Subgroups — C₂×#20242981 #26203922

Order 12032.

1046 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689500	C₂	1	1	yes	maximal, characteristic
2	C₂×#15689497	C₂	1	2	yes	maximal, complemented
3	C₂×#15689471	C₂	1	2	yes	maximal, complemented
4	#20242981	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₈×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄₇×#500	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#10625336	—	2	4	no	—
8	C₂²×D₃₇₆	—	2	2	no	—
9	C₂×#13094021	C₂²	1	2	yes	—
10	#15689500	—	2	2	no	—
11	#15689500	C₂²	1	4	yes	—
12	#15689497	—	2	4	no	—
13	#15689497	C₂²	1	8	yes	complemented
14	#15689471	—	2	4	no	—
15	#15689471	C₂²	1	8	yes	complemented
16	C₂×#13093944	—	2	4	no	—
17	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
19	C₈×C₄₇⋊C₈	C₂²	1	4	yes	complemented
20	C₂×C₄⋊Q₈×C₄₇	C₂²	1	2	yes	—
21	C₄₇×#500	—	2	2	no	—
22	C₄₇×#500	C₂²	1	4	yes	complemented
23	C₂×#107×C₄₇	—	2	2	no	—
24	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
25	C₂²×D₁₈₈	—	2	2	no	—
26	C₂²×D₁₈₈	—	2	2	no	—
27	C₂²×D₁₈₈	—	2	2	no	—
28	C₂×#10625336	—	2	16	no	—
29	C₂×#10625336	—	4	16	no	—
30	C₂×D₃₇₆	—	2	8	no	—
31	C₂×D₃₇₆	—	4	8	no	—
32	C₂²×C₄₇⋊C₈	D₄	1	2	yes	—
33	C₂²×C₄₇⋊C₈	D₄	1	2	yes	—
34	#13094021	—	2	2	no	—
35	#13094021	—	2	2	no	—
36	#13094021	—	2	2	no	—
37	#13094021	C₂³	1	4	yes	—
38	#13093944	—	2	16	no	—
39	C₄×C₄₇⋊C₈	D₄	1	2	yes	—
40	C₄×C₄₇⋊C₈	D₄	1	2	yes	—
41	C₄×C₄₇⋊C₈	C₂³	1	2	yes	—
42	C₄×C₄₇⋊C₈	C₂³	1	2	yes	—
43	C₂²×Q₈×C₄₇	—	2	2	no	—
44	C₂²×C₈×C₄₇	D₄	1	2	yes	—
45	C₄⋊Q₈×C₄₇	—	2	2	no	—
46	C₄⋊Q₈×C₄₇	—	2	2	no	—
47	C₄⋊Q₈×C₄₇	—	2	2	no	—
48	C₄⋊Q₈×C₄₇	C₂³	1	4	yes	—
49	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
50	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
51	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
52	#107×C₄₇	—	2	8	no	—
53	C₄×C₈×C₄₇	D₄	1	2	yes	—
54	C₄×C₈×C₄₇	C₂³	1	2	yes	—
55	C₂³×D₄₇	—	4	2	no	—
56	C₂×Q₈×C₄₇	—	2	4	no	—
57	C₂×Q₈×C₄₇	—	4	2	no	—
58	C₂×Q₈×C₄₇	—	4	8	no	—
59	C₂²×C₄×C₄₇	—	4	2	no	—
60	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
63	C₂×D₁₈₈	—	2	4	no	—
64	C₂×D₁₈₈	—	2	4	no	—
65	C₂×D₁₈₈	—	2	4	no	—
66	C₂×D₁₈₈	—	4	8	no	—
67	C₂×D₁₈₈	—	4	8	no	—
68	C₂×D₁₈₈	—	4	8	no	—
69	C₂×D₁₈₈	—	4	2	no	—
70	C₂×D₁₈₈	—	4	2	no	—
71	C₂×D₁₈₈	—	4	2	no	—
72	C₂×C₈×C₄₇	SD₁₆	1	8	yes	complemented
73	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
74	C₄⋊C₄×C₄₇	—	2	4	no	—
75	C₄⋊C₄×C₄₇	—	2	4	no	—
76	C₄⋊C₄×C₄₇	—	4	2	no	—
77	C₄⋊C₄×C₄₇	—	4	2	no	—
78	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
82	#10625336	—	4	32	no	—
83	C₂×C₄₇⋊C₈	—	2	4	no	—
84	C₂×C₄₇⋊C₈	D₈	1	8	yes	complemented
85	C₂×C₄₇⋊C₈	C₂×D₄	1	4	yes	—
86	C₂×C₄₇⋊C₈	C₂×D₄	1	4	yes	—
87	D₃₇₆	—	4	16	no	—
88	C₂³×C₄₇	#127	1	1	yes	characteristic
89	C₂²×D₄₇	—	4	4	no	—
90	C₂²×D₄₇	—	8	4	no	—
91	C₂²×D₄₇	—	8	4	no	—
92	C₂²×D₄₇	—	8	4	no	—
93	Q₈×C₄₇	—	4	8	no	—
94	Q₈×C₄₇	—	8	4	no	—
95	C₂×C₄×C₄₇	—	4	4	no	—
96	C₂×C₄×C₄₇	—	8	4	no	—
97	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
98	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
99	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
100	C₂×C₄×C₄₇	C₂×D₈	1	4	yes	—
101	C₂×C₄×C₄₇	C₂×SD₁₆	1	4	yes	—
102	C₂×C₄×C₄₇	#135	1	4	yes	—
103	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
104	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
105	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
106	D₁₈₈	—	4	8	no	—
107	D₁₈₈	—	4	8	no	—
108	D₁₈₈	—	4	8	no	—
109	D₁₈₈	—	8	4	no	—
110	D₁₈₈	—	8	4	no	—
111	D₁₈₈	—	8	4	no	—
112	C₈×C₄₇	C₂×SD₁₆	1	8	yes	complemented
113	C₄₇⋊C₈	—	2	4	no	—
114	C₄₇⋊C₈	C₂×D₈	1	8	yes	complemented
115	C₂×#1706	—	47	1	no	maximal, Sylow
116	C₂²×C₄₇	#492	1	2	yes	—
117	C₂²×C₄₇	#493	1	2	yes	—
118	C₂²×C₄₇	#495	1	2	yes	—
119	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
120	C₂×D₄₇	—	8	4	no	—
121	C₂×D₄₇	—	8	4	no	—
122	C₂×D₄₇	—	8	4	no	—
123	C₂×D₄₇	—	16	8	no	—
124	C₄×C₄₇	—	8	4	no	—
125	C₄×C₄₇	#492	1	2	yes	—
126	C₄×C₄₇	#493	1	2	yes	—
127	C₄×C₄₇	#495	1	2	yes	—
128	C₄×C₄₇	C₂²×D₈	1	2	yes	—
129	C₄×C₄₇	C₂²×SD₁₆	1	2	yes	—
130	C₄×C₄₇	C₂×#135	1	2	yes	—
131	C₂×#500	D₄₇	1	1	yes	characteristic, complemented
132	C₂×#493	—	47	1	no	—
133	C₂×#492	—	47	2	no	—
134	C₂×#486	—	47	2	no	—
135	#1706	—	47	8	no	—
136	C₂×C₈²	—	47	1	no	—
137	C₂×C₄₇	#1706	1	4	yes	complemented
138	C₂×C₄₇	C₂×#492	1	1	yes	characteristic
139	C₂×C₄₇	C₂×#493	1	1	yes	characteristic
140	C₂×C₄₇	C₂×#495	1	1	yes	characteristic
141	D₄₇	—	16	4	no	—
142	C₂²×SD₁₆	—	94	4	no	—
143	C₂²×D₈	—	94	2	no	—
144	C₂×C₄⋊Q₈	C₂×D₄₇	1	2	yes	—
145	C₂×#127	—	47	2	no	—
146	#500	—	2	2	no	—
147	#500	C₂×D₄₇	1	4	yes	complemented
148	#493	—	47	4	no	—
149	#493	—	94	2	no	—
150	#492	—	47	8	no	—
151	#492	—	94	4	no	—
152	#486	—	47	8	no	—
153	#486	—	94	4	no	—
154	C₂×#107	—	2	2	no	—
155	C₂×#102	—	94	4	no	—
156	C₂×C₄×C₈	—	47	1	no	—
157	C₂×C₄×C₈	—	47	1	no	—
158	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
159	C₈²	—	47	4	no	—
160	C₄₇	C₂×#1706	1	1	yes	characteristic, complemented, Sylow
161	C₂²×Q₈	—	2	2	no	—
162	C₂²×D₄	—	94	2	no	—
163	C₂²×D₄	—	94	2	no	—
164	C₂²×D₄	—	94	2	no	—
165	C₂×SD₁₆	—	94	16	no	—
166	C₂×SD₁₆	—	188	16	no	—
167	C₂×D₈	—	94	8	no	—
168	C₂×D₈	—	188	8	no	—
169	C₂²×C₈	—	47	2	no	—
170	C₂²×C₈	—	47	2	no	—
171	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
172	C₄⋊Q₈	—	2	2	no	—
173	C₄⋊Q₈	—	2	2	no	—
174	C₄⋊Q₈	—	2	2	no	—
175	C₄⋊Q₈	C₂²×D₄₇	1	4	yes	—
176	#127	—	47	4	no	—
177	#127	—	94	2	no	—
178	#127	—	94	2	no	—
179	#127	—	94	2	no	—
180	C₂×C₄⋊C₄	—	2	2	no	—
181	C₂×C₄⋊C₄	—	2	2	no	—
182	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
183	#107	—	2	8	no	—
184	#102	—	94	16	no	—
185	C₄×C₈	—	47	2	no	—
186	C₄×C₈	—	47	2	no	—
187	C₄×C₈	—	47	2	no	—
188	C₄×C₈	—	47	2	no	—
189	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
190	C₄×C₈	C₂²×D₄₇	1	2	yes	—
191	C₂⁴	—	188	2	no	—
192	C₂×Q₈	—	2	4	no	—
193	C₂×Q₈	—	4	2	no	—
194	C₂×Q₈	—	4	8	no	—
195	C₂×D₄	—	94	4	no	—
196	C₂×D₄	—	94	4	no	—
197	C₂×D₄	—	94	4	no	—
198	C₂×D₄	—	188	2	no	—
199	C₂×D₄	—	188	2	no	—
200	C₂×D₄	—	188	2	no	—
201	C₂×D₄	—	188	8	no	—
202	C₂×D₄	—	188	8	no	—
203	C₂×D₄	—	188	8	no	—
204	C₂²×C₄	—	4	2	no	—
205	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
206	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
207	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
208	SD₁₆	—	188	32	no	—
209	D₈	—	188	16	no	—
210	C₂×C₈	—	47	8	no	—
211	C₂×C₈	—	47	4	no	—
212	C₂×C₈	—	47	4	no	—
213	C₂×C₈	—	94	4	no	—
214	C₂×C₈	#10625336	1	8	yes	complemented
215	C₂×C₈	C₂×C₂²⋊D₄₇	1	4	yes	—
216	C₄⋊C₄	—	2	4	no	—
217	C₄⋊C₄	—	2	4	no	—
218	C₄⋊C₄	—	4	2	no	—
219	C₄⋊C₄	—	4	2	no	—
220	C₄²	D₄×D₄₇	1	1	yes	characteristic
221	C₄²	D₄×D₄₇	1	1	yes	characteristic
222	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
223	C₄²	C₂³×D₄₇	1	1	yes	characteristic
224	C₂³	—	376	4	no	—
225	C₂³	—	376	4	no	—
226	C₂³	—	376	4	no	—
227	C₂³	—	188	4	no	—
228	C₂³	#13094020	1	1	yes	characteristic
229	Q₈	—	4	8	no	—
230	Q₈	—	8	4	no	—
231	D₄	—	376	4	no	—
232	D₄	—	376	4	no	—
233	D₄	—	376	4	no	—
234	D₄	—	188	8	no	—
235	D₄	—	188	8	no	—
236	D₄	—	188	8	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	—	8	4	no	—
239	C₂×C₄	#13094020	1	1	yes	characteristic
240	C₂×C₄	#13094020	1	1	yes	characteristic
241	C₂×C₄	#13094020	1	1	yes	characteristic
242	C₂×C₄	D₈×D₄₇	1	4	yes	—
243	C₂×C₄	C₂×#10625336	1	4	yes	—
244	C₂×C₄	#13094047	1	4	yes	—
245	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
246	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
247	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
248	C₈	—	47	8	no	—
249	C₈	—	94	4	no	—
250	C₈	C₂×#10625336	1	8	yes	complemented
251	C₂²	—	376	4	no	—
252	C₂²	—	376	4	no	—
253	C₂²	—	376	4	no	—
254	C₂²	—	752	8	no	—
255	C₂²	#15689497	1	2	yes	—
256	C₂²	#15689498	1	2	yes	—
257	C₂²	#15689508	1	2	yes	—
258	C₂²	C₂×#13094020	1	1	yes	characteristic
259	C₄	—	8	4	no	—
260	C₄	#15689497	1	2	yes	—
261	C₄	#15689498	1	2	yes	—
262	C₄	#15689508	1	2	yes	—
263	C₄	C₂×D₈×D₄₇	1	2	yes	—
264	C₄	C₂²×#10625336	1	2	yes	—
265	C₄	C₂×#13094047	1	2	yes	—
266	C₂	—	752	4	no	—
267	C₂	#20242981	1	4	yes	direct product factor
268	C₂	C₂×#15689497	1	1	yes	characteristic
269	C₂	C₂×#15689498	1	1	yes	characteristic
270	C₂	C₂×#15689508	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13094020
commutator, cosocle	C₄²×C₄₇	C₂⁴
Frattini	C₄²	C₂³×D₄₇
Fitting	C₂×C₄₇×#500	C₂
socle	C₂³×C₄₇	#127
prime radical	C₂×#15689500	C₂
hypercenter	C₂×#500	D₄₇
hypocenter	C₄₇	C₂×#1706
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
2. C₂×#15689497	C₂	semidirect
3. C₂×#15689471	C₂	semidirect
4. #20242981	C₂	direct (N × H)
5. C₂×C₈×C₄₇⋊C₈	C₂	semidirect
6. C₂×C₄₇×#500	C₂	semidirect
13. #15689497	C₂²	semidirect
15. #15689471	C₂²	semidirect
19. C₈×C₄₇⋊C₈	C₂²	semidirect
22. C₄₇×#500	C₂²	semidirect
72. C₂×C₈×C₄₇	SD₁₆	semidirect
84. C₂×C₄₇⋊C₈	D₈	semidirect
112. C₈×C₄₇	C₂×SD₁₆	semidirect
114. C₄₇⋊C₈	C₂×D₈	semidirect
131. C₂×#500	D₄₇	semidirect
137. C₂×C₄₇	#1706	semidirect
147. #500	C₂×D₄₇	semidirect
160. C₄₇	C₂×#1706	semidirect
214. C₂×C₈	#10625336	semidirect
250. C₈	C₂×#10625336	semidirect
267. C₂	#20242981	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×#15689500	2
C₂×#15689497	2
C₂×#15689471	2
#20242981	2
C₂×C₈×C₄₇⋊C₈	2
C₂×C₄₇×#500	2
C₂×#1706	47

Subgroups — C2×#20242981 #26203922

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20242981 #26203922