Subgroups — C₂×#20241211 #26104212

Order 12032.

714 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15688878	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688878	C₂	1	1	yes	maximal, characteristic
3	#20241211	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₄₇×#389	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₄₇⋊C₈	—	2	1	no	—
6	C₂³×C₄₇⋊C₈	—	2	1	no	—
7	#15688878	—	2	1	no	—
8	#15688878	—	2	1	no	—
9	#15688878	C₂²	1	2	yes	—
10	#15688878	C₂²	1	2	yes	—
11	C₂²×C₄⋊C₄×C₄₇	—	2	1	no	—
12	C₂²×C₄⋊C₄×C₄₇	C₄	1	1	yes	characteristic
13	C₂²×C₄⋊C₄×C₄₇	C₄	1	1	yes	characteristic
14	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
15	C₄₇×#389	—	2	1	no	—
16	C₄₇×#389	—	2	1	no	—
17	C₄₇×#389	C₄	1	2	yes	—
18	C₄₇×#389	C₂²	1	2	yes	—
19	C₂×#95×C₄₇	—	2	1	no	—
20	C₂²×C₄₇⋊C₈	—	2	4	no	—
21	C₂²×C₄₇⋊C₈	—	2	4	no	—
22	C₂²×C₄₇⋊C₈	—	2	2	no	—
23	C₂²×C₄₇⋊C₈	—	2	2	no	—
24	C₂²×C₄₇⋊C₈	—	4	2	no	—
25	C₂²×C₄₇⋊C₈	—	4	2	no	—
26	C₂²×C₄₇⋊C₈	—	4	2	no	—
27	C₂²×C₄₇⋊C₈	—	4	2	no	—
28	C₂³×C₄×C₄₇	—	2	1	no	—
29	C₂³×C₄×C₄₇	—	2	1	no	—
30	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
31	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
32	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
33	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
34	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
35	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
36	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
37	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
38	C₂×C₄⋊C₄×C₄₇	—	4	2	no	—
39	C₂×C₄⋊C₄×C₄₇	—	4	2	no	—
40	C₂×C₄⋊C₄×C₄₇	—	4	2	no	—
41	C₂×C₄⋊C₄×C₄₇	C₈	1	4	yes	complemented
42	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
43	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₄₇	—	4	2	no	—
45	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
49	#95×C₄₇	—	2	2	no	—
50	#95×C₄₇	—	2	2	no	—
51	C₂⁴×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	—	2	2	no	—
53	C₂²×C₄×C₄₇	—	2	2	no	—
54	C₂²×C₄×C₄₇	—	2	2	no	—
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	4	no	—
59	C₂²×C₄×C₄₇	—	2	2	no	—
60	C₂²×C₄×C₄₇	—	2	2	no	—
61	C₂²×C₄×C₄₇	—	4	2	no	—
62	C₂²×C₄×C₄₇	—	4	2	no	—
63	C₂²×C₄×C₄₇	—	4	2	no	—
64	C₂²×C₄×C₄₇	—	4	2	no	—
65	C₂²×C₄×C₄₇	—	4	2	no	—
66	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	C₂×C₈	1	2	yes	—
70	C₂²×C₄×C₄₇	M₁₆	1	2	yes	—
71	C₂²×C₄×C₄₇	D₈	1	2	yes	—
72	C₂²×C₄×C₄₇	SD₁₆	1	2	yes	—
73	C₂²×C₄×C₄₇	SD₁₆	1	2	yes	—
74	C₂²×C₄×C₄₇	Q₁₆	1	2	yes	—
75	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
78	C₄⋊C₄×C₄₇	—	2	4	no	—
79	C₄⋊C₄×C₄₇	—	2	2	no	—
80	C₄⋊C₄×C₄₇	—	4	2	no	—
81	C₄⋊C₄×C₄₇	—	4	2	no	—
82	C₄⋊C₄×C₄₇	—	4	4	no	—
83	C₄⋊C₄×C₄₇	—	4	2	no	—
84	C₄⋊C₄×C₄₇	—	4	2	no	—
85	C₄⋊C₄×C₄₇	C₂×C₈	1	4	yes	complemented
86	C₄²×C₄₇	—	4	1	no	—
87	C₄²×C₄₇	—	4	1	no	—
88	C₄²×C₄₇	—	4	1	no	—
89	C₄²×C₄₇	—	4	1	no	—
90	C₂×C₄₇⋊C₈	—	2	4	no	—
91	C₂×C₄₇⋊C₈	—	2	4	no	—
92	C₂×C₄₇⋊C₈	—	4	8	no	—
93	C₂×C₄₇⋊C₈	—	4	8	no	—
94	C₂×C₄₇⋊C₈	—	4	2	no	—
95	C₂×C₄₇⋊C₈	—	4	2	no	—
96	C₂×C₄₇⋊C₈	—	4	2	no	—
97	C₂×C₄₇⋊C₈	—	4	2	no	—
98	C₂³×C₄₇	C₂²⋊C₈	1	2	yes	—
99	C₂³×C₄₇	#99	1	2	yes	—
100	C₂³×C₄₇	#101	1	2	yes	—
101	C₂³×C₄₇	#102	1	2	yes	—
102	C₂³×C₄₇	#103	1	2	yes	—
103	C₂³×C₄₇	#104	1	2	yes	—
104	C₂³×C₄₇	#104	1	2	yes	—
105	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₄₇	—	2	4	no	—
107	C₂×C₄×C₄₇	—	2	4	no	—
108	C₂×C₄×C₄₇	—	2	4	no	—
109	C₂×C₄×C₄₇	—	2	4	no	—
110	C₂×C₄×C₄₇	—	2	4	no	—
111	C₂×C₄×C₄₇	—	2	4	no	—
112	C₂×C₄×C₄₇	—	2	1	no	—
113	C₂×C₄×C₄₇	—	2	1	no	—
114	C₂×C₄×C₄₇	—	2	1	no	—
115	C₂×C₄×C₄₇	—	2	1	no	—
116	C₂×C₄×C₄₇	—	2	1	no	—
117	C₂×C₄×C₄₇	—	2	1	no	—
118	C₂×C₄×C₄₇	—	2	1	no	—
119	C₂×C₄×C₄₇	—	2	1	no	—
120	C₂×C₄×C₄₇	—	2	1	no	—
121	C₂×C₄×C₄₇	—	2	1	no	—
122	C₂×C₄×C₄₇	—	2	1	no	—
123	C₂×C₄×C₄₇	—	2	1	no	—
124	C₂×C₄×C₄₇	—	2	4	no	—
125	C₂×C₄×C₄₇	—	4	8	no	—
126	C₂×C₄×C₄₇	—	4	2	no	—
127	C₂×C₄×C₄₇	—	4	2	no	—
128	C₂×C₄×C₄₇	—	4	2	no	—
129	C₂×C₄×C₄₇	—	4	2	no	—
130	C₂×C₄×C₄₇	—	4	2	no	—
131	C₂×C₄×C₄₇	—	8	4	no	—
132	C₂×C₄×C₄₇	C₂²⋊C₈	1	2	yes	—
133	C₂×C₄×C₄₇	#102	1	2	yes	—
134	C₂×C₄×C₄₇	#103	1	2	yes	—
135	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
136	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
137	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
138	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
139	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
140	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
141	C₄₇⋊C₈	—	4	4	no	—
142	C₄₇⋊C₈	—	4	4	no	—
143	C₂×#1271	—	47	1	no	maximal, Sylow
144	C₂²×C₄₇	#324	1	4	yes	—
145	C₂²×C₄₇	D₄⋊C₈	1	4	yes	—
146	C₂²×C₄₇	Q₈⋊C₈	1	4	yes	—
147	C₂²×C₄₇	#327	1	4	yes	—
148	C₂²×C₄₇	#328	1	4	yes	—
149	C₂²×C₄₇	#330	1	4	yes	—
150	C₂²×C₄₇	#332	1	4	yes	—
151	C₂²×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
152	C₂²×C₄₇	C₂×#99	1	1	yes	characteristic
153	C₂²×C₄₇	C₂×#101	1	1	yes	characteristic
154	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
155	C₂²×C₄₇	C₂×#103	1	1	yes	characteristic
156	C₂²×C₄₇	C₂×#104	1	1	yes	characteristic
157	C₂²×C₄₇	C₂×#104	1	1	yes	characteristic
158	C₄×C₄₇	—	2	2	no	—
159	C₄×C₄₇	—	2	1	no	—
160	C₄×C₄₇	—	2	1	no	—
161	C₄×C₄₇	—	2	1	no	—
162	C₄×C₄₇	—	2	2	no	—
163	C₄×C₄₇	—	2	1	no	—
164	C₄×C₄₇	—	2	1	no	—
165	C₄×C₄₇	—	2	2	no	—
166	C₄×C₄₇	—	2	1	no	—
167	C₄×C₄₇	—	4	4	no	—
168	C₄×C₄₇	—	8	2	no	—
169	C₂×#389	D₄₇	1	1	yes	characteristic
170	C₂×#336	—	47	1	no	—
171	C₂×#336	—	47	1	no	—
172	#1271	—	47	4	no	—
173	C₂×C₄₇	#1271	1	8	yes	complemented
174	C₂×C₄₇	C₂×#324	1	1	yes	characteristic
175	C₂×C₄₇	C₂×D₄⋊C₈	1	1	yes	characteristic
176	C₂×C₄₇	C₂×Q₈⋊C₈	1	1	yes	characteristic
177	C₂×C₄₇	C₂×#327	1	1	yes	characteristic
178	C₂×C₄₇	C₂×#328	1	1	yes	characteristic
179	C₂×C₄₇	C₂×#330	1	1	yes	characteristic
180	C₂×C₄₇	C₂×#332	1	1	yes	characteristic
181	C₂³×C₈	—	94	1	no	—
182	C₂³×C₈	—	94	1	no	—
183	C₂²×C₄⋊C₄	—	2	1	no	—
184	C₂²×C₄⋊C₄	Dic₄₇	1	1	yes	characteristic
185	C₂²×C₄⋊C₄	Dic₄₇	1	1	yes	characteristic
186	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
187	#389	—	2	1	no	—
188	#389	—	2	1	no	—
189	#389	Dic₄₇	1	2	yes	—
190	#389	C₂×D₄₇	1	2	yes	—
191	C₂×#95	—	2	1	no	—
192	#336	—	47	2	no	—
193	#336	—	47	2	no	—
194	#336	—	94	1	no	—
195	#336	—	94	1	no	—
196	C₄₇	C₂×#1271	1	1	yes	characteristic, complemented, Sylow
197	C₂³×C₄	—	2	1	no	—
198	C₂³×C₄	—	2	1	no	—
199	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
200	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
201	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
202	C₂²×C₈	—	94	4	no	—
203	C₂²×C₈	—	94	4	no	—
204	C₂²×C₈	—	94	2	no	—
205	C₂²×C₈	—	94	2	no	—
206	C₂²×C₈	—	188	2	no	—
207	C₂²×C₈	—	188	2	no	—
208	C₂²×C₈	—	188	2	no	—
209	C₂²×C₈	—	188	2	no	—
210	C₂×C₄⋊C₄	—	2	2	no	—
211	C₂×C₄⋊C₄	—	2	1	no	—
212	C₂×C₄⋊C₄	—	2	2	no	—
213	C₂×C₄⋊C₄	—	2	1	no	—
214	C₂×C₄⋊C₄	—	2	2	no	—
215	C₂×C₄⋊C₄	—	4	2	no	—
216	C₂×C₄⋊C₄	—	4	2	no	—
217	C₂×C₄⋊C₄	—	4	2	no	—
218	C₂×C₄⋊C₄	C₄₇⋊C₈	1	4	yes	complemented
219	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
220	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
221	C₂×C₄²	—	4	2	no	—
222	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
223	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
224	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
225	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
226	#95	—	2	2	no	—
227	#95	—	2	2	no	—
228	C₂⁴	C₂²⋊Dic₄₇	1	1	yes	characteristic
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	4	no	—
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	4	2	no	—
239	C₂²×C₄	—	4	2	no	—
240	C₂²×C₄	—	4	2	no	—
241	C₂²×C₄	—	4	2	no	—
242	C₂²×C₄	—	4	2	no	—
243	C₂²×C₄	C₂×C₄₇⋊C₈	1	2	yes	—
244	C₂²×C₄	#10625329	1	2	yes	—
245	C₂²×C₄	#10625334	1	2	yes	—
246	C₂²×C₄	#10625335	1	2	yes	—
247	C₂²×C₄	#10625336	1	2	yes	—
248	C₂²×C₄	#10625337	1	2	yes	—
249	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
250	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
251	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
252	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
253	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
254	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
255	C₂×C₈	—	94	4	no	—
256	C₂×C₈	—	94	4	no	—
257	C₂×C₈	—	188	2	no	—
258	C₂×C₈	—	188	2	no	—
259	C₂×C₈	—	188	8	no	—
260	C₂×C₈	—	188	8	no	—
261	C₂×C₈	—	188	2	no	—
262	C₂×C₈	—	188	2	no	—
263	C₄⋊C₄	—	2	4	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	4	2	no	—
266	C₄⋊C₄	—	4	2	no	—
267	C₄⋊C₄	—	4	4	no	—
268	C₄⋊C₄	—	4	2	no	—
269	C₄⋊C₄	—	4	2	no	—
270	C₄⋊C₄	C₂×C₄₇⋊C₈	1	4	yes	complemented
271	C₄²	—	4	1	no	—
272	C₄²	—	4	1	no	—
273	C₄²	—	4	1	no	—
274	C₄²	—	4	1	no	—
275	C₂³	C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
276	C₂³	#13093939	1	2	yes	—
277	C₂³	#13093943	1	2	yes	—
278	C₂³	#13093945	1	2	yes	—
279	C₂³	#13093948	1	2	yes	—
280	C₂³	#13093951	1	2	yes	—
281	C₂³	#13093951	1	2	yes	—
282	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	4	no	—
292	C₂×C₄	—	2	4	no	—
293	C₂×C₄	—	2	4	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	2	4	no	—
296	C₂×C₄	—	2	4	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	4	no	—
302	C₂×C₄	—	4	8	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	8	4	no	—
309	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
310	C₂×C₄	#13093945	1	2	yes	—
311	C₂×C₄	#13093948	1	2	yes	—
312	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
313	C₂×C₄	C₂×#10625329	1	1	yes	characteristic
314	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
315	C₂×C₄	C₂×#10625336	1	1	yes	characteristic
316	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
317	C₂×C₄	C₂×#10625337	1	1	yes	characteristic
318	C₈	—	188	4	no	—
319	C₈	—	188	4	no	—
320	C₂²	#15688849	1	4	yes	—
321	C₂²	D₄⋊(C₄₇⋊C₈)	1	4	yes	—
322	C₂²	Q₈⋊(C₄₇⋊C₈)	1	4	yes	—
323	C₂²	#15688857	1	4	yes	—
324	C₂²	#15688860	1	4	yes	—
325	C₂²	#15688865	1	4	yes	—
326	C₂²	#15688869	1	4	yes	—
327	C₂²	C₂×C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
328	C₂²	C₂×#13093939	1	1	yes	characteristic
329	C₂²	C₂×#13093943	1	1	yes	characteristic
330	C₂²	C₂×#13093945	1	1	yes	characteristic
331	C₂²	C₂×#13093948	1	1	yes	characteristic
332	C₂²	C₂×#13093951	1	1	yes	characteristic
333	C₂²	C₂×#13093951	1	1	yes	characteristic
334	C₄	—	2	2	no	—
335	C₄	—	2	1	no	—
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	2	2	no	—
339	C₄	—	2	1	no	—
340	C₄	—	2	1	no	—
341	C₄	—	2	2	no	—
342	C₄	—	2	1	no	—
343	C₄	—	4	4	no	—
344	C₄	—	8	2	no	—
345	C₂	#20241211	1	8	yes	direct product factor
346	C₂	C₂×#15688849	1	1	yes	characteristic
347	C₂	C₂×D₄⋊(C₄₇⋊C₈)	1	1	yes	characteristic
348	C₂	C₂×Q₈⋊(C₄₇⋊C₈)	1	1	yes	characteristic
349	C₂	C₂×#15688857	1	1	yes	characteristic
350	C₂	C₂×#15688860	1	1	yes	characteristic
351	C₂	C₂×#15688865	1	1	yes	characteristic
352	C₂	C₂×#15688869	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂²⋊Dic₄₇
commutator	C₂×C₄×C₄₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₂×C₄₇×#389	C₂
socle, prime radical	C₂⁴×C₄₇	C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂×#389	D₄₇
hypocenter	C₄₇	C₂×#1271
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂×#389

Complements

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

N	H ≅ G/N	type
3. #20241211	C₂	direct (N × H)
41. C₂×C₄⋊C₄×C₄₇	C₈	semidirect
85. C₄⋊C₄×C₄₇	C₂×C₈	semidirect
173. C₂×C₄₇	#1271	semidirect
196. C₄₇	C₂×#1271	semidirect
218. C₂×C₄⋊C₄	C₄₇⋊C₈	semidirect
270. C₄⋊C₄	C₂×C₄₇⋊C₈	semidirect
345. C₂	#20241211	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×#15688878	2
#20241211	2
C₂×C₄₇×#389	2
C₂×#1271	47

Subgroups — C2×#20241211 #26104212

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20241211 #26104212