Subgroups — C₂×#20242053 #26184586

Order 12032.

994 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689176	C₂	1	2	yes	maximal
2	#20242053	C₂	1	8	yes	maximal, direct product factor
3	C₂×#15688869	C₂	1	4	yes	maximal, complemented
4	C₂×C₄₇×#533	C₂	1	1	yes	maximal, characteristic
5	#15689176	—	2	4	no	—
6	#15689176	C₂²	1	8	yes	—
7	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
8	C₂×C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
9	C₂×C₄×C₄₇⋊C₈	—	2	2	no	—
10	#15688869	C₂²	1	16	yes	complemented
11	C₄₇×#533	—	2	4	no	—
12	C₄₇×#533	C₄	1	4	yes	—
13	C₄₇×#533	C₂²	1	4	yes	—
14	C₂×C₄⋊Q₈×C₄₇	C₂²	1	2	yes	—
15	C₂×#125×C₄₇	C₄	1	2	yes	—
16	C₂×C₂²⋊Q₈×C₄₇	—	2	2	no	—
17	C₂×C₄×Q₈×C₄₇	—	2	2	no	—
18	C₂×#117×C₄₇	C₄	1	2	yes	—
19	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
20	C₂²×C₄₇⋊C₈	—	2	4	no	—
21	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
22	C₄⋊(C₄₇⋊C₈)	C₂³	1	8	yes	—
23	C₂²⋊(C₄₇⋊C₈)	—	2	8	no	—
24	C₄×C₄₇⋊C₈	—	2	8	no	—
25	C₂²×Q₈×C₄₇	—	2	2	no	—
26	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₄₇	D₄	1	2	yes	—
28	C₄⋊Q₈×C₄₇	—	2	2	no	—
29	C₄⋊Q₈×C₄₇	—	2	2	no	—
30	C₄⋊Q₈×C₄₇	C₂×C₄	1	4	yes	—
31	C₄⋊Q₈×C₄₇	C₂³	1	4	yes	—
32	#125×C₄₇	—	2	4	no	—
33	#125×C₄₇	C₂×C₄	1	8	yes	—
34	C₂²⋊Q₈×C₄₇	—	2	8	no	—
35	C₂²⋊Q₈×C₄₇	—	2	8	no	—
36	C₄×Q₈×C₄₇	—	2	8	no	—
37	C₄×Q₈×C₄₇	—	2	8	no	—
38	#117×C₄₇	—	2	4	no	—
39	#117×C₄₇	C₂×C₄	1	8	yes	—
40	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
41	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
42	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
43	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
44	C₂×C₄²×C₄₇	—	2	2	no	—
45	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₄₇	D₄	1	2	yes	—
48	C₂×C₄²×C₄₇	D₄	1	4	yes	—
49	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
51	C₂⁴×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×Q₈×C₄₇	—	2	4	no	—
53	C₂×Q₈×C₄₇	—	2	4	no	—
54	C₂×Q₈×C₄₇	—	4	8	no	—
55	C₂²×C₄×C₄₇	—	2	4	no	—
56	C₂²×C₄×C₄₇	—	2	8	no	—
57	C₂²×C₄×C₄₇	—	2	4	no	—
58	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
59	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
60	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
61	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
65	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
66	C₄⋊C₄×C₄₇	—	2	16	no	—
67	C₄⋊C₄×C₄₇	—	2	4	no	—
68	C₄⋊C₄×C₄₇	—	2	4	no	—
69	C₄⋊C₄×C₄₇	—	2	4	no	—
70	C₄⋊C₄×C₄₇	C₂²×C₄	1	8	yes	—
71	C₂²⋊C₄×C₄₇	—	2	8	no	—
72	C₄²×C₄₇	—	2	8	no	—
73	C₄²×C₄₇	C₂²⋊C₄	1	4	yes	—
74	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
75	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
76	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
77	C₄²×C₄₇	C₂×D₄	1	4	yes	—
78	C₄²×C₄₇	C₂×D₄	1	2	yes	—
79	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
80	C₂×C₄₇⋊C₈	—	2	8	no	—
81	C₂×C₄₇⋊C₈	—	4	8	no	—
82	C₂³×C₄₇	—	2	2	no	—
83	C₂³×C₄₇	—	2	4	no	—
84	C₂³×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
85	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	Q₈×C₄₇	—	4	8	no	—
87	Q₈×C₄₇	—	4	8	no	—
88	C₂×C₄×C₄₇	—	2	2	no	—
89	C₂×C₄×C₄₇	—	2	4	no	—
90	C₂×C₄×C₄₇	—	2	8	no	—
91	C₂×C₄×C₄₇	—	2	4	no	—
92	C₂×C₄×C₄₇	—	2	4	no	—
93	C₂×C₄×C₄₇	—	2	8	no	—
94	C₂×C₄×C₄₇	—	2	8	no	—
95	C₂×C₄×C₄₇	—	4	8	no	—
96	C₂×C₄×C₄₇	#101	1	4	yes	—
97	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
98	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
100	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
101	C₂×C₄×C₄₇	#135	1	8	yes	—
102	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
103	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
104	C₄₇⋊C₈	—	4	8	no	—
105	C₂×#1544	—	47	1	no	maximal, Sylow
106	C₂²×C₄₇	—	2	2	no	—
107	C₂²×C₄₇	—	2	4	no	—
108	C₂²×C₄₇	—	4	4	no	—
109	C₂²×C₄₇	C₂×#101	1	2	yes	—
110	C₂²×C₄₇	#416	1	4	yes	—
111	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
112	C₄×C₄₇	—	2	1	no	—
113	C₄×C₄₇	—	2	1	no	—
114	C₄×C₄₇	—	2	4	no	—
115	C₄×C₄₇	—	4	8	no	—
116	C₄×C₄₇	C₂×#101	1	4	yes	—
117	C₄×C₄₇	#416	1	4	yes	—
118	C₄×C₄₇	C₂×#135	1	4	yes	—
119	C₂×#533	D₄₇	1	1	yes	characteristic
120	C₂×#431	—	47	2	no	—
121	#1544	—	47	8	no	—
122	C₂×#332	—	47	4	no	—
123	C₂×C₄₇	—	4	2	no	—
124	C₂×C₄₇	#1544	1	4	yes	complemented
125	C₂×C₄₇	C₂²×#101	1	1	yes	characteristic
126	C₂×C₄₇	C₂×#416	1	2	yes	—
127	#533	—	2	4	no	—
128	#533	Dic₄₇	1	4	yes	—
129	#533	C₂×D₄₇	1	4	yes	—
130	C₂×C₄⋊Q₈	C₂×D₄₇	1	2	yes	—
131	C₂×#125	Dic₄₇	1	2	yes	—
132	C₂×C₂²⋊Q₈	—	2	2	no	—
133	C₂×C₄×Q₈	—	2	2	no	—
134	C₂×#117	Dic₄₇	1	2	yes	—
135	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
136	#431	—	47	8	no	—
137	#431	—	94	4	no	—
138	C₂×C₄⋊C₈	—	47	4	no	—
139	C₂×C₂²⋊C₈	—	94	2	no	—
140	C₂×C₄×C₈	—	94	2	no	—
141	#332	—	47	16	no	—
142	C₄₇	C₂×#1544	1	1	yes	characteristic, complemented, Sylow
143	C₂²×Q₈	—	2	2	no	—
144	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
145	C₂³×C₄	C₂²⋊D₄₇	1	2	yes	—
146	C₂²×C₈	—	94	4	no	—
147	C₄⋊Q₈	—	2	2	no	—
148	C₄⋊Q₈	—	2	2	no	—
149	C₄⋊Q₈	C₂×Dic₄₇	1	4	yes	—
150	C₄⋊Q₈	C₂²×D₄₇	1	4	yes	—
151	#125	—	2	4	no	—
152	#125	C₂×Dic₄₇	1	8	yes	—
153	C₂²⋊Q₈	—	2	8	no	—
154	C₂²⋊Q₈	—	2	8	no	—
155	C₄×Q₈	—	2	8	no	—
156	C₄×Q₈	—	2	8	no	—
157	#117	—	2	4	no	—
158	#117	C₂×Dic₄₇	1	8	yes	—
159	C₂×C₄⋊C₄	—	2	4	no	—
160	C₂×C₄⋊C₄	—	2	2	no	—
161	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
162	C₂×C₂²⋊C₄	—	2	2	no	—
163	C₂×C₄²	—	2	2	no	—
164	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
165	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
166	C₂×C₄²	C₂²⋊D₄₇	1	4	yes	—
167	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
168	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
169	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
170	C₄⋊C₈	—	47	8	no	—
171	C₄⋊C₈	—	94	4	no	—
172	C₂²⋊C₈	—	94	8	no	—
173	C₄×C₈	—	94	8	no	—
174	C₂⁴	C₂²⋊Dic₄₇	1	1	yes	characteristic
175	C₂×Q₈	—	2	4	no	—
176	C₂×Q₈	—	2	4	no	—
177	C₂×Q₈	—	4	8	no	—
178	C₂²×C₄	—	2	4	no	—
179	C₂²×C₄	—	2	8	no	—
180	C₂²×C₄	—	2	4	no	—
181	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
182	C₂²×C₄	C₂²⋊Dic₄₇	1	4	yes	—
183	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
184	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
185	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
186	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
187	C₂²×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
188	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
189	C₂×C₈	—	94	8	no	—
190	C₂×C₈	—	188	8	no	—
191	C₄⋊C₄	—	2	16	no	—
192	C₄⋊C₄	—	2	4	no	—
193	C₄⋊C₄	—	2	4	no	—
194	C₄⋊C₄	—	2	4	no	—
195	C₄⋊C₄	C₂²×Dic₄₇	1	8	yes	—
196	C₂²⋊C₄	—	2	8	no	—
197	C₄²	—	2	8	no	—
198	C₄²	C₂²⋊Dic₄₇	1	4	yes	—
199	C₄²	C₂²⋊Dic₄₇	1	2	yes	—
200	C₄²	C₂²×Dic₄₇	1	2	yes	—
201	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
202	C₄²	C₂×C₂²⋊D₄₇	1	4	yes	—
203	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
204	C₄²	C₂³×D₄₇	1	1	yes	characteristic
205	C₂³	—	2	2	no	—
206	C₂³	—	2	4	no	—
207	C₂³	C₂×C₂²⋊Dic₄₇	1	2	yes	—
208	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
209	Q₈	—	4	8	no	—
210	Q₈	—	4	8	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	4	no	—
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	—	2	8	no	—
216	C₂×C₄	—	2	8	no	—
217	C₂×C₄	—	2	8	no	—
218	C₂×C₄	—	4	8	no	—
219	C₂×C₄	#13093943	1	4	yes	—
220	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	4	yes	—
221	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
222	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
223	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
224	C₂×C₄	#13094044	1	8	yes	—
225	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
226	C₂×C₄	C₂²×C₂²⋊D₄₇	1	2	yes	—
227	C₈	—	188	8	no	—
228	C₂²	—	2	2	no	—
229	C₂²	—	2	4	no	—
230	C₂²	—	4	4	no	—
231	C₂²	C₂×#13093943	1	2	yes	—
232	C₂²	#15689119	1	4	yes	—
233	C₂²	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
234	C₄	—	2	1	no	—
235	C₄	—	2	1	no	—
236	C₄	—	2	4	no	—
237	C₄	—	4	8	no	—
238	C₄	C₂×#13093943	1	4	yes	—
239	C₄	#15689119	1	4	yes	—
240	C₄	C₂×#13094044	1	4	yes	—
241	C₂	—	4	2	no	—
242	C₂	#20242053	1	4	yes	direct product factor
243	C₂	C₂²×#13093943	1	1	yes	characteristic
244	C₂	C₂×#15689119	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²⋊Dic₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₄²	C₂³×D₄₇
Fitting	C₂×C₄₇×#533	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂²⋊C₄
cosocle	C₄²×C₄₇	C₂⁴
hypercenter	C₂×#533	D₄₇
hypocenter	C₄₇	C₂×#1544
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#533

Complements

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

N	H ≅ G/N	type
2. #20242053	C₂	direct (N × H)
3. C₂×#15688869	C₂	semidirect
10. #15688869	C₂²	semidirect
124. C₂×C₄₇	#1544	semidirect
142. C₄₇	C₂×#1544	semidirect
242. C₂	#20242053	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×#15689176	2
#20242053	2
C₂×#15688869	2
C₂×C₄₇×#533	2
C₂×#1544	47

Subgroups — C2×#20242053 #26184586

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20242053 #26184586