Subgroups — C₂×#20299306 #28404106

Order 14464.

1200 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20299306	C₂	1	16	yes	maximal, direct product factor
2	C₂×#16813621	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16813612	C₂	1	2	yes	maximal, complemented
4	C₂×#16813602	C₂	1	2	yes	maximal, complemented
5	C₂×#16813599	C₂	1	2	yes	maximal, complemented
6	C₂×#16813598	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×Q₈×Dic₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×C₂²⋊D₁₁₃	C₂	1	1	yes	maximal, characteristic
9	C₂×#16813582	C₂	1	2	yes	maximal
10	C₂²×C₄⋊Dic₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄⋊C₄×D₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₂²⋊Q₈×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×Dic₂₂₆	C₂²	1	2	yes	—
14	C₂²×C₂²⋊D₁₁₃	C₂²	1	1	yes	characteristic
15	C₂³×Dic₁₁₃	C₂²	1	2	yes	—
16	C₂²×C₄×D₁₁₃	C₂²	1	1	yes	characteristic
17	C₂²×C₄×D₁₁₃	C₂²	1	2	yes	—
18	#16813621	C₂²	1	8	yes	complemented
19	#16813612	C₂²	1	16	yes	complemented
20	#16813602	C₂²	1	16	yes	complemented
21	#16813599	C₂²	1	16	yes	complemented
22	#16813598	C₂²	1	8	yes	complemented
23	Q₈×Dic₁₁₃	C₂²	1	8	yes	complemented
24	C₄×C₂²⋊D₁₁₃	C₂²	1	8	yes	—
25	#16813582	C₂²	1	16	yes	—
26	C₂×C₄⋊Dic₁₁₃	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊Dic₁₁₃	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊Dic₁₁₃	C₂²	1	2	yes	—
29	C₂×C₄⋊Dic₁₁₃	C₂²	1	2	yes	—
30	C₂×C₄⋊Dic₁₁₃	C₂²	1	8	yes	complemented
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	2	yes	—
35	C₄⋊C₄×D₁₁₃	C₂²	1	8	yes	complemented
36	C₂×C₂²⋊Dic₁₁₃	C₂²	1	1	yes	characteristic
37	C₂×#14268624	C₂²	1	2	yes	—
38	C₂×#14268624	C₂²	1	1	yes	characteristic
39	C₂×C₄×Dic₁₁₃	C₂²	1	2	yes	—
40	C₂×C₄×Dic₁₁₃	C₂²	1	1	yes	characteristic
41	C₂²×Q₈×C₁₁₃	C₂²	1	1	yes	characteristic
42	C₂³×C₄×C₁₁₃	C₂²	1	1	yes	characteristic
43	C₂²⋊Q₈×C₁₁₃	C₂²	1	8	yes	complemented
44	C₂×C₄⋊C₄×C₁₁₃	C₂²	1	1	yes	characteristic
45	C₂×C₄⋊C₄×C₁₁₃	C₂²	1	2	yes	—
46	C₂×C₂²⋊C₄×C₁₁₃	C₂²	1	2	yes	—
47	C₂³×D₁₁₃	C₂³	1	1	yes	characteristic
48	C₂×Dic₂₂₆	—	4	4	no	—
49	C₂×Dic₂₂₆	C₂³	1	8	yes	—
50	C₂×C₂²⋊D₁₁₃	Q₈	1	8	yes	complemented
51	C₂×C₂²⋊D₁₁₃	C₂³	1	4	yes	—
52	C₂²×Dic₁₁₃	—	4	4	no	—
53	C₂²×Dic₁₁₃	C₂³	1	1	yes	characteristic
54	C₂²×Dic₁₁₃	C₂³	1	2	yes	—
55	C₂²×Dic₁₁₃	C₂³	1	2	yes	—
56	C₂²×Dic₁₁₃	C₂³	1	1	yes	characteristic
57	C₂²×Dic₁₁₃	C₂³	1	8	yes	—
58	C₂²×Dic₁₁₃	C₂³	1	1	yes	characteristic
59	C₂×C₄×D₁₁₃	—	2	4	no	—
60	C₂×C₄×D₁₁₃	—	4	4	no	—
61	C₂×C₄×D₁₁₃	C₂³	1	4	yes	—
62	C₂×C₄×D₁₁₃	C₂³	1	8	yes	—
63	C₄⋊Dic₁₁₃	C₂³	1	4	yes	—
64	C₄⋊Dic₁₁₃	C₂³	1	4	yes	—
65	C₄⋊Dic₁₁₃	C₂³	1	8	yes	—
66	C₄⋊Dic₁₁₃	C₂³	1	8	yes	—
67	#14268626	C₂³	1	8	yes	—
68	#14268626	C₂³	1	8	yes	—
69	#14268626	C₂³	1	4	yes	—
70	#14268626	C₂³	1	8	yes	—
71	C₂²⋊Dic₁₁₃	C₂³	1	4	yes	—
72	#14268624	C₂³	1	8	yes	—
73	#14268624	C₂³	1	4	yes	—
74	C₄×Dic₁₁₃	C₂³	1	8	yes	—
75	C₄×Dic₁₁₃	C₂³	1	4	yes	—
76	C₂⁴×C₁₁₃	C₂³	1	1	yes	characteristic
77	C₂×Q₈×C₁₁₃	—	2	4	no	—
78	C₂×Q₈×C₁₁₃	C₂³	1	4	yes	—
79	C₂²×C₄×C₁₁₃	—	2	4	no	—
80	C₂²×C₄×C₁₁₃	C₂³	1	2	yes	—
81	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
82	C₂²×C₄×C₁₁₃	C₂³	1	2	yes	—
83	C₂²×C₄×C₁₁₃	C₂³	1	4	yes	—
84	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
85	C₄⋊C₄×C₁₁₃	C₂³	1	4	yes	—
86	C₄⋊C₄×C₁₁₃	C₂³	1	8	yes	—
87	C₂²⋊C₄×C₁₁₃	C₂³	1	8	yes	—
88	C₂³×C₁₁₃	—	2	2	no	—
89	C₂³×C₁₁₃	—	2	2	no	—
90	C₂³×C₁₁₃	C₂×Q₈	1	4	yes	—
91	C₂³×C₁₁₃	C₂⁴	1	2	yes	—
92	C₂³×C₁₁₃	C₂⁴	1	1	yes	characteristic
93	C₂²×D₁₁₃	—	2	2	no	—
94	C₂²×D₁₁₃	—	2	2	no	—
95	C₂²×D₁₁₃	C₂×Q₈	1	4	yes	—
96	C₂²×D₁₁₃	C₂⁴	1	2	yes	—
97	Q₈×C₁₁₃	—	2	8	no	—
98	C₂×C₄×C₁₁₃	—	2	2	no	—
99	C₂×C₄×C₁₁₃	—	2	4	no	—
100	C₂×C₄×C₁₁₃	—	2	4	no	—
101	C₂×C₄×C₁₁₃	—	2	8	no	—
102	C₂×C₄×C₁₁₃	#41	1	4	yes	—
103	C₂×C₄×C₁₁₃	C₂⁴	1	4	yes	—
104	C₂×C₄×C₁₁₃	C₂⁴	1	4	yes	—
105	C₂×C₄×C₁₁₃	C₂⁴	1	2	yes	—
106	C₂×C₄×C₁₁₃	C₂⁴	1	2	yes	—
107	C₂²⋊D₁₁₃	C₂×Q₈	1	16	yes	complemented
108	C₂×Dic₁₁₃	—	2	2	no	—
109	C₂×Dic₁₁₃	—	2	4	no	—
110	C₂×Dic₁₁₃	—	2	4	no	—
111	C₂×Dic₁₁₃	—	2	2	no	—
112	C₂×Dic₁₁₃	—	4	8	no	—
113	C₂×Dic₁₁₃	C₂×Q₈	1	4	yes	—
114	C₂×Dic₁₁₃	C₂⁴	1	2	yes	—
115	C₂×Dic₁₁₃	C₂⁴	1	4	yes	—
116	C₂×Dic₁₁₃	C₂⁴	1	4	yes	—
117	C₂×Dic₁₁₃	C₂⁴	1	2	yes	—
118	C₂×Dic₁₁₃	C₂⁴	1	2	yes	—
119	C₄×D₁₁₃	—	2	8	no	—
120	C₄×D₁₁₃	—	4	8	no	—
121	Dic₂₂₆	—	4	8	no	—
122	C₂²×C₁₁₃	—	2	8	no	—
123	C₂²×C₁₁₃	—	2	2	no	—
124	C₂²×C₁₁₃	—	2	2	no	—
125	C₂²×C₁₁₃	C₂²×Q₈	1	4	yes	—
126	C₂²×C₁₁₃	C₂²×Q₈	1	2	yes	—
127	C₂²×C₁₁₃	C₂×#41	1	2	yes	—
128	C₂²×C₁₁₃	H(5,𝔽₂)	1	2	yes	—
129	C₂²×C₁₁₃	C₂⁵	1	1	yes	characteristic
130	C₂×D₁₁₃	—	2	8	no	—
131	C₂×D₁₁₃	—	2	2	no	—
132	C₂×D₁₁₃	—	2	2	no	—
133	C₂×D₁₁₃	C₂²×Q₈	1	4	yes	—
134	C₄×C₁₁₃	—	2	2	no	—
135	C₄×C₁₁₃	—	2	4	no	—
136	C₄×C₁₁₃	—	2	4	no	—
137	C₄×C₁₁₃	C₂×#41	1	4	yes	—
138	Dic₁₁₃	—	2	2	no	—
139	Dic₁₁₃	—	2	2	no	—
140	Dic₁₁₃	—	2	4	no	—
141	Dic₁₁₃	—	2	4	no	—
142	Dic₁₁₃	C₂²×Q₈	1	4	yes	—
143	C₂×C₁₁₃	—	2	4	no	—
144	C₂×C₁₁₃	#554	1	4	yes	complemented
145	C₂×C₁₁₃	C₂³×Q₈	1	1	yes	characteristic
146	C₂×C₁₁₃	C₂²×#41	1	1	yes	characteristic
147	C₂×C₁₁₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
148	D₁₁₃	—	2	4	no	—
149	C₂×#554	—	113	1	no	maximal, Sylow
150	C₁₁₃	C₂×#554	1	1	yes	characteristic, complemented, Sylow
151	#554	—	113	16	no	—
152	C₂×C₄⋊Q₈	—	113	1	no	—
153	C₂×#125	—	113	2	no	—
154	C₂×C₂²⋊Q₈	—	113	1	no	—
155	C₂×C₂²⋊Q₈	—	113	2	no	—
156	C₂×C₂²⋊Q₈	—	113	2	no	—
157	C₂×C₂²⋊Q₈	D₁₁₃	1	1	yes	characteristic, complemented
158	C₂×C₄×Q₈	—	113	1	no	—
159	C₂×C₄×D₄	—	113	1	no	—
160	C₂×C₄×D₄	—	113	2	no	—
161	C₂²×C₄⋊C₄	—	113	1	no	—
162	C₂²×C₄⋊C₄	—	113	1	no	—
163	C₂²×Q₈	—	113	2	no	—
164	C₂²×Q₈	C₂×D₁₁₃	1	1	yes	characteristic
165	C₂²×D₄	—	113	1	no	—
166	C₂³×C₄	—	113	2	no	—
167	C₂³×C₄	—	113	2	no	—
168	C₂³×C₄	—	113	1	no	—
169	C₂³×C₄	C₂×D₁₁₃	1	1	yes	characteristic
170	C₄⋊Q₈	—	113	8	no	—
171	#125	—	113	16	no	—
172	C₂²⋊Q₈	—	113	8	no	—
173	C₂²⋊Q₈	—	113	16	no	—
174	C₂²⋊Q₈	—	113	16	no	—
175	C₂²⋊Q₈	C₂×D₁₁₃	1	8	yes	complemented
176	C₄×Q₈	—	113	8	no	—
177	C₄×D₄	—	113	8	no	—
178	C₄×D₄	—	113	16	no	—
179	C₂×C₄⋊C₄	—	113	1	no	—
180	C₂×C₄⋊C₄	—	113	2	no	—
181	C₂×C₄⋊C₄	—	113	2	no	—
182	C₂×C₄⋊C₄	—	113	1	no	—
183	C₂×C₄⋊C₄	—	113	8	no	—
184	C₂×C₄⋊C₄	—	113	1	no	—
185	C₂×C₄⋊C₄	—	113	2	no	—
186	C₂×C₄⋊C₄	—	113	2	no	—
187	C₂×C₄⋊C₄	—	113	2	no	—
188	C₂×C₄⋊C₄	—	113	8	no	—
189	C₂×C₄⋊C₄	C₂×D₁₁₃	1	1	yes	characteristic
190	C₂×C₄⋊C₄	C₂×D₁₁₃	1	2	yes	—
191	C₂×C₂²⋊C₄	—	113	1	no	—
192	C₂×C₂²⋊C₄	—	113	1	no	—
193	C₂×C₂²⋊C₄	—	113	2	no	—
194	C₂×C₂²⋊C₄	C₂×D₁₁₃	1	2	yes	—
195	C₂×C₄²	—	113	2	no	—
196	C₂×C₄²	—	113	1	no	—
197	C₂⁴	—	113	1	no	—
198	C₂⁴	C₂²×D₁₁₃	1	1	yes	characteristic
199	C₂×Q₈	—	2	4	no	—
200	C₂×Q₈	—	113	8	no	—
201	C₂×Q₈	—	452	4	no	—
202	C₂×Q₈	C₂²×D₁₁₃	1	4	yes	—
203	C₂×D₄	—	113	8	no	—
204	C₂×D₄	—	113	4	no	—
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	113	1	no	—
207	C₂²×C₄	—	113	8	no	—
208	C₂²×C₄	—	113	1	no	—
209	C₂²×C₄	—	113	2	no	—
210	C₂²×C₄	—	113	1	no	—
211	C₂²×C₄	—	113	2	no	—
212	C₂²×C₄	—	113	8	no	—
213	C₂²×C₄	—	113	4	no	—
214	C₂²×C₄	—	452	4	no	—
215	C₂²×C₄	—	452	4	no	—
216	C₂²×C₄	—	226	4	no	—
217	C₂²×C₄	C₂²×D₁₁₃	1	4	yes	—
218	C₂²×C₄	C₂²×D₁₁₃	1	2	yes	—
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	C₄⋊C₄	—	113	4	no	—
223	C₄⋊C₄	—	113	8	no	—
224	C₄⋊C₄	—	113	8	no	—
225	C₄⋊C₄	—	113	4	no	—
226	C₄⋊C₄	—	113	4	no	—
227	C₄⋊C₄	—	113	8	no	—
228	C₄⋊C₄	—	113	8	no	—
229	C₄⋊C₄	—	113	8	no	—
230	C₄⋊C₄	C₂²×D₁₁₃	1	4	yes	—
231	C₄⋊C₄	C₂²×D₁₁₃	1	8	yes	—
232	C₂²⋊C₄	—	113	4	no	—
233	C₂²⋊C₄	—	113	4	no	—
234	C₂²⋊C₄	—	113	8	no	—
235	C₂²⋊C₄	C₂²×D₁₁₃	1	8	yes	—
236	C₄²	—	113	8	no	—
237	C₄²	—	113	4	no	—
238	C₂³	—	2	2	no	—
239	C₂³	—	2	2	no	—
240	C₂³	—	113	4	no	—
241	C₂³	—	113	2	no	—
242	C₂³	—	226	2	no	—
243	C₂³	—	226	2	no	—
244	C₂³	Q₈×D₁₁₃	1	4	yes	—
245	C₂³	C₂³×D₁₁₃	1	2	yes	—
246	C₂³	C₂³×D₁₁₃	1	1	yes	characteristic
247	Q₈	—	2	8	no	—
248	Q₈	—	452	8	no	—
249	D₄	—	113	16	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	2	8	no	—
254	C₂×C₄	—	113	4	no	—
255	C₂×C₄	—	113	2	no	—
256	C₂×C₄	—	113	2	no	—
257	C₂×C₄	—	113	4	no	—
258	C₂×C₄	—	113	2	no	—
259	C₂×C₄	—	113	4	no	—
260	C₂×C₄	—	452	8	no	—
261	C₂×C₄	—	452	8	no	—
262	C₂×C₄	—	226	2	no	—
263	C₂×C₄	—	226	4	no	—
264	C₂×C₄	—	226	4	no	—
265	C₂×C₄	—	226	2	no	—
266	C₂×C₄	—	226	8	no	—
267	C₂×C₄	#14268646	1	4	yes	—
268	C₂×C₄	C₂³×D₁₁₃	1	4	yes	—
269	C₂×C₄	C₂³×D₁₁₃	1	2	yes	—
270	C₂×C₄	C₂³×D₁₁₃	1	4	yes	—
271	C₂×C₄	C₂³×D₁₁₃	1	2	yes	—
272	C₂²	—	2	8	no	—
273	C₂²	—	2	2	no	—
274	C₂²	—	2	2	no	—
275	C₂²	—	113	4	no	—
276	C₂²	—	226	8	no	—
277	C₂²	—	226	2	no	—
278	C₂²	—	226	2	no	—
279	C₂²	C₂×Q₈×D₁₁₃	1	4	yes	—
280	C₂²	C₂×Q₈×D₁₁₃	1	2	yes	—
281	C₂²	C₂×#14268646	1	2	yes	—
282	C₂²	#16813668	1	2	yes	—
283	C₂²	C₂⁴×D₁₁₃	1	1	yes	characteristic
284	C₄	—	2	2	no	—
285	C₄	—	2	4	no	—
286	C₄	—	2	4	no	—
287	C₄	—	113	4	no	—
288	C₄	—	226	2	no	—
289	C₄	—	226	4	no	—
290	C₄	—	226	4	no	—
291	C₄	—	226	2	no	—
292	C₄	C₂×#14268646	1	4	yes	—
293	C₂	—	2	4	no	—
294	C₂	—	226	4	no	—
295	C₂	#20299306	1	4	yes	direct product factor
296	C₂	C₂²×Q₈×D₁₁₃	1	1	yes	characteristic
297	C₂	C₂²×#14268646	1	1	yes	characteristic
298	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×C₂²⋊Q₈×C₁₁₃	C₂
socle	C₂³×C₁₁₃	C₂⁴
prime radical	C₂²×C₂²⋊D₁₁₃	C₂²
hypercenter	C₂×C₂²⋊Q₈	D₁₁₃
hypocenter	C₁₁₃	C₂×#554
LCS	G ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
1. #20299306	C₂	direct (N × H)
2. C₂×#16813621	C₂	semidirect
3. C₂×#16813612	C₂	semidirect
4. C₂×#16813602	C₂	semidirect
5. C₂×#16813599	C₂	semidirect
6. C₂×#16813598	C₂	semidirect
7. C₂×Q₈×Dic₁₁₃	C₂	semidirect
10. C₂²×C₄⋊Dic₁₁₃	C₂	semidirect
11. C₂×C₄⋊C₄×D₁₁₃	C₂	semidirect
12. C₂×C₂²⋊Q₈×C₁₁₃	C₂	semidirect
18. #16813621	C₂²	semidirect
19. #16813612	C₂²	semidirect
20. #16813602	C₂²	semidirect
21. #16813599	C₂²	semidirect
22. #16813598	C₂²	semidirect
23. Q₈×Dic₁₁₃	C₂²	semidirect
30. C₂×C₄⋊Dic₁₁₃	C₂²	semidirect
35. C₄⋊C₄×D₁₁₃	C₂²	semidirect
43. C₂²⋊Q₈×C₁₁₃	C₂²	semidirect
50. C₂×C₂²⋊D₁₁₃	Q₈	semidirect
107. C₂²⋊D₁₁₃	C₂×Q₈	semidirect
144. C₂×C₁₁₃	#554	semidirect
150. C₁₁₃	C₂×#554	semidirect
157. C₂×C₂²⋊Q₈	D₁₁₃	semidirect
175. C₂²⋊Q₈	C₂×D₁₁₃	semidirect
295. C₂	#20299306	direct (N × H)

Maximal subgroups

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

subgroup	index
#20299306	2
C₂×#16813621	2
C₂×#16813612	2
C₂×#16813602	2
C₂×#16813599	2
C₂×#16813598	2
C₂×Q₈×Dic₁₁₃	2
C₂×C₄×C₂²⋊D₁₁₃	2
C₂×#16813582	2
C₂²×C₄⋊Dic₁₁₃	2
C₂×C₄⋊C₄×D₁₁₃	2
C₂×C₂²⋊Q₈×C₁₁₃	2
C₂×#554	113

Subgroups — C2×#20299306 #28404106

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20299306 #28404106