Subgroups — C₂×#20245204 #26369405

Order 12032.

538 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20245204	C₂	1	8	yes	maximal, direct product factor
2	C₂×#15688944	C₂	1	2	yes	maximal, complemented
3	C₂×#15688905	C₂	1	2	yes	maximal
4	C₂×#15688901	C₂	1	2	yes	maximal, complemented
5	C₂×C₄₇×#405	C₂	1	1	yes	maximal, characteristic
6	C₂×#13093964	C₂²	1	4	yes	—
7	#15688944	C₂²	1	8	yes	complemented
8	#15688905	C₂²	1	8	yes	—
9	#15688901	C₂²	1	8	yes	complemented
10	C₂²×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
11	C₂³×C₈×C₄₇	C₄	1	1	yes	characteristic
12	C₂×#117×C₄₇	C₄	1	1	yes	characteristic
13	C₄₇×#405	C₄	1	4	yes	—
14	C₄₇×#405	C₂²	1	4	yes	—
15	C₂×#97×C₄₇	C₄	1	1	yes	characteristic
16	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₄₇	C₄	1	1	yes	characteristic
18	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
19	#13093964	—	2	4	no	—
20	#13093964	C₂³	1	8	yes	—
21	C₂×C₄₇⋊C₁₆	—	2	4	no	—
22	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
23	C₂×M₁₆×C₄₇	—	2	2	no	—
24	C₂×M₁₆×C₄₇	—	2	2	no	—
25	C₂×M₁₆×C₄₇	C₂×C₄	1	2	yes	—
26	C₂×M₁₆×C₄₇	C₂³	1	2	yes	—
27	C₂²×C₈×C₄₇	—	2	2	no	—
28	C₂²×C₈×C₄₇	—	2	2	no	—
29	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
31	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
34	#117×C₄₇	—	2	2	no	—
35	#117×C₄₇	C₂×C₄	1	4	yes	—
36	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
37	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
38	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
40	#97×C₄₇	C₂×C₄	1	4	yes	—
41	#97×C₄₇	C₂×C₄	1	2	yes	—
42	#97×C₄₇	C₂³	1	2	yes	—
43	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
44	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
45	C₄×C₈×C₄₇	C₂³	1	2	yes	—
46	C₂⁴×C₄₇	M₁₆	1	1	yes	characteristic
47	C₂²×C₄×C₄₇	—	2	1	no	—
48	C₂²×C₄×C₄₇	—	2	1	no	—
49	C₂²×C₄×C₄₇	—	2	1	no	—
50	C₂²×C₄×C₄₇	—	2	1	no	—
51	C₂²×C₄×C₄₇	—	2	1	no	—
52	C₂²×C₄×C₄₇	—	2	1	no	—
53	C₂²×C₄×C₄₇	M₁₆	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	M₁₆	1	2	yes	—
55	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
56	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
57	M₁₆×C₄₇	—	2	4	no	—
58	M₁₆×C₄₇	—	2	4	no	—
59	C₂×C₈×C₄₇	—	2	1	no	—
60	C₂×C₈×C₄₇	—	2	1	no	—
61	C₂×C₈×C₄₇	—	2	1	no	—
62	C₂×C₈×C₄₇	—	2	1	no	—
63	C₂×C₈×C₄₇	—	2	8	no	—
64	C₂×C₈×C₄₇	M₁₆	1	2	yes	—
65	C₂×C₈×C₄₇	M₁₆	1	2	yes	—
66	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
67	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
68	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
70	C₂×C₈×C₄₇	#41	1	2	yes	—
71	C₂×C₈×C₄₇	#41	1	2	yes	—
72	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
73	C₄⋊C₄×C₄₇	—	2	4	no	—
74	C₂²⋊C₄×C₄₇	—	2	4	no	—
75	C₄²×C₄₇	—	2	1	no	—
76	C₄²×C₄₇	—	2	1	no	—
77	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
78	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
79	C₄₇⋊C₁₆	—	2	8	no	—
80	C₂³×C₄₇	—	2	1	no	—
81	C₂³×C₄₇	—	2	1	no	—
82	C₂³×C₄₇	—	4	2	no	—
83	C₂³×C₄₇	C₂×M₁₆	1	1	yes	characteristic
84	C₂³×C₄₇	C₂×M₁₆	1	2	yes	—
85	C₂×C₄×C₄₇	—	2	1	no	—
86	C₂×C₄×C₄₇	—	2	1	no	—
87	C₂×C₄×C₄₇	—	2	2	no	—
88	C₂×C₄×C₄₇	—	2	2	no	—
89	C₂×C₄×C₄₇	—	2	2	no	—
90	C₂×C₄×C₄₇	—	2	2	no	—
91	C₂×C₄×C₄₇	—	2	2	no	—
92	C₂×C₄×C₄₇	—	2	2	no	—
93	C₂×C₄×C₄₇	—	4	2	no	—
94	C₂×C₄×C₄₇	—	4	2	no	—
95	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
96	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
97	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
98	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
99	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
100	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
101	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
102	C₈×C₄₇	—	2	2	no	—
103	C₈×C₄₇	—	2	2	no	—
104	C₈×C₄₇	#117	1	2	yes	—
105	C₈×C₄₇	C₂×M₁₆	1	4	yes	—
106	C₈×C₄₇	C₂×#41	1	2	yes	—
107	C₂×#2166	—	47	1	no	maximal, Sylow
108	C₂²×C₄₇	—	2	2	no	—
109	C₂²×C₄₇	—	2	2	no	—
110	C₂²×C₄₇	—	4	2	no	—
111	C₂²×C₄₇	—	4	2	no	—
112	C₂²×C₄₇	—	4	2	no	—
113	C₂²×C₄₇	#432	1	1	yes	characteristic
114	C₂²×C₄₇	#432	1	1	yes	characteristic
115	C₂²×C₄₇	C₂²×M₁₆	1	1	yes	characteristic
116	C₄×C₄₇	—	2	2	no	—
117	C₄×C₄₇	—	4	2	no	—
118	C₄×C₄₇	—	4	2	no	—
119	C₄×C₄₇	#432	1	1	yes	characteristic
120	C₄×C₄₇	#432	1	1	yes	characteristic
121	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
122	C₄×C₄₇	C₂²×M₁₆	1	1	yes	characteristic
123	C₂×#405	D₄₇	1	1	yes	characteristic
124	#2166	—	47	8	no	—
125	C₂×#364	—	47	2	no	—
126	C₂×#349	—	47	2	no	—
127	C₂×#347	—	47	2	no	—
128	C₂×C₄₇	—	2	1	no	—
129	C₂×C₄₇	—	2	1	no	—
130	C₂×C₄₇	—	4	2	no	—
131	C₂×C₄₇	#2166	1	2	yes	complemented
132	C₂×C₄₇	C₂×#432	1	1	yes	characteristic
133	C₂²×M₁₆	C₂×D₄₇	1	1	yes	characteristic
134	C₂³×C₈	Dic₄₇	1	1	yes	characteristic
135	C₂×#117	Dic₄₇	1	1	yes	characteristic
136	C₂×M₃₂	—	47	4	no	—
137	#405	Dic₄₇	1	4	yes	—
138	#405	C₂×D₄₇	1	4	yes	—
139	C₂×#97	Dic₄₇	1	1	yes	characteristic
140	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
141	C₂×C₄×C₈	Dic₄₇	1	1	yes	characteristic
142	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
143	#364	—	47	8	no	—
144	#349	—	47	8	no	—
145	#347	—	47	8	no	—
146	C₄₇	C₂×#2166	1	1	yes	characteristic, complemented, Sylow
147	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
148	C₂×M₁₆	—	2	2	no	—
149	C₂×M₁₆	—	2	2	no	—
150	C₂×M₁₆	C₂×Dic₄₇	1	2	yes	—
151	C₂×M₁₆	C₂²×D₄₇	1	2	yes	—
152	C₂²×C₈	—	2	2	no	—
153	C₂²×C₈	—	2	2	no	—
154	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
155	C₂²×C₈	C₂×Dic₄₇	1	4	yes	—
156	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
157	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
158	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
159	#117	—	2	2	no	—
160	#117	C₂×Dic₄₇	1	4	yes	—
161	C₂×C₄⋊C₄	—	2	1	no	—
162	C₂×C₂²⋊C₄	—	2	1	no	—
163	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
164	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
165	M₃₂	—	47	8	no	—
166	M₃₂	—	94	4	no	—
167	C₂×C₁₆	—	94	4	no	—
168	#97	C₂×Dic₄₇	1	4	yes	—
169	#97	C₂×Dic₄₇	1	2	yes	—
170	#97	C₂²×D₄₇	1	2	yes	—
171	C₄×C₈	C₂×Dic₄₇	1	4	yes	—
172	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
173	C₄×C₈	C₂²×D₄₇	1	2	yes	—
174	C₂⁴	#10625329	1	1	yes	characteristic
175	C₂²×C₄	—	2	1	no	—
176	C₂²×C₄	—	2	1	no	—
177	C₂²×C₄	—	2	1	no	—
178	C₂²×C₄	—	2	1	no	—
179	C₂²×C₄	—	2	1	no	—
180	C₂²×C₄	—	2	1	no	—
181	C₂²×C₄	#10625329	1	1	yes	characteristic
182	C₂²×C₄	#10625329	1	2	yes	—
183	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
184	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
185	M₁₆	—	2	4	no	—
186	M₁₆	—	2	4	no	—
187	C₂×C₈	—	2	1	no	—
188	C₂×C₈	—	2	1	no	—
189	C₂×C₈	—	2	1	no	—
190	C₂×C₈	—	2	1	no	—
191	C₂×C₈	—	2	8	no	—
192	C₂×C₈	#10625329	1	2	yes	—
193	C₂×C₈	#10625329	1	2	yes	—
194	C₂×C₈	#10625350	1	2	yes	—
195	C₂×C₈	#10625350	1	2	yes	—
196	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
197	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
198	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
199	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
200	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
201	C₄⋊C₄	—	2	4	no	—
202	C₂²⋊C₄	—	2	4	no	—
203	C₄²	—	2	1	no	—
204	C₄²	—	2	1	no	—
205	C₄²	C₂²×Dic₄₇	1	2	yes	—
206	C₄²	C₂²×Dic₄₇	1	2	yes	—
207	C₁₆	—	94	8	no	—
208	C₂³	—	2	1	no	—
209	C₂³	—	2	1	no	—
210	C₂³	—	4	2	no	—
211	C₂³	C₂×#10625329	1	1	yes	characteristic
212	C₂³	C₂×#10625329	1	2	yes	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	#13093983	1	1	yes	characteristic
224	C₂×C₄	#13093983	1	1	yes	characteristic
225	C₂×C₄	C₂×#10625329	1	2	yes	—
226	C₂×C₄	C₂×#10625329	1	1	yes	characteristic
227	C₂×C₄	C₂×#10625329	1	1	yes	characteristic
228	C₂×C₄	C₂×#10625329	1	1	yes	characteristic
229	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
230	C₈	—	2	2	no	—
231	C₈	—	2	2	no	—
232	C₈	#13093983	1	2	yes	—
233	C₈	C₂×#10625329	1	4	yes	—
234	C₈	C₂×#10625350	1	2	yes	—
235	C₂²	—	2	2	no	—
236	C₂²	—	2	2	no	—
237	C₂²	—	4	2	no	—
238	C₂²	—	4	2	no	—
239	C₂²	—	4	2	no	—
240	C₂²	#15689180	1	1	yes	characteristic
241	C₂²	#15689180	1	1	yes	characteristic
242	C₂²	C₂²×#10625329	1	1	yes	characteristic
243	C₄	—	2	2	no	—
244	C₄	—	4	2	no	—
245	C₄	—	4	2	no	—
246	C₄	#15689180	1	1	yes	characteristic
247	C₄	#15689180	1	1	yes	characteristic
248	C₄	C₂×#13093983	1	1	yes	characteristic
249	C₄	C₂²×#10625329	1	1	yes	characteristic
250	C₂	—	2	1	no	—
251	C₂	—	2	1	no	—
252	C₂	—	4	2	no	—
253	C₂	#20245204	1	2	yes	direct product factor
254	C₂	C₂×#15689180	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13093983
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₂×C₄₇×#405	C₂
socle	C₂²×C₄₇	#432
prime radical	C₂⁴×C₄₇	M₁₆
cosocle	C₂×C₈×C₄₇	C₂⁴
hypercenter	C₂×#405	D₄₇
hypocenter	C₄₇	C₂×#2166
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
1. #20245204	C₂	direct (N × H)
2. C₂×#15688944	C₂	semidirect
4. C₂×#15688901	C₂	semidirect
7. #15688944	C₂²	semidirect
9. #15688901	C₂²	semidirect
131. C₂×C₄₇	#2166	semidirect
146. C₄₇	C₂×#2166	semidirect
253. C₂	#20245204	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20245204	2
C₂×#15688944	2
C₂×#15688905	2
C₂×#15688901	2
C₂×C₄₇×#405	2
C₂×#2166	47

Subgroups — C2×#20245204 #26369405

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20245204 #26369405