Subgroups — C₂×#20253900 #27085881

Order 12032.

1942 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689676	C₂	1	2	yes	maximal
2	#20253900	C₂	1	16	yes	maximal, direct product factor
3	C₂×#15689527	C₂	1	2	yes	maximal, complemented
4	C₂×#15689524	C₂	1	2	yes	maximal, complemented
5	C₂×#15689523	C₂	1	2	yes	maximal, complemented
6	C₂×#15689520	C₂	1	2	yes	maximal, complemented
7	C₂×#15689161	C₂	1	1	yes	maximal, characteristic
8	C₂×#15689159	C₂	1	2	yes	maximal
9	C₂²×C₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄₇×#427	C₂	1	1	yes	maximal, characteristic
11	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
12	#15689676	—	2	8	no	—
13	#15689676	C₂²	1	16	yes	—
14	C₂×#13094023	C₂²	1	4	yes	—
15	C₂×#13094015	C₂²	1	4	yes	—
16	C₂×#13094001	—	2	4	no	—
17	C₂×C₄×Dic₉₄	—	2	4	no	—
18	C₂×#13093983	C₂²	1	2	yes	—
19	C₂²×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
20	#15689527	C₂²	1	16	yes	complemented
21	#15689524	C₂²	1	16	yes	complemented
22	#15689523	C₂²	1	16	yes	complemented
23	#15689520	C₂²	1	16	yes	complemented
24	#15689161	C₂²	1	8	yes	—
25	#15689159	C₂²	1	16	yes	—
26	C₂×#13093958	C₂²	1	2	yes	—
27	C₂×#13093957	C₂²	1	4	yes	—
28	C₂×#13093956	C₂²	1	2	yes	—
29	C₂×#13093955	C₂²	1	4	yes	—
30	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
31	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
32	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
33	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
34	C₂×C₈×Dic₄₇	C₂²	1	8	yes	—
35	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
36	C₂×#117×C₄₇	C₂²	1	2	yes	—
37	C₄₇×#427	C₂²	1	8	yes	—
38	C₂×#107×C₄₇	C₂²	1	2	yes	—
39	C₂×#106×C₄₇	C₂²	1	2	yes	—
40	C₂²×Dic₉₄	—	2	4	no	—
41	C₂³×Dic₄₇	D₄	1	2	yes	—
42	C₂²×C₄₇⋊C₈	Q₈	1	8	yes	complemented
43	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
44	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
45	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
46	#13094023	—	2	8	no	—
47	#13094023	C₂³	1	16	yes	—
48	#13094015	—	2	8	no	—
49	#13094015	C₂³	1	16	yes	—
50	#13094001	—	2	32	no	—
51	C₄×Dic₉₄	—	2	32	no	—
52	#13093983	—	2	4	no	—
53	#13093983	C₂³	1	8	yes	—
54	C₂×C₄⋊Dic₄₇	—	2	4	no	—
55	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
56	C₂×Dic₄₇⋊C₄	—	2	4	no	—
57	C₂×Dic₄₇⋊C₄	—	2	4	no	—
58	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
59	C₂×C₄×Dic₄₇	—	2	2	no	—
60	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
61	C₂×C₄×Dic₄₇	D₄	1	4	yes	—
62	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
63	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
64	C₂×C₄×Dic₄₇	C₂³	1	4	yes	—
65	#13093958	C₂³	1	8	yes	—
66	#13093957	C₂³	1	16	yes	—
67	#13093956	C₂³	1	8	yes	—
68	#13093955	C₂³	1	16	yes	—
69	C₈×Dic₄₇	C₂³	1	4	yes	—
70	C₈×Dic₄₇	C₂³	1	4	yes	—
71	C₈×Dic₄₇	C₂³	1	4	yes	—
72	C₈×Dic₄₇	C₂³	1	4	yes	—
73	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
74	C₂²×C₈×C₄₇	Q₈	1	8	yes	complemented
75	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
76	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
77	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
78	#117×C₄₇	—	2	4	no	—
79	#117×C₄₇	C₂³	1	8	yes	—
80	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
81	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
82	C₂×C₄²×C₄₇	—	2	2	no	—
83	#107×C₄₇	C₂³	1	8	yes	—
84	#106×C₄₇	C₂³	1	8	yes	—
85	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₄₇	—	2	4	no	—
87	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
88	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
90	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
91	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
92	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
93	C₂²×Dic₄₇	—	2	8	no	—
94	C₂²×Dic₄₇	—	2	4	no	—
95	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
96	C₂²×Dic₄₇	C₂×D₄	1	8	yes	—
97	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
98	C₂×Dic₉₄	—	2	16	no	—
99	C₂×Dic₉₄	—	4	16	no	—
100	C₂×C₈×C₄₇	C₂×Q₈	1	4	yes	—
101	C₂×C₈×C₄₇	C₂×Q₈	1	4	yes	—
102	C₂×C₈×C₄₇	C₂×Q₈	1	16	yes	complemented
103	C₂×C₈×C₄₇	C₂⁴	1	2	yes	—
104	C₂×C₈×C₄₇	C₂⁴	1	2	yes	—
105	C₄⋊C₄×C₄₇	—	2	4	no	—
106	C₄⋊C₄×C₄₇	C₂⁴	1	8	yes	—
107	C₂²⋊C₄×C₄₇	—	2	8	no	—
108	C₄²×C₄₇	—	2	8	no	—
109	C₂²⋊Dic₄₇	—	2	8	no	—
110	C₄⋊Dic₄₇	—	2	4	no	—
111	C₄⋊Dic₄₇	—	2	16	no	—
112	C₄⋊Dic₄₇	C₂⁴	1	8	yes	—
113	Dic₄₇⋊C₄	—	2	16	no	—
114	Dic₄₇⋊C₄	—	2	16	no	—
115	C₄×Dic₄₇	—	2	8	no	—
116	C₄×Dic₄₇	C₂×D₄	1	8	yes	—
117	C₄×Dic₄₇	C₂×D₄	1	2	yes	—
118	C₄×Dic₄₇	C₂×D₄	1	2	yes	—
119	C₄×Dic₄₇	C₂⁴	1	2	yes	—
120	C₄×Dic₄₇	C₂⁴	1	2	yes	—
121	C₂×C₄₇⋊C₈	C₂×Q₈	1	4	yes	—
122	C₂×C₄₇⋊C₈	C₂×Q₈	1	4	yes	—
123	C₂×C₄₇⋊C₈	C₂×Q₈	1	16	yes	complemented
124	C₂×C₄₇⋊C₈	C₂⁴	1	2	yes	—
125	C₂×C₄₇⋊C₈	C₂⁴	1	2	yes	—
126	C₂³×C₄₇	—	2	4	no	—
127	C₂³×C₄₇	C₄⋊Q₈	1	4	yes	—
128	C₂³×C₄₇	C₂²×D₄	1	2	yes	—
129	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
130	C₂×C₄×C₄₇	—	2	8	no	—
131	C₂×C₄×C₄₇	—	4	8	no	—
132	C₂×C₄×C₄₇	C₄⋊Q₈	1	8	yes	—
133	C₂×C₄×C₄₇	C₄⋊Q₈	1	4	yes	—
134	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
135	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
137	C₂×C₄×C₄₇	C₂²×Q₈	1	4	yes	—
138	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
139	C₂×C₄×C₄₇	C₂²×Q₈	1	4	yes	—
140	C₂×C₄×C₄₇	C₂⁵	1	1	yes	characteristic
141	C₂×Dic₄₇	—	2	4	no	—
142	C₂×Dic₄₇	—	2	16	no	—
143	C₂×Dic₄₇	—	2	8	no	—
144	C₂×Dic₄₇	—	4	8	no	—
145	C₂×Dic₄₇	#135	1	8	yes	—
146	C₂×Dic₄₇	C₂²×D₄	1	4	yes	—
147	C₂×Dic₄₇	C₂²×D₄	1	4	yes	—
148	Dic₉₄	—	4	32	no	—
149	C₈×C₄₇	C₂²×Q₈	1	4	yes	—
150	C₈×C₄₇	C₂²×Q₈	1	4	yes	—
151	C₄₇⋊C₈	C₂²×Q₈	1	4	yes	—
152	C₄₇⋊C₈	C₂²×Q₈	1	4	yes	—
153	C₂×#3160	—	47	1	no	maximal, Sylow
154	C₂²×C₄₇	—	2	8	no	—
155	C₂²×C₄₇	—	2	4	no	—
156	C₂²×C₄₇	C₂×C₄⋊Q₈	1	4	yes	—
157	C₂²×C₄₇	C₂×C₄⋊Q₈	1	2	yes	—
158	C₂²×C₄₇	C₂×#135	1	4	yes	—
159	C₂²×C₄₇	C₂³×D₄	1	1	yes	characteristic
160	C₄×C₄₇	—	4	8	no	—
161	C₄×C₄₇	C₂×C₄⋊Q₈	1	4	yes	—
162	C₄×C₄₇	C₂×C₄⋊Q₈	1	2	yes	—
163	C₄×C₄₇	C₂³×Q₈	1	1	yes	characteristic
164	C₄×C₄₇	C₂³×Q₈	1	1	yes	characteristic
165	Dic₄₇	—	2	4	no	—
166	Dic₄₇	—	4	8	no	—
167	Dic₄₇	C₂×#135	1	8	yes	—
168	C₂×#533	—	47	2	no	—
169	#3160	—	47	16	no	—
170	C₂×#500	—	47	2	no	—
171	C₂×#499	—	47	2	no	—
172	C₂×#499	—	47	2	no	—
173	C₂×#498	—	47	2	no	—
174	C₂×#427	—	47	1	no	—
175	C₂×#427	—	47	2	no	—
176	C₂×#427	D₄₇	1	1	yes	characteristic
177	C₂²×C₄×C₈	—	47	1	no	—
178	C₂×C₄₇	—	2	4	no	—
179	C₂×C₄₇	#3160	1	4	yes	complemented
180	C₂×C₄₇	C₂²×C₄⋊Q₈	1	1	yes	characteristic
181	C₂×C₄₇	C₂²×#135	1	2	yes	—
182	C₂³×C₈	—	47	1	no	—
183	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
184	#533	—	47	16	no	—
185	#533	—	94	8	no	—
186	C₂×C₄⋊Q₈	—	47	4	no	—
187	C₂×#125	—	47	4	no	—
188	C₂×C₂²⋊Q₈	—	94	4	no	—
189	C₂×C₄×Q₈	—	94	4	no	—
190	C₂×#117	—	47	2	no	—
191	C₂×#117	C₂×D₄₇	1	2	yes	—
192	C₂²×C₄²	—	47	1	no	—
193	#500	—	47	16	no	—
194	#499	—	47	16	no	—
195	#499	—	47	16	no	—
196	#498	—	47	16	no	—
197	#427	—	47	8	no	—
198	#427	—	47	16	no	—
199	#427	C₂×D₄₇	1	8	yes	—
200	C₂×#107	—	47	2	no	—
201	C₂×#107	—	47	4	no	—
202	C₂×#107	C₂×D₄₇	1	2	yes	—
203	C₂×#106	—	47	2	no	—
204	C₂×#106	—	47	4	no	—
205	C₂×#106	C₂×D₄₇	1	2	yes	—
206	C₂×C₄×C₈	—	47	1	no	—
207	C₂×C₄×C₈	—	47	1	no	—
208	C₂×C₄×C₈	—	47	1	no	—
209	C₂×C₄×C₈	—	47	1	no	—
210	C₂×C₄×C₈	—	47	8	no	—
211	C₄₇	C₂×#3160	1	1	yes	characteristic, complemented, Sylow
212	C₂²×Q₈	—	94	4	no	—
213	C₂³×C₄	—	47	2	no	—
214	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
215	C₂²×C₈	—	47	8	no	—
216	C₂²×C₈	—	47	1	no	—
217	C₂²×C₈	—	47	1	no	—
218	C₂²×C₈	—	47	4	no	—
219	C₂²×C₈	Dic₉₄	1	8	yes	complemented
220	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
221	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
222	C₂²×C₈	C₂²×D₄₇	1	4	yes	—
223	C₄⋊Q₈	—	47	16	no	—
224	C₄⋊Q₈	—	94	8	no	—
225	#125	—	47	16	no	—
226	#125	—	94	8	no	—
227	C₂²⋊Q₈	—	94	32	no	—
228	C₄×Q₈	—	94	32	no	—
229	#117	—	2	4	no	—
230	#117	—	47	8	no	—
231	#117	—	94	4	no	—
232	#117	C₂²×D₄₇	1	8	yes	—
233	C₂×C₄⋊C₄	—	47	4	no	—
234	C₂×C₄⋊C₄	—	94	4	no	—
235	C₂×C₄⋊C₄	—	94	4	no	—
236	C₂×C₄⋊C₄	—	94	4	no	—
237	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
238	C₂×C₂²⋊C₄	—	2	2	no	—
239	C₂×C₂²⋊C₄	—	94	2	no	—
240	C₂×C₄²	—	2	2	no	—
241	C₂×C₄²	—	47	4	no	—
242	C₂×C₄²	—	47	2	no	—
243	C₂×C₄²	—	47	1	no	—
244	C₂×C₄²	—	47	1	no	—
245	C₂×C₄²	—	47	4	no	—
246	C₂×C₄²	—	94	2	no	—
247	#107	—	47	8	no	—
248	#107	—	47	16	no	—
249	#107	C₂²×D₄₇	1	8	yes	—
250	#106	—	47	8	no	—
251	#106	—	47	16	no	—
252	#106	C₂²×D₄₇	1	8	yes	—
253	C₄×C₈	—	47	4	no	—
254	C₄×C₈	—	47	4	no	—
255	C₄×C₈	—	47	4	no	—
256	C₄×C₈	—	47	4	no	—
257	C₂⁴	D₄×D₄₇	1	1	yes	characteristic
258	C₂×Q₈	—	94	16	no	—
259	C₂×Q₈	—	188	16	no	—
260	C₂²×C₄	—	2	4	no	—
261	C₂²×C₄	—	47	2	no	—
262	C₂²×C₄	—	47	2	no	—
263	C₂²×C₄	—	47	8	no	—
264	C₂²×C₄	—	94	8	no	—
265	C₂²×C₄	—	94	4	no	—
266	C₂²×C₄	C₂×Dic₉₄	1	4	yes	—
267	C₂²×C₄	D₄×D₄₇	1	2	yes	—
268	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
269	C₂²×C₄	Q₈×D₄₇	1	4	yes	—
270	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
271	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
272	C₂×C₈	—	47	4	no	—
273	C₂×C₈	—	47	4	no	—
274	C₂×C₈	—	47	16	no	—
275	C₂×C₈	—	47	2	no	—
276	C₂×C₈	—	47	2	no	—
277	C₂×C₈	C₂×Dic₉₄	1	4	yes	—
278	C₂×C₈	C₂×Dic₉₄	1	4	yes	—
279	C₂×C₈	C₂×Dic₉₄	1	16	yes	complemented
280	C₂×C₈	C₂³×D₄₇	1	2	yes	—
281	C₂×C₈	C₂³×D₄₇	1	2	yes	—
282	C₄⋊C₄	—	2	4	no	—
283	C₄⋊C₄	—	47	8	no	—
284	C₄⋊C₄	—	94	16	no	—
285	C₄⋊C₄	—	94	16	no	—
286	C₄⋊C₄	—	94	16	no	—
287	C₄⋊C₄	—	94	4	no	—
288	C₄⋊C₄	C₂³×D₄₇	1	8	yes	—
289	C₂²⋊C₄	—	2	8	no	—
290	C₂²⋊C₄	—	94	8	no	—
291	C₄²	—	2	8	no	—
292	C₄²	—	47	8	no	—
293	C₄²	—	47	2	no	—
294	C₄²	—	47	2	no	—
295	C₄²	—	47	2	no	—
296	C₄²	—	47	2	no	—
297	C₄²	—	94	8	no	—
298	C₂³	—	2	4	no	—
299	C₂³	#13094023	1	4	yes	—
300	C₂³	C₂×D₄×D₄₇	1	2	yes	—
301	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
302	Q₈	—	188	32	no	—
303	C₂×C₄	—	2	8	no	—
304	C₂×C₄	—	4	8	no	—
305	C₂×C₄	—	47	8	no	—
306	C₂×C₄	—	47	4	no	—
307	C₂×C₄	—	47	4	no	—
308	C₂×C₄	—	94	16	no	—
309	C₂×C₄	—	94	4	no	—
310	C₂×C₄	—	94	8	no	—
311	C₂×C₄	—	188	8	no	—
312	C₂×C₄	#13094023	1	8	yes	—
313	C₂×C₄	#13094023	1	4	yes	—
314	C₂×C₄	C₂×D₄×D₄₇	1	2	yes	—
315	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
316	C₂×C₄	C₂×Q₈×D₄₇	1	4	yes	—
317	C₂×C₄	C₂×Q₈×D₄₇	1	2	yes	—
318	C₂×C₄	C₂²×Dic₉₄	1	2	yes	—
319	C₂×C₄	C₂²×Dic₉₄	1	4	yes	—
320	C₂×C₄	C₂⁴×D₄₇	1	1	yes	characteristic
321	C₈	—	47	4	no	—
322	C₈	—	47	4	no	—
323	C₈	C₂²×Dic₉₄	1	4	yes	—
324	C₈	C₂²×Dic₉₄	1	4	yes	—
325	C₂²	—	2	4	no	—
326	C₂²	—	2	8	no	—
327	C₂²	C₂×#13094023	1	4	yes	—
328	C₂²	C₂×#13094023	1	2	yes	—
329	C₂²	#135×D₄₇	1	4	yes	—
330	C₂²	C₂²×D₄×D₄₇	1	1	yes	characteristic
331	C₄	—	4	8	no	—
332	C₄	—	47	8	no	—
333	C₄	—	94	4	no	—
334	C₄	—	188	8	no	—
335	C₄	C₂×#13094023	1	2	yes	—
336	C₄	C₂×#13094023	1	4	yes	—
337	C₄	C₂²×Q₈×D₄₇	1	1	yes	characteristic
338	C₄	C₂³×Dic₉₄	1	1	yes	characteristic
339	C₂	—	2	4	no	—
340	C₂	#20253900	1	4	yes	direct product factor
341	C₂	C₂²×#13094023	1	1	yes	characteristic
342	C₂	C₂×#135×D₄₇	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	D₄×D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂×C₄₇×#427	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂⁴×C₄₇	C₂×D₄
hypercenter	C₂×#427	D₄₇
hypocenter	C₄₇	C₂×#3160
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#427

Complements

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

N	H ≅ G/N	type
2. #20253900	C₂	direct (N × H)
3. C₂×#15689527	C₂	semidirect
4. C₂×#15689524	C₂	semidirect
5. C₂×#15689523	C₂	semidirect
6. C₂×#15689520	C₂	semidirect
20. #15689527	C₂²	semidirect
21. #15689524	C₂²	semidirect
22. #15689523	C₂²	semidirect
23. #15689520	C₂²	semidirect
42. C₂²×C₄₇⋊C₈	Q₈	semidirect
74. C₂²×C₈×C₄₇	Q₈	semidirect
102. C₂×C₈×C₄₇	C₂×Q₈	semidirect
123. C₂×C₄₇⋊C₈	C₂×Q₈	semidirect
179. C₂×C₄₇	#3160	semidirect
211. C₄₇	C₂×#3160	semidirect
219. C₂²×C₈	Dic₉₄	semidirect
279. C₂×C₈	C₂×Dic₉₄	semidirect
340. C₂	#20253900	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×#15689676	2
#20253900	2
C₂×#15689527	2
C₂×#15689524	2
C₂×#15689523	2
C₂×#15689520	2
C₂×#15689161	2
C₂×#15689159	2
C₂²×C₈×Dic₄₇	2
C₂×C₄₇×#427	2
C₂×#3160	47

Subgroups — C2×#20253900 #27085881

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20253900 #27085881