Subgroups — C₂×#20244121 #26219605

Order 12032.

1182 conjugacy classes of subgroups.

The subgroup lattice has 336 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×#13093958	C₂	1	1	yes	maximal, characteristic
3	C₂×#15689021	C₂	1	2	yes	maximal
4	#20244121	C₂	1	8	yes	maximal, direct product factor
5	C₂×#15688886	C₂	1	2	yes	maximal
6	C₂²×C₄×C₈×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	C₄	1	8	yes	complemented
12	C₂×#13093958	C₂²	1	4	yes	—
13	C₂×#13093958	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×C₄×C₈×C₄₇	—	2	4	no	—
22	C₂×C₄×C₈×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	#13093958	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	C₄×C₈×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	2	no	—
58	C₂²×C₄×C₄₇	—	2	2	no	—
59	C₂²×C₄×C₄₇	D₈	1	4	yes	—
60	C₂²×C₄×C₄₇	Q₁₆	1	4	yes	—
61	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
62	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
67	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
68	C₂²×C₄×C₄₇	#41	1	2	yes	—
69	C₂²×C₄×C₄₇	#41	1	2	yes	—
70	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
71	C₂²×Dic₄₇	—	2	8	no	—
72	C₂²×Dic₄₇	—	2	4	no	—
73	C₂²×Dic₄₇	—	2	4	no	—
74	C₂²×Dic₄₇	—	4	4	no	—
75	C₂²×Dic₄₇	—	4	4	no	—
76	C₂²×Dic₄₇	—	4	4	no	—
77	C₂²×Dic₄₇	—	4	4	no	—
78	C₂²×Dic₄₇	—	4	4	no	—
79	C₂×C₈×C₄₇	—	2	8	no	—
80	C₂×C₈×C₄₇	—	2	2	no	—
81	C₂×C₈×C₄₇	—	2	2	no	—
82	C₂×C₈×C₄₇	C₄⋊C₄	1	16	yes	complemented
83	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
84	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
85	C₂×C₈×C₄₇	C₂×Q₈	1	4	yes	—
86	C₂×C₈×C₄₇	C₂×Q₈	1	4	yes	—
87	C₄²×C₄₇	—	2	4	no	—
88	C₄²×C₄₇	—	2	2	no	—
89	C₄²×C₄₇	—	2	2	no	—
90	C₄⋊Dic₄₇	—	2	8	no	—
91	C₄⋊Dic₄₇	—	2	4	no	—
92	C₄⋊Dic₄₇	—	2	4	no	—
93	C₄⋊Dic₄₇	—	2	4	no	—
94	C₄⋊Dic₄₇	—	4	4	no	—
95	C₄⋊Dic₄₇	C₂²×C₄	1	8	yes	—
96	Dic₄₇⋊C₄	—	4	16	no	—
97	C₂³×C₄₇	C₄×D₄	1	2	yes	—
98	C₂³×C₄₇	#121	1	2	yes	—
99	C₂³×C₄₇	C₄⋊Q₈	1	2	yes	—
100	C₂³×C₄₇	C₂×D₈	1	2	yes	—
101	C₂³×C₄₇	C₂×Q₁₆	1	2	yes	—
102	C₂³×C₄₇	#135	1	2	yes	—
103	C₂³×C₄₇	#135	1	2	yes	—
104	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
105	C₂×C₄×C₄₇	—	2	2	no	—
106	C₂×C₄×C₄₇	—	2	2	no	—
107	C₂×C₄×C₄₇	—	2	2	no	—
108	C₂×C₄×C₄₇	—	2	2	no	—
109	C₂×C₄×C₄₇	—	2	8	no	—
110	C₂×C₄×C₄₇	—	2	8	no	—
111	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
112	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₄₇	C₄×Q₈	1	4	yes	—
114	C₂×C₄×C₄₇	#121	1	2	yes	—
115	C₂×C₄×C₄₇	C₂²⋊Q₈	1	4	yes	—
116	C₂×C₄×C₄₇	#125	1	4	yes	—
117	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
118	C₂×C₄×C₄₇	C₂×D₈	1	4	yes	—
119	C₂×C₄×C₄₇	C₂×Q₁₆	1	4	yes	—
120	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
121	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
123	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
124	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
125	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
126	C₂×Dic₄₇	—	2	8	no	—
127	C₂×Dic₄₇	—	4	4	no	—
128	C₂×Dic₄₇	—	4	4	no	—
129	C₂×Dic₄₇	—	4	16	no	—
130	C₂×Dic₄₇	—	4	4	no	—
131	C₂×Dic₄₇	—	4	4	no	—
132	C₂×Dic₄₇	—	4	4	no	—
133	C₂×Dic₄₇	—	8	8	no	—
134	C₈×C₄₇	—	2	4	no	—
135	C₈×C₄₇	C₂×C₄⋊C₄	1	8	yes	complemented
136	C₂×#1942	—	47	1	no	maximal, Sylow
137	C₂²×C₄₇	#389	1	4	yes	—
138	C₂²×C₄₇	C₄×D₈	1	4	yes	—
139	C₂²×C₄₇	C₄×Q₁₆	1	4	yes	—
140	C₂²×C₄₇	#466	1	4	yes	—
141	C₂²×C₄₇	#467	1	4	yes	—
142	C₂²×C₄₇	#499	1	4	yes	—
143	C₂²×C₄₇	#500	1	4	yes	—
144	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
145	C₂²×C₄₇	C₂×#121	1	1	yes	characteristic
146	C₂²×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
147	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
148	C₂²×C₄₇	C₂²×Q₁₆	1	1	yes	characteristic
149	C₂²×C₄₇	C₂×#135	1	1	yes	characteristic
150	C₂²×C₄₇	C₂×#135	1	1	yes	characteristic
151	C₄×C₄₇	—	2	4	no	—
152	C₄×C₄₇	—	2	4	no	—
153	C₄×C₄₇	#389	1	4	yes	—
154	C₄×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
155	C₄×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
156	C₄×C₄₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
157	C₄×C₄₇	C₂×#125	1	1	yes	characteristic
158	Dic₄₇	—	4	8	no	—
159	Dic₄₇	—	8	4	no	—
160	C₂²×#107	—	47	1	no	—
161	C₂²×#107	—	47	1	no	—
162	C₂²×C₄×C₈	D₄₇	1	1	yes	characteristic
163	C₂×#389	—	47	2	no	—
164	#1942	—	47	8	no	—
165	C₂×#340	—	47	2	no	—
166	C₂×C₄₇	#1942	1	8	yes	complemented
167	C₂×C₄₇	C₂×#389	1	1	yes	characteristic
168	C₂×C₄₇	C₂×C₄×D₈	1	1	yes	characteristic
169	C₂×C₄₇	C₂×C₄×Q₁₆	1	1	yes	characteristic
170	C₂×C₄₇	C₂×#466	1	1	yes	characteristic
171	C₂×C₄₇	C₂×#467	1	1	yes	characteristic
172	C₂×C₄₇	C₂×#499	1	1	yes	characteristic
173	C₂×C₄₇	C₂×#500	1	1	yes	characteristic
174	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
175	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
176	C₂²×C₄⋊C₄	—	47	2	no	—
177	C₂²×C₄⋊C₄	—	47	2	no	—
178	C₂²×C₄⋊C₄	—	94	2	no	—
179	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
180	C₂×#107	—	47	8	no	—
181	C₂×#107	—	47	4	no	—
182	C₂×#107	—	47	4	no	—
183	C₂×#107	—	94	4	no	—
184	C₂×C₄×C₈	—	2	4	no	—
185	C₂×C₄×C₈	C₂×D₄₇	1	4	yes	—
186	#389	—	47	8	no	—
187	#389	—	94	4	no	—
188	C₂×#95	—	94	2	no	—
189	#340	—	47	8	no	—
190	C₄₇	C₂×#1942	1	1	yes	characteristic, complemented, Sylow
191	C₂³×C₄	—	94	2	no	—
192	C₂³×C₄	—	94	2	no	—
193	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
194	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
195	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
196	C₂²×C₈	—	2	2	no	—
197	C₂²×C₈	—	2	2	no	—
198	C₂²×C₈	Dic₉₄	1	4	yes	—
199	C₂²×C₈	Dic₉₄	1	4	yes	—
200	C₂²×C₈	C₄×D₄₇	1	4	yes	—
201	C₂²×C₈	C₂²⋊D₄₇	1	4	yes	—
202	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
203	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
204	C₂×C₄⋊C₄	—	47	8	no	—
205	C₂×C₄⋊C₄	—	47	4	no	—
206	C₂×C₄⋊C₄	—	47	4	no	—
207	C₂×C₄⋊C₄	—	94	2	no	—
208	C₂×C₄⋊C₄	—	94	4	no	—
209	C₂×C₄⋊C₄	—	94	4	no	—
210	C₂×C₄⋊C₄	—	94	4	no	—
211	C₂×C₄⋊C₄	—	94	2	no	—
212	C₂×C₄⋊C₄	—	94	8	no	—
213	C₂×C₄⋊C₄	—	188	8	no	—
214	C₂×C₄²	—	2	2	no	—
215	C₂×C₄²	—	2	2	no	—
216	C₂×C₄²	D₁₈₈	1	2	yes	—
217	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
218	#107	—	47	16	no	—
219	#107	—	94	8	no	—
220	C₄×C₈	—	2	8	no	—
221	#95	—	94	8	no	—
222	C₂⁴	C₂×D₁₈₈	1	1	yes	characteristic
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	94	8	no	—
228	C₂²×C₄	—	94	4	no	—
229	C₂²×C₄	—	94	4	no	—
230	C₂²×C₄	—	188	4	no	—
231	C₂²×C₄	—	188	4	no	—
232	C₂²×C₄	—	188	4	no	—
233	C₂²×C₄	—	188	4	no	—
234	C₂²×C₄	—	188	4	no	—
235	C₂²×C₄	D₃₇₆	1	4	yes	—
236	C₂²×C₄	Dic₁₈₈	1	4	yes	—
237	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
238	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
239	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
240	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
241	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
242	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
243	C₂²×C₄	#10625350	1	2	yes	—
244	C₂²×C₄	#10625350	1	2	yes	—
245	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
246	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
247	C₂×C₈	—	2	8	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	Dic₄₇⋊C₄	1	16	yes	complemented
251	C₂×C₈	C₂×Dic₉₄	1	4	yes	—
252	C₂×C₈	C₂×Dic₉₄	1	4	yes	—
253	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
254	C₂×C₈	C₂×C₂²⋊D₄₇	1	4	yes	—
255	C₄⋊C₄	—	47	8	no	—
256	C₄⋊C₄	—	94	8	no	—
257	C₄⋊C₄	—	94	4	no	—
258	C₄⋊C₄	—	94	4	no	—
259	C₄⋊C₄	—	94	4	no	—
260	C₄⋊C₄	—	188	4	no	—
261	C₄⋊C₄	—	188	16	no	—
262	C₄²	—	2	4	no	—
263	C₄²	—	2	2	no	—
264	C₄²	—	2	2	no	—
265	C₂³	C₄×D₁₈₈	1	2	yes	—
266	C₂³	#13093998	1	2	yes	—
267	C₂³	#13094024	1	2	yes	—
268	C₂³	C₂×D₃₇₆	1	2	yes	—
269	C₂³	C₂×Dic₁₈₈	1	2	yes	—
270	C₂³	#13094046	1	2	yes	—
271	C₂³	#13094046	1	2	yes	—
272	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
273	C₂×C₄	—	2	8	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	8	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	94	8	no	—
280	C₂×C₄	—	376	8	no	—
281	C₂×C₄	—	188	4	no	—
282	C₂×C₄	—	188	4	no	—
283	C₂×C₄	—	188	4	no	—
284	C₂×C₄	—	188	4	no	—
285	C₂×C₄	—	188	4	no	—
286	C₂×C₄	—	188	16	no	—
287	C₂×C₄	C₂×Dic₄₇⋊C₄	1	4	yes	—
288	C₂×C₄	C₄×D₁₈₈	1	2	yes	—
289	C₂×C₄	C₄×Dic₉₄	1	4	yes	—
290	C₂×C₄	#13093998	1	2	yes	—
291	C₂×C₄	#13094003	1	4	yes	—
292	C₂×C₄	#13094016	1	4	yes	—
293	C₂×C₄	#13094024	1	2	yes	—
294	C₂×C₄	C₂×D₃₇₆	1	4	yes	—
295	C₂×C₄	C₂×Dic₁₈₈	1	4	yes	—
296	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
297	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
298	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
299	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
300	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
301	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
302	C₈	—	2	4	no	—
303	C₈	C₂×Dic₄₇⋊C₄	1	8	yes	complemented
304	C₂²	#15689021	1	4	yes	—
305	C₂²	C₄×D₃₇₆	1	4	yes	—
306	C₂²	C₄×Dic₁₈₈	1	4	yes	—
307	C₂²	#15689359	1	4	yes	—
308	C₂²	#15689364	1	4	yes	—
309	C₂²	#15689525	1	4	yes	—
310	C₂²	#15689528	1	4	yes	—
311	C₂²	C₂×C₄×D₁₈₈	1	1	yes	characteristic
312	C₂²	C₂×#13093998	1	1	yes	characteristic
313	C₂²	C₂×#13094024	1	1	yes	characteristic
314	C₂²	C₂²×D₃₇₆	1	1	yes	characteristic
315	C₂²	C₂²×Dic₁₈₈	1	1	yes	characteristic
316	C₂²	C₂×#13094046	1	1	yes	characteristic
317	C₂²	C₂×#13094046	1	1	yes	characteristic
318	C₄	—	2	4	no	—
319	C₄	—	2	4	no	—
320	C₄	—	376	4	no	—
321	C₄	—	188	8	no	—
322	C₄	#15689021	1	4	yes	—
323	C₄	C₂²×Dic₄₇⋊C₄	1	1	yes	characteristic
324	C₄	C₂×C₄×Dic₉₄	1	1	yes	characteristic
325	C₄	C₂×#13094003	1	1	yes	characteristic
326	C₄	C₂×#13094016	1	1	yes	characteristic
327	C₂	#20244121	1	8	yes	direct product factor
328	C₂	C₂×#15689021	1	1	yes	characteristic
329	C₂	C₂×C₄×D₃₇₆	1	1	yes	characteristic
330	C₂	C₂×C₄×Dic₁₈₈	1	1	yes	characteristic
331	C₂	C₂×#15689359	1	1	yes	characteristic
332	C₂	C₂×#15689364	1	1	yes	characteristic
333	C₂	C₂×#15689525	1	1	yes	characteristic
334	C₂	C₂×#15689528	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂²×C₄×C₈×C₄₇	C₂
socle, prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂²×C₄×C₈	D₄₇
hypocenter	C₄₇	C₂×#1942
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
4. #20244121	C₂	direct (N × H)
11. C₂×#13093958	C₄	semidirect
36. #13093958	C₂×C₄	semidirect
82. C₂×C₈×C₄₇	C₄⋊C₄	semidirect
135. C₈×C₄₇	C₂×C₄⋊C₄	semidirect
166. C₂×C₄₇	#1942	semidirect
190. C₄₇	C₂×#1942	semidirect
250. C₂×C₈	Dic₄₇⋊C₄	semidirect
303. C₈	C₂×Dic₄₇⋊C₄	semidirect
327. C₂	#20244121	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
C₂×#15689021	2
#20244121	2
C₂×#15688886	2
C₂²×C₄×C₈×C₄₇	2
C₂×#1942	47

Subgroups — C2×#20244121 #26219605

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20244121 #26219605