Subgroups — C₂×#20243541 #26216659

Order 12032.

1230 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₂²×Dic₄₇⋊C₈	C₂	1	4	yes	maximal
3	C₂×#15688969	C₂	1	1	yes	maximal, characteristic
4	#20243541	C₂	1	8	yes	maximal, direct product factor
5	C₂²×C₄⋊C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₄₇⋊C₈	C₂²	1	2	yes	—
7	C₂²×C₄×Dic₄₇	C₂²	1	2	yes	—
8	C₂×C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
9	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
10	C₂×Dic₄₇⋊C₈	—	2	16	no	—
11	C₂×Dic₄₇⋊C₈	C₂²	1	16	yes	—
12	#15688969	C₄	1	4	yes	—
13	#15688969	C₂²	1	4	yes	—
14	C₂×C₄₇⋊#95	C₄	1	2	yes	—
15	C₂×C₄₇⋊#95	C₄	1	2	yes	—
16	C₂³×C₈×C₄₇	C₂²	1	2	yes	—
17	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₈×C₄₇	—	2	4	no	—
19	C₂×C₄⋊C₈×C₄₇	C₂²	1	4	yes	—
20	C₂³×Dic₄₇	C₂×C₄	1	4	yes	—
21	C₂²×C₄₇⋊C₈	—	2	8	no	—
22	C₂²×C₄₇⋊C₈	—	2	4	no	—
23	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
24	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
25	C₂×C₄×Dic₄₇	D₄	1	8	yes	—
26	C₂×C₄×Dic₄₇	Q₈	1	8	yes	—
27	C₂×C₄×Dic₄₇	C₂³	1	4	yes	—
28	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
29	Dic₄₇⋊C₈	—	2	32	no	—
30	C₄₇⋊#95	C₂×C₄	1	8	yes	—
31	C₄₇⋊#95	C₂×C₄	1	8	yes	—
32	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
33	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
34	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
35	C₂²×C₈×C₄₇	—	2	8	no	—
36	C₂²×C₈×C₄₇	—	2	4	no	—
37	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
38	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₄₇	D₄	1	4	yes	—
40	C₂×C₄²×C₄₇	Q₈	1	4	yes	—
41	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
42	C₄⋊C₈×C₄₇	—	2	8	no	—
43	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₄₇	—	2	4	no	—
45	C₂²×C₄×C₄₇	—	2	4	no	—
46	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
47	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
48	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
49	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
50	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
52	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
53	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
54	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
55	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
56	C₂²×Dic₄₇	—	2	8	no	—
57	C₂²×Dic₄₇	—	2	8	no	—
58	C₂²×Dic₄₇	C₄⋊C₄	1	16	yes	—
59	C₂²×Dic₄₇	C₂²×C₄	1	8	yes	—
60	C₂×C₈×C₄₇	—	2	4	no	—
61	C₂×C₈×C₄₇	—	2	8	no	—
62	C₂×C₈×C₄₇	—	4	8	no	—
63	C₄²×C₄₇	C₄⋊C₄	1	8	yes	—
64	C₄²×C₄₇	C₂×D₄	1	4	yes	—
65	C₄²×C₄₇	C₂×Q₈	1	4	yes	—
66	C₄×Dic₄₇	C₄⋊C₄	1	16	yes	—
67	C₄×Dic₄₇	C₂×D₄	1	8	yes	—
68	C₄×Dic₄₇	C₂×Q₈	1	8	yes	—
69	C₂×C₄₇⋊C₈	—	2	8	no	—
70	C₂×C₄₇⋊C₈	—	2	4	no	—
71	C₂×C₄₇⋊C₈	—	4	8	no	—
72	C₂³×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
73	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
74	C₂³×C₄₇	#131	1	4	yes	—
75	C₂³×C₄₇	#131	1	4	yes	—
76	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
77	C₂×C₄×C₄₇	—	2	4	no	—
78	C₂×C₄×C₄₇	—	2	4	no	—
79	C₂×C₄×C₄₇	—	2	8	no	—
80	C₂×C₄×C₄₇	—	2	8	no	—
81	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
82	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
83	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
84	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
85	C₂×C₄×C₄₇	#127	1	4	yes	—
86	C₂×C₄×C₄₇	C₄⋊Q₈	1	4	yes	—
87	C₂×C₄×C₄₇	C₄⋊Q₈	1	8	yes	—
88	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
89	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
90	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
91	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
92	C₂×Dic₄₇	—	2	8	no	—
93	C₂×Dic₄₇	—	2	8	no	—
94	C₂×Dic₄₇	—	2	32	no	—
95	C₂×Dic₄₇	C₂×C₄⋊C₄	1	16	yes	—
96	C₈×C₄₇	—	4	4	no	—
97	C₄₇⋊C₈	—	4	4	no	—
98	C₂×#1843	—	47	1	no	maximal, Sylow
99	C₂²×C₄₇	#384	1	4	yes	—
100	C₂²×C₄₇	#424	1	16	yes	—
101	C₂²×C₄₇	#424	1	4	yes	—
102	C₂²×C₄₇	#424	1	4	yes	—
103	C₂²×C₄₇	C₂²×C₄⋊C₄	1	2	yes	—
104	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
105	C₂²×C₄₇	C₂×#131	1	2	yes	—
106	C₂²×C₄₇	C₂×#131	1	2	yes	—
107	C₄×C₄₇	—	2	4	no	—
108	C₄×C₄₇	—	2	4	no	—
109	C₄×C₄₇	#384	1	4	yes	—
110	C₄×C₄₇	C₂×#127	1	1	yes	characteristic
111	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
112	C₄×C₄₇	C₂×C₄⋊Q₈	1	2	yes	—
113	Dic₄₇	—	2	16	no	—
114	C₂²×C₄⋊C₈	—	47	1	no	—
115	C₂²×C₄⋊C₈	—	47	4	no	—
116	C₂²×C₄⋊C₈	D₄₇	1	1	yes	characteristic
117	C₂×#376	—	47	1	no	—
118	#1843	—	47	8	no	—
119	C₂×C₄₇	#1843	1	8	yes	complemented
120	C₂×C₄₇	C₂×#384	1	1	yes	characteristic
121	C₂×C₄₇	C₂×#424	1	4	yes	—
122	C₂×C₄₇	C₂×#424	1	1	yes	characteristic
123	C₂×C₄₇	C₂×#424	1	1	yes	characteristic
124	C₂³×C₈	—	47	2	no	—
125	C₂³×C₈	C₂×D₄₇	1	2	yes	—
126	C₂²×C₄²	—	47	2	no	—
127	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
128	C₂×C₄⋊C₈	—	2	4	no	—
129	C₂×C₄⋊C₈	—	47	16	no	—
130	C₂×C₄⋊C₈	—	47	4	no	—
131	C₂×C₄⋊C₈	—	94	16	no	—
132	C₂×C₄⋊C₈	—	94	4	no	—
133	C₂×C₄⋊C₈	C₂×D₄₇	1	4	yes	—
134	#376	—	47	4	no	—
135	#376	—	47	4	no	—
136	C₂×#95	—	47	2	no	—
137	C₂×#95	—	47	2	no	—
138	C₄₇	C₂×#1843	1	1	yes	characteristic, complemented, Sylow
139	C₂³×C₄	—	47	4	no	—
140	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
141	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
142	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
143	C₂²×C₈	—	2	8	no	—
144	C₂²×C₈	—	2	4	no	—
145	C₂²×C₈	—	47	4	no	—
146	C₂²×C₈	—	94	8	no	—
147	C₂²×C₈	—	94	4	no	—
148	C₂²×C₈	C₂²×D₄₇	1	4	yes	—
149	C₂×C₄²	—	47	4	no	—
150	C₂×C₄²	—	47	8	no	—
151	C₂×C₄²	—	47	8	no	—
152	C₂×C₄²	—	47	4	no	—
153	C₂×C₄²	Dic₉₄	1	4	yes	—
154	C₂×C₄²	C₄×D₄₇	1	2	yes	—
155	C₂×C₄²	C₂²⋊D₄₇	1	4	yes	—
156	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
157	C₄⋊C₈	—	2	8	no	—
158	C₄⋊C₈	—	94	32	no	—
159	C₄⋊C₈	—	94	8	no	—
160	#95	—	47	8	no	—
161	#95	—	47	8	no	—
162	C₂⁴	C₂×C₄×D₄₇	1	1	yes	characteristic
163	C₂²×C₄	—	2	4	no	—
164	C₂²×C₄	—	2	4	no	—
165	C₂²×C₄	—	47	16	no	—
166	C₂²×C₄	—	47	8	no	—
167	C₂²×C₄	—	94	8	no	—
168	C₂²×C₄	—	94	8	no	—
169	C₂²×C₄	Dic₄₇⋊C₄	1	4	yes	—
170	C₂²×C₄	Dic₄₇⋊C₄	1	4	yes	—
171	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
172	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
173	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
174	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
175	C₂²×C₄	D₄×D₄₇	1	4	yes	—
176	C₂²×C₄	Q₈×D₄₇	1	4	yes	—
177	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
178	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
179	C₂×C₈	—	2	8	no	—
180	C₂×C₈	—	2	4	no	—
181	C₂×C₈	—	4	8	no	—
182	C₂×C₈	—	94	8	no	—
183	C₂×C₈	—	94	4	no	—
184	C₂×C₈	—	188	8	no	—
185	C₄²	—	47	16	no	—
186	C₄²	—	47	8	no	—
187	C₄²	—	47	8	no	—
188	C₄²	Dic₄₇⋊C₄	1	8	yes	—
189	C₄²	C₂×Dic₉₄	1	4	yes	—
190	C₄²	C₂×C₂²⋊D₄₇	1	4	yes	—
191	C₂³	C₄⋊C₄×D₄₇	1	4	yes	—
192	C₂³	C₂×Dic₄₇⋊C₄	1	2	yes	—
193	C₂³	#13094030	1	4	yes	—
194	C₂³	#13094032	1	4	yes	—
195	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	2	8	no	—
199	C₂×C₄	—	2	8	no	—
200	C₂×C₄	—	47	16	no	—
201	C₂×C₄	—	94	8	no	—
202	C₂×C₄	—	94	8	no	—
203	C₂×C₄	—	94	32	no	—
204	C₂×C₄	C₄⋊C₄×D₄₇	1	4	yes	—
205	C₂×C₄	C₂×Dic₄₇⋊C₄	1	4	yes	—
206	C₂×C₄	C₂×Dic₄₇⋊C₄	1	4	yes	—
207	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
208	C₂×C₄	#13094020	1	4	yes	—
209	C₂×C₄	#13094022	1	4	yes	—
210	C₂×C₄	#13094023	1	8	yes	—
211	C₂×C₄	C₂×D₄×D₄₇	1	2	yes	—
212	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
213	C₂×C₄	C₂×Q₈×D₄₇	1	2	yes	—
214	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
215	C₈	—	4	4	no	—
216	C₈	—	188	4	no	—
217	C₂²	#15688992	1	4	yes	—
218	C₂²	#15689148	1	16	yes	—
219	C₂²	#15689149	1	4	yes	—
220	C₂²	#15689151	1	4	yes	—
221	C₂²	C₂×C₄⋊C₄×D₄₇	1	2	yes	—
222	C₂²	C₂²×Dic₄₇⋊C₄	1	1	yes	characteristic
223	C₂²	C₂×#13094030	1	2	yes	—
224	C₂²	C₂×#13094032	1	2	yes	—
225	C₄	—	2	4	no	—
226	C₄	—	2	4	no	—
227	C₄	—	94	16	no	—
228	C₄	#15688992	1	4	yes	—
229	C₄	C₂×#13094020	1	1	yes	characteristic
230	C₄	C₂×#13094022	1	1	yes	characteristic
231	C₄	C₂×#13094023	1	2	yes	—
232	C₂	#20243541	1	8	yes	direct product factor
233	C₂	C₂×#15688992	1	1	yes	characteristic
234	C₂	C₂×#15689148	1	4	yes	—
235	C₂	C₂×#15689149	1	1	yes	characteristic
236	C₂	C₂×#15689151	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂²×D₄₇
commutator	C₂³×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂²×C₄⋊C₈×C₄₇	C₂
socle, prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂²×C₄⋊C₈	D₄₇
hypocenter	C₄₇	C₂×#1843
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
4. #20243541	C₂	direct (N × H)
119. C₂×C₄₇	#1843	semidirect
138. C₄₇	C₂×#1843	semidirect
232. C₂	#20243541	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×C₄⋊(C₄₇⋊C₈)	2
C₂²×Dic₄₇⋊C₈	2
C₂×#15688969	2
#20243541	2
C₂²×C₄⋊C₈×C₄₇	2
C₂×#1843	47

Subgroups — C2×#20243541 #26216659

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20243541 #26216659