Subgroups — C₂²×#14583452 #26003299

Order 11008.

8238 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583452	C₂	1	48	yes	maximal, direct product factor
2	C₂²×#13063324	C₂	1	2	yes	maximal, complemented
3	C₂²×#13063316	C₂	1	2	yes	maximal, complemented
4	C₂²×#13063302	C₂	1	4	yes	maximal
5	C₂²×C₄×Dic₈₆	C₂	1	4	yes	maximal, complemented
6	C₂²×#13063284	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₄×Dic₄₃	C₂	1	1	yes	maximal, characteristic
8	C₂²×#117×C₄₃	C₂	1	1	yes	maximal, characteristic
9	C₂³×Dic₈₆	C₂²	1	4	yes	—
10	C₂⁴×Dic₄₃	C₂²	1	2	yes	—
11	#14583452	C₂²	1	256	yes	direct product factor
12	C₂×#13063324	C₂²	1	48	yes	complemented
13	C₂×#13063316	C₂²	1	48	yes	complemented
14	C₂×#13063302	C₂²	1	96	yes	—
15	C₂×C₄×Dic₈₆	C₂²	1	96	yes	complemented
16	C₂×#13063284	C₂²	1	24	yes	—
17	C₂²×C₄⋊Dic₄₃	C₂²	1	4	yes	—
18	C₂²×C₄⋊Dic₄₃	C₂²	1	2	yes	—
19	C₂²×Dic₄₃⋊C₄	C₂²	1	4	yes	—
20	C₂²×Dic₄₃⋊C₄	C₂²	1	4	yes	—
21	C₂²×C₂²⋊Dic₄₃	C₂²	1	2	yes	—
22	C₂²×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
23	C₂²×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
24	C₂²×C₄×Dic₄₃	C₂²	1	24	yes	—
25	C₂²×C₄×Dic₄₃	C₂²	1	2	yes	—
26	C₂²×C₄×Dic₄₃	C₂²	1	2	yes	—
27	C₂⁴×C₄×C₄₃	C₂²	1	1	yes	characteristic
28	C₂×#117×C₄₃	C₂²	1	24	yes	—
29	C₂²×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
30	C₂²×C₂²⋊C₄×C₄₃	C₂²	1	2	yes	—
31	C₂²×C₄²×C₄₃	C₂²	1	2	yes	—
32	C₂²×Dic₈₆	—	2	32	no	—
33	C₂²×Dic₈₆	C₂³	1	48	yes	—
34	C₂³×Dic₄₃	—	2	16	no	—
35	C₂³×Dic₄₃	C₂³	1	2	yes	—
36	C₂³×Dic₄₃	C₂³	1	24	yes	—
37	C₂³×Dic₄₃	C₂³	1	4	yes	—
38	C₂³×Dic₄₃	C₂³	1	2	yes	—
39	#13063324	C₂³	1	128	yes	complemented
40	#13063316	C₂³	1	128	yes	complemented
41	#13063302	C₂³	1	256	yes	—
42	C₄×Dic₈₆	C₂³	1	256	yes	complemented
43	#13063284	C₂³	1	64	yes	—
44	C₂×C₄⋊Dic₄₃	C₂³	1	24	yes	—
45	C₂×C₄⋊Dic₄₃	C₂³	1	48	yes	—
46	C₂×Dic₄₃⋊C₄	C₂³	1	48	yes	—
47	C₂×Dic₄₃⋊C₄	C₂³	1	48	yes	—
48	C₂×C₂²⋊Dic₄₃	C₂³	1	24	yes	—
49	C₂×C₄×Dic₄₃	C₂³	1	24	yes	—
50	C₂×C₄×Dic₄₃	C₂³	1	12	yes	—
51	C₂×C₄×Dic₄₃	C₂³	1	12	yes	—
52	C₂×C₄×Dic₄₃	C₂³	1	64	yes	—
53	C₂×C₄×Dic₄₃	C₂³	1	24	yes	—
54	C₂⁵×C₄₃	C₂³	1	1	yes	characteristic
55	C₂³×C₄×C₄₃	Q₈	1	16	yes	complemented
56	C₂³×C₄×C₄₃	C₂³	1	4	yes	—
57	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
58	C₂³×C₄×C₄₃	C₂³	1	12	yes	—
59	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
60	#117×C₄₃	C₂³	1	64	yes	—
61	C₂×C₄⋊C₄×C₄₃	C₂³	1	24	yes	—
62	C₂×C₂²⋊C₄×C₄₃	C₂³	1	24	yes	—
63	C₂×C₄²×C₄₃	C₂³	1	24	yes	—
64	C₂⁴×C₄₃	—	2	8	no	—
65	C₂⁴×C₄₃	C₂×Q₈	1	8	yes	—
66	C₂⁴×C₄₃	C₂⁴	1	6	yes	—
67	C₂⁴×C₄₃	C₂⁴	1	1	yes	characteristic
68	C₂²×C₄×C₄₃	—	2	16	no	—
69	C₂²×C₄×C₄₃	C₂×Q₈	1	8	yes	—
70	C₂²×C₄×C₄₃	C₂×Q₈	1	8	yes	—
71	C₂²×C₄×C₄₃	C₂×Q₈	1	96	yes	complemented
72	C₂²×C₄×C₄₃	C₂⁴	1	24	yes	—
73	C₂²×C₄×C₄₃	C₂⁴	1	6	yes	—
74	C₂²×C₄×C₄₃	C₂⁴	1	6	yes	—
75	C₂²×C₄×C₄₃	C₂⁴	1	16	yes	—
76	C₂²×Dic₄₃	—	2	96	no	—
77	C₂²×Dic₄₃	—	2	8	no	—
78	C₂²×Dic₄₃	—	2	16	no	—
79	C₂²×Dic₄₃	#41	1	16	yes	—
80	C₂²×Dic₄₃	C₂⁴	1	12	yes	—
81	C₂²×Dic₄₃	C₂⁴	1	32	yes	—
82	C₂²×Dic₄₃	C₂⁴	1	24	yes	—
83	C₂²×Dic₄₃	C₂⁴	1	12	yes	—
84	C₂×Dic₈₆	—	2	192	no	—
85	C₂×Dic₈₆	C₂⁴	1	64	yes	—
86	C₄⋊C₄×C₄₃	C₂⁴	1	32	yes	—
87	C₂²⋊C₄×C₄₃	C₂⁴	1	32	yes	—
88	C₄²×C₄₃	C₂⁴	1	32	yes	—
89	C₂²⋊Dic₄₃	C₂⁴	1	32	yes	—
90	C₄⋊Dic₄₃	C₂⁴	1	32	yes	—
91	C₄⋊Dic₄₃	C₂⁴	1	64	yes	—
92	Dic₄₃⋊C₄	C₂⁴	1	64	yes	—
93	Dic₄₃⋊C₄	C₂⁴	1	64	yes	—
94	C₄×Dic₄₃	C₂⁴	1	32	yes	—
95	C₄×Dic₄₃	C₂⁴	1	16	yes	—
96	C₄×Dic₄₃	C₂⁴	1	16	yes	—
97	C₄×Dic₄₃	C₂⁴	1	32	yes	—
98	C₂³×C₄₃	—	2	32	no	—
99	C₂³×C₄₃	—	2	24	no	—
100	C₂³×C₄₃	C₂²×Q₈	1	24	yes	—
101	C₂³×C₄₃	C₂²×Q₈	1	4	yes	—
102	C₂³×C₄₃	C₂×#41	1	8	yes	—
103	C₂³×C₄₃	C₂⁵	1	4	yes	—
104	C₂³×C₄₃	C₂⁵	1	3	yes	—
105	C₂×C₄×C₄₃	—	2	48	no	—
106	C₂×C₄×C₄₃	C₂²×Q₈	1	24	yes	—
107	C₂×C₄×C₄₃	C₂²×Q₈	1	24	yes	—
108	C₂×C₄×C₄₃	C₂²×Q₈	1	64	yes	complemented
109	C₂×C₄×C₄₃	C₂⁵	1	16	yes	—
110	C₂×C₄×C₄₃	C₂⁵	1	4	yes	—
111	C₂×C₄×C₄₃	C₂⁵	1	4	yes	—
112	C₂×Dic₄₃	—	2	48	no	—
113	C₂×Dic₄₃	—	2	24	no	—
114	C₂×Dic₄₃	—	2	64	no	—
115	C₂×Dic₄₃	C₂×#41	1	48	yes	—
116	C₂×Dic₄₃	C₂⁵	1	8	yes	—
117	C₂×Dic₄₃	C₂⁵	1	16	yes	—
118	C₂×Dic₄₃	C₂⁵	1	8	yes	—
119	Dic₈₆	—	2	128	no	—
120	C₂²×#533	—	43	1	no	maximal, Sylow
121	C₂²×C₄₃	—	2	8	no	—
122	C₂²×C₄₃	—	2	48	no	—
123	C₂²×C₄₃	#533	1	16	yes	complemented
124	C₂²×C₄₃	C₂³×Q₈	1	8	yes	—
125	C₂²×C₄₃	C₂³×Q₈	1	6	yes	—
126	C₂²×C₄₃	C₂²×#41	1	12	yes	—
127	C₂²×C₄₃	C₂⁶	1	1	yes	characteristic
128	C₄×C₄₃	—	2	16	no	—
129	C₄×C₄₃	C₂³×Q₈	1	8	yes	—
130	C₄×C₄₃	C₂³×Q₈	1	8	yes	—
131	Dic₄₃	—	2	16	no	—
132	Dic₄₃	—	2	8	no	—
133	Dic₄₃	C₂²×#41	1	16	yes	—
134	C₂×#533	—	43	48	no	—
135	C₂²×C₄⋊Q₈	—	43	2	no	—
136	C₂²×#125	—	43	2	no	—
137	C₂²×C₂²⋊Q₈	—	43	4	no	—
138	C₂²×C₄×Q₈	—	43	4	no	—
139	C₂²×#117	—	43	1	no	—
140	C₂²×#117	D₄₃	1	1	yes	characteristic
141	C₂³×C₄²	—	43	1	no	—
142	C₂×C₄₃	—	2	8	no	—
143	C₂×C₄₃	C₂×#533	1	12	yes	complemented
144	C₂×C₄₃	C₂⁴×Q₈	1	1	yes	characteristic
145	C₂×C₄₃	C₂³×#41	1	2	yes	—
146	C₂³×Q₈	—	43	4	no	—
147	C₂⁴×C₄	—	43	2	no	—
148	C₂⁴×C₄	C₂×D₄₃	1	1	yes	characteristic
149	#533	—	43	256	no	—
150	C₂×C₄⋊Q₈	—	43	48	no	—
151	C₂×#125	—	43	48	no	—
152	C₂×C₂²⋊Q₈	—	43	96	no	—
153	C₂×C₄×Q₈	—	43	96	no	—
154	C₂×#117	—	43	24	no	—
155	C₂×#117	C₂×D₄₃	1	24	yes	—
156	C₂²×C₄⋊C₄	—	43	4	no	—
157	C₂²×C₄⋊C₄	—	43	4	no	—
158	C₂²×C₄⋊C₄	—	43	2	no	—
159	C₂²×C₄⋊C₄	—	43	4	no	—
160	C₂²×C₄⋊C₄	C₂×D₄₃	1	2	yes	—
161	C₂²×C₂²⋊C₄	—	43	2	no	—
162	C₂²×C₂²⋊C₄	C₂×D₄₃	1	2	yes	—
163	C₂²×C₄²	—	43	1	no	—
164	C₂²×C₄²	—	43	1	no	—
165	C₂²×C₄²	—	43	24	no	—
166	C₂²×C₄²	—	43	2	no	—
167	C₂²×C₄²	—	43	2	no	—
168	C₂²×C₄²	C₂×D₄₃	1	2	yes	—
169	C₄₃	C₂²×#533	1	1	yes	characteristic, complemented, Sylow
170	C₂⁵	C₂²×D₄₃	1	1	yes	characteristic
171	C₂²×Q₈	—	43	48	no	—
172	C₂²×Q₈	—	86	32	no	—
173	C₂³×C₄	—	43	2	no	—
174	C₂³×C₄	—	43	4	no	—
175	C₂³×C₄	—	43	2	no	—
176	C₂³×C₄	—	43	24	no	—
177	C₂³×C₄	—	86	16	no	—
178	C₂³×C₄	Dic₈₆	1	16	yes	complemented
179	C₂³×C₄	C₂²×D₄₃	1	4	yes	—
180	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
181	C₂³×C₄	C₂²×D₄₃	1	12	yes	—
182	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
183	C₄⋊Q₈	—	43	128	no	—
184	#125	—	43	128	no	—
185	C₂²⋊Q₈	—	43	256	no	—
186	C₄×Q₈	—	43	256	no	—
187	#117	—	43	64	no	—
188	#117	C₂²×D₄₃	1	64	yes	—
189	C₂×C₄⋊C₄	—	43	24	no	—
190	C₂×C₄⋊C₄	—	43	48	no	—
191	C₂×C₄⋊C₄	—	43	48	no	—
192	C₂×C₄⋊C₄	—	43	48	no	—
193	C₂×C₄⋊C₄	C₂²×D₄₃	1	24	yes	—
194	C₂×C₂²⋊C₄	—	43	24	no	—
195	C₂×C₂²⋊C₄	C₂²×D₄₃	1	24	yes	—
196	C₂×C₄²	—	43	24	no	—
197	C₂×C₄²	—	43	24	no	—
198	C₂×C₄²	—	43	12	no	—
199	C₂×C₄²	—	43	12	no	—
200	C₂×C₄²	—	43	64	no	—
201	C₂×C₄²	C₂²×D₄₃	1	24	yes	—
202	C₂⁴	—	2	8	no	—
203	C₂⁴	C₂×Dic₈₆	1	8	yes	—
204	C₂⁴	C₂³×D₄₃	1	6	yes	—
205	C₂⁴	C₂³×D₄₃	1	1	yes	characteristic
206	C₂×Q₈	—	43	64	no	—
207	C₂×Q₈	—	86	192	no	—
208	C₂²×C₄	—	2	16	no	—
209	C₂²×C₄	—	43	16	no	—
210	C₂²×C₄	—	43	12	no	—
211	C₂²×C₄	—	43	12	no	—
212	C₂²×C₄	—	43	32	no	—
213	C₂²×C₄	—	43	24	no	—
214	C₂²×C₄	—	86	96	no	—
215	C₂²×C₄	—	86	8	no	—
216	C₂²×C₄	—	86	16	no	—
217	C₂²×C₄	C₂×Dic₈₆	1	8	yes	—
218	C₂²×C₄	C₂×Dic₈₆	1	8	yes	—
219	C₂²×C₄	C₂×Dic₈₆	1	96	yes	complemented
220	C₂²×C₄	C₂³×D₄₃	1	24	yes	—
221	C₂²×C₄	C₂³×D₄₃	1	6	yes	—
222	C₂²×C₄	C₂³×D₄₃	1	6	yes	—
223	C₂²×C₄	C₂³×D₄₃	1	16	yes	—
224	C₄⋊C₄	—	43	32	no	—
225	C₄⋊C₄	—	43	64	no	—
226	C₄⋊C₄	—	43	64	no	—
227	C₄⋊C₄	—	43	64	no	—
228	C₄⋊C₄	C₂³×D₄₃	1	32	yes	—
229	C₂²⋊C₄	—	43	32	no	—
230	C₂²⋊C₄	C₂³×D₄₃	1	32	yes	—
231	C₄²	—	43	32	no	—
232	C₄²	—	43	16	no	—
233	C₄²	—	43	16	no	—
234	C₄²	—	43	32	no	—
235	C₄²	C₂³×D₄₃	1	32	yes	—
236	C₂³	—	2	32	no	—
237	C₂³	—	2	24	no	—
238	C₂³	C₂²×Dic₈₆	1	24	yes	—
239	C₂³	C₂²×Dic₈₆	1	4	yes	—
240	C₂³	#41×D₄₃	1	8	yes	—
241	C₂³	C₂⁴×D₄₃	1	4	yes	—
242	C₂³	C₂⁴×D₄₃	1	3	yes	—
243	Q₈	—	86	128	no	—
244	C₂×C₄	—	2	48	no	—
245	C₂×C₄	—	43	48	no	—
246	C₂×C₄	—	43	8	no	—
247	C₂×C₄	—	43	8	no	—
248	C₂×C₄	—	43	16	no	—
249	C₂×C₄	—	86	64	no	—
250	C₂×C₄	—	86	24	no	—
251	C₂×C₄	—	86	48	no	—
252	C₂×C₄	C₂²×Dic₈₆	1	24	yes	—
253	C₂×C₄	C₂²×Dic₈₆	1	24	yes	—
254	C₂×C₄	C₂²×Dic₈₆	1	64	yes	complemented
255	C₂×C₄	C₂⁴×D₄₃	1	16	yes	—
256	C₂×C₄	C₂⁴×D₄₃	1	4	yes	—
257	C₂×C₄	C₂⁴×D₄₃	1	4	yes	—
258	C₂²	—	2	8	no	—
259	C₂²	—	2	48	no	—
260	C₂²	#14583452	1	16	yes	direct product factor
261	C₂²	C₂³×Dic₈₆	1	8	yes	—
262	C₂²	C₂³×Dic₈₆	1	6	yes	—
263	C₂²	C₂×#41×D₄₃	1	12	yes	—
264	C₂²	C₂⁵×D₄₃	1	1	yes	characteristic
265	C₄	—	2	16	no	—
266	C₄	—	43	16	no	—
267	C₄	—	86	8	no	—
268	C₄	—	86	16	no	—
269	C₄	C₂³×Dic₈₆	1	8	yes	—
270	C₄	C₂³×Dic₈₆	1	8	yes	—
271	C₂	—	2	8	no	—
272	C₂	C₂×#14583452	1	12	yes	direct product factor
273	C₂	C₂⁴×Dic₈₆	1	1	yes	characteristic
274	C₂	C₂²×#41×D₄₃	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₄₃
commutator, cosocle	C₂²×C₄₃	C₂⁶
Frattini	C₂²	C₂⁵×D₄₃
Fitting	C₂²×#117×C₄₃	C₂
socle	C₂⁴×C₄₃	C₂⁴
prime radical	C₂⁵×C₄₃	C₂³
hypercenter	C₂²×#117	D₄₃
hypocenter	C₄₃	C₂²×#533
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×#117

Complements

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

N	H ≅ G/N	type
1. C₂×#14583452	C₂	direct (N × H)
2. C₂²×#13063324	C₂	semidirect
3. C₂²×#13063316	C₂	semidirect
5. C₂²×C₄×Dic₈₆	C₂	semidirect
11. #14583452	C₂²	direct (N × H)
12. C₂×#13063324	C₂²	semidirect
13. C₂×#13063316	C₂²	semidirect
15. C₂×C₄×Dic₈₆	C₂²	semidirect
39. #13063324	C₂³	semidirect
40. #13063316	C₂³	semidirect
42. C₄×Dic₈₆	C₂³	semidirect
55. C₂³×C₄×C₄₃	Q₈	semidirect
71. C₂²×C₄×C₄₃	C₂×Q₈	semidirect
108. C₂×C₄×C₄₃	C₂²×Q₈	semidirect
123. C₂²×C₄₃	#533	semidirect
143. C₂×C₄₃	C₂×#533	semidirect
169. C₄₃	C₂²×#533	semidirect
178. C₂³×C₄	Dic₈₆	semidirect
219. C₂²×C₄	C₂×Dic₈₆	semidirect
254. C₂×C₄	C₂²×Dic₈₆	semidirect
260. C₂²	#14583452	direct (N × H)
272. C₂	C₂×#14583452	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#14583452	2
C₂²×#13063324	2
C₂²×#13063316	2
C₂²×#13063302	2
C₂²×C₄×Dic₈₆	2
C₂²×#13063284	2
C₂³×C₄×Dic₄₃	2
C₂²×#117×C₄₃	2
C₂²×#533	43

Subgroups — C22×#14583452 #26003299

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#14583452 #26003299