Subgroups — C₂×#20251852 #27079752

Order 12032.

4818 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	C₂×#15689601	C₂	1	2	yes	maximal, complemented
2	#20251852	C₂	1	16	yes	maximal, direct product factor
3	C₂×#15689097	C₂	1	12	yes	maximal, complemented
4	C₂²×2₋¹⁺⁴×C₄₇	C₂	1	1	yes	maximal, characteristic
5	#15689601	—	2	8	no	—
6	#15689601	C₂²	1	16	yes	complemented
7	C₂×Q₈×Dic₄₇	—	2	8	no	—
8	C₂×#13093983	C₂²	1	12	yes	—
9	#15689097	C₂²	1	96	yes	complemented
10	C₂×#13093939	C₂²	1	16	yes	—
11	C₂×2₋¹⁺⁴×C₄₇	—	2	8	no	—
12	C₂×2₋¹⁺⁴×C₄₇	—	2	8	no	—
13	C₂×2₋¹⁺⁴×C₄₇	C₄	1	8	yes	complemented
14	C₂×2₋¹⁺⁴×C₄₇	C₂²	1	8	yes	—
15	C₂²×#41×C₄₇	C₄	1	4	yes	complemented
16	C₂²×#41×C₄₇	C₂²	1	6	yes	—
17	C₂³×Q₈×C₄₇	C₄	1	4	yes	complemented
18	C₂³×Q₈×C₄₇	C₂²	1	1	yes	characteristic
19	Q₈×Dic₄₇	—	2	64	no	—
20	#13093983	—	2	24	no	—
21	#13093983	C₂³	1	48	yes	—
22	C₂×C₄⋊Dic₄₇	—	2	12	no	—
23	C₂×C₂²⋊Dic₄₇	C₂³	1	8	yes	—
24	C₂×C₄×Dic₄₇	—	2	12	no	—
25	#13093939	C₂³	1	64	yes	—
26	2₋¹⁺⁴×C₄₇	—	2	64	no	—
27	2₋¹⁺⁴×C₄₇	—	2	64	no	—
28	C₂×#41×C₄₇	—	2	16	no	—
29	C₂×#41×C₄₇	—	2	16	no	—
30	C₂×#41×C₄₇	—	2	24	no	—
31	C₂×#41×C₄₇	—	2	24	no	—
32	C₂×#41×C₄₇	C₂×C₄	1	24	yes	—
33	C₂×#41×C₄₇	C₂×C₄	1	32	yes	complemented
34	C₂×#41×C₄₇	C₂³	1	24	yes	—
35	C₂²×Q₈×C₄₇	—	2	16	no	—
36	C₂²×Q₈×C₄₇	—	2	16	no	—
37	C₂²×Q₈×C₄₇	C₂×C₄	1	6	yes	—
38	C₂²×Q₈×C₄₇	C₂×C₄	1	4	yes	—
39	C₂²×Q₈×C₄₇	C₂×C₄	1	4	yes	—
40	C₂²×Q₈×C₄₇	C₂×C₄	1	32	yes	complemented
41	C₂²×Q₈×C₄₇	D₄	1	8	yes	—
42	C₂²×Q₈×C₄₇	D₄	1	8	yes	—
43	C₂²×Q₈×C₄₇	C₂³	1	4	yes	—
44	C₂²×D₄×C₄₇	C₂×C₄	1	6	yes	—
45	C₂²×D₄×C₄₇	C₂³	1	4	yes	—
46	C₂³×C₄×C₄₇	C₂×C₄	1	12	yes	—
47	C₂³×C₄×C₄₇	C₂³	1	3	yes	—
48	C₂⁴×C₄₇	C₂²×C₄	1	4	yes	—
49	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
50	#41×C₄₇	—	2	128	no	—
51	#41×C₄₇	—	2	96	no	—
52	#41×C₄₇	—	2	96	no	—
53	C₂×Q₈×C₄₇	—	2	128	no	—
54	C₂×Q₈×C₄₇	—	2	12	no	—
55	C₂×Q₈×C₄₇	—	2	12	no	—
56	C₂×Q₈×C₄₇	C₂²⋊C₄	1	16	yes	—
57	C₂×Q₈×C₄₇	C₂²⋊C₄	1	16	yes	—
58	C₂×Q₈×C₄₇	C₂²⋊C₄	1	32	yes	—
59	C₂×Q₈×C₄₇	C₂²×C₄	1	24	yes	—
60	C₂×Q₈×C₄₇	C₂²×C₄	1	16	yes	—
61	C₂×Q₈×C₄₇	C₂×D₄	1	32	yes	—
62	C₂×D₄×C₄₇	—	2	8	no	—
63	C₂×D₄×C₄₇	—	2	8	no	—
64	C₂×D₄×C₄₇	—	2	12	no	—
65	C₂×D₄×C₄₇	—	2	12	no	—
66	C₂×D₄×C₄₇	C₂²×C₄	1	24	yes	—
67	C₂×D₄×C₄₇	C₂²×C₄	1	8	yes	—
68	C₂×D₄×C₄₇	C₂⁴	1	8	yes	—
69	C₂²×C₄×C₄₇	—	2	24	no	—
70	C₂²×C₄×C₄₇	—	2	24	no	—
71	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
72	C₂²×C₄×C₄₇	C₂²×C₄	1	48	yes	—
73	C₂²×C₄×C₄₇	C₂²×C₄	1	6	yes	—
74	C₂²×C₄×C₄₇	C₂²×C₄	1	6	yes	—
75	C₂²×C₄×C₄₇	C₂×D₄	1	12	yes	—
76	C₂²×C₄×C₄₇	C₂×D₄	1	12	yes	—
77	C₂²×C₄×C₄₇	C₂⁴	1	6	yes	—
78	C₂²×Dic₄₇	—	2	8	no	—
79	C₂²⋊Dic₄₇	—	2	8	no	—
80	C₂²⋊Dic₄₇	C₂⁴	1	16	yes	—
81	C₄⋊Dic₄₇	—	2	48	no	—
82	C₄×Dic₄₇	—	2	48	no	—
83	C₂³×C₄₇	—	2	4	no	—
84	C₂³×C₄₇	—	2	4	no	—
85	C₂³×C₄₇	—	4	2	no	—
86	C₂³×C₄₇	—	4	8	no	—
87	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
88	C₂³×C₄₇	C₂³×C₄	1	8	yes	—
89	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
90	C₂³×C₄₇	C₂²×D₄	1	2	yes	—
91	C₂³×C₄₇	C₂²×D₄	1	2	yes	—
92	C₂³×C₄₇	C₂⁵	1	1	yes	characteristic
93	Q₈×C₄₇	—	2	48	no	—
94	Q₈×C₄₇	C₂×C₂²⋊C₄	1	64	yes	—
95	D₄×C₄₇	—	2	16	no	—
96	D₄×C₄₇	—	2	16	no	—
97	D₄×C₄₇	—	2	48	no	—
98	C₂×C₄×C₄₇	—	2	4	no	—
99	C₂×C₄×C₄₇	—	2	4	no	—
100	C₂×C₄×C₄₇	—	2	96	no	—
101	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	12	yes	—
102	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	12	yes	—
103	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	24	yes	—
104	C₂×C₄×C₄₇	C₂³×C₄	1	8	yes	—
105	C₂×C₄×C₄₇	C₂³×C₄	1	12	yes	—
106	C₂×C₄×C₄₇	C₂²×D₄	1	24	yes	—
107	C₂×Dic₄₇	—	2	16	no	—
108	C₂×Dic₄₇	—	4	16	no	—
109	C₂×#2884	—	47	1	no	maximal, Sylow
110	C₂²×C₄₇	—	2	4	no	—
111	C₂²×C₄₇	—	2	2	no	—
112	C₂²×C₄₇	—	2	8	no	—
113	C₂²×C₄₇	—	4	1	no	—
114	C₂²×C₄₇	—	4	2	no	—
115	C₂²×C₄₇	—	4	1	no	—
116	C₂²×C₄₇	—	4	8	no	—
117	C₂²×C₄₇	—	4	8	no	—
118	C₂²×C₄₇	—	4	8	no	—
119	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
120	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	2	yes	—
121	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
122	C₂²×C₄₇	C₂⁴×C₄	1	1	yes	characteristic
123	C₂²×C₄₇	C₂³×D₄	1	2	yes	—
124	C₄×C₄₇	—	2	8	no	—
125	C₄×C₄₇	C₂²×C₂²⋊C₄	1	24	yes	—
126	Dic₄₇	—	4	16	no	—
127	C₂²×2₋¹⁺⁴	D₄₇	1	1	yes	characteristic
128	C₂×#519	—	47	2	no	—
129	#2884	—	47	16	no	—
130	C₂×#410	—	47	12	no	—
131	C₂×C₄₇	—	2	2	no	—
132	C₂×C₄₇	—	2	1	no	—
133	C₂×C₄₇	—	2	1	no	—
134	C₂×C₄₇	—	2	2	no	—
135	C₂×C₄₇	—	4	8	no	—
136	C₂×C₄₇	#2884	1	2	yes	complemented
137	C₂×C₄₇	C₂³×C₂²⋊C₄	1	1	yes	characteristic
138	C₂×2₋¹⁺⁴	—	2	8	no	—
139	C₂×2₋¹⁺⁴	—	2	8	no	—
140	C₂×2₋¹⁺⁴	Dic₄₇	1	8	yes	complemented
141	C₂×2₋¹⁺⁴	C₂×D₄₇	1	8	yes	—
142	C₂²×#41	Dic₄₇	1	4	yes	complemented
143	C₂²×#41	C₂×D₄₇	1	6	yes	—
144	C₂³×Q₈	Dic₄₇	1	4	yes	complemented
145	C₂³×Q₈	C₂×D₄₇	1	1	yes	characteristic
146	#519	—	47	16	no	—
147	#519	—	94	8	no	—
148	C₂×C₄×Q₈	—	94	8	no	—
149	C₂×#117	—	47	12	no	—
150	#410	—	47	96	no	—
151	C₂×#99	—	47	16	no	—
152	C₄₇	C₂×#2884	1	1	yes	characteristic, complemented, Sylow
153	2₋¹⁺⁴	—	2	64	no	—
154	2₋¹⁺⁴	—	2	64	no	—
155	C₂×#41	—	2	16	no	—
156	C₂×#41	—	2	16	no	—
157	C₂×#41	—	2	24	no	—
158	C₂×#41	—	2	24	no	—
159	C₂×#41	C₂×Dic₄₇	1	24	yes	—
160	C₂×#41	C₂×Dic₄₇	1	32	yes	complemented
161	C₂×#41	C₂²×D₄₇	1	24	yes	—
162	C₂²×Q₈	—	2	16	no	—
163	C₂²×Q₈	—	2	16	no	—
164	C₂²×Q₈	C₂×Dic₄₇	1	6	yes	—
165	C₂²×Q₈	C₂×Dic₄₇	1	4	yes	—
166	C₂²×Q₈	C₂×Dic₄₇	1	4	yes	—
167	C₂²×Q₈	C₂×Dic₄₇	1	32	yes	complemented
168	C₂²×Q₈	C₂²⋊D₄₇	1	8	yes	—
169	C₂²×Q₈	C₂²⋊D₄₇	1	8	yes	—
170	C₂²×Q₈	C₂²×D₄₇	1	4	yes	—
171	C₂²×D₄	C₂×Dic₄₇	1	6	yes	—
172	C₂²×D₄	C₂²×D₄₇	1	4	yes	—
173	C₂³×C₄	C₂×Dic₄₇	1	12	yes	—
174	C₂³×C₄	C₂²×D₄₇	1	3	yes	—
175	C₄×Q₈	—	94	64	no	—
176	#117	—	47	48	no	—
177	#117	—	94	24	no	—
178	C₂×C₄⋊C₄	—	94	12	no	—
179	C₂×C₂²⋊C₄	—	47	8	no	—
180	C₂×C₄²	—	94	12	no	—
181	#99	—	47	64	no	—
182	C₂⁴	C₂²×Dic₄₇	1	4	yes	—
183	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
184	#41	—	2	128	no	—
185	#41	—	2	96	no	—
186	#41	—	2	96	no	—
187	C₂×Q₈	—	2	128	no	—
188	C₂×Q₈	—	2	12	no	—
189	C₂×Q₈	—	2	12	no	—
190	C₂×Q₈	C₂²⋊Dic₄₇	1	16	yes	—
191	C₂×Q₈	C₂²⋊Dic₄₇	1	16	yes	—
192	C₂×Q₈	C₂²⋊Dic₄₇	1	32	yes	—
193	C₂×Q₈	C₂²×Dic₄₇	1	24	yes	—
194	C₂×Q₈	C₂²×Dic₄₇	1	16	yes	—
195	C₂×Q₈	C₂×C₂²⋊D₄₇	1	32	yes	—
196	C₂×D₄	—	2	8	no	—
197	C₂×D₄	—	2	8	no	—
198	C₂×D₄	—	2	12	no	—
199	C₂×D₄	—	2	12	no	—
200	C₂×D₄	C₂²×Dic₄₇	1	24	yes	—
201	C₂×D₄	C₂²×Dic₄₇	1	8	yes	—
202	C₂×D₄	C₂³×D₄₇	1	8	yes	—
203	C₂²×C₄	—	2	24	no	—
204	C₂²×C₄	—	2	24	no	—
205	C₂²×C₄	—	94	8	no	—
206	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
207	C₂²×C₄	C₂²×Dic₄₇	1	48	yes	—
208	C₂²×C₄	C₂²×Dic₄₇	1	6	yes	—
209	C₂²×C₄	C₂²×Dic₄₇	1	6	yes	—
210	C₂²×C₄	C₂×C₂²⋊D₄₇	1	12	yes	—
211	C₂²×C₄	C₂×C₂²⋊D₄₇	1	12	yes	—
212	C₂²×C₄	C₂³×D₄₇	1	6	yes	—
213	C₄⋊C₄	—	94	48	no	—
214	C₂²⋊C₄	—	47	16	no	—
215	C₂²⋊C₄	—	94	8	no	—
216	C₄²	—	94	48	no	—
217	C₂³	—	2	4	no	—
218	C₂³	—	2	4	no	—
219	C₂³	—	4	2	no	—
220	C₂³	—	4	8	no	—
221	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
222	C₂³	C₂³×Dic₄₇	1	8	yes	—
223	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
224	C₂³	C₂²×C₂²⋊D₄₇	1	2	yes	—
225	C₂³	C₂²×C₂²⋊D₄₇	1	2	yes	—
226	C₂³	C₂⁴×D₄₇	1	1	yes	characteristic
227	Q₈	—	2	48	no	—
228	Q₈	C₂×C₂²⋊Dic₄₇	1	64	yes	—
229	D₄	—	2	16	no	—
230	D₄	—	2	16	no	—
231	D₄	—	2	48	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	96	no	—
235	C₂×C₄	—	94	16	no	—
236	C₂×C₄	—	188	16	no	—
237	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	12	yes	—
238	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	24	yes	—
239	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	12	yes	—
240	C₂×C₄	C₂³×Dic₄₇	1	8	yes	—
241	C₂×C₄	C₂³×Dic₄₇	1	12	yes	—
242	C₂×C₄	C₂²×C₂²⋊D₄₇	1	24	yes	—
243	C₂²	—	2	4	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	8	no	—
246	C₂²	—	4	1	no	—
247	C₂²	—	4	2	no	—
248	C₂²	—	4	1	no	—
249	C₂²	—	4	8	no	—
250	C₂²	—	4	8	no	—
251	C₂²	—	4	8	no	—
252	C₂²	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
253	C₂²	C₂²×C₂²⋊Dic₄₇	1	2	yes	—
254	C₂²	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
255	C₂²	C₂⁴×Dic₄₇	1	1	yes	characteristic
256	C₂²	C₂³×C₂²⋊D₄₇	1	2	yes	—
257	C₄	—	2	8	no	—
258	C₄	—	188	16	no	—
259	C₄	C₂²×C₂²⋊Dic₄₇	1	24	yes	—
260	C₂	—	2	2	no	—
261	C₂	—	2	1	no	—
262	C₂	—	2	1	no	—
263	C₂	—	2	2	no	—
264	C₂	—	4	8	no	—
265	C₂	#20251852	1	2	yes	direct product factor
266	C₂	C₂³×C₂²⋊Dic₄₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊Dic₄₇
commutator	C₂²×C₄₇	C₂⁴×C₄
Frattini	C₂³	C₂⁴×D₄₇
Fitting, prime radical	C₂²×2₋¹⁺⁴×C₄₇	C₂
socle	C₂²×C₄₇	C₂²×C₂²⋊C₄
cosocle	C₂³×C₄₇	C₂⁵
hypercenter	C₂²×2₋¹⁺⁴	D₄₇
hypocenter	C₄₇	C₂×#2884
LCS	G ▷ C₂²×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂³×Q₈ ◁ C₂²×2₋¹⁺⁴

Complements

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

N	H ≅ G/N	type
1. C₂×#15689601	C₂	semidirect
2. #20251852	C₂	direct (N × H)
3. C₂×#15689097	C₂	semidirect
6. #15689601	C₂²	semidirect
9. #15689097	C₂²	semidirect
13. C₂×2₋¹⁺⁴×C₄₇	C₄	semidirect
15. C₂²×#41×C₄₇	C₄	semidirect
17. C₂³×Q₈×C₄₇	C₄	semidirect
33. C₂×#41×C₄₇	C₂×C₄	semidirect
40. C₂²×Q₈×C₄₇	C₂×C₄	semidirect
136. C₂×C₄₇	#2884	semidirect
140. C₂×2₋¹⁺⁴	Dic₄₇	semidirect
142. C₂²×#41	Dic₄₇	semidirect
144. C₂³×Q₈	Dic₄₇	semidirect
152. C₄₇	C₂×#2884	semidirect
160. C₂×#41	C₂×Dic₄₇	semidirect
167. C₂²×Q₈	C₂×Dic₄₇	semidirect
265. C₂	#20251852	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×#15689601	2
#20251852	2
C₂×#15689097	2
C₂²×2₋¹⁺⁴×C₄₇	2
C₂×#2884	47

Subgroups — C2×#20251852 #27079752

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20251852 #27079752