Subgroups — C₂×#20244131 #26219624

Order 12032.

1142 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093958	C₂	1	1	yes	maximal, characteristic
2	C₂²×#13093956	C₂	1	1	yes	maximal, characteristic
3	C₂×#15689021	C₂	1	2	yes	maximal
4	#20244131	C₂	1	8	yes	maximal, direct product factor
5	C₂×#15688886	C₂	1	2	yes	maximal
6	C₂²×#97×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
8	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
9	C₂²×Dic₄₇⋊C₄	—	2	2	no	—
10	C₂×#13093958	—	2	4	no	—
11	C₂×#13093958	C₂²	1	4	yes	—
12	C₂×#13093956	C₄	1	8	yes	complemented
13	C₂×#13093956	C₂²	1	4	yes	—
14	#15689021	—	2	4	no	—
15	#15689021	C₂²	1	8	yes	—
16	C₂×C₄₇⋊#95	—	2	2	no	—
17	#15688886	C₂²	1	8	yes	—
18	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
19	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
20	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
21	C₂×#97×C₄₇	—	2	4	no	—
22	C₂×#97×C₄₇	C₂²	1	4	yes	—
23	C₂³×Dic₄₇	—	2	2	no	—
24	C₂³×Dic₄₇	—	2	2	no	—
25	C₂×C₄⋊Dic₄₇	—	2	2	no	—
26	C₂×C₄⋊Dic₄₇	—	2	4	no	—
27	C₂×C₄⋊Dic₄₇	—	2	4	no	—
28	C₂×C₄⋊Dic₄₇	—	2	4	no	—
29	C₂×C₄⋊Dic₄₇	—	2	2	no	—
30	C₂×C₄⋊Dic₄₇	C₂×C₄	1	8	yes	—
31	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
32	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
33	C₂×Dic₄₇⋊C₄	—	2	8	no	—
34	C₂×Dic₄₇⋊C₄	—	4	8	no	—
35	#13093958	—	2	8	no	—
36	#13093956	C₂×C₄	1	16	yes	complemented
37	C₄₇⋊#95	—	2	8	no	—
38	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
39	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
40	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₄₇	—	2	2	no	—
42	C₂²×C₈×C₄₇	—	2	2	no	—
43	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
44	C₂²×C₈×C₄₇	D₄	1	4	yes	—
45	C₂²×C₈×C₄₇	Q₈	1	4	yes	—
46	C₂²×C₈×C₄₇	Q₈	1	4	yes	—
47	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
48	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
49	C₂×C₄²×C₄₇	—	2	2	no	—
50	C₂×C₄²×C₄₇	—	2	2	no	—
51	C₂×C₄²×C₄₇	D₄	1	2	yes	—
52	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
53	#97×C₄₇	—	2	8	no	—
54	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	—	2	2	no	—
56	C₂²×C₄×C₄₇	—	2	2	no	—
57	C₂²×C₄×C₄₇	—	2	4	no	—
58	C₂²×C₄×C₄₇	—	2	4	no	—
59	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
65	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
66	C₂²×C₄×C₄₇	#41	1	2	yes	—
67	C₂²×C₄×C₄₇	#41	1	2	yes	—
68	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
69	C₂²×Dic₄₇	—	2	8	no	—
70	C₂²×Dic₄₇	—	2	4	no	—
71	C₂²×Dic₄₇	—	2	4	no	—
72	C₂²×Dic₄₇	—	4	4	no	—
73	C₂²×Dic₄₇	—	4	4	no	—
74	C₂²×Dic₄₇	—	4	4	no	—
75	C₂²×Dic₄₇	—	4	4	no	—
76	C₂²×Dic₄₇	—	4	4	no	—
77	C₂×C₈×C₄₇	—	2	8	no	—
78	C₂×C₈×C₄₇	—	2	2	no	—
79	C₂×C₈×C₄₇	—	2	2	no	—
80	C₂×C₈×C₄₇	C₄⋊C₄	1	16	yes	complemented
81	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
82	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
83	C₂×C₈×C₄₇	C₂×Q₈	1	4	yes	—
84	C₂×C₈×C₄₇	C₂×Q₈	1	4	yes	—
85	C₄²×C₄₇	—	2	4	no	—
86	C₄²×C₄₇	—	2	2	no	—
87	C₄²×C₄₇	—	2	2	no	—
88	C₄⋊Dic₄₇	—	2	8	no	—
89	C₄⋊Dic₄₇	—	2	4	no	—
90	C₄⋊Dic₄₇	—	2	4	no	—
91	C₄⋊Dic₄₇	—	2	4	no	—
92	C₄⋊Dic₄₇	—	4	4	no	—
93	C₄⋊Dic₄₇	C₂²×C₄	1	8	yes	—
94	Dic₄₇⋊C₄	—	4	16	no	—
95	C₂³×C₄₇	C₄×D₄	1	2	yes	—
96	C₂³×C₄₇	#121	1	2	yes	—
97	C₂³×C₄₇	C₄⋊Q₈	1	2	yes	—
98	C₂³×C₄₇	AGL(1,ℤ/8)	1	4	yes	—
99	C₂³×C₄₇	#137	1	4	yes	—
100	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
101	C₂×C₄×C₄₇	—	2	2	no	—
102	C₂×C₄×C₄₇	—	2	2	no	—
103	C₂×C₄×C₄₇	—	2	4	no	—
104	C₂×C₄×C₄₇	—	2	4	no	—
105	C₂×C₄×C₄₇	—	4	4	no	—
106	C₂×C₄×C₄₇	—	4	4	no	—
107	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
108	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
109	C₂×C₄×C₄₇	C₄×Q₈	1	4	yes	—
110	C₂×C₄×C₄₇	#121	1	2	yes	—
111	C₂×C₄×C₄₇	C₂²⋊Q₈	1	4	yes	—
112	C₂×C₄×C₄₇	#125	1	4	yes	—
113	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
114	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
115	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
116	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
117	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
118	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
119	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
120	C₂×Dic₄₇	—	2	8	no	—
121	C₂×Dic₄₇	—	4	4	no	—
122	C₂×Dic₄₇	—	4	4	no	—
123	C₂×Dic₄₇	—	4	16	no	—
124	C₂×Dic₄₇	—	4	4	no	—
125	C₂×Dic₄₇	—	4	4	no	—
126	C₂×Dic₄₇	—	4	4	no	—
127	C₂×Dic₄₇	—	8	8	no	—
128	C₈×C₄₇	—	2	4	no	—
129	C₈×C₄₇	C₂×C₄⋊C₄	1	8	yes	complemented
130	C₂×#1944	—	47	1	no	maximal, Sylow
131	C₂²×C₄₇	#389	1	4	yes	—
132	C₂²×C₄₇	#440	1	8	yes	—
133	C₂²×C₄₇	#469	1	4	yes	—
134	C₂²×C₄₇	#470	1	4	yes	—
135	C₂²×C₄₇	#501	1	8	yes	—
136	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
137	C₂²×C₄₇	C₂×#121	1	1	yes	characteristic
138	C₂²×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
139	C₂²×C₄₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
140	C₂²×C₄₇	C₂×#137	1	2	yes	—
141	C₄×C₄₇	—	4	2	no	—
142	C₄×C₄₇	—	4	2	no	—
143	C₄×C₄₇	#389	1	4	yes	—
144	C₄×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
145	C₄×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
146	C₄×C₄₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
147	C₄×C₄₇	C₂×#125	1	1	yes	characteristic
148	Dic₄₇	—	4	8	no	—
149	Dic₄₇	—	8	4	no	—
150	C₂²×#107	—	47	1	no	—
151	C₂²×#106	—	47	1	no	—
152	C₂²×#97	D₄₇	1	1	yes	characteristic
153	C₂×#389	—	47	2	no	—
154	#1944	—	47	8	no	—
155	C₂×#340	—	47	2	no	—
156	C₂×C₄₇	#1944	1	8	yes	complemented
157	C₂×C₄₇	C₂×#389	1	1	yes	characteristic
158	C₂×C₄₇	C₂×#440	1	2	yes	—
159	C₂×C₄₇	C₂×#469	1	1	yes	characteristic
160	C₂×C₄₇	C₂×#470	1	1	yes	characteristic
161	C₂×C₄₇	C₂×#501	1	2	yes	—
162	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
163	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
164	C₂²×C₄⋊C₄	—	47	2	no	—
165	C₂²×C₄⋊C₄	—	47	2	no	—
166	C₂²×C₄⋊C₄	—	94	2	no	—
167	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
168	C₂×#107	—	47	4	no	—
169	C₂×#107	—	94	4	no	—
170	C₂×#106	—	47	8	no	—
171	C₂×#106	—	47	4	no	—
172	C₂×#97	—	2	4	no	—
173	C₂×#97	C₂×D₄₇	1	4	yes	—
174	#389	—	47	8	no	—
175	#389	—	94	4	no	—
176	C₂×#95	—	94	2	no	—
177	#340	—	47	8	no	—
178	C₄₇	C₂×#1944	1	1	yes	characteristic, complemented, Sylow
179	C₂³×C₄	—	94	2	no	—
180	C₂³×C₄	—	94	2	no	—
181	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
182	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
183	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
184	C₂²×C₈	—	2	2	no	—
185	C₂²×C₈	—	2	2	no	—
186	C₂²×C₈	Dic₉₄	1	4	yes	—
187	C₂²×C₈	Dic₉₄	1	4	yes	—
188	C₂²×C₈	C₄×D₄₇	1	4	yes	—
189	C₂²×C₈	C₂²⋊D₄₇	1	4	yes	—
190	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
191	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
192	C₂×C₄⋊C₄	—	47	8	no	—
193	C₂×C₄⋊C₄	—	47	4	no	—
194	C₂×C₄⋊C₄	—	47	4	no	—
195	C₂×C₄⋊C₄	—	94	2	no	—
196	C₂×C₄⋊C₄	—	94	4	no	—
197	C₂×C₄⋊C₄	—	94	4	no	—
198	C₂×C₄⋊C₄	—	94	4	no	—
199	C₂×C₄⋊C₄	—	94	2	no	—
200	C₂×C₄⋊C₄	—	94	8	no	—
201	C₂×C₄⋊C₄	—	188	8	no	—
202	C₂×C₄²	—	2	2	no	—
203	C₂×C₄²	—	2	2	no	—
204	C₂×C₄²	D₁₈₈	1	2	yes	—
205	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
206	#107	—	94	8	no	—
207	#106	—	47	16	no	—
208	#97	—	2	8	no	—
209	#95	—	94	8	no	—
210	C₂⁴	C₂×D₁₈₈	1	1	yes	characteristic
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	4	no	—
214	C₂²×C₄	—	2	4	no	—
215	C₂²×C₄	—	94	8	no	—
216	C₂²×C₄	—	94	4	no	—
217	C₂²×C₄	—	94	4	no	—
218	C₂²×C₄	—	188	4	no	—
219	C₂²×C₄	—	188	4	no	—
220	C₂²×C₄	—	188	4	no	—
221	C₂²×C₄	—	188	4	no	—
222	C₂²×C₄	—	188	4	no	—
223	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
224	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
225	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
226	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
227	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
228	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
229	C₂²×C₄	#10625350	1	2	yes	—
230	C₂²×C₄	#10625350	1	2	yes	—
231	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
232	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
233	C₂×C₈	—	2	8	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	Dic₄₇⋊C₄	1	16	yes	complemented
237	C₂×C₈	C₂×Dic₉₄	1	4	yes	—
238	C₂×C₈	C₂×Dic₉₄	1	4	yes	—
239	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
240	C₂×C₈	C₂×C₂²⋊D₄₇	1	4	yes	—
241	C₄⋊C₄	—	47	8	no	—
242	C₄⋊C₄	—	94	8	no	—
243	C₄⋊C₄	—	94	4	no	—
244	C₄⋊C₄	—	94	4	no	—
245	C₄⋊C₄	—	94	4	no	—
246	C₄⋊C₄	—	188	16	no	—
247	C₄⋊C₄	—	188	4	no	—
248	C₄²	—	2	4	no	—
249	C₄²	—	2	2	no	—
250	C₄²	—	2	2	no	—
251	C₂³	C₄×D₁₈₈	1	2	yes	—
252	C₂³	#13093998	1	2	yes	—
253	C₂³	#13094024	1	2	yes	—
254	C₂³	#13094052	1	4	yes	—
255	C₂³	#13094057	1	4	yes	—
256	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	4	4	no	—
262	C₂×C₄	—	4	4	no	—
263	C₂×C₄	—	94	8	no	—
264	C₂×C₄	—	376	8	no	—
265	C₂×C₄	—	188	4	no	—
266	C₂×C₄	—	188	4	no	—
267	C₂×C₄	—	188	4	no	—
268	C₂×C₄	—	188	16	no	—
269	C₂×C₄	—	188	4	no	—
270	C₂×C₄	—	188	4	no	—
271	C₂×C₄	C₂×Dic₄₇⋊C₄	1	4	yes	—
272	C₂×C₄	C₄×D₁₈₈	1	2	yes	—
273	C₂×C₄	C₄×Dic₉₄	1	4	yes	—
274	C₂×C₄	#13093998	1	2	yes	—
275	C₂×C₄	#13094003	1	4	yes	—
276	C₂×C₄	#13094016	1	4	yes	—
277	C₂×C₄	#13094024	1	2	yes	—
278	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
279	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
280	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
281	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
282	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
283	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
284	C₈	—	2	4	no	—
285	C₈	C₂×Dic₄₇⋊C₄	1	8	yes	complemented
286	C₂²	#15689021	1	4	yes	—
287	C₂²	#15689225	1	8	yes	—
288	C₂²	#15689376	1	4	yes	—
289	C₂²	#15689381	1	4	yes	—
290	C₂²	#15689532	1	8	yes	—
291	C₂²	C₂×C₄×D₁₈₈	1	1	yes	characteristic
292	C₂²	C₂×#13093998	1	1	yes	characteristic
293	C₂²	C₂×#13094024	1	1	yes	characteristic
294	C₂²	C₂×#13094052	1	2	yes	—
295	C₂²	C₂×#13094057	1	2	yes	—
296	C₄	—	4	2	no	—
297	C₄	—	4	2	no	—
298	C₄	—	376	4	no	—
299	C₄	—	188	8	no	—
300	C₄	#15689021	1	4	yes	—
301	C₄	C₂²×Dic₄₇⋊C₄	1	1	yes	characteristic
302	C₄	C₂×C₄×Dic₉₄	1	1	yes	characteristic
303	C₄	C₂×#13094003	1	1	yes	characteristic
304	C₄	C₂×#13094016	1	1	yes	characteristic
305	C₂	#20244131	1	8	yes	direct product factor
306	C₂	C₂×#15689021	1	1	yes	characteristic
307	C₂	C₂×#15689225	1	2	yes	—
308	C₂	C₂×#15689376	1	1	yes	characteristic
309	C₂	C₂×#15689381	1	1	yes	characteristic
310	C₂	C₂×#15689532	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂²×#97×C₄₇	C₂
socle, prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂²×#97	D₄₇
hypocenter	C₄₇	C₂×#1944
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×#97

Complements

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

N	H ≅ G/N	type
4. #20244131	C₂	direct (N × H)
12. C₂×#13093956	C₄	semidirect
36. #13093956	C₂×C₄	semidirect
80. C₂×C₈×C₄₇	C₄⋊C₄	semidirect
129. C₈×C₄₇	C₂×C₄⋊C₄	semidirect
156. C₂×C₄₇	#1944	semidirect
178. C₄₇	C₂×#1944	semidirect
236. C₂×C₈	Dic₄₇⋊C₄	semidirect
285. C₈	C₂×Dic₄₇⋊C₄	semidirect
305. C₂	#20244131	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×#13093958	2
C₂²×#13093956	2
C₂×#15689021	2
#20244131	2
C₂×#15688886	2
C₂²×#97×C₄₇	2
C₂×#1944	47

Subgroups — C2×#20244131 #26219624

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20244131 #26219624