Subgroups — C₂×#20251988 #27079951

Order 12032.

2766 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689606	C₂	1	2	yes	maximal
2	#20251988	C₂	1	16	yes	maximal, direct product factor
3	C₂×#15689165	C₂	1	6	yes	maximal, complemented
4	C₂×#15689161	C₂	1	3	yes	maximal, complemented
5	C₂²×#13093958	C₂	1	3	yes	maximal, complemented
6	C₂²×#131×C₄₇	C₂	1	1	yes	maximal, characteristic
7	#15689606	—	2	8	no	—
8	#15689606	C₂²	1	16	yes	—
9	C₂×Q₈×Dic₄₇	—	2	2	no	—
10	C₂×D₄×Dic₄₇	—	2	6	no	—
11	C₂×#13093983	C₂²	1	6	yes	—
12	C₂²×C₄⋊Dic₄₇	C₂²	1	6	yes	—
13	#15689165	C₂²	1	48	yes	complemented
14	#15689161	C₂²	1	24	yes	complemented
15	C₂×#13093958	C₂²	1	3	yes	—
16	C₂×#13093958	C₂²	1	6	yes	—
17	C₂×#13093958	C₂²	1	1	yes	characteristic
18	C₂×#13093958	C₂²	1	24	yes	complemented
19	C₂×#13093956	C₂²	1	6	yes	—
20	C₂²×#41×C₄₇	C₂²	1	1	yes	characteristic
21	C₂×#131×C₄₇	C₄	1	16	yes	complemented
22	C₂×#131×C₄₇	C₂²	1	8	yes	—
23	C₂²×M₁₆×C₄₇	C₂²	1	3	yes	—
24	C₂³×C₈×C₄₇	C₂²	1	3	yes	—
25	C₂³×Dic₄₇	—	2	6	no	—
26	Q₈×Dic₄₇	—	2	16	no	—
27	D₄×Dic₄₇	—	2	48	no	—
28	#13093983	—	2	12	no	—
29	#13093983	C₂³	1	24	yes	—
30	C₂×C₄⋊Dic₄₇	—	2	6	no	—
31	C₂×C₄⋊Dic₄₇	—	2	12	no	—
32	C₂×C₄⋊Dic₄₇	C₂³	1	6	yes	—
33	C₂×C₄⋊Dic₄₇	C₂³	1	24	yes	—
34	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
35	C₂×C₂²⋊Dic₄₇	—	2	6	no	—
36	C₂×C₄×Dic₄₇	—	2	6	no	—
37	#13093958	C₂³	1	12	yes	—
38	#13093958	C₂³	1	24	yes	—
39	#13093958	C₂³	1	4	yes	—
40	#13093956	C₂³	1	24	yes	—
41	C₂×#41×C₄₇	C₂×C₄	1	8	yes	—
42	C₂×#41×C₄₇	D₄	1	4	yes	—
43	C₂×#41×C₄₇	Q₈	1	8	yes	—
44	C₂×#41×C₄₇	C₂³	1	4	yes	—
45	C₂²×Q₈×C₄₇	D₄	1	1	yes	characteristic
46	C₂²×D₄×C₄₇	D₄	1	3	yes	—
47	C₂³×C₄×C₄₇	C₂³	1	3	yes	—
48	#131×C₄₇	C₂×C₄	1	64	yes	complemented
49	C₂×M₁₆×C₄₇	C₂×C₄	1	24	yes	complemented
50	C₂×M₁₆×C₄₇	C₂³	1	12	yes	—
51	C₂²×C₈×C₄₇	C₂×C₄	1	24	yes	complemented
52	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₄₇	C₂³	1	3	yes	—
54	C₂²×C₈×C₄₇	C₂³	1	12	yes	—
55	C₂⁴×C₄₇	C₂×D₄	1	3	yes	—
56	#41×C₄₇	C₄⋊C₄	1	32	yes	—
57	#41×C₄₇	C₂²×C₄	1	16	yes	—
58	#41×C₄₇	C₂×Q₈	1	16	yes	—
59	C₂×Q₈×C₄₇	C₄⋊C₄	1	8	yes	—
60	C₂×Q₈×C₄₇	C₂×D₄	1	4	yes	—
61	C₂×D₄×C₄₇	C₄⋊C₄	1	24	yes	—
62	C₂×D₄×C₄₇	C₂×D₄	1	12	yes	—
63	C₂²×C₄×C₄₇	C₂²×C₄	1	12	yes	—
64	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
65	C₂²×C₄×C₄₇	C₂×D₄	1	6	yes	—
66	C₂²×C₄×C₄₇	C₂×Q₈	1	12	yes	—
67	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	C₂⁴	1	6	yes	—
69	C₂²×Dic₄₇	—	2	2	no	—
70	C₂²×Dic₄₇	—	2	6	no	—
71	C₂²×Dic₄₇	—	2	24	no	—
72	C₂²×Dic₄₇	—	8	12	no	—
73	M₁₆×C₄₇	C₂²×C₄	1	48	yes	complemented
74	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
75	C₂×C₈×C₄₇	C₂²×C₄	1	12	yes	—
76	C₂×C₈×C₄₇	C₂²×C₄	1	48	yes	complemented
77	C₂×C₈×C₄₇	C₂⁴	1	2	yes	—
78	C₂×C₈×C₄₇	C₂⁴	1	6	yes	—
79	C₂²⋊Dic₄₇	—	2	24	no	—
80	C₄⋊Dic₄₇	—	2	6	no	—
81	C₄⋊Dic₄₇	—	2	24	no	—
82	C₄⋊Dic₄₇	—	2	2	no	—
83	C₄⋊Dic₄₇	C₂⁴	1	4	yes	—
84	C₄⋊Dic₄₇	C₂⁴	1	12	yes	—
85	C₄×Dic₄₇	—	2	24	no	—
86	C₂³×C₄₇	—	2	6	no	—
87	C₂³×C₄₇	—	2	6	no	—
88	C₂³×C₄₇	C₂×C₄⋊C₄	1	12	yes	—
89	C₂³×C₄₇	C₂²×D₄	1	6	yes	—
90	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
91	Q₈×C₄₇	C₂×C₄⋊C₄	1	16	yes	—
92	D₄×C₄₇	C₂×C₄⋊C₄	1	48	yes	—
93	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	12	yes	—
94	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	24	yes	—
95	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
96	C₂×C₄×C₄₇	C₂³×C₄	1	12	yes	—
97	C₂×C₄×C₄₇	C₂²×D₄	1	6	yes	—
98	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
99	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
100	C₂×C₄×C₄₇	C₂²×Q₈	1	12	yes	—
101	C₂×C₄×C₄₇	C₂⁵	1	1	yes	characteristic
102	C₂×Dic₄₇	—	2	4	no	—
103	C₂×Dic₄₇	—	2	12	no	—
104	C₂×Dic₄₇	—	4	12	no	—
105	C₂×Dic₄₇	—	4	4	no	—
106	C₂×Dic₄₇	—	8	24	no	—
107	C₈×C₄₇	C₂³×C₄	1	4	yes	—
108	C₈×C₄₇	C₂³×C₄	1	12	yes	—
109	C₂×#2913	—	47	1	no	maximal, Sylow
110	C₂²×C₄₇	—	2	24	no	—
111	C₂²×C₄₇	—	2	6	no	—
112	C₂²×C₄₇	—	2	6	no	—
113	C₂²×C₄₇	C₂²×C₄⋊C₄	1	12	yes	—
114	C₂²×C₄₇	C₂²×C₄⋊C₄	1	2	yes	—
115	C₂²×C₄₇	#576	1	2	yes	—
116	C₂²×C₄₇	#578	1	2	yes	—
117	C₂²×C₄₇	C₂³×D₄	1	1	yes	characteristic
118	C₄×C₄₇	C₂²×C₄⋊C₄	1	2	yes	—
119	C₄×C₄₇	C₂²×C₄⋊C₄	1	12	yes	—
120	C₄×C₄₇	C₂⁴×C₄	1	1	yes	characteristic
121	C₄×C₄₇	C₂³×Q₈	1	1	yes	characteristic
122	Dic₄₇	—	4	12	no	—
123	Dic₄₇	—	4	4	no	—
124	C₂²×#131	D₄₇	1	1	yes	characteristic
125	C₂×#520	—	47	2	no	—
126	#2913	—	47	16	no	—
127	C₂×#428	—	47	6	no	—
128	C₂×#427	—	47	3	no	—
129	C₂²×#107	—	47	3	no	—
130	C₂×C₄₇	—	2	12	no	—
131	C₂×C₄₇	#2913	1	4	yes	complemented
132	C₂×C₄₇	C₂³×C₄⋊C₄	1	1	yes	characteristic
133	C₂×C₄₇	C₂×#576	1	1	yes	characteristic
134	C₂×C₄₇	C₂×#578	1	1	yes	characteristic
135	C₂²×#41	C₂×D₄₇	1	1	yes	characteristic
136	C₂×#131	Dic₄₇	1	16	yes	complemented
137	C₂×#131	C₂×D₄₇	1	8	yes	—
138	C₂²×M₁₆	C₂×D₄₇	1	3	yes	—
139	C₂³×C₈	C₂×D₄₇	1	3	yes	—
140	#520	—	47	16	no	—
141	#520	—	94	8	no	—
142	C₂×C₄×Q₈	—	94	2	no	—
143	C₂×C₄×D₄	—	94	6	no	—
144	C₂×#117	—	47	6	no	—
145	C₂²×C₄⋊C₄	—	47	6	no	—
146	#428	—	47	48	no	—
147	#427	—	47	24	no	—
148	C₂×#107	—	47	3	no	—
149	C₂×#107	—	47	6	no	—
150	C₂×#107	—	47	24	no	—
151	C₂×#107	—	47	1	no	—
152	C₂×#106	—	47	6	no	—
153	C₄₇	C₂×#2913	1	1	yes	characteristic, complemented, Sylow
154	C₂×#41	Dic₉₄	1	8	yes	—
155	C₂×#41	D₁₈₈	1	4	yes	—
156	C₂×#41	C₂×Dic₄₇	1	8	yes	—
157	C₂×#41	C₂²×D₄₇	1	4	yes	—
158	C₂²×Q₈	D₁₈₈	1	1	yes	characteristic
159	C₂²×D₄	D₁₈₈	1	3	yes	—
160	C₂³×C₄	—	94	6	no	—
161	C₂³×C₄	C₂²×D₄₇	1	3	yes	—
162	#131	C₂×Dic₄₇	1	64	yes	complemented
163	C₂×M₁₆	C₂×Dic₄₇	1	24	yes	complemented
164	C₂×M₁₆	C₂²×D₄₇	1	12	yes	—
165	C₂²×C₈	C₂×Dic₄₇	1	24	yes	complemented
166	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
167	C₂²×C₈	C₂²×D₄₇	1	3	yes	—
168	C₂²×C₈	C₂²×D₄₇	1	12	yes	—
169	C₄×Q₈	—	94	16	no	—
170	C₄×D₄	—	94	48	no	—
171	#117	—	47	24	no	—
172	#117	—	94	12	no	—
173	C₂×C₄⋊C₄	—	47	6	no	—
174	C₂×C₄⋊C₄	—	47	24	no	—
175	C₂×C₄⋊C₄	—	47	2	no	—
176	C₂×C₄⋊C₄	—	94	6	no	—
177	C₂×C₄⋊C₄	—	94	12	no	—
178	C₂×C₂²⋊C₄	—	94	6	no	—
179	C₂×C₄²	—	94	6	no	—
180	#107	—	47	12	no	—
181	#107	—	47	24	no	—
182	#107	—	47	4	no	—
183	#106	—	47	24	no	—
184	C₂⁴	C₂×D₁₈₈	1	3	yes	—
185	#41	C₄⋊Dic₄₇	1	32	yes	—
186	#41	C₂×Dic₉₄	1	16	yes	—
187	#41	C₂²×Dic₄₇	1	16	yes	—
188	C₂×Q₈	C₄⋊Dic₄₇	1	8	yes	—
189	C₂×Q₈	C₂×D₁₈₈	1	4	yes	—
190	C₂×D₄	C₄⋊Dic₄₇	1	24	yes	—
191	C₂×D₄	C₂×D₁₈₈	1	12	yes	—
192	C₂²×C₄	—	94	24	no	—
193	C₂²×C₄	—	94	2	no	—
194	C₂²×C₄	—	94	6	no	—
195	C₂²×C₄	—	376	12	no	—
196	C₂²×C₄	C₂×Dic₉₄	1	12	yes	—
197	C₂²×C₄	C₂×D₁₈₈	1	6	yes	—
198	C₂²×C₄	C₂×D₁₈₈	1	3	yes	—
199	C₂²×C₄	C₂²×Dic₄₇	1	12	yes	—
200	C₂²×C₄	C₂³×D₄₇	1	6	yes	—
201	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
202	M₁₆	C₂²×Dic₄₇	1	48	yes	complemented
203	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
204	C₂×C₈	C₂²×Dic₄₇	1	12	yes	—
205	C₂×C₈	C₂²×Dic₄₇	1	48	yes	complemented
206	C₂×C₈	C₂³×D₄₇	1	6	yes	—
207	C₂×C₈	C₂³×D₄₇	1	2	yes	—
208	C₄⋊C₄	—	47	4	no	—
209	C₄⋊C₄	—	47	12	no	—
210	C₄⋊C₄	—	94	6	no	—
211	C₄⋊C₄	—	94	24	no	—
212	C₄⋊C₄	—	94	2	no	—
213	C₂²⋊C₄	—	94	24	no	—
214	C₄²	—	94	24	no	—
215	C₂³	—	2	6	no	—
216	C₂³	—	2	6	no	—
217	C₂³	C₂×C₄⋊Dic₄₇	1	12	yes	—
218	C₂³	C₂²×D₁₈₈	1	6	yes	—
219	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
220	Q₈	C₂×C₄⋊Dic₄₇	1	16	yes	—
221	D₄	C₂×C₄⋊Dic₄₇	1	48	yes	—
222	C₂×C₄	—	94	4	no	—
223	C₂×C₄	—	94	12	no	—
224	C₂×C₄	—	376	24	no	—
225	C₂×C₄	—	188	12	no	—
226	C₂×C₄	—	188	4	no	—
227	C₂×C₄	C₂×C₄⋊Dic₄₇	1	12	yes	—
228	C₂×C₄	C₂×C₄⋊Dic₄₇	1	24	yes	—
229	C₂×C₄	C₂³×Dic₄₇	1	2	yes	—
230	C₂×C₄	C₂³×Dic₄₇	1	12	yes	—
231	C₂×C₄	C₂²×D₁₈₈	1	6	yes	—
232	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
233	C₂×C₄	C₂²×Dic₉₄	1	2	yes	—
234	C₂×C₄	C₂²×Dic₉₄	1	12	yes	—
235	C₂×C₄	C₂⁴×D₄₇	1	1	yes	characteristic
236	C₈	C₂³×Dic₄₇	1	4	yes	—
237	C₈	C₂³×Dic₄₇	1	12	yes	—
238	C₂²	—	2	24	no	—
239	C₂²	—	2	6	no	—
240	C₂²	—	2	6	no	—
241	C₂²	C₂²×C₄⋊Dic₄₇	1	12	yes	—
242	C₂²	C₂²×C₄⋊Dic₄₇	1	2	yes	—
243	C₂²	#15689928	1	2	yes	—
244	C₂²	#15689940	1	2	yes	—
245	C₂²	C₂³×D₁₈₈	1	1	yes	characteristic
246	C₄	—	188	12	no	—
247	C₄	—	188	4	no	—
248	C₄	C₂²×C₄⋊Dic₄₇	1	2	yes	—
249	C₄	C₂²×C₄⋊Dic₄₇	1	12	yes	—
250	C₄	C₂⁴×Dic₄₇	1	1	yes	characteristic
251	C₄	C₂³×Dic₉₄	1	1	yes	characteristic
252	C₂	—	2	12	no	—
253	C₂	#20251988	1	4	yes	direct product factor
254	C₂	C₂³×C₄⋊Dic₄₇	1	1	yes	characteristic
255	C₂	C₂×#15689928	1	1	yes	characteristic
256	C₂	C₂×#15689940	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₁₈₈
commutator	C₄×C₄₇	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂²×#131×C₄₇	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂²×#41×C₄₇	C₂²
cosocle	C₂×C₄×C₄₇	C₂⁵
hypercenter	C₂²×#131	D₄₇
hypocenter	C₄₇	C₂×#2913
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×#41 ◁ C₂²×#131

Complements

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

N	H ≅ G/N	type
2. #20251988	C₂	direct (N × H)
3. C₂×#15689165	C₂	semidirect
4. C₂×#15689161	C₂	semidirect
5. C₂²×#13093958	C₂	semidirect
13. #15689165	C₂²	semidirect
14. #15689161	C₂²	semidirect
18. C₂×#13093958	C₂²	semidirect
21. C₂×#131×C₄₇	C₄	semidirect
48. #131×C₄₇	C₂×C₄	semidirect
49. C₂×M₁₆×C₄₇	C₂×C₄	semidirect
51. C₂²×C₈×C₄₇	C₂×C₄	semidirect
73. M₁₆×C₄₇	C₂²×C₄	semidirect
76. C₂×C₈×C₄₇	C₂²×C₄	semidirect
131. C₂×C₄₇	#2913	semidirect
136. C₂×#131	Dic₄₇	semidirect
153. C₄₇	C₂×#2913	semidirect
162. #131	C₂×Dic₄₇	semidirect
163. C₂×M₁₆	C₂×Dic₄₇	semidirect
165. C₂²×C₈	C₂×Dic₄₇	semidirect
202. M₁₆	C₂²×Dic₄₇	semidirect
205. C₂×C₈	C₂²×Dic₄₇	semidirect
253. C₂	#20251988	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×#15689606	2
#20251988	2
C₂×#15689165	2
C₂×#15689161	2
C₂²×#13093958	2
C₂²×#131×C₄₇	2
C₂×#2913	47

Subgroups — C2×#20251988 #27079951

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20251988 #27079951