Subgroups — C₂×#20245199 #26369399

Order 12032.

570 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#10625324	C₂	1	1	yes	maximal, characteristic
2	#20245199	C₂	1	8	yes	maximal, direct product factor
3	C₂×#15688944	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#15688904	C₂	1	2	yes	maximal
5	C₂×#15688900	C₂	1	2	yes	maximal, complemented
6	C₂×C₄₇×#364	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#10625324	C₄	1	1	yes	characteristic
8	C₂²×#10625324	C₂²	1	1	yes	characteristic
9	C₂×C₄²×D₄₇	C₄	1	1	yes	characteristic
10	C₂×#13093964	C₂²	1	2	yes	—
11	C₄×#10625324	C₄	1	4	yes	—
12	C₄×#10625324	C₂²	1	4	yes	—
13	C₂×#13093934	C₄	1	1	yes	characteristic
14	C₂×#13093934	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₄₇⋊C₈	C₄	1	1	yes	characteristic
16	#15688944	C₂²	1	4	yes	complemented
17	#15688904	C₂²	1	8	yes	—
18	#15688900	C₂²	1	8	yes	complemented
19	C₂×M₃₂×C₄₇	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
21	C₄₇×#364	C₂²	1	4	yes	complemented
22	C₂²×C₄×D₄₇	—	2	1	no	—
23	C₂²×C₄×D₄₇	C₂×C₄	1	1	yes	characteristic
24	C₂×#10625324	—	2	2	no	—
25	C₂×#10625324	—	2	2	no	—
26	C₂×#10625324	—	2	1	no	—
27	C₂×#10625324	—	2	1	no	—
28	C₂×#10625324	—	2	1	no	—
29	C₂×#10625324	—	2	1	no	—
30	C₂×#10625324	C₂×C₄	1	2	yes	—
31	C₂×#10625324	C₂×C₄	1	4	yes	—
32	C₂×#10625324	C₂³	1	2	yes	—
33	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×Dic₄₇	—	2	1	no	—
36	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
37	C₄²×D₄₇	—	2	2	no	—
38	C₄²×D₄₇	C₂×C₄	1	4	yes	—
39	#13093964	—	2	2	no	—
40	#13093964	C₂³	1	4	yes	—
41	C₂×C₄₇⋊C₁₆	—	2	2	no	—
42	#13093934	C₂×C₄	1	4	yes	—
43	#13093934	C₂×C₄	1	2	yes	—
44	#13093934	C₂³	1	2	yes	—
45	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
46	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₄₇	C₂×C₄	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	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
53	C₄×C₈×C₄₇	C₂³	1	2	yes	—
54	C₂³×D₄₇	M₁₆	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	—	2	1	no	—
56	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
57	C₂²×Dic₄₇	—	2	1	no	—
58	C₂²×Dic₄₇	M₁₆	1	1	yes	characteristic
59	C₂×C₄×D₄₇	—	2	1	no	—
60	C₂×C₄×D₄₇	—	2	1	no	—
61	C₂×C₄×D₄₇	—	2	1	no	—
62	C₂×C₄×D₄₇	—	2	1	no	—
63	C₂×C₄×D₄₇	—	2	4	no	—
64	C₂×C₄×D₄₇	—	4	4	no	—
65	C₂×C₄×D₄₇	M₁₆	1	2	yes	—
66	C₂×C₄×D₄₇	C₂²×C₄	1	2	yes	—
67	C₂×C₈×C₄₇	M₁₆	1	2	yes	—
68	C₂×C₈×C₄₇	M₁₆	1	2	yes	—
69	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
71	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
72	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
73	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
74	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
75	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
76	C₄²×C₄₇	—	2	1	no	—
77	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
78	C₄×Dic₄₇	—	2	1	no	—
79	C₄×Dic₄₇	—	2	4	no	—
80	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
81	C₂×C₄₇⋊C₈	—	2	1	no	—
82	C₂×C₄₇⋊C₈	—	2	1	no	—
83	C₂×C₄₇⋊C₈	—	2	1	no	—
84	C₂×C₄₇⋊C₈	—	2	1	no	—
85	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
86	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
87	#10625324	—	2	2	no	—
88	#10625324	—	2	2	no	—
89	#10625324	—	2	8	no	—
90	#10625324	—	2	2	no	—
91	#10625324	—	2	2	no	—
92	C₁₆×C₄₇	—	2	4	no	—
93	C₄₇⋊C₁₆	—	2	4	no	—
94	C₂³×C₄₇	C₂×M₁₆	1	1	yes	characteristic
95	C₂²×D₄₇	—	2	1	no	—
96	C₂²×D₄₇	—	2	1	no	—
97	C₂²×D₄₇	—	4	2	no	—
98	C₂²×D₄₇	C₂×M₁₆	1	2	yes	—
99	C₂×C₄×C₄₇	—	2	4	no	—
100	C₂×C₄×C₄₇	—	2	2	no	—
101	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
102	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
103	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
104	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
105	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
106	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
107	C₂×Dic₄₇	—	2	1	no	—
108	C₂×Dic₄₇	—	2	1	no	—
109	C₂×Dic₄₇	—	2	2	no	—
110	C₂×Dic₄₇	—	4	2	no	—
111	C₂×Dic₄₇	C₂×M₁₆	1	2	yes	—
112	C₄×D₄₇	—	2	2	no	—
113	C₄×D₄₇	—	2	2	no	—
114	C₄×D₄₇	—	2	2	no	—
115	C₄×D₄₇	—	2	2	no	—
116	C₄×D₄₇	—	4	8	no	—
117	C₈×C₄₇	#117	1	1	yes	characteristic
118	C₈×C₄₇	#117	1	1	yes	characteristic
119	C₈×C₄₇	C₂×M₁₆	1	4	yes	—
120	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
121	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
122	C₄₇⋊C₈	—	2	2	no	—
123	C₄₇⋊C₈	—	2	2	no	—
124	C₂×#2165	—	47	1	no	maximal, Sylow
125	C₂²×C₄₇	—	2	2	no	—
126	C₂²×C₄₇	#432	1	1	yes	characteristic
127	C₂²×C₄₇	#432	1	1	yes	characteristic
128	C₂²×C₄₇	C₂²×M₁₆	1	1	yes	characteristic
129	C₂×D₄₇	—	2	2	no	—
130	C₂×D₄₇	—	4	2	no	—
131	C₂×D₄₇	—	4	2	no	—
132	C₂×D₄₇	—	4	2	no	—
133	C₄×C₄₇	—	2	4	no	—
134	C₄×C₄₇	#432	1	1	yes	characteristic
135	C₄×C₄₇	#432	1	1	yes	characteristic
136	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
137	C₄×C₄₇	C₂²×M₁₆	1	1	yes	characteristic
138	Dic₄₇	—	2	2	no	—
139	Dic₄₇	—	4	2	no	—
140	C₂×C₄×M₁₆	—	47	1	no	—
141	#2165	—	47	8	no	—
142	C₂×#364	—	47	1	no	—
143	C₂×#364	D₄₇	1	1	yes	characteristic, complemented
144	C₂×#349	—	47	2	no	—
145	C₂×#347	—	47	2	no	—
146	C₂×C₄₇	—	2	1	no	—
147	C₂×C₄₇	—	2	1	no	—
148	C₂×C₄₇	#2165	1	2	yes	complemented
149	C₂×C₄₇	C₂×#432	1	1	yes	characteristic
150	D₄₇	—	4	2	no	—
151	C₂²×M₁₆	—	47	1	no	—
152	C₂²×M₁₆	—	47	1	no	—
153	C₂²×C₄²	—	47	1	no	—
154	C₂×M₃₂	—	47	2	no	—
155	C₂×M₃₂	C₂×D₄₇	1	2	yes	—
156	C₄×M₁₆	—	47	4	no	—
157	C₄×M₁₆	—	47	4	no	—
158	C₂×#97	—	47	1	no	—
159	C₂×#97	—	47	1	no	—
160	C₂×C₄×C₈	—	47	1	no	—
161	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
162	#364	—	47	4	no	—
163	#364	C₂×D₄₇	1	4	yes	complemented
164	#349	—	47	8	no	—
165	#347	—	47	8	no	—
166	C₄₇	C₂×#2165	1	1	yes	characteristic, complemented, Sylow
167	C₂³×C₄	—	47	1	no	—
168	C₂³×C₄	—	94	1	no	—
169	C₂×M₁₆	—	47	2	no	—
170	C₂×M₁₆	—	47	4	no	—
171	C₂×M₁₆	—	47	2	no	—
172	C₂×M₁₆	—	94	2	no	—
173	C₂×M₁₆	—	94	2	no	—
174	C₂×M₁₆	—	94	1	no	—
175	C₂×M₁₆	—	94	1	no	—
176	C₂×M₁₆	—	94	1	no	—
177	C₂×M₁₆	—	94	1	no	—
178	C₂²×C₈	—	47	1	no	—
179	C₂²×C₈	—	47	1	no	—
180	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
181	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
182	C₂×C₄²	—	47	1	no	—
183	C₂×C₄²	—	47	4	no	—
184	C₂×C₄²	—	94	2	no	—
185	C₂×C₄²	—	94	1	no	—
186	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
187	M₃₂	—	2	2	no	—
188	M₃₂	—	47	4	no	—
189	M₃₂	—	94	2	no	—
190	M₃₂	C₂²×D₄₇	1	4	yes	—
191	C₂×C₁₆	—	2	2	no	—
192	C₂×C₁₆	—	94	2	no	—
193	#97	—	47	2	no	—
194	#97	—	47	4	no	—
195	#97	—	47	2	no	—
196	C₄×C₈	—	47	4	no	—
197	C₄×C₈	C₄×D₄₇	1	2	yes	—
198	C₄×C₈	C₂²×D₄₇	1	2	yes	—
199	C₂⁴	—	47	1	no	—
200	C₂²×C₄	—	2	1	no	—
201	C₂²×C₄	—	47	1	no	—
202	C₂²×C₄	—	47	2	no	—
203	C₂²×C₄	—	47	2	no	—
204	C₂²×C₄	—	94	4	no	—
205	C₂²×C₄	—	94	1	no	—
206	C₂²×C₄	—	94	1	no	—
207	C₂²×C₄	—	94	1	no	—
208	C₂²×C₄	—	94	1	no	—
209	C₂²×C₄	—	94	1	no	—
210	C₂²×C₄	—	188	4	no	—
211	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
212	M₁₆	—	94	2	no	—
213	M₁₆	—	94	2	no	—
214	M₁₆	—	94	8	no	—
215	M₁₆	—	94	2	no	—
216	M₁₆	—	94	2	no	—
217	C₂×C₈	—	47	2	no	—
218	C₂×C₈	—	47	2	no	—
219	C₂×C₈	—	94	1	no	—
220	C₂×C₈	—	94	1	no	—
221	C₂×C₈	—	94	1	no	—
222	C₂×C₈	—	94	1	no	—
223	C₂×C₈	#10625324	1	2	yes	—
224	C₂×C₈	#10625324	1	2	yes	—
225	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
226	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
227	C₂×C₈	#10625352	1	1	yes	characteristic
228	C₂×C₈	#10625352	1	1	yes	characteristic
229	C₂×C₈	#10625354	1	1	yes	characteristic
230	C₂×C₈	#10625354	1	1	yes	characteristic
231	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
232	C₄²	—	2	1	no	—
233	C₄²	—	47	2	no	—
234	C₄²	—	94	4	no	—
235	C₄²	—	94	1	no	—
236	C₄²	C₂×C₄×D₄₇	1	2	yes	—
237	C₁₆	—	2	4	no	—
238	C₁₆	—	94	4	no	—
239	C₂³	—	47	2	no	—
240	C₂³	—	94	1	no	—
241	C₂³	—	94	1	no	—
242	C₂³	—	188	2	no	—
243	C₂³	M₁₆×D₄₇	1	1	yes	characteristic
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	47	2	no	—
247	C₂×C₄	—	94	1	no	—
248	C₂×C₄	—	94	1	no	—
249	C₂×C₄	—	94	2	no	—
250	C₂×C₄	—	94	2	no	—
251	C₂×C₄	—	94	2	no	—
252	C₂×C₄	—	94	2	no	—
253	C₂×C₄	—	94	2	no	—
254	C₂×C₄	—	188	8	no	—
255	C₂×C₄	—	188	2	no	—
256	C₂×C₄	#13093982	1	1	yes	characteristic
257	C₂×C₄	#13093982	1	1	yes	characteristic
258	C₂×C₄	M₁₆×D₄₇	1	1	yes	characteristic
259	C₂×C₄	C₂×#10625324	1	1	yes	characteristic
260	C₂×C₄	C₂×#10625324	1	1	yes	characteristic
261	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
262	C₈	—	94	2	no	—
263	C₈	—	94	2	no	—
264	C₈	#13093982	1	1	yes	characteristic
265	C₈	#13093982	1	1	yes	characteristic
266	C₈	C₂×#10625324	1	4	yes	—
267	C₈	C₂×#10625352	1	1	yes	characteristic
268	C₈	C₂×#10625354	1	1	yes	characteristic
269	C₂²	—	2	2	no	—
270	C₂²	—	94	2	no	—
271	C₂²	—	188	2	no	—
272	C₂²	—	188	2	no	—
273	C₂²	—	188	2	no	—
274	C₂²	#15689179	1	1	yes	characteristic
275	C₂²	#15689179	1	1	yes	characteristic
276	C₂²	C₂×M₁₆×D₄₇	1	1	yes	characteristic
277	C₄	—	2	4	no	—
278	C₄	—	94	2	no	—
279	C₄	—	188	2	no	—
280	C₄	#15689179	1	1	yes	characteristic
281	C₄	#15689179	1	1	yes	characteristic
282	C₄	C₂×#13093982	1	1	yes	characteristic
283	C₄	C₂²×#10625324	1	1	yes	characteristic
284	C₂	—	2	1	no	—
285	C₂	—	2	1	no	—
286	C₂	—	188	2	no	—
287	C₂	#20245199	1	2	yes	direct product factor
288	C₂	C₂×#15689179	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13093982
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₂×C₄₇×#364	C₂
socle	C₂²×C₄₇	#432
prime radical	C₂³×D₄₇	M₁₆
cosocle	C₂×C₈×C₄₇	C₂⁴
hypercenter	C₂×#364	D₄₇
hypocenter	C₄₇	C₂×#2165
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
2. #20245199	C₂	direct (N × H)
3. C₂×#15688944	C₂	semidirect
5. C₂×#15688900	C₂	semidirect
6. C₂×C₄₇×#364	C₂	semidirect
16. #15688944	C₂²	semidirect
18. #15688900	C₂²	semidirect
21. C₄₇×#364	C₂²	semidirect
143. C₂×#364	D₄₇	semidirect
148. C₂×C₄₇	#2165	semidirect
163. #364	C₂×D₄₇	semidirect
166. C₄₇	C₂×#2165	semidirect
287. C₂	#20245199	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×C₄×#10625324	2
#20245199	2
C₂×#15688944	2
C₂×#15688904	2
C₂×#15688900	2
C₂×C₄₇×#364	2
C₂×#2165	47

Subgroups — C2×#20245199 #26369399

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20245199 #26369399