Subgroups — C₂×#20244117 #26219600

Order 12032.

1182 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093956	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#15689022	C₂	1	2	yes	maximal
4	#20244117	C₂	1	8	yes	maximal, direct product factor
5	C₂×#15688885	C₂	1	2	yes	maximal
6	C₂²×#106×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₄⋊Dic₄₇	—	2	2	no	—
9	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
10	C₂²×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
11	C₂×#13093956	—	2	4	no	—
12	C₂×#13093956	C₂²	1	4	yes	—
13	C₂×C₈×Dic₄₇	—	2	4	no	—
14	C₂×C₈×Dic₄₇	C₂²	1	4	yes	—
15	#15689022	—	2	4	no	—
16	#15689022	C₂²	1	8	yes	—
17	C₂×C₄₇⋊#95	—	2	2	no	—
18	#15688885	C₂²	1	8	yes	—
19	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
20	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
21	C₂×#106×C₄₇	C₄	1	8	yes	complemented
22	C₂×#106×C₄₇	C₂²	1	4	yes	—
23	C₂³×Dic₄₇	—	2	2	no	—
24	C₂³×Dic₄₇	D₄	1	2	yes	—
25	C₂²×C₄₇⋊C₈	—	2	2	no	—
26	C₂²×C₄₇⋊C₈	—	2	2	no	—
27	C₂²×C₄₇⋊C₈	Q₈	1	4	yes	—
28	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
29	C₂×C₄⋊Dic₄₇	—	2	4	no	—
30	C₂×C₄⋊Dic₄₇	—	2	4	no	—
31	C₂×C₄⋊Dic₄₇	—	2	2	no	—
32	C₂×C₄⋊Dic₄₇	—	2	8	no	—
33	C₂×C₄⋊Dic₄₇	—	4	8	no	—
34	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
35	C₂×C₄×Dic₄₇	—	2	2	no	—
36	C₂×C₄×Dic₄₇	—	2	2	no	—
37	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
38	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
39	#13093956	—	2	8	no	—
40	C₈×Dic₄₇	—	2	8	no	—
41	C₄₇⋊#95	—	2	8	no	—
42	C₂³×C₄×C₄₇	—	2	2	no	—
43	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
45	C₂²×C₈×C₄₇	D₄	1	4	yes	—
46	C₂²×C₈×C₄₇	Q₈	1	4	yes	—
47	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
48	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
49	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
50	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	8	yes	—
51	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
52	#106×C₄₇	C₂×C₄	1	16	yes	complemented
53	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	—	2	8	no	—
55	C₂²×C₄×C₄₇	—	2	4	no	—
56	C₂²×C₄×C₄₇	—	4	4	no	—
57	C₂²×C₄×C₄₇	—	4	4	no	—
58	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
60	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
62	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
63	C₂²×C₄×C₄₇	#41	1	2	yes	—
64	C₂²×C₄×C₄₇	#41	1	2	yes	—
65	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
66	C₂²×Dic₄₇	—	2	4	no	—
67	C₂²×Dic₄₇	—	2	4	no	—
68	C₂²×Dic₄₇	—	2	4	no	—
69	C₂²×Dic₄₇	—	4	4	no	—
70	C₂²×Dic₄₇	—	4	4	no	—
71	C₂²×Dic₄₇	—	4	4	no	—
72	C₂²×Dic₄₇	SD₁₆	1	8	yes	—
73	C₂²×Dic₄₇	C₂×D₄	1	4	yes	—
74	C₂×C₈×C₄₇	C₄⋊C₄	1	16	yes	complemented
75	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
76	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
77	C₂×C₈×C₄₇	C₂×Q₈	1	4	yes	—
78	C₄⋊C₄×C₄₇	—	2	8	no	—
79	C₄⋊C₄×C₄₇	—	2	4	no	—
80	C₄⋊C₄×C₄₇	C₂²×C₄	1	8	yes	—
81	C₄⋊Dic₄₇	—	2	4	no	—
82	C₄⋊Dic₄₇	—	2	4	no	—
83	C₄⋊Dic₄₇	—	4	4	no	—
84	C₄⋊Dic₄₇	—	4	16	no	—
85	C₄×Dic₄₇	—	2	4	no	—
86	C₄×Dic₄₇	—	2	2	no	—
87	C₄×Dic₄₇	—	2	2	no	—
88	C₂×C₄₇⋊C₈	—	2	8	no	—
89	C₂×C₄₇⋊C₈	—	2	2	no	—
90	C₂×C₄₇⋊C₈	—	2	2	no	—
91	C₂×C₄₇⋊C₈	C₂×Q₈	1	4	yes	—
92	C₂³×C₄₇	C₄×D₄	1	2	yes	—
93	C₂³×C₄₇	#121	1	2	yes	—
94	C₂³×C₄₇	C₄⋊Q₈	1	2	yes	—
95	C₂³×C₄₇	C₂×SD₁₆	1	4	yes	—
96	C₂³×C₄₇	#135	1	4	yes	—
97	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
98	C₂×C₄×C₄₇	—	2	8	no	—
99	C₂×C₄×C₄₇	—	4	4	no	—
100	C₂×C₄×C₄₇	—	4	4	no	—
101	C₂×C₄×C₄₇	—	4	16	no	—
102	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
103	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
104	C₂×C₄×C₄₇	C₄×Q₈	1	4	yes	—
105	C₂×C₄×C₄₇	#121	1	2	yes	—
106	C₂×C₄×C₄₇	C₂²⋊Q₈	1	4	yes	—
107	C₂×C₄×C₄₇	#125	1	4	yes	—
108	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
109	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
110	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
111	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
112	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
113	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
114	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
115	C₂×Dic₄₇	—	2	4	no	—
116	C₂×Dic₄₇	—	2	4	no	—
117	C₂×Dic₄₇	—	2	16	no	—
118	C₂×Dic₄₇	—	4	4	no	—
119	C₂×Dic₄₇	—	4	4	no	—
120	C₂×Dic₄₇	—	4	4	no	—
121	C₂×Dic₄₇	—	8	8	no	—
122	C₂×Dic₄₇	C₂×SD₁₆	1	8	yes	—
123	C₈×C₄₇	C₂×C₄⋊C₄	1	8	yes	complemented
124	C₄₇⋊C₈	—	2	4	no	—
125	C₂×#1941	—	47	1	no	maximal, Sylow
126	C₂²×C₄₇	#389	1	4	yes	—
127	C₂²×C₄₇	C₄×SD₁₆	1	8	yes	—
128	C₂²×C₄₇	#465	1	8	yes	—
129	C₂²×C₄₇	#498	1	4	yes	—
130	C₂²×C₄₇	#499	1	4	yes	—
131	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
132	C₂²×C₄₇	C₂×#121	1	1	yes	characteristic
133	C₂²×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
134	C₂²×C₄₇	C₂²×SD₁₆	1	2	yes	—
135	C₂²×C₄₇	C₂×#135	1	2	yes	—
136	C₄×C₄₇	—	4	8	no	—
137	C₄×C₄₇	#389	1	4	yes	—
138	C₄×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
139	C₄×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
140	C₄×C₄₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
141	C₄×C₄₇	C₂×#125	1	1	yes	characteristic
142	Dic₄₇	—	2	8	no	—
143	Dic₄₇	—	8	4	no	—
144	C₂²×#106	—	47	1	no	—
145	C₂²×#106	D₄₇	1	1	yes	characteristic
146	C₂²×C₄×C₈	—	47	1	no	—
147	C₂×#389	—	47	2	no	—
148	#1941	—	47	8	no	—
149	C₂×#340	—	47	2	no	—
150	C₂×C₄₇	#1941	1	8	yes	complemented
151	C₂×C₄₇	C₂×#389	1	1	yes	characteristic
152	C₂×C₄₇	C₂×C₄×SD₁₆	1	2	yes	—
153	C₂×C₄₇	C₂×#465	1	2	yes	—
154	C₂×C₄₇	C₂×#498	1	1	yes	characteristic
155	C₂×C₄₇	C₂×#499	1	1	yes	characteristic
156	C₂³×C₈	—	47	1	no	—
157	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
158	C₂²×C₄⋊C₄	—	47	2	no	—
159	C₂²×C₄⋊C₄	—	94	2	no	—
160	C₂²×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
161	C₂²×C₄²	—	47	1	no	—
162	C₂×#106	—	47	4	no	—
163	C₂×#106	—	94	4	no	—
164	C₂×#106	Dic₄₇	1	8	yes	complemented
165	C₂×#106	C₂×D₄₇	1	4	yes	—
166	C₂×C₄×C₈	—	47	4	no	—
167	C₂×C₄×C₈	—	94	4	no	—
168	#389	—	47	8	no	—
169	#389	—	94	4	no	—
170	C₂×#95	—	94	2	no	—
171	#340	—	47	8	no	—
172	C₄₇	C₂×#1941	1	1	yes	characteristic, complemented, Sylow
173	C₂³×C₄	—	2	2	no	—
174	C₂³×C₄	—	47	2	no	—
175	C₂³×C₄	—	94	2	no	—
176	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
177	C₂²×C₈	—	47	4	no	—
178	C₂²×C₈	—	47	2	no	—
179	C₂²×C₈	—	94	2	no	—
180	C₂²×C₈	—	94	2	no	—
181	C₂²×C₈	Dic₉₄	1	4	yes	—
182	C₂²×C₈	D₁₈₈	1	4	yes	—
183	C₂²×C₈	C₂×Dic₄₇	1	4	yes	—
184	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
185	C₂×C₄⋊C₄	—	2	2	no	—
186	C₂×C₄⋊C₄	—	2	4	no	—
187	C₂×C₄⋊C₄	—	47	4	no	—
188	C₂×C₄⋊C₄	—	94	4	no	—
189	C₂×C₄⋊C₄	—	94	4	no	—
190	C₂×C₄⋊C₄	—	94	8	no	—
191	C₂×C₄⋊C₄	—	94	2	no	—
192	C₂×C₄⋊C₄	—	188	8	no	—
193	C₂×C₄⋊C₄	C₂×Dic₄₇	1	8	yes	—
194	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
195	C₂×C₄²	—	47	2	no	—
196	C₂×C₄²	—	47	2	no	—
197	C₂×C₄²	—	94	2	no	—
198	C₂×C₄²	—	94	2	no	—
199	#106	—	94	8	no	—
200	#106	C₂×Dic₄₇	1	16	yes	complemented
201	C₄×C₈	—	94	8	no	—
202	#95	—	94	8	no	—
203	C₂⁴	D₄×D₄₇	1	1	yes	characteristic
204	C₂²×C₄	—	2	8	no	—
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	4	4	no	—
207	C₂²×C₄	—	4	4	no	—
208	C₂²×C₄	—	47	8	no	—
209	C₂²×C₄	—	47	4	no	—
210	C₂²×C₄	—	94	4	no	—
211	C₂²×C₄	—	94	4	no	—
212	C₂²×C₄	—	94	4	no	—
213	C₂²×C₄	—	188	4	no	—
214	C₂²×C₄	—	188	4	no	—
215	C₂²×C₄	—	188	4	no	—
216	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
217	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
218	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
219	C₂²×C₄	#10625352	1	2	yes	—
220	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
221	C₂²×C₄	#10625354	1	2	yes	—
222	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
223	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
224	C₂×C₈	—	47	4	no	—
225	C₂×C₈	—	94	8	no	—
226	C₂×C₈	—	94	2	no	—
227	C₂×C₈	—	94	2	no	—
228	C₂×C₈	C₄⋊Dic₄₇	1	16	yes	complemented
229	C₂×C₈	C₂×Dic₉₄	1	4	yes	—
230	C₂×C₈	C₂×D₁₈₈	1	4	yes	—
231	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
232	C₄⋊C₄	—	2	8	no	—
233	C₄⋊C₄	—	2	4	no	—
234	C₄⋊C₄	—	94	4	no	—
235	C₄⋊C₄	—	94	4	no	—
236	C₄⋊C₄	—	188	16	no	—
237	C₄⋊C₄	—	188	4	no	—
238	C₄⋊C₄	C₂²×Dic₄₇	1	8	yes	—
239	C₄²	—	94	4	no	—
240	C₄²	—	94	2	no	—
241	C₄²	—	94	2	no	—
242	C₂³	D₄×Dic₄₇	1	2	yes	—
243	C₂³	#13093999	1	2	yes	—
244	C₂³	#13094023	1	2	yes	—
245	C₂³	SD₁₆×D₄₇	1	4	yes	—
246	C₂³	#13094045	1	4	yes	—
247	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
248	C₂×C₄	—	2	8	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	—	4	4	no	—
251	C₂×C₄	—	4	16	no	—
252	C₂×C₄	—	47	8	no	—
253	C₂×C₄	—	94	16	no	—
254	C₂×C₄	—	94	4	no	—
255	C₂×C₄	—	94	4	no	—
256	C₂×C₄	—	376	8	no	—
257	C₂×C₄	—	188	4	no	—
258	C₂×C₄	—	188	4	no	—
259	C₂×C₄	—	188	4	no	—
260	C₂×C₄	C₂×C₄⋊Dic₄₇	1	4	yes	—
261	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
262	C₂×C₄	Q₈×Dic₄₇	1	4	yes	—
263	C₂×C₄	#13093999	1	2	yes	—
264	C₂×C₄	#13094004	1	4	yes	—
265	C₂×C₄	#13094015	1	4	yes	—
266	C₂×C₄	#13094023	1	2	yes	—
267	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
268	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
269	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
270	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
271	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
272	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
273	C₈	—	94	4	no	—
274	C₈	C₂×C₄⋊Dic₄₇	1	8	yes	complemented
275	C₂²	#15689022	1	4	yes	—
276	C₂²	SD₁₆×Dic₄₇	1	8	yes	—
277	C₂²	#15689353	1	8	yes	—
278	C₂²	#15689520	1	4	yes	—
279	C₂²	#15689523	1	4	yes	—
280	C₂²	C₂×D₄×Dic₄₇	1	1	yes	characteristic
281	C₂²	C₂×#13093999	1	1	yes	characteristic
282	C₂²	C₂×#13094023	1	1	yes	characteristic
283	C₂²	C₂×SD₁₆×D₄₇	1	2	yes	—
284	C₂²	C₂×#13094045	1	2	yes	—
285	C₄	—	4	8	no	—
286	C₄	—	94	8	no	—
287	C₄	—	376	4	no	—
288	C₄	#15689022	1	4	yes	—
289	C₄	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
290	C₄	C₂×Q₈×Dic₄₇	1	1	yes	characteristic
291	C₄	C₂×#13094004	1	1	yes	characteristic
292	C₄	C₂×#13094015	1	1	yes	characteristic
293	C₂	#20244117	1	8	yes	direct product factor
294	C₂	C₂×#15689022	1	1	yes	characteristic
295	C₂	C₂×SD₁₆×Dic₄₇	1	2	yes	—
296	C₂	C₂×#15689353	1	2	yes	—
297	C₂	C₂×#15689520	1	1	yes	characteristic
298	C₂	C₂×#15689523	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂²×#106×C₄₇	C₂
socle, prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂²×#106	D₄₇
hypocenter	C₄₇	C₂×#1941
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂²×#106

Complements

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

N	H ≅ G/N	type
4. #20244117	C₂	direct (N × H)
21. C₂×#106×C₄₇	C₄	semidirect
52. #106×C₄₇	C₂×C₄	semidirect
74. C₂×C₈×C₄₇	C₄⋊C₄	semidirect
123. C₈×C₄₇	C₂×C₄⋊C₄	semidirect
150. C₂×C₄₇	#1941	semidirect
164. C₂×#106	Dic₄₇	semidirect
172. C₄₇	C₂×#1941	semidirect
200. #106	C₂×Dic₄₇	semidirect
228. C₂×C₈	C₄⋊Dic₄₇	semidirect
274. C₈	C₂×C₄⋊Dic₄₇	semidirect
293. C₂	#20244117	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×#13093956	2
C₂²×C₈×Dic₄₇	2
C₂×#15689022	2
#20244117	2
C₂×#15688885	2
C₂²×#106×C₄₇	2
C₂×#1941	47

Subgroups — C2×#20244117 #26219600

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20244117 #26219600