Subgroups — C₂×#20245201 #26369402

Order 12032.

538 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689081	C₂	1	1	yes	maximal, characteristic
2	#20245201	C₂	1	8	yes	maximal, direct product factor
3	C₂×#15688943	C₂	1	2	yes	maximal, complemented
4	C₂×#15688905	C₂	1	1	yes	maximal, characteristic
5	C₂×#15688900	C₂	1	2	yes	maximal, complemented
6	C₂×C₄₇×#349	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₄₇⋊C₈	C₄	1	1	yes	characteristic
8	C₂×#13093983	C₄	1	1	yes	characteristic
9	C₂×#13093964	C₂²	1	2	yes	—
10	#15689081	C₄	1	4	yes	—
11	#15689081	C₂²	1	4	yes	—
12	C₂×#13093934	C₄	1	1	yes	characteristic
13	C₂×#13093934	C₂²	1	1	yes	characteristic
14	C₂×C₈×Dic₄₇	C₄	1	1	yes	characteristic
15	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
16	#15688943	C₂²	1	8	yes	complemented
17	#15688905	C₂²	1	4	yes	—
18	#15688900	C₂²	1	8	yes	complemented
19	C₂²×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
20	C₂×M₃₂×C₄₇	C₂²	1	2	yes	—
21	C₄₇×#349	C₂²	1	4	yes	—
22	C₂²×C₄₇⋊C₈	—	2	2	no	—
23	C₂²×C₄₇⋊C₈	—	2	2	no	—
24	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
26	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
27	#13093983	—	2	2	no	—
28	#13093983	C₂×C₄	1	4	yes	—
29	C₂×C₄⋊Dic₄₇	—	2	1	no	—
30	C₂×C₂²⋊Dic₄₇	—	2	1	no	—
31	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
33	#13093964	—	2	2	no	—
34	#13093964	C₂³	1	4	yes	—
35	C₂×C₄₇⋊C₁₆	—	2	2	no	—
36	#13093934	C₂×C₄	1	4	yes	—
37	#13093934	C₂×C₄	1	2	yes	—
38	#13093934	C₂³	1	2	yes	—
39	C₈×Dic₄₇	C₂×C₄	1	4	yes	—
40	C₈×Dic₄₇	C₂×C₄	1	2	yes	—
41	C₈×Dic₄₇	C₂³	1	2	yes	—
42	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
43	C₂×M₁₆×C₄₇	—	2	2	no	—
44	C₂×M₁₆×C₄₇	—	2	2	no	—
45	C₂×M₁₆×C₄₇	C₂×C₄	1	2	yes	—
46	C₂×M₁₆×C₄₇	C₂³	1	2	yes	—
47	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
49	M₃₂×C₄₇	—	2	2	no	—
50	M₃₂×C₄₇	C₂³	1	4	yes	—
51	C₂×C₁₆×C₄₇	—	2	2	no	—
52	C₂⁴×C₄₇	M₁₆	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	—	2	1	no	—
54	C₂²×C₄×C₄₇	—	2	1	no	—
55	C₂²×C₄×C₄₇	—	2	1	no	—
56	C₂²×C₄×C₄₇	—	2	1	no	—
57	C₂²×C₄×C₄₇	M₁₆	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	M₁₆	1	2	yes	—
59	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
61	C₂²×Dic₄₇	—	2	1	no	—
62	C₂²×Dic₄₇	—	2	1	no	—
63	M₁₆×C₄₇	—	2	4	no	—
64	M₁₆×C₄₇	—	2	4	no	—
65	C₂×C₈×C₄₇	—	2	1	no	—
66	C₂×C₈×C₄₇	—	2	1	no	—
67	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
68	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₄₇	#41	1	2	yes	—
70	C₂×C₈×C₄₇	#41	1	2	yes	—
71	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
72	C₂²⋊Dic₄₇	—	2	4	no	—
73	C₄⋊Dic₄₇	—	2	4	no	—
74	C₄×Dic₄₇	—	2	1	no	—
75	C₄×Dic₄₇	—	2	1	no	—
76	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
77	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
78	C₂×C₄₇⋊C₈	—	2	1	no	—
79	C₂×C₄₇⋊C₈	—	2	1	no	—
80	C₂×C₄₇⋊C₈	—	2	8	no	—
81	C₂×C₄₇⋊C₈	M₁₆	1	2	yes	—
82	C₂×C₄₇⋊C₈	M₁₆	1	2	yes	—
83	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
84	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
85	C₁₆×C₄₇	—	2	4	no	—
86	C₄₇⋊C₁₆	—	2	4	no	—
87	C₂³×C₄₇	—	2	1	no	—
88	C₂³×C₄₇	—	2	1	no	—
89	C₂³×C₄₇	—	4	2	no	—
90	C₂³×C₄₇	C₂×M₁₆	1	1	yes	characteristic
91	C₂³×C₄₇	C₂×M₁₆	1	2	yes	—
92	C₂×C₄×C₄₇	—	2	1	no	—
93	C₂×C₄×C₄₇	—	2	1	no	—
94	C₂×C₄×C₄₇	—	2	2	no	—
95	C₂×C₄×C₄₇	—	2	2	no	—
96	C₂×C₄×C₄₇	—	2	2	no	—
97	C₂×C₄×C₄₇	—	2	2	no	—
98	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
99	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
100	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
101	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
102	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
103	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
104	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
105	C₂×Dic₄₇	—	2	2	no	—
106	C₂×Dic₄₇	—	2	2	no	—
107	C₂×Dic₄₇	—	4	2	no	—
108	C₂×Dic₄₇	—	4	2	no	—
109	C₈×C₄₇	—	2	2	no	—
110	C₈×C₄₇	#117	1	2	yes	—
111	C₈×C₄₇	C₂×#41	1	2	yes	—
112	C₄₇⋊C₈	—	2	2	no	—
113	C₄₇⋊C₈	C₂×M₁₆	1	4	yes	—
114	C₂×#2166	—	47	1	no	maximal, Sylow
115	C₂²×C₄₇	—	2	2	no	—
116	C₂²×C₄₇	—	2	2	no	—
117	C₂²×C₄₇	—	4	2	no	—
118	C₂²×C₄₇	—	4	2	no	—
119	C₂²×C₄₇	—	4	2	no	—
120	C₂²×C₄₇	#432	1	1	yes	characteristic
121	C₂²×C₄₇	#432	1	1	yes	characteristic
122	C₂²×C₄₇	C₂²×M₁₆	1	1	yes	characteristic
123	C₄×C₄₇	—	2	2	no	—
124	C₄×C₄₇	#432	1	1	yes	characteristic
125	C₄×C₄₇	#432	1	1	yes	characteristic
126	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
127	C₄×C₄₇	C₂²×M₁₆	1	1	yes	characteristic
128	Dic₄₇	—	4	2	no	—
129	Dic₄₇	—	4	2	no	—
130	C₂×#405	—	47	1	no	—
131	#2166	—	47	8	no	—
132	C₂×#364	—	47	2	no	—
133	C₂×#349	—	47	1	no	—
134	C₂×#349	D₄₇	1	1	yes	characteristic
135	C₂×#347	—	47	2	no	—
136	C₂×C₄₇	—	2	1	no	—
137	C₂×C₄₇	—	2	1	no	—
138	C₂×C₄₇	—	4	2	no	—
139	C₂×C₄₇	#2166	1	2	yes	complemented
140	C₂×C₄₇	C₂×#432	1	1	yes	characteristic
141	C₂²×M₁₆	C₂×D₄₇	1	1	yes	characteristic
142	C₂³×C₈	—	47	1	no	—
143	C₂×#117	—	47	1	no	—
144	C₂×M₃₂	—	47	2	no	—
145	C₂×M₃₂	C₂×D₄₇	1	2	yes	—
146	#405	—	47	4	no	—
147	#405	—	47	4	no	—
148	C₂×#97	—	47	1	no	—
149	C₂×#97	—	47	1	no	—
150	C₂×C₄×C₈	—	47	1	no	—
151	C₂×C₄×C₈	—	47	1	no	—
152	#364	—	47	8	no	—
153	#349	—	47	4	no	—
154	#349	C₂×D₄₇	1	4	yes	—
155	#347	—	47	8	no	—
156	C₄₇	C₂×#2166	1	1	yes	characteristic, complemented, Sylow
157	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
158	C₂×M₁₆	—	2	2	no	—
159	C₂×M₁₆	—	2	2	no	—
160	C₂×M₁₆	C₄×D₄₇	1	2	yes	—
161	C₂×M₁₆	C₂²×D₄₇	1	2	yes	—
162	C₂²×C₈	—	47	1	no	—
163	C₂²×C₈	—	47	1	no	—
164	C₂²×C₈	—	47	4	no	—
165	C₂²×C₈	—	94	2	no	—
166	C₂²×C₈	—	94	2	no	—
167	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
168	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
169	#117	—	47	4	no	—
170	#117	—	94	2	no	—
171	C₂×C₄⋊C₄	—	94	1	no	—
172	C₂×C₂²⋊C₄	—	94	1	no	—
173	C₂×C₄²	—	47	1	no	—
174	C₂×C₄²	—	47	1	no	—
175	M₃₂	—	2	2	no	—
176	M₃₂	—	47	4	no	—
177	M₃₂	—	94	2	no	—
178	M₃₂	C₂²×D₄₇	1	4	yes	—
179	C₂×C₁₆	—	2	2	no	—
180	C₂×C₁₆	—	94	2	no	—
181	#97	—	47	4	no	—
182	#97	—	47	2	no	—
183	#97	—	47	2	no	—
184	C₄×C₈	—	47	4	no	—
185	C₄×C₈	—	47	2	no	—
186	C₄×C₈	—	47	2	no	—
187	C₂⁴	#10625324	1	1	yes	characteristic
188	C₂²×C₄	—	2	1	no	—
189	C₂²×C₄	—	2	1	no	—
190	C₂²×C₄	—	2	1	no	—
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	—	94	1	no	—
193	C₂²×C₄	—	94	1	no	—
194	C₂²×C₄	#10625324	1	1	yes	characteristic
195	C₂²×C₄	#10625324	1	2	yes	—
196	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
197	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
198	M₁₆	—	2	4	no	—
199	M₁₆	—	2	4	no	—
200	C₂×C₈	—	2	1	no	—
201	C₂×C₈	—	2	1	no	—
202	C₂×C₈	—	47	2	no	—
203	C₂×C₈	—	47	2	no	—
204	C₂×C₈	—	47	2	no	—
205	C₂×C₈	—	47	2	no	—
206	C₂×C₈	—	94	8	no	—
207	C₂×C₈	—	94	1	no	—
208	C₂×C₈	—	94	1	no	—
209	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
210	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
211	C₂×C₈	#10625352	1	2	yes	—
212	C₂×C₈	#10625352	1	2	yes	—
213	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
214	C₄⋊C₄	—	94	4	no	—
215	C₂²⋊C₄	—	94	4	no	—
216	C₄²	—	47	2	no	—
217	C₄²	—	47	2	no	—
218	C₄²	—	94	1	no	—
219	C₄²	—	94	1	no	—
220	C₁₆	—	2	4	no	—
221	C₁₆	—	94	4	no	—
222	C₂³	—	2	1	no	—
223	C₂³	—	2	1	no	—
224	C₂³	—	4	2	no	—
225	C₂³	C₂×#10625324	1	1	yes	characteristic
226	C₂³	C₂×#10625324	1	2	yes	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	94	2	no	—
234	C₂×C₄	—	94	2	no	—
235	C₂×C₄	—	188	2	no	—
236	C₂×C₄	—	188	2	no	—
237	C₂×C₄	#13093980	1	1	yes	characteristic
238	C₂×C₄	#13093980	1	1	yes	characteristic
239	C₂×C₄	M₁₆×D₄₇	1	1	yes	characteristic
240	C₂×C₄	M₁₆×D₄₇	1	1	yes	characteristic
241	C₂×C₄	C₂×#10625324	1	2	yes	—
242	C₂×C₄	C₂×#10625324	1	1	yes	characteristic
243	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
244	C₈	—	2	2	no	—
245	C₈	—	47	4	no	—
246	C₈	—	94	2	no	—
247	C₈	#13093980	1	2	yes	—
248	C₈	C₂×#10625352	1	2	yes	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	4	2	no	—
252	C₂²	—	4	2	no	—
253	C₂²	—	4	2	no	—
254	C₂²	#15689177	1	1	yes	characteristic
255	C₂²	#15689177	1	1	yes	characteristic
256	C₂²	C₂²×#10625324	1	1	yes	characteristic
257	C₄	—	2	2	no	—
258	C₄	—	188	2	no	—
259	C₄	—	188	2	no	—
260	C₄	#15689177	1	1	yes	characteristic
261	C₄	#15689177	1	1	yes	characteristic
262	C₄	C₂×#13093980	1	1	yes	characteristic
263	C₄	C₂×M₁₆×D₄₇	1	1	yes	characteristic
264	C₂	—	2	1	no	—
265	C₂	—	2	1	no	—
266	C₂	—	4	2	no	—
267	C₂	#20245201	1	2	yes	direct product factor
268	C₂	C₂×#15689177	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13093980
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₂×C₄₇×#349	C₂
socle	C₂²×C₄₇	#432
prime radical	C₂⁴×C₄₇	M₁₆
cosocle	C₂×C₈×C₄₇	C₂⁴
hypercenter	C₂×#349	D₄₇
hypocenter	C₄₇	C₂×#2166
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#349

Complements

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

N	H ≅ G/N	type
2. #20245201	C₂	direct (N × H)
3. C₂×#15688943	C₂	semidirect
5. C₂×#15688900	C₂	semidirect
16. #15688943	C₂²	semidirect
18. #15688900	C₂²	semidirect
139. C₂×C₄₇	#2166	semidirect
156. C₄₇	C₂×#2166	semidirect
267. C₂	#20245201	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×#15689081	2
#20245201	2
C₂×#15688943	2
C₂×#15688905	2
C₂×#15688900	2
C₂×C₄₇×#349	2
C₂×#2166	47

Subgroups — C2×#20245201 #26369402

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20245201 #26369402