁₁₃	C₂³	semidirect
76. M₁₆×C₁₁₃	C₂³	semidirect
148. C₂×C₁₁₃	#421	semidirect
152. C₁₁₃	C₂×#421	semidirect
154. C₂²×M₁₆	D₁₁₃	semidirect
173. C₂×M₁₆	C₂×D₁₁₃	semidirect
174. C₂×M₁₆	C₂×D₁₁₃	semidirect
224. M₁₆	C₂²×D₁₁₃	semidirect
301. C₂	#20298614	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
C₂×#16813581	2
#20298614	2
C₂×#16813548	2
C₂²×M₁₆×C₁₁₃	2
C₂×#421	113

Subgroups — C2×#20298614 #28399115

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20298614 #28399115