Subgroups — C₂×#20242786 #26198540

Order 12032.

726 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689526	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#15689414	C₂	1	1	yes	maximal, characteristic
3	C₂×#15689410	C₂	1	1	yes	maximal, characteristic
4	C₂×#15689188	C₂	1	1	yes	maximal, characteristic
5	#20242786	C₂	1	8	yes	maximal, direct product factor
6	C₂×#15688875	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#15688871	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄₇×#333	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₈×D₄₇	—	2	1	no	—
10	C₂×#13094024	C₂²	1	1	yes	characteristic
11	C₂×C₄×D₁₈₈	C₂²	1	1	yes	characteristic
12	#15689526	—	2	2	no	—
13	#15689526	C₂²	1	4	yes	complemented
14	#15689414	—	2	2	no	—
15	#15689414	C₂²	1	4	yes	—
16	#15689410	—	2	2	no	—
17	#15689410	C₂²	1	4	yes	—
18	#15689188	—	2	2	no	—
19	#15689188	C₂²	1	4	yes	—
20	C₂×#13093957	—	2	1	no	—
21	C₂×#13093957	—	2	1	no	—
22	C₂×#13093956	—	2	1	no	—
23	C₂×#13093955	—	2	1	no	—
24	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
25	C₂×#13093946	—	2	1	no	—
26	C₂×#13093944	—	2	1	no	—
27	C₂×(C₂×D₄₇)⋊C₈	—	2	1	no	—
28	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
29	#15688875	C₂²	1	4	yes	complemented
30	#15688871	C₂²	1	4	yes	complemented
31	C₂×C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
32	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
33	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
34	C₄₇×#333	C₂²	1	4	yes	complemented
35	C₂²×Dic₉₄	—	2	1	no	—
36	C₂²×D₁₈₈	D₄	1	1	yes	characteristic
37	C₂²×C₄×D₄₇	—	2	1	no	—
38	C₂²×C₄₇⋊C₈	—	2	1	no	—
39	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
40	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
41	C₂×C₈×D₄₇	—	2	4	no	—
42	C₂×C₈×D₄₇	—	4	4	no	—
43	#13094024	—	2	1	no	—
44	#13094024	—	2	1	no	—
45	#13094024	—	2	1	no	—
46	#13094024	C₂³	1	2	yes	—
47	C₄×D₁₈₈	—	2	1	no	—
48	C₄×D₁₈₈	—	2	1	no	—
49	C₄×D₁₈₈	D₄	1	2	yes	—
50	C₄×D₁₈₈	C₂³	1	2	yes	—
51	C₂×C₄⋊Dic₄₇	—	2	1	no	—
52	C₂×C₄⋊Dic₄₇	—	2	1	no	—
53	C₂×C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
54	C₂×(C₂×D₄₇)⋊C₄	—	2	1	no	—
55	#13093957	—	2	4	no	—
56	#13093957	—	2	4	no	—
57	#13093956	—	2	4	no	—
58	#13093955	—	2	4	no	—
59	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
60	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
61	#13093946	—	2	4	no	—
62	#13093944	—	2	4	no	—
63	(C₂×D₄₇)⋊C₈	—	2	4	no	—
64	C₄×C₄₇⋊C₈	D₄	1	2	yes	—
65	C₄×C₄₇⋊C₈	C₂³	1	2	yes	—
66	C₂²×Q₈×C₄₇	—	2	1	no	—
67	C₂²×C₈×C₄₇	—	2	1	no	—
68	C₂²×C₈×C₄₇	—	2	1	no	—
69	C₄⋊Q₈×C₄₇	—	2	1	no	—
70	C₄⋊Q₈×C₄₇	—	2	1	no	—
71	C₄⋊Q₈×C₄₇	—	2	1	no	—
72	C₄⋊Q₈×C₄₇	C₂³	1	2	yes	—
73	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
74	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
75	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
76	C₄⋊C₈×C₄₇	—	2	1	no	—
77	C₄⋊C₈×C₄₇	—	2	1	no	—
78	C₄⋊C₈×C₄₇	C₂³	1	2	yes	—
79	C₄⋊C₈×C₄₇	C₂³	1	2	yes	—
80	C₂³×D₄₇	—	2	1	no	—
81	C₂×Q₈×C₄₇	—	2	2	no	—
82	C₂×Q₈×C₄₇	—	4	1	no	—
83	C₂×Q₈×C₄₇	—	8	2	no	—
84	C₂²×C₄×C₄₇	—	4	1	no	—
85	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
88	C₂²×Dic₄₇	—	2	1	no	—
89	C₂²×Dic₄₇	—	4	1	no	—
90	C₂×D₁₈₈	—	2	1	no	—
91	C₂×D₁₈₈	—	2	2	no	—
92	C₂×D₁₈₈	Q₁₆	1	4	yes	—
93	C₂×D₁₈₈	C₂×D₄	1	2	yes	—
94	C₂×C₄×D₄₇	—	2	2	no	—
95	C₂×C₄×D₄₇	—	2	2	no	—
96	C₂×C₄×D₄₇	—	4	2	no	—
97	C₂×C₄×D₄₇	—	4	2	no	—
98	C₂×Dic₉₄	—	2	2	no	—
99	C₂×Dic₉₄	—	4	1	no	—
100	C₂×Dic₉₄	—	8	2	no	—
101	C₂×C₈×C₄₇	—	2	2	no	—
102	C₂×C₈×C₄₇	—	2	2	no	—
103	C₂×C₈×C₄₇	—	4	2	no	—
104	C₂×C₈×C₄₇	—	4	2	no	—
105	C₄⋊C₄×C₄₇	—	2	2	no	—
106	C₄⋊C₄×C₄₇	—	2	2	no	—
107	C₄⋊C₄×C₄₇	—	4	1	no	—
108	C₄⋊C₄×C₄₇	—	4	1	no	—
109	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₄₇	#41	1	1	yes	characteristic
112	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
113	(C₂×D₄₇)⋊C₄	—	2	2	no	—
114	(C₂×D₄₇)⋊C₄	—	2	2	no	—
115	C₄⋊Dic₄₇	—	2	1	no	—
116	C₄⋊Dic₄₇	—	2	2	no	—
117	C₄⋊Dic₄₇	—	2	2	no	—
118	C₄⋊Dic₄₇	—	4	1	no	—
119	C₄⋊Dic₄₇	—	4	1	no	—
120	C₄⋊Dic₄₇	C₂×D₄	1	2	yes	—
121	C₂×C₄₇⋊C₈	—	2	2	no	—
122	C₂×C₄₇⋊C₈	—	2	2	no	—
123	C₂×C₄₇⋊C₈	—	4	2	no	—
124	C₂×C₄₇⋊C₈	SD₁₆	1	4	yes	—
125	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
126	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
127	C₈×D₄₇	—	4	8	no	—
128	C₂³×C₄₇	#121	1	1	yes	characteristic
129	C₂²×D₄₇	—	2	2	no	—
130	C₂²×D₄₇	—	2	4	no	—
131	C₂²×D₄₇	—	4	2	no	—
132	C₂²×D₄₇	—	4	2	no	—
133	Q₈×C₄₇	—	8	2	no	—
134	Q₈×C₄₇	—	8	2	no	—
135	C₂×C₄×C₄₇	—	2	2	no	—
136	C₂×C₄×C₄₇	—	4	2	no	—
137	C₂×C₄×C₄₇	—	8	2	no	—
138	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
139	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
140	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
141	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
142	C₂×C₄×C₄₇	C₂×Q₁₆	1	2	yes	—
143	C₂×C₄×C₄₇	#135	1	2	yes	—
144	C₂×C₄×C₄₇	#137	1	2	yes	—
145	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
148	C₂×Dic₄₇	—	2	2	no	—
149	C₂×Dic₄₇	—	4	2	no	—
150	C₂×Dic₄₇	—	4	2	no	—
151	C₂×Dic₄₇	—	8	2	no	—
152	D₁₈₈	—	2	2	no	—
153	D₁₈₈	—	2	4	no	—
154	D₁₈₈	C₂×Q₁₆	1	4	yes	—
155	C₄×D₄₇	—	4	2	no	—
156	C₄×D₄₇	—	4	2	no	—
157	C₄×D₄₇	—	4	4	no	—
158	Dic₉₄	—	8	2	no	—
159	Dic₉₄	—	8	2	no	—
160	C₈×C₄₇	—	4	2	no	—
161	C₈×C₄₇	—	4	2	no	—
162	C₄₇⋊C₈	—	2	2	no	—
163	C₄₇⋊C₈	—	4	2	no	—
164	C₄₇⋊C₈	C₂×SD₁₆	1	4	yes	—
165	C₂×#1675	—	47	1	no	maximal, Sylow
166	C₂²×C₄₇	#461	1	2	yes	—
167	C₂²×C₄₇	#467	1	2	yes	—
168	C₂²×C₄₇	#472	1	2	yes	—
169	C₂²×C₄₇	C₂×#121	1	1	yes	characteristic
170	C₂×D₄₇	—	2	4	no	—
171	C₂×D₄₇	—	4	8	no	—
172	C₂×D₄₇	—	4	2	no	—
173	C₂×D₄₇	—	4	2	no	—
174	C₄×C₄₇	—	2	1	no	—
175	C₄×C₄₇	—	2	1	no	—
176	C₄×C₄₇	—	8	2	no	—
177	C₄×C₄₇	#461	1	2	yes	—
178	C₄×C₄₇	#467	1	2	yes	—
179	C₄×C₄₇	C₂²×SD₁₆	1	1	yes	characteristic
180	C₄×C₄₇	C₂²×Q₁₆	1	1	yes	characteristic
181	C₄×C₄₇	C₂×#135	1	1	yes	characteristic
182	C₄×C₄₇	C₂×#137	1	1	yes	characteristic
183	Dic₄₇	—	4	2	no	—
184	Dic₄₇	—	8	2	no	—
185	C₂×#500	—	47	1	no	—
186	C₂×#476	—	47	1	no	—
187	C₂×#476	—	47	1	no	—
188	C₂×C₈×D₄	—	47	1	no	—
189	#1675	—	47	8	no	—
190	C₂×#335	—	47	1	no	—
191	C₂×#333	—	47	1	no	—
192	C₂×#333	D₄₇	1	1	yes	characteristic, complemented
193	C₂×C₄₇	#1675	1	4	yes	complemented
194	C₂×C₄₇	C₂×#461	1	1	yes	characteristic
195	C₂×C₄₇	C₂×#467	1	1	yes	characteristic
196	C₂×C₄₇	C₂×#472	1	1	yes	characteristic
197	D₄₇	—	4	4	no	—
198	C₂³×C₈	—	94	1	no	—
199	C₂×C₄⋊Q₈	—	47	1	no	—
200	C₂×C₄⋊Q₈	C₂×D₄₇	1	1	yes	characteristic
201	C₂×C₄×D₄	—	47	1	no	—
202	#500	—	47	4	no	—
203	#500	—	94	2	no	—
204	#476	—	47	4	no	—
205	#476	—	47	4	no	—
206	#476	—	94	2	no	—
207	#476	—	94	2	no	—
208	C₈×D₄	—	47	4	no	—
209	C₈×D₄	—	94	2	no	—
210	C₂×#107	—	94	1	no	—
211	C₂×#107	—	94	1	no	—
212	C₂×#106	—	94	1	no	—
213	C₂×#106	—	94	1	no	—
214	C₂×C₄⋊C₈	—	47	1	no	—
215	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
216	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
217	C₂×#102	—	94	1	no	—
218	C₂×#102	—	94	1	no	—
219	C₂×C₂²⋊C₈	—	94	1	no	—
220	C₂×C₄×C₈	—	47	1	no	—
221	#335	—	47	4	no	—
222	#333	—	47	4	no	—
223	#333	C₂×D₄₇	1	4	yes	complemented
224	C₄₇	C₂×#1675	1	1	yes	characteristic, complemented, Sylow
225	C₂²×Q₈	—	2	1	no	—
226	C₂²×Q₈	—	94	1	no	—
227	C₂²×D₄	—	47	1	no	—
228	C₂³×C₄	—	94	1	no	—
229	C₂²×C₈	—	2	1	no	—
230	C₂²×C₈	—	2	1	no	—
231	C₂²×C₈	—	47	1	no	—
232	C₂²×C₈	—	47	1	no	—
233	C₂²×C₈	—	94	4	no	—
234	C₂²×C₈	—	94	1	no	—
235	C₂²×C₈	—	188	4	no	—
236	C₄⋊Q₈	—	2	1	no	—
237	C₄⋊Q₈	—	2	1	no	—
238	C₄⋊Q₈	—	2	1	no	—
239	C₄⋊Q₈	—	47	2	no	—
240	C₄⋊Q₈	—	94	1	no	—
241	C₄⋊Q₈	—	94	1	no	—
242	C₄⋊Q₈	—	94	1	no	—
243	C₄⋊Q₈	C₂²×D₄₇	1	2	yes	—
244	C₄×D₄	—	47	2	no	—
245	C₄×D₄	—	47	2	no	—
246	C₄×D₄	—	94	1	no	—
247	C₄×D₄	—	94	1	no	—
248	C₂×C₄⋊C₄	—	2	1	no	—
249	C₂×C₄⋊C₄	—	2	1	no	—
250	C₂×C₄⋊C₄	—	47	1	no	—
251	C₂×C₄⋊C₄	—	94	1	no	—
252	C₂×C₄⋊C₄	—	94	1	no	—
253	C₂×C₂²⋊C₄	—	94	1	no	—
254	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
255	#107	—	94	4	no	—
256	#107	—	94	4	no	—
257	#106	—	94	4	no	—
258	#106	—	94	4	no	—
259	C₄⋊C₈	—	2	1	no	—
260	C₄⋊C₈	—	2	1	no	—
261	C₄⋊C₈	—	47	2	no	—
262	C₄⋊C₈	—	94	1	no	—
263	C₄⋊C₈	C₂²×D₄₇	1	2	yes	—
264	C₄⋊C₈	C₂²×D₄₇	1	2	yes	—
265	#102	—	94	4	no	—
266	#102	—	94	4	no	—
267	C₂²⋊C₈	—	94	4	no	—
268	C₄×C₈	—	47	2	no	—
269	C₄×C₈	—	47	2	no	—
270	C₂⁴	—	94	1	no	—
271	C₂×Q₈	—	2	2	no	—
272	C₂×Q₈	—	4	1	no	—
273	C₂×Q₈	—	8	2	no	—
274	C₂×Q₈	—	94	2	no	—
275	C₂×Q₈	—	376	2	no	—
276	C₂×Q₈	—	188	1	no	—
277	C₂×D₄	—	47	4	no	—
278	C₂×D₄	—	47	2	no	—
279	C₂×D₄	—	94	1	no	—
280	C₂×D₄	—	94	2	no	—
281	C₂²×C₄	—	4	1	no	—
282	C₂²×C₄	—	94	1	no	—
283	C₂²×C₄	—	94	2	no	—
284	C₂²×C₄	—	94	2	no	—
285	C₂²×C₄	—	188	1	no	—
286	C₂²×C₄	—	188	2	no	—
287	C₂²×C₄	—	188	2	no	—
288	C₂²×C₄	#10625350	1	1	yes	characteristic
289	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
290	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
291	C₂×C₈	—	2	2	no	—
292	C₂×C₈	—	2	2	no	—
293	C₂×C₈	—	4	2	no	—
294	C₂×C₈	—	4	2	no	—
295	C₂×C₈	—	47	4	no	—
296	C₂×C₈	—	47	2	no	—
297	C₂×C₈	—	47	2	no	—
298	C₂×C₈	—	94	2	no	—
299	C₂×C₈	—	94	2	no	—
300	C₂×C₈	—	188	8	no	—
301	C₂×C₈	—	188	2	no	—
302	C₄⋊C₄	—	2	2	no	—
303	C₄⋊C₄	—	2	2	no	—
304	C₄⋊C₄	—	4	1	no	—
305	C₄⋊C₄	—	4	1	no	—
306	C₄⋊C₄	—	47	2	no	—
307	C₄⋊C₄	—	94	1	no	—
308	C₄⋊C₄	—	94	2	no	—
309	C₄⋊C₄	—	94	2	no	—
310	C₄⋊C₄	—	188	1	no	—
311	C₄⋊C₄	—	188	1	no	—
312	C₂²⋊C₄	—	94	2	no	—
313	C₂²⋊C₄	—	94	2	no	—
314	C₄²	#10625350	1	1	yes	characteristic
315	C₄²	D₄×D₄₇	1	1	yes	characteristic
316	C₄²	D₄×D₄₇	1	1	yes	characteristic
317	C₄²	C₂³×D₄₇	1	1	yes	characteristic
318	C₂³	—	94	2	no	—
319	C₂³	—	94	4	no	—
320	C₂³	—	188	2	no	—
321	C₂³	—	188	2	no	—
322	C₂³	#13093997	1	1	yes	characteristic
323	Q₈	—	8	2	no	—
324	Q₈	—	8	2	no	—
325	Q₈	—	376	2	no	—
326	Q₈	—	376	2	no	—
327	D₄	—	47	4	no	—
328	D₄	—	94	4	no	—
329	D₄	—	94	2	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	4	2	no	—
332	C₂×C₄	—	8	2	no	—
333	C₂×C₄	—	94	2	no	—
334	C₂×C₄	—	376	2	no	—
335	C₂×C₄	—	188	2	no	—
336	C₂×C₄	—	188	2	no	—
337	C₂×C₄	—	188	4	no	—
338	C₂×C₄	—	188	2	no	—
339	C₂×C₄	—	188	2	no	—
340	C₂×C₄	#13093997	1	1	yes	characteristic
341	C₂×C₄	#13093997	1	1	yes	characteristic
342	C₂×C₄	#13093997	1	1	yes	characteristic
343	C₂×C₄	SD₁₆×D₄₇	1	2	yes	—
344	C₂×C₄	Q₁₆×D₄₇	1	2	yes	—
345	C₂×C₄	#13094045	1	2	yes	—
346	C₂×C₄	#13094056	1	2	yes	—
347	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
348	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
349	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
350	C₈	—	4	2	no	—
351	C₈	—	4	2	no	—
352	C₈	—	47	4	no	—
353	C₈	—	94	2	no	—
354	C₈	—	188	2	no	—
355	C₂²	—	94	4	no	—
356	C₂²	—	188	8	no	—
357	C₂²	—	188	2	no	—
358	C₂²	—	188	2	no	—
359	C₂²	#15689327	1	2	yes	—
360	C₂²	#15689363	1	2	yes	—
361	C₂²	#15689392	1	2	yes	—
362	C₂²	C₂×#13093997	1	1	yes	characteristic
363	C₄	—	2	1	no	—
364	C₄	—	2	1	no	—
365	C₄	—	8	2	no	—
366	C₄	—	376	2	no	—
367	C₄	—	188	2	no	—
368	C₄	#15689327	1	2	yes	—
369	C₄	#15689363	1	2	yes	—
370	C₄	C₂×SD₁₆×D₄₇	1	1	yes	characteristic
371	C₄	C₂×Q₁₆×D₄₇	1	1	yes	characteristic
372	C₄	C₂×#13094045	1	1	yes	characteristic
373	C₄	C₂×#13094056	1	1	yes	characteristic
374	C₂	—	188	4	no	—
375	C₂	#20242786	1	4	yes	direct product factor
376	C₂	C₂×#15689327	1	1	yes	characteristic
377	C₂	C₂×#15689363	1	1	yes	characteristic
378	C₂	C₂×#15689392	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13093997
commutator, cosocle	C₄²×C₄₇	C₂⁴
Frattini	C₄²	C₂³×D₄₇
Fitting	C₂×C₄₇×#333	C₂
socle	C₂³×C₄₇	#121
prime radical	C₂²×D₁₈₈	D₄
hypercenter	C₂×#333	D₄₇
hypocenter	C₄₇	C₂×#1675
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#333

Complements

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

N	H ≅ G/N	type
1. C₂×#15689526	C₂	semidirect
5. #20242786	C₂	direct (N × H)
6. C₂×#15688875	C₂	semidirect
7. C₂×#15688871	C₂	semidirect
8. C₂×C₄₇×#333	C₂	semidirect
13. #15689526	C₂²	semidirect
29. #15688875	C₂²	semidirect
30. #15688871	C₂²	semidirect
34. C₄₇×#333	C₂²	semidirect
192. C₂×#333	D₄₇	semidirect
193. C₂×C₄₇	#1675	semidirect
223. #333	C₂×D₄₇	semidirect
224. C₄₇	C₂×#1675	semidirect
375. C₂	#20242786	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×#15689526	2
C₂×#15689414	2
C₂×#15689410	2
C₂×#15689188	2
#20242786	2
C₂×#15688875	2
C₂×#15688871	2
C₂×C₄₇×#333	2
C₂×#1675	47

Subgroups — C2×#20242786 #26198540

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20242786 #26198540