Subgroups — C₂×#20251733 #26422206

Order 12032.

5694 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13094021	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#13093998	C₂	1	6	yes	maximal, complemented
3	#20251733	C₂	1	16	yes	maximal, direct product factor
4	C₂×#15689025	C₂	1	6	yes	maximal, complemented
5	C₂×#15688993	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×C₄²×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₁₈₈	C₂²	1	6	yes	—
8	C₂³×C₂²⋊D₄₇	C₂²	1	4	yes	—
9	C₂×#13094021	—	2	8	no	—
10	C₂×#13094021	C₂²	1	8	yes	complemented
11	C₂×#13093998	—	2	48	no	—
12	C₂×#13093998	C₂²	1	48	yes	complemented
13	C₂²×C₄⋊Dic₄₇	C₂²	1	6	yes	—
14	C₂²×(C₂×D₄₇)⋊C₄	C₂²	1	12	yes	—
15	#15689025	C₂²	1	48	yes	complemented
16	#15688993	C₂²	1	8	yes	complemented
17	C₂⁴×C₄×C₄₇	C₂²	1	3	yes	—
18	C₂²×C₄²×C₄₇	—	2	8	no	—
19	C₂²×C₄²×C₄₇	C₂²	1	3	yes	—
20	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
21	C₂²×C₄²×C₄₇	C₂²	1	8	yes	complemented
22	C₂⁴×D₄₇	C₂³	1	4	yes	—
23	C₂²×D₁₈₈	—	2	24	no	—
24	C₂²×D₁₈₈	—	2	24	no	—
25	C₂²×D₁₈₈	—	2	24	no	—
26	C₂²×D₁₈₈	C₂³	1	24	yes	—
27	C₂²×C₂²⋊D₄₇	—	2	48	no	—
28	C₂²×C₂²⋊D₄₇	C₂³	1	16	yes	—
29	C₂³×Dic₄₇	C₂³	1	4	yes	—
30	#13094021	—	2	32	no	—
31	#13093998	—	2	192	no	—
32	C₂×C₄⋊Dic₄₇	—	2	24	no	—
33	C₂×C₄⋊Dic₄₇	C₂³	1	24	yes	—
34	C₂×(C₂×D₄₇)⋊C₄	—	2	48	no	—
35	C₂×(C₂×D₄₇)⋊C₄	C₂³	1	48	yes	—
36	C₂⁵×C₄₇	C₂³	1	1	yes	characteristic
37	C₂³×C₄×C₄₇	—	2	12	no	—
38	C₂³×C₄×C₄₇	—	2	12	no	—
39	C₂³×C₄×C₄₇	D₄	1	24	yes	complemented
40	C₂³×C₄×C₄₇	C₂³	1	3	yes	—
41	C₂³×C₄×C₄₇	C₂³	1	12	yes	—
42	C₂³×C₄×C₄₇	C₂³	1	3	yes	—
43	C₂×C₄²×C₄₇	—	2	12	no	—
44	C₂×C₄²×C₄₇	—	2	32	no	—
45	C₂×C₄²×C₄₇	D₄	1	8	yes	complemented
46	C₂×C₄²×C₄₇	C₂³	1	12	yes	—
47	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
48	C₂⁴×C₄₇	—	2	2	no	—
49	C₂⁴×C₄₇	—	2	6	no	—
50	C₂⁴×C₄₇	C₂×D₄	1	12	yes	—
51	C₂⁴×C₄₇	C₂⁴	1	2	yes	—
52	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
53	C₂³×D₄₇	—	2	24	no	—
54	C₂³×D₄₇	—	2	8	no	—
55	C₂³×D₄₇	—	2	24	no	—
56	C₂³×D₄₇	C₂⁴	1	8	yes	—
57	C₂²×C₄×C₄₇	—	2	6	no	—
58	C₂²×C₄×C₄₇	—	2	6	no	—
59	C₂²×C₄×C₄₇	—	2	96	no	—
60	C₂²×C₄×C₄₇	—	2	24	no	—
61	C₂²×C₄×C₄₇	—	2	24	no	—
62	C₂²×C₄×C₄₇	C₂×D₄	1	12	yes	—
63	C₂²×C₄×C₄₇	C₂×D₄	1	12	yes	—
64	C₂²×C₄×C₄₇	C₂×D₄	1	12	yes	—
65	C₂²×C₄×C₄₇	C₂×D₄	1	48	yes	complemented
66	C₂²×C₄×C₄₇	#41	1	12	yes	—
67	C₂²×C₄×C₄₇	C₂⁴	1	6	yes	—
68	C₂²×C₄×C₄₇	C₂⁴	1	6	yes	—
69	C₂×C₂²⋊D₄₇	—	2	96	no	—
70	C₂×C₂²⋊D₄₇	—	4	64	no	—
71	C₂²×Dic₄₇	—	2	24	no	—
72	C₂²×Dic₄₇	C₂⁴	1	8	yes	—
73	C₂×D₁₈₈	—	2	48	no	—
74	C₂×D₁₈₈	—	2	48	no	—
75	C₂×D₁₈₈	—	2	48	no	—
76	C₂×D₁₈₈	—	4	96	no	—
77	C₄²×C₄₇	—	2	24	no	—
78	C₄²×C₄₇	C₂×D₄	1	16	yes	complemented
79	(C₂×D₄₇)⋊C₄	—	2	96	no	—
80	C₄⋊Dic₄₇	—	2	48	no	—
81	C₂³×C₄₇	—	2	24	no	—
82	C₂³×C₄₇	—	2	24	no	—
83	C₂³×C₄₇	—	2	2	no	—
84	C₂³×C₄₇	—	2	6	no	—
85	C₂³×C₄₇	#127	1	16	yes	complemented
86	C₂³×C₄₇	C₂²×D₄	1	2	yes	—
87	C₂³×C₄₇	C₂²×D₄	1	12	yes	—
88	C₂³×C₄₇	C₂²×D₄	1	6	yes	—
89	C₂³×C₄₇	C₂×#41	1	6	yes	—
90	C₂³×C₄₇	C₂⁵	1	1	yes	characteristic
91	C₂²×D₄₇	—	2	24	no	—
92	C₂²×D₄₇	—	2	8	no	—
93	C₂²×D₄₇	—	2	24	no	—
94	C₂²×D₄₇	—	4	48	no	—
95	C₂²×D₄₇	—	4	16	no	—
96	C₂²×D₄₇	—	4	48	no	—
97	C₂×C₄×C₄₇	—	2	24	no	—
98	C₂×C₄×C₄₇	—	2	6	no	—
99	C₂×C₄×C₄₇	—	2	6	no	—
100	C₂×C₄×C₄₇	—	2	96	no	—
101	C₂×C₄×C₄₇	#121	1	48	yes	complemented
102	C₂×C₄×C₄₇	C₂²×D₄	1	12	yes	—
103	C₂×C₄×C₄₇	C₂²×D₄	1	12	yes	—
104	C₂×C₄×C₄₇	C₂²×D₄	1	12	yes	—
105	C₂×C₄×C₄₇	C₂×#41	1	12	yes	—
106	C₂²⋊D₄₇	—	4	64	no	—
107	C₂×Dic₄₇	—	2	24	no	—
108	C₂×Dic₄₇	—	4	16	no	—
109	D₁₈₈	—	4	96	no	—
110	C₂×#2856	—	47	1	no	maximal, Sylow
111	C₂²×C₄₇	—	2	32	no	—
112	C₂²×C₄₇	—	2	12	no	—
113	C₂²×C₄₇	—	2	12	no	—
114	C₂²×C₄₇	C₂×#121	1	12	yes	—
115	C₂²×C₄₇	C₂×#127	1	4	yes	—
116	C₂²×C₄₇	C₂×#127	1	8	yes	complemented
117	C₂²×C₄₇	#532	1	12	yes	—
118	C₂²×C₄₇	C₂³×D₄	1	1	yes	characteristic
119	C₂²×C₄₇	C₂³×D₄	1	3	yes	—
120	C₂²×C₄₇	C₂²×#41	1	3	yes	—
121	C₂×D₄₇	—	4	24	no	—
122	C₂×D₄₇	—	4	8	no	—
123	C₂×D₄₇	—	4	24	no	—
124	C₂×D₄₇	—	8	32	no	—
125	C₄×C₄₇	—	2	12	no	—
126	C₄×C₄₇	C₂×#121	1	24	yes	complemented
127	Dic₄₇	—	4	8	no	—
128	C₂²×#127	—	47	1	no	—
129	C₂²×#121	—	47	6	no	—
130	C₂³×C₄²	D₄₇	1	1	yes	characteristic, complemented
131	#2856	—	47	16	no	—
132	C₂×#390	—	47	6	no	—
133	C₂×#384	—	47	1	no	—
134	C₂×C₄₇	—	2	8	no	—
135	C₂×C₄₇	#2856	1	8	yes	complemented
136	C₂×C₄₇	C₂²×#121	1	3	yes	—
137	C₂×C₄₇	C₂²×#127	1	1	yes	characteristic
138	C₂×C₄₇	C₂×#532	1	3	yes	—
139	D₄₇	—	8	8	no	—
140	C₂³×D₄	—	47	4	no	—
141	C₂³×D₄	—	47	6	no	—
142	C₂⁴×C₄	C₂×D₄₇	1	3	yes	—
143	C₂×#127	—	47	8	no	—
144	C₂×#127	—	94	8	no	—
145	C₂×#121	—	47	48	no	—
146	C₂×#121	—	94	48	no	—
147	C₂²×C₄⋊C₄	—	47	6	no	—
148	C₂²×C₂²⋊C₄	—	47	12	no	—
149	C₂²×C₄²	—	2	8	no	—
150	C₂²×C₄²	C₂×D₄₇	1	3	yes	—
151	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
152	C₂²×C₄²	C₂×D₄₇	1	8	yes	complemented
153	#390	—	47	48	no	—
154	#384	—	47	8	no	—
155	C₄₇	C₂×#2856	1	1	yes	characteristic, complemented, Sylow
156	C₂⁵	—	47	4	no	—
157	C₂⁵	C₂²×D₄₇	1	1	yes	characteristic
158	C₂²×D₄	—	47	16	no	—
159	C₂²×D₄	—	47	24	no	—
160	C₂²×D₄	—	94	48	no	—
161	C₂²×D₄	—	94	24	no	—
162	C₂²×D₄	—	94	24	no	—
163	C₂²×D₄	—	94	24	no	—
164	C₂³×C₄	—	2	12	no	—
165	C₂³×C₄	—	2	12	no	—
166	C₂³×C₄	—	47	4	no	—
167	C₂³×C₄	D₁₈₈	1	24	yes	complemented
168	C₂³×C₄	C₂²×D₄₇	1	3	yes	—
169	C₂³×C₄	C₂²×D₄₇	1	12	yes	—
170	C₂³×C₄	C₂²×D₄₇	1	3	yes	—
171	#127	—	94	32	no	—
172	#121	—	94	192	no	—
173	C₂×C₄⋊C₄	—	47	24	no	—
174	C₂×C₄⋊C₄	—	94	24	no	—
175	C₂×C₂²⋊C₄	—	47	48	no	—
176	C₂×C₂²⋊C₄	—	94	48	no	—
177	C₂×C₄²	—	2	12	no	—
178	C₂×C₄²	—	2	32	no	—
179	C₂×C₄²	C₂²⋊D₄₇	1	8	yes	complemented
180	C₂×C₄²	C₂²×D₄₇	1	12	yes	—
181	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
182	C₂⁴	—	2	2	no	—
183	C₂⁴	—	2	6	no	—
184	C₂⁴	—	47	8	no	—
185	C₂⁴	—	94	8	no	—
186	C₂⁴	—	94	24	no	—
187	C₂⁴	—	94	24	no	—
188	C₂⁴	C₂×D₁₈₈	1	12	yes	—
189	C₂⁴	C₂³×D₄₇	1	2	yes	—
190	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
191	C₂×D₄	—	94	96	no	—
192	C₂×D₄	—	94	48	no	—
193	C₂×D₄	—	94	48	no	—
194	C₂×D₄	—	94	48	no	—
195	C₂×D₄	—	188	64	no	—
196	C₂×D₄	—	188	96	no	—
197	C₂²×C₄	—	2	6	no	—
198	C₂²×C₄	—	2	6	no	—
199	C₂²×C₄	—	2	96	no	—
200	C₂²×C₄	—	2	24	no	—
201	C₂²×C₄	—	2	24	no	—
202	C₂²×C₄	—	47	8	no	—
203	C₂²×C₄	—	94	24	no	—
204	C₂²×C₄	C₂×D₁₈₈	1	12	yes	—
205	C₂²×C₄	C₂×D₁₈₈	1	12	yes	—
206	C₂²×C₄	C₂×D₁₈₈	1	48	yes	complemented
207	C₂²×C₄	#10625350	1	12	yes	—
208	C₂²×C₄	C₂×C₂²⋊D₄₇	1	12	yes	—
209	C₂²×C₄	C₂³×D₄₇	1	6	yes	—
210	C₂²×C₄	C₂³×D₄₇	1	6	yes	—
211	C₄⋊C₄	—	94	48	no	—
212	C₂²⋊C₄	—	94	96	no	—
213	C₄²	—	2	24	no	—
214	C₄²	C₂×C₂²⋊D₄₇	1	16	yes	complemented
215	C₂³	—	2	24	no	—
216	C₂³	—	2	24	no	—
217	C₂³	—	2	2	no	—
218	C₂³	—	2	6	no	—
219	C₂³	—	94	8	no	—
220	C₂³	—	94	24	no	—
221	C₂³	—	94	24	no	—
222	C₂³	—	188	48	no	—
223	C₂³	—	188	16	no	—
224	C₂³	—	188	48	no	—
225	C₂³	#13094021	1	16	yes	complemented
226	C₂³	C₂²×C₂²⋊D₄₇	1	2	yes	—
227	C₂³	C₂²×D₁₈₈	1	12	yes	—
228	C₂³	C₂²×D₁₈₈	1	6	yes	—
229	C₂³	C₂×#10625350	1	6	yes	—
230	C₂³	C₂⁴×D₄₇	1	1	yes	characteristic
231	D₄	—	188	64	no	—
232	D₄	—	188	96	no	—
233	C₂×C₄	—	2	24	no	—
234	C₂×C₄	—	2	6	no	—
235	C₂×C₄	—	2	6	no	—
236	C₂×C₄	—	2	96	no	—
237	C₂×C₄	—	94	24	no	—
238	C₂×C₄	—	188	16	no	—
239	C₂×C₄	#13093998	1	48	yes	complemented
240	C₂×C₄	C₂²×C₂²⋊D₄₇	1	12	yes	—
241	C₂×C₄	C₂²×D₁₈₈	1	12	yes	—
242	C₂×C₄	C₂²×D₁₈₈	1	12	yes	—
243	C₂×C₄	C₂×#10625350	1	12	yes	—
244	C₂²	—	2	32	no	—
245	C₂²	—	2	12	no	—
246	C₂²	—	2	12	no	—
247	C₂²	—	376	32	no	—
248	C₂²	—	188	24	no	—
249	C₂²	—	188	8	no	—
250	C₂²	—	188	24	no	—
251	C₂²	C₂×#13093998	1	12	yes	—
252	C₂²	C₂×#13094021	1	4	yes	—
253	C₂²	C₂×#13094021	1	8	yes	complemented
254	C₂²	#15689673	1	12	yes	—
255	C₂²	C₂³×C₂²⋊D₄₇	1	1	yes	characteristic
256	C₂²	C₂³×D₁₈₈	1	3	yes	—
257	C₂²	C₂²×#10625350	1	3	yes	—
258	C₄	—	2	12	no	—
259	C₄	—	188	8	no	—
260	C₄	C₂×#13093998	1	24	yes	complemented
261	C₂	—	2	8	no	—
262	C₂	—	376	8	no	—
263	C₂	#20251733	1	8	yes	direct product factor
264	C₂	C₂²×#13093998	1	3	yes	—
265	C₂	C₂²×#13094021	1	1	yes	characteristic
266	C₂	C₂×#15689673	1	3	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₄₇
commutator, cosocle	C₂³×C₄₇	C₂⁵
Frattini	C₂³	C₂⁴×D₄₇
Fitting	C₂³×C₄²×C₄₇	C₂
socle	C₂⁴×C₄₇	C₂⁴
hypercenter	C₂³×C₄²	D₄₇
hypocenter	C₄₇	C₂×#2856
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄²

Complements

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

N	H ≅ G/N	type
1. C₂²×#13094021	C₂	semidirect
2. C₂²×#13093998	C₂	semidirect
3. #20251733	C₂	direct (N × H)
4. C₂×#15689025	C₂	semidirect
5. C₂×#15688993	C₂	semidirect
6. C₂³×C₄²×C₄₇	C₂	semidirect
10. C₂×#13094021	C₂²	semidirect
12. C₂×#13093998	C₂²	semidirect
15. #15689025	C₂²	semidirect
16. #15688993	C₂²	semidirect
21. C₂²×C₄²×C₄₇	C₂²	semidirect
39. C₂³×C₄×C₄₇	D₄	semidirect
45. C₂×C₄²×C₄₇	D₄	semidirect
65. C₂²×C₄×C₄₇	C₂×D₄	semidirect
78. C₄²×C₄₇	C₂×D₄	semidirect
85. C₂³×C₄₇	#127	semidirect
101. C₂×C₄×C₄₇	#121	semidirect
116. C₂²×C₄₇	C₂×#127	semidirect
126. C₄×C₄₇	C₂×#121	semidirect
130. C₂³×C₄²	D₄₇	semidirect
135. C₂×C₄₇	#2856	semidirect
152. C₂²×C₄²	C₂×D₄₇	semidirect
155. C₄₇	C₂×#2856	semidirect
167. C₂³×C₄	D₁₈₈	semidirect
179. C₂×C₄²	C₂²⋊D₄₇	semidirect
206. C₂²×C₄	C₂×D₁₈₈	semidirect
214. C₄²	C₂×C₂²⋊D₄₇	semidirect
225. C₂³	#13094021	semidirect
239. C₂×C₄	#13093998	semidirect
253. C₂²	C₂×#13094021	semidirect
260. C₄	C₂×#13093998	semidirect
263. C₂	#20251733	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×#13094021	2
C₂²×#13093998	2
#20251733	2
C₂×#15689025	2
C₂×#15688993	2
C₂³×C₄²×C₄₇	2
C₂×#2856	47

Subgroups — C2×#20251733 #26422206

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20251733 #26422206