Subgroups — C₂×#20253764 #27085547

Order 12032.

2806 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13094021	C₂	1	1	yes	maximal, characteristic
2	#20253764	C₂	1	16	yes	maximal, direct product factor
3	C₂×#15689489	C₂	1	4	yes	maximal, complemented
4	C₂×#15689471	C₂	1	4	yes	maximal, complemented
5	C₂×#15689127	C₂	1	4	yes	maximal, complemented
6	C₂×C₄×#10625329	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄₇×#533	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×D₁₈₈	—	2	2	no	—
9	C₂³×D₁₈₈	C₂²	1	2	yes	—
10	C₂²×#10625329	C₂²	1	2	yes	—
11	C₂×#13094021	—	2	2	no	—
12	C₂×#13094021	—	2	4	no	—
13	C₂×#13094021	C₂²	1	2	yes	—
14	C₂×#13094021	C₂²	1	8	yes	—
15	C₂×#13094021	C₂²	1	2	yes	—
16	#15689489	C₂²	1	32	yes	complemented
17	#15689471	C₂²	1	32	yes	complemented
18	#15689127	C₂²	1	32	yes	complemented
19	C₂×#13093944	C₂²	1	16	yes	complemented
20	C₄×#10625329	C₂²	1	8	yes	complemented
21	C₂×#13093935	C₂²	1	2	yes	complemented
22	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	complemented
23	C₄₇×#533	—	2	4	no	—
24	C₄₇×#533	C₂²	1	8	yes	complemented
25	C₂×C₄⋊Q₈×C₄₇	C₂²	1	2	yes	complemented
26	C₂×#125×C₄₇	C₂²	1	2	yes	complemented
27	C₂×C₂²⋊Q₈×C₄₇	—	2	2	no	—
28	C₂×C₄×Q₈×C₄₇	—	2	2	no	—
29	C₂×#117×C₄₇	C₂²	1	2	yes	—
30	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
31	C₂⁴×D₄₇	—	2	2	no	—
32	C₂²×D₁₈₈	—	2	16	no	—
33	C₂²×D₁₈₈	—	2	2	no	—
34	C₂²×D₁₈₈	—	2	4	no	—
35	C₂²×D₁₈₈	—	2	4	no	—
36	C₂²×D₁₈₈	—	2	16	no	—
37	C₂²×D₁₈₈	—	2	4	no	—
38	C₂²×D₁₈₈	—	4	16	no	—
39	C₂²×D₁₈₈	C₂³	1	4	yes	—
40	C₂²×D₁₈₈	C₂³	1	8	yes	—
41	C₂×#10625329	—	2	8	no	—
42	C₂×#10625329	C₂³	1	8	yes	—
43	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
44	#13094021	—	2	16	no	—
45	#13094021	—	2	4	no	—
46	#13094021	—	2	4	no	—
47	#13094021	C₂³	1	8	yes	—
48	#13094021	C₂³	1	8	yes	—
49	#13093944	C₂³	1	64	yes	complemented
50	#13093935	C₂³	1	8	yes	complemented
51	C₄×C₄₇⋊C₈	C₂³	1	8	yes	complemented
52	C₂²×Q₈×C₄₇	—	2	2	no	—
53	C₂³×C₄×C₄₇	D₄	1	2	yes	—
54	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
55	C₄⋊Q₈×C₄₇	—	2	4	no	—
56	C₄⋊Q₈×C₄₇	C₂³	1	8	yes	complemented
57	#125×C₄₇	—	2	4	no	—
58	#125×C₄₇	C₂³	1	8	yes	complemented
59	C₂²⋊Q₈×C₄₇	—	2	16	no	—
60	C₄×Q₈×C₄₇	—	2	16	no	—
61	#117×C₄₇	—	2	4	no	—
62	#117×C₄₇	C₂³	1	8	yes	—
63	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
64	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
65	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
66	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
67	C₂×C₄²×C₄₇	—	2	2	no	—
68	C₂×C₄²×C₄₇	D₄	1	2	yes	—
69	C₂×C₄²×C₄₇	D₄	1	4	yes	—
70	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
71	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
72	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
73	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
74	C₂³×D₄₇	—	2	8	no	—
75	C₂³×D₄₇	—	2	4	no	—
76	C₂³×D₄₇	—	4	16	no	—
77	C₂³×D₄₇	—	4	8	no	—
78	C₂×Q₈×C₄₇	—	2	8	no	—
79	C₂×Q₈×C₄₇	—	4	8	no	—
80	C₂²×C₄×C₄₇	—	2	8	no	—
81	C₂²×C₄×C₄₇	—	2	4	no	—
82	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₄₇	C₂×D₄	1	8	yes	—
84	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
87	C₂²×C₄×C₄₇	#41	1	8	yes	—
88	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
89	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
90	C₂×D₁₈₈	—	2	16	no	—
91	C₂×D₁₈₈	—	2	32	no	—
92	C₂×D₁₈₈	—	2	8	no	—
93	C₂×D₁₈₈	—	2	4	no	—
94	C₂×D₁₈₈	—	2	16	no	—
95	C₂×D₁₈₈	—	2	16	no	—
96	C₂×D₁₈₈	—	4	16	no	—
97	C₂×D₁₈₈	—	4	64	no	—
98	C₂×D₁₈₈	—	4	16	no	—
99	C₂×D₁₈₈	—	4	8	no	—
100	C₂×D₁₈₈	—	4	16	no	—
101	C₂×D₁₈₈	C₂⁴	1	8	yes	—
102	C₄⋊C₄×C₄₇	—	2	16	no	—
103	C₄⋊C₄×C₄₇	—	2	8	no	—
104	C₄⋊C₄×C₄₇	—	2	4	no	—
105	C₄⋊C₄×C₄₇	C₂⁴	1	8	yes	—
106	C₂²⋊C₄×C₄₇	—	2	8	no	—
107	C₄²×C₄₇	—	2	8	no	—
108	C₄²×C₄₇	C₂×D₄	1	8	yes	—
109	C₄²×C₄₇	C₂×D₄	1	2	yes	—
110	C₄²×C₄₇	C₂×D₄	1	2	yes	—
111	C₄²×C₄₇	C₂⁴	1	2	yes	—
112	C₄²×C₄₇	C₂⁴	1	2	yes	—
113	#10625329	—	2	16	no	—
114	C₂×C₄₇⋊C₈	—	2	8	no	—
115	C₂×C₄₇⋊C₈	C₂⁴	1	8	yes	—
116	C₂³×C₄₇	—	2	4	no	—
117	C₂³×C₄₇	#124	1	4	yes	—
118	C₂³×C₄₇	C₂²×D₄	1	2	yes	—
119	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
120	C₂²×D₄₇	—	2	8	no	—
121	C₂²×D₄₇	—	4	32	no	—
122	C₂²×D₄₇	—	4	16	no	—
123	C₂²×D₄₇	—	4	8	no	—
124	C₂²×D₄₇	—	4	16	no	—
125	C₂²×D₄₇	—	8	32	no	—
126	Q₈×C₄₇	—	4	16	no	—
127	C₂×C₄×C₄₇	—	2	4	no	—
128	C₂×C₄×C₄₇	—	2	16	no	—
129	C₂×C₄×C₄₇	—	2	8	no	—
130	C₂×C₄×C₄₇	—	4	8	no	—
131	C₂×C₄×C₄₇	#124	1	8	yes	—
132	C₂×C₄×C₄₇	#124	1	4	yes	—
133	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	8	yes	—
134	C₂×C₄×C₄₇	C₂²×D₄	1	4	yes	—
135	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
136	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₄₇	C₂²×D₄	1	4	yes	—
138	C₂×C₄×C₄₇	C₂×#41	1	4	yes	—
139	C₂×C₄×C₄₇	C₂×#41	1	8	yes	—
140	C₂×C₄×C₄₇	C₂⁵	1	1	yes	characteristic
141	D₁₈₈	—	2	16	no	—
142	D₁₈₈	—	4	32	no	—
143	D₁₈₈	—	4	16	no	—
144	D₁₈₈	—	4	8	no	—
145	D₁₈₈	—	4	32	no	—
146	C₄₇⋊C₈	—	2	8	no	—
147	C₂×#3138	—	47	1	no	maximal, Sylow
148	C₂²×C₄₇	—	2	8	no	—
149	C₂²×C₄₇	—	2	4	no	—
150	C₂²×C₄₇	C₂×#124	1	4	yes	—
151	C₂²×C₄₇	C₂×#124	1	2	yes	—
152	C₂²×C₄₇	C₂×AGL(1,ℤ/8)	1	4	yes	—
153	C₂²×C₄₇	C₂³×D₄	1	1	yes	characteristic
154	C₂×D₄₇	—	4	8	no	—
155	C₂×D₄₇	—	4	16	no	—
156	C₂×D₄₇	—	8	32	no	—
157	C₂×D₄₇	—	8	16	no	—
158	C₄×C₄₇	—	2	4	no	—
159	C₄×C₄₇	—	4	8	no	—
160	C₄×C₄₇	C₂×#124	1	4	yes	—
161	C₄×C₄₇	C₂×#124	1	2	yes	—
162	C₄×C₄₇	C₂×AGL(1,ℤ/8)	1	8	yes	—
163	C₄×C₄₇	C₂²×#41	1	2	yes	—
164	C₂×#533	D₄₇	1	1	yes	characteristic, complemented
165	C₂²×#127	—	47	1	no	—
166	#3138	—	47	16	no	—
167	C₂×#490	—	47	4	no	—
168	C₂×#486	—	47	4	no	—
169	C₂×#418	—	47	4	no	—
170	C₂×C₄×M₁₆	—	47	1	no	—
171	C₂×C₄₇	—	2	4	no	—
172	C₂×C₄₇	#3138	1	4	yes	complemented
173	C₂×C₄₇	C₂²×#124	1	1	yes	characteristic
174	C₂×C₄₇	C₂²×AGL(1,ℤ/8)	1	2	yes	—
175	D₄₇	—	8	8	no	—
176	C₂³×D₄	—	47	2	no	—
177	C₂³×D₄	—	94	2	no	—
178	C₂²×M₁₆	—	47	2	no	—
179	#533	—	2	4	no	—
180	#533	C₂×D₄₇	1	8	yes	complemented
181	C₂×C₄⋊Q₈	C₂×D₄₇	1	2	yes	complemented
182	C₂×#127	—	47	2	no	—
183	C₂×#127	—	47	2	no	—
184	C₂×#127	—	47	8	no	—
185	C₂×#127	—	94	2	no	—
186	C₂×#127	—	94	4	no	—
187	C₂×#125	C₂×D₄₇	1	2	yes	complemented
188	C₂×C₂²⋊Q₈	—	2	2	no	—
189	C₂×C₄×Q₈	—	2	2	no	—
190	C₂×#117	C₂×D₄₇	1	2	yes	—
191	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
192	#490	—	47	32	no	—
193	#486	—	47	32	no	—
194	#418	—	47	32	no	—
195	C₂×#102	—	47	16	no	—
196	C₄×M₁₆	—	47	8	no	—
197	C₂×#97	—	47	2	no	—
198	C₂×C₄×C₈	—	47	2	no	—
199	C₄₇	C₂×#3138	1	1	yes	characteristic, complemented, Sylow
200	C₂⁵	—	94	2	no	—
201	C₂²×Q₈	—	2	2	no	—
202	C₂²×D₄	—	47	4	no	—
203	C₂²×D₄	—	47	8	no	—
204	C₂²×D₄	—	94	16	no	—
205	C₂²×D₄	—	94	16	no	—
206	C₂²×D₄	—	94	4	no	—
207	C₂²×D₄	—	94	4	no	—
208	C₂²×D₄	—	94	4	no	—
209	C₂²×D₄	—	94	2	no	—
210	C₂²×D₄	—	188	16	no	—
211	C₂³×C₄	C₂²⋊D₄₇	1	2	yes	—
212	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
213	C₂×M₁₆	—	47	8	no	—
214	C₂×M₁₆	—	94	8	no	—
215	C₂²×C₈	—	47	4	no	—
216	C₄⋊Q₈	—	2	4	no	—
217	C₄⋊Q₈	C₂²×D₄₇	1	8	yes	complemented
218	#127	—	47	8	no	—
219	#127	—	47	8	no	—
220	#127	—	94	16	no	—
221	#127	—	94	4	no	—
222	#127	—	94	4	no	—
223	#125	—	2	4	no	—
224	#125	C₂²×D₄₇	1	8	yes	complemented
225	C₂²⋊Q₈	—	2	16	no	—
226	C₄×Q₈	—	2	16	no	—
227	#117	—	2	4	no	—
228	#117	C₂²×D₄₇	1	8	yes	—
229	C₂×C₄⋊C₄	—	2	4	no	—
230	C₂×C₄⋊C₄	—	2	2	no	—
231	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
232	C₂×C₂²⋊C₄	—	2	2	no	—
233	C₂×C₄²	—	2	2	no	—
234	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
235	C₂×C₄²	C₂²⋊D₄₇	1	4	yes	—
236	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
237	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
238	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
239	#102	—	47	64	no	—
240	#97	—	47	8	no	—
241	C₄×C₈	—	47	8	no	—
242	C₂⁴	—	94	8	no	—
243	C₂⁴	—	94	4	no	—
244	C₂⁴	—	188	16	no	—
245	C₂⁴	—	188	8	no	—
246	C₂⁴	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
247	C₂×Q₈	—	2	8	no	—
248	C₂×Q₈	—	4	8	no	—
249	C₂×D₄	—	47	8	no	—
250	C₂×D₄	—	94	16	no	—
251	C₂×D₄	—	94	16	no	—
252	C₂×D₄	—	94	8	no	—
253	C₂×D₄	—	94	4	no	—
254	C₂×D₄	—	94	16	no	—
255	C₂×D₄	—	94	32	no	—
256	C₂×D₄	—	188	16	no	—
257	C₂×D₄	—	188	64	no	—
258	C₂×D₄	—	188	16	no	—
259	C₂×D₄	—	188	8	no	—
260	C₂×D₄	—	188	16	no	—
261	C₂²×C₄	—	2	8	no	—
262	C₂²×C₄	—	2	4	no	—
263	C₂²×C₄	#10625354	1	8	yes	—
264	C₂²×C₄	C₂×C₂²⋊D₄₇	1	8	yes	—
265	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
266	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
267	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
268	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
269	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
270	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
271	M₁₆	—	94	16	no	—
272	C₂×C₈	—	47	8	no	—
273	C₂×C₈	—	94	8	no	—
274	C₄⋊C₄	—	2	16	no	—
275	C₄⋊C₄	—	2	8	no	—
276	C₄⋊C₄	—	2	4	no	—
277	C₄⋊C₄	C₂³×D₄₇	1	8	yes	—
278	C₂²⋊C₄	—	2	8	no	—
279	C₄²	—	2	8	no	—
280	C₄²	C₂×C₂²⋊D₄₇	1	8	yes	—
281	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
282	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
283	C₄²	C₂³×D₄₇	1	2	yes	—
284	C₄²	C₂³×D₄₇	1	2	yes	—
285	C₂³	—	2	4	no	—
286	C₂³	—	94	8	no	—
287	C₂³	—	376	32	no	—
288	C₂³	—	188	32	no	—
289	C₂³	—	188	16	no	—
290	C₂³	—	188	8	no	—
291	C₂³	—	188	16	no	—
292	C₂³	#13094013	1	4	yes	—
293	C₂³	C₂²×C₂²⋊D₄₇	1	2	yes	—
294	C₂³	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
295	Q₈	—	4	16	no	—
296	D₄	—	94	16	no	—
297	D₄	—	188	32	no	—
298	D₄	—	188	16	no	—
299	D₄	—	188	8	no	—
300	D₄	—	188	32	no	—
301	C₂×C₄	—	2	4	no	—
302	C₂×C₄	—	2	16	no	—
303	C₂×C₄	—	2	8	no	—
304	C₂×C₄	—	4	8	no	—
305	C₂×C₄	#13094013	1	8	yes	—
306	C₂×C₄	#13094013	1	4	yes	—
307	C₂×C₄	#13094049	1	8	yes	—
308	C₂×C₄	C₂²×C₂²⋊D₄₇	1	4	yes	—
309	C₂×C₄	C₂²×C₂²⋊D₄₇	1	2	yes	—
310	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
311	C₂×C₄	C₂²×C₂²⋊D₄₇	1	4	yes	—
312	C₂×C₄	C₂×#10625354	1	4	yes	—
313	C₂×C₄	C₂×#10625354	1	8	yes	—
314	C₂×C₄	C₂⁴×D₄₇	1	1	yes	characteristic
315	C₈	—	94	8	no	—
316	C₂²	—	2	8	no	—
317	C₂²	—	2	4	no	—
318	C₂²	—	376	32	no	—
319	C₂²	—	376	16	no	—
320	C₂²	—	188	16	no	—
321	C₂²	—	188	8	no	—
322	C₂²	C₂×#13094013	1	4	yes	—
323	C₂²	C₂×#13094013	1	2	yes	—
324	C₂²	C₂×#13094049	1	4	yes	—
325	C₂²	C₂³×C₂²⋊D₄₇	1	1	yes	characteristic
326	C₄	—	2	4	no	—
327	C₄	—	4	8	no	—
328	C₄	C₂×#13094013	1	4	yes	—
329	C₄	C₂×#13094013	1	2	yes	—
330	C₄	C₂×#13094049	1	8	yes	—
331	C₄	C₂²×#10625354	1	2	yes	—
332	C₂	—	2	4	no	—
333	C₂	—	376	8	no	—
334	C₂	#20253764	1	4	yes	direct product factor
335	C₂	C₂²×#13094013	1	1	yes	characteristic
336	C₂	C₂²×#13094049	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂×C₄₇×#533	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂²×#13094021	C₂
hypercenter	C₂×#533	D₄₇
hypocenter	C₄₇	C₂×#3138
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×#533

Complements

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

N	H ≅ G/N	type
2. #20253764	C₂	direct (N × H)
3. C₂×#15689489	C₂	semidirect
4. C₂×#15689471	C₂	semidirect
5. C₂×#15689127	C₂	semidirect
6. C₂×C₄×#10625329	C₂	semidirect
7. C₂×C₄₇×#533	C₂	semidirect
16. #15689489	C₂²	semidirect
17. #15689471	C₂²	semidirect
18. #15689127	C₂²	semidirect
19. C₂×#13093944	C₂²	semidirect
20. C₄×#10625329	C₂²	semidirect
21. C₂×#13093935	C₂²	semidirect
22. C₂×C₄×C₄₇⋊C₈	C₂²	semidirect
24. C₄₇×#533	C₂²	semidirect
25. C₂×C₄⋊Q₈×C₄₇	C₂²	semidirect
26. C₂×#125×C₄₇	C₂²	semidirect
49. #13093944	C₂³	semidirect
50. #13093935	C₂³	semidirect
51. C₄×C₄₇⋊C₈	C₂³	semidirect
56. C₄⋊Q₈×C₄₇	C₂³	semidirect
58. #125×C₄₇	C₂³	semidirect
164. C₂×#533	D₄₇	semidirect
172. C₂×C₄₇	#3138	semidirect
180. #533	C₂×D₄₇	semidirect
181. C₂×C₄⋊Q₈	C₂×D₄₇	semidirect
187. C₂×#125	C₂×D₄₇	semidirect
199. C₄₇	C₂×#3138	semidirect
217. C₄⋊Q₈	C₂²×D₄₇	semidirect
224. #125	C₂²×D₄₇	semidirect
334. C₂	#20253764	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×#13094021	2
#20253764	2
C₂×#15689489	2
C₂×#15689471	2
C₂×#15689127	2
C₂×C₄×#10625329	2
C₂×C₄₇×#533	2
C₂×#3138	47

Subgroups — C2×#20253764 #27085547

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20253764 #27085547