Subgroups — C₂×#20241213 #26104220

Order 12032.

778 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689029	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688878	C₂	1	1	yes	maximal, characteristic
3	#20241213	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₄₇×#336	C₂	1	1	yes	maximal, characteristic
5	C₂³×Dic₉₄	C₄	1	1	yes	characteristic
6	C₂³×C₄₇⋊C₈	—	2	1	no	—
7	C₂²×C₄⋊Dic₄₇	C₄	1	1	yes	characteristic
8	#15689029	—	2	1	no	—
9	#15689029	—	2	1	no	—
10	#15689029	C₄	1	2	yes	—
11	#15689029	C₂²	1	2	yes	—
12	C₂×C₄₇⋊#95	—	2	1	no	—
13	C₂×C₄₇⋊#95	—	2	1	no	—
14	#15688878	—	2	1	no	—
15	#15688878	C₂²	1	2	yes	—
16	C₂³×C₈×C₄₇	—	2	1	no	—
17	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
18	C₄₇×#336	—	2	1	no	—
19	C₄₇×#336	C₂²	1	2	yes	—
20	C₂²×Dic₉₄	—	2	1	no	—
21	C₂²×Dic₉₄	—	2	2	no	—
22	C₂²×Dic₉₄	—	4	2	no	—
23	C₂²×Dic₉₄	—	4	2	no	—
24	C₂²×Dic₉₄	C₈	1	4	yes	complemented
25	C₂²×Dic₉₄	C₂×C₄	1	2	yes	—
26	C₂³×Dic₄₇	—	2	1	no	—
27	C₂³×Dic₄₇	—	2	1	no	—
28	C₂²×C₄₇⋊C₈	—	2	4	no	—
29	C₂²×C₄₇⋊C₈	—	2	2	no	—
30	C₂²×C₄₇⋊C₈	—	4	2	no	—
31	C₂²×C₄₇⋊C₈	—	4	2	no	—
32	C₂×C₄⋊Dic₄₇	—	2	1	no	—
33	C₂×C₄⋊Dic₄₇	—	4	2	no	—
34	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
35	C₄₇⋊#95	—	2	2	no	—
36	C₄₇⋊#95	—	2	2	no	—
37	C₄₇⋊#95	—	2	2	no	—
38	C₄₇⋊#95	—	2	2	no	—
39	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
40	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
41	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₄₇	—	2	4	no	—
43	C₂²×C₈×C₄₇	—	2	2	no	—
44	C₂²×C₈×C₄₇	—	4	2	no	—
45	C₂²×C₈×C₄₇	—	4	2	no	—
46	C₂×C₄²×C₄₇	—	4	2	no	—
47	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
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	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	C₂×C₈	1	2	yes	—
62	C₂²×C₄×C₄₇	M₁₆	1	2	yes	—
63	C₂²×C₄×C₄₇	SD₁₆	1	2	yes	—
64	C₂²×C₄×C₄₇	SD₁₆	1	2	yes	—
65	C₂²×C₄×C₄₇	Q₁₆	1	2	yes	—
66	C₂²×C₄×C₄₇	Q₁₆	1	2	yes	—
67	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×Dic₄₇	—	2	4	no	—
71	C₂²×Dic₄₇	—	2	2	no	—
72	C₂²×Dic₄₇	—	2	2	no	—
73	C₂²×Dic₄₇	—	4	2	no	—
74	C₂²×Dic₄₇	—	4	2	no	—
75	C₂²×Dic₄₇	—	4	2	no	—
76	C₂²×Dic₄₇	—	4	2	no	—
77	C₂²×Dic₄₇	—	4	2	no	—
78	C₂×Dic₉₄	—	2	4	no	—
79	C₂×Dic₉₄	—	2	2	no	—
80	C₂×Dic₉₄	—	4	8	no	—
81	C₂×Dic₉₄	—	4	2	no	—
82	C₂×Dic₉₄	—	4	2	no	—
83	C₂×Dic₉₄	—	4	8	no	—
84	C₂×Dic₉₄	—	4	2	no	—
85	C₂×Dic₉₄	—	4	2	no	—
86	C₂×Dic₉₄	C₂×C₈	1	4	yes	complemented
87	C₂×C₈×C₄₇	—	2	4	no	—
88	C₂×C₈×C₄₇	—	4	8	no	—
89	C₂×C₈×C₄₇	—	4	2	no	—
90	C₂×C₈×C₄₇	—	4	2	no	—
91	C₄²×C₄₇	—	4	1	no	—
92	C₄²×C₄₇	—	4	1	no	—
93	C₄²×C₄₇	—	4	1	no	—
94	C₄²×C₄₇	—	4	1	no	—
95	C₄⋊Dic₄₇	—	4	2	no	—
96	C₄⋊Dic₄₇	—	4	2	no	—
97	C₂×C₄₇⋊C₈	—	2	4	no	—
98	C₂×C₄₇⋊C₈	—	4	8	no	—
99	C₂×C₄₇⋊C₈	—	4	2	no	—
100	C₂×C₄₇⋊C₈	—	4	2	no	—
101	C₂³×C₄₇	C₂²⋊C₈	1	2	yes	—
102	C₂³×C₄₇	#99	1	2	yes	—
103	C₂³×C₄₇	#100	1	2	yes	—
104	C₂³×C₄₇	#103	1	2	yes	—
105	C₂³×C₄₇	#103	1	2	yes	—
106	C₂³×C₄₇	#104	1	2	yes	—
107	C₂³×C₄₇	#104	1	2	yes	—
108	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	4	no	—
113	C₂×C₄×C₄₇	—	2	4	no	—
114	C₂×C₄×C₄₇	—	2	4	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	1	no	—
125	C₂×C₄×C₄₇	—	2	1	no	—
126	C₂×C₄×C₄₇	—	2	1	no	—
127	C₂×C₄×C₄₇	C₂²⋊C₈	1	2	yes	—
128	C₂×C₄×C₄₇	#103	1	2	yes	—
129	C₂×C₄×C₄₇	#103	1	2	yes	—
130	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
131	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
132	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
133	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
134	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
135	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
136	C₂×Dic₄₇	—	2	4	no	—
137	C₂×Dic₄₇	—	4	8	no	—
138	C₂×Dic₄₇	—	4	2	no	—
139	C₂×Dic₄₇	—	4	2	no	—
140	C₂×Dic₄₇	—	4	2	no	—
141	C₂×Dic₄₇	—	4	2	no	—
142	C₂×Dic₄₇	—	4	2	no	—
143	C₂×Dic₄₇	—	8	4	no	—
144	Dic₉₄	—	4	8	no	—
145	Dic₉₄	—	4	4	no	—
146	Dic₉₄	—	4	4	no	—
147	C₈×C₄₇	—	4	4	no	—
148	C₄₇⋊C₈	—	4	4	no	—
149	C₂×#1272	—	47	1	no	maximal, Sylow
150	C₂²×C₄₇	#323	1	4	yes	—
151	C₂²×C₄₇	Q₈⋊C₈	1	4	yes	—
152	C₂²×C₄₇	Q₈⋊C₈	1	4	yes	—
153	C₂²×C₄₇	#328	1	4	yes	—
154	C₂²×C₄₇	#328	1	4	yes	—
155	C₂²×C₄₇	#329	1	4	yes	—
156	C₂²×C₄₇	#333	1	4	yes	—
157	C₂²×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
158	C₂²×C₄₇	C₂×#99	1	1	yes	characteristic
159	C₂²×C₄₇	C₂×#100	1	1	yes	characteristic
160	C₂²×C₄₇	C₂×#103	1	1	yes	characteristic
161	C₂²×C₄₇	C₂×#103	1	1	yes	characteristic
162	C₂²×C₄₇	C₂×#104	1	1	yes	characteristic
163	C₂²×C₄₇	C₂×#104	1	1	yes	characteristic
164	C₄×C₄₇	—	2	2	no	—
165	C₄×C₄₇	—	2	1	no	—
166	C₄×C₄₇	—	2	1	no	—
167	C₄×C₄₇	—	2	1	no	—
168	C₄×C₄₇	—	2	1	no	—
169	C₄×C₄₇	—	2	2	no	—
170	C₄×C₄₇	—	2	2	no	—
171	C₄×C₄₇	—	2	1	no	—
172	C₄×C₄₇	—	2	1	no	—
173	Dic₄₇	—	4	4	no	—
174	Dic₄₇	—	8	2	no	—
175	C₂×#391	—	47	1	no	—
176	C₂×#336	—	47	1	no	—
177	C₂×#336	D₄₇	1	1	yes	characteristic
178	#1272	—	47	4	no	—
179	C₂×C₄₇	#1272	1	8	yes	complemented
180	C₂×C₄₇	C₂×#323	1	1	yes	characteristic
181	C₂×C₄₇	C₂×Q₈⋊C₈	1	1	yes	characteristic
182	C₂×C₄₇	C₂×Q₈⋊C₈	1	1	yes	characteristic
183	C₂×C₄₇	C₂×#328	1	1	yes	characteristic
184	C₂×C₄₇	C₂×#328	1	1	yes	characteristic
185	C₂×C₄₇	C₂×#329	1	1	yes	characteristic
186	C₂×C₄₇	C₂×#333	1	1	yes	characteristic
187	C₂³×Q₈	—	47	1	no	—
188	C₂³×C₈	—	2	1	no	—
189	C₂³×C₈	—	94	1	no	—
190	C₂²×C₄⋊C₄	—	47	1	no	—
191	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
192	#391	—	47	2	no	—
193	#391	—	47	2	no	—
194	#391	—	94	1	no	—
195	#391	—	94	1	no	—
196	C₂×#95	—	94	1	no	—
197	C₂×#95	—	94	1	no	—
198	#336	—	2	1	no	—
199	#336	—	47	2	no	—
200	#336	—	94	1	no	—
201	#336	C₂×D₄₇	1	2	yes	—
202	C₄₇	C₂×#1272	1	1	yes	characteristic, complemented, Sylow
203	C₂²×Q₈	—	47	4	no	—
204	C₂²×Q₈	—	47	2	no	—
205	C₂²×Q₈	—	94	2	no	—
206	C₂²×Q₈	—	94	1	no	—
207	C₂²×Q₈	—	188	2	no	—
208	C₂²×Q₈	—	188	2	no	—
209	C₂³×C₄	—	94	1	no	—
210	C₂³×C₄	—	94	1	no	—
211	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
212	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
213	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
214	C₂²×C₈	—	2	4	no	—
215	C₂²×C₈	—	2	2	no	—
216	C₂²×C₈	—	4	2	no	—
217	C₂²×C₈	—	4	2	no	—
218	C₂²×C₈	—	94	4	no	—
219	C₂²×C₈	—	94	2	no	—
220	C₂²×C₈	—	188	2	no	—
221	C₂²×C₈	—	188	2	no	—
222	C₂×C₄⋊C₄	—	47	2	no	—
223	C₂×C₄⋊C₄	—	94	1	no	—
224	C₂×C₄⋊C₄	—	188	2	no	—
225	C₂×C₄²	—	4	2	no	—
226	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
227	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
228	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
229	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
230	#95	—	94	2	no	—
231	#95	—	94	2	no	—
232	#95	—	94	2	no	—
233	#95	—	94	2	no	—
234	C₂⁴	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
235	C₂×Q₈	—	47	4	no	—
236	C₂×Q₈	—	94	4	no	—
237	C₂×Q₈	—	94	2	no	—
238	C₂×Q₈	—	188	2	no	—
239	C₂×Q₈	—	188	2	no	—
240	C₂×Q₈	—	188	8	no	—
241	C₂×Q₈	—	188	2	no	—
242	C₂×Q₈	—	188	2	no	—
243	C₂×Q₈	—	188	8	no	—
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	—	2	2	no	—
246	C₂²×C₄	—	2	2	no	—
247	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	—	2	2	no	—
249	C₂²×C₄	—	2	2	no	—
250	C₂²×C₄	—	94	4	no	—
251	C₂²×C₄	—	94	2	no	—
252	C₂²×C₄	—	94	2	no	—
253	C₂²×C₄	—	188	2	no	—
254	C₂²×C₄	—	188	2	no	—
255	C₂²×C₄	—	188	2	no	—
256	C₂²×C₄	—	188	2	no	—
257	C₂²×C₄	—	188	2	no	—
258	C₂²×C₄	C₈×D₄₇	1	2	yes	—
259	C₂²×C₄	#10625324	1	2	yes	—
260	C₂²×C₄	#10625325	1	2	yes	—
261	C₂²×C₄	Dic₁₈₈	1	2	yes	—
262	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
263	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
264	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
265	C₂²×C₄	#10625335	1	2	yes	—
266	C₂²×C₄	#10625337	1	2	yes	—
267	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
268	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
269	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
270	C₂×C₈	—	2	4	no	—
271	C₂×C₈	—	4	2	no	—
272	C₂×C₈	—	4	8	no	—
273	C₂×C₈	—	4	2	no	—
274	C₂×C₈	—	94	4	no	—
275	C₂×C₈	—	188	2	no	—
276	C₂×C₈	—	188	8	no	—
277	C₂×C₈	—	188	2	no	—
278	C₄⋊C₄	—	188	2	no	—
279	C₄⋊C₄	—	188	2	no	—
280	C₄²	—	4	1	no	—
281	C₄²	—	4	1	no	—
282	C₄²	—	4	1	no	—
283	C₄²	—	4	1	no	—
284	C₂³	(C₂×D₄₇)⋊C₈	1	2	yes	—
285	C₂³	#13093938	1	2	yes	—
286	C₂³	#13093940	1	2	yes	—
287	C₂³	#13093947	1	2	yes	—
288	C₂³	#13093949	1	2	yes	—
289	C₂³	#13093950	1	2	yes	—
290	C₂³	#13093952	1	2	yes	—
291	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
292	Q₈	—	188	4	no	—
293	Q₈	—	188	8	no	—
294	Q₈	—	188	4	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	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	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	4	no	—
304	C₂×C₄	—	2	4	no	—
305	C₂×C₄	—	2	4	no	—
306	C₂×C₄	—	2	4	no	—
307	C₂×C₄	—	2	4	no	—
308	C₂×C₄	—	2	4	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	94	4	no	—
314	C₂×C₄	—	376	4	no	—
315	C₂×C₄	—	188	2	no	—
316	C₂×C₄	—	188	2	no	—
317	C₂×C₄	—	188	2	no	—
318	C₂×C₄	—	188	2	no	—
319	C₂×C₄	—	188	2	no	—
320	C₂×C₄	—	188	8	no	—
321	C₂×C₄	(C₂×D₄₇)⋊C₈	1	2	yes	—
322	C₂×C₄	#13093947	1	2	yes	—
323	C₂×C₄	#13093949	1	2	yes	—
324	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
325	C₂×C₄	C₂×#10625324	1	1	yes	characteristic
326	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
327	C₂×C₄	C₂×#10625325	1	1	yes	characteristic
328	C₂×C₄	C₂×#10625337	1	1	yes	characteristic
329	C₂×C₄	C₂×Dic₁₈₈	1	1	yes	characteristic
330	C₈	—	4	4	no	—
331	C₈	—	188	4	no	—
332	C₂²	#15688846	1	4	yes	—
333	C₂²	#15688853	1	4	yes	—
334	C₂²	#15688855	1	4	yes	—
335	C₂²	#15688859	1	4	yes	—
336	C₂²	#15688861	1	4	yes	—
337	C₂²	#15688862	1	4	yes	—
338	C₂²	#15688871	1	4	yes	—
339	C₂²	C₂×(C₂×D₄₇)⋊C₈	1	1	yes	characteristic
340	C₂²	C₂×#13093938	1	1	yes	characteristic
341	C₂²	C₂×#13093940	1	1	yes	characteristic
342	C₂²	C₂×#13093947	1	1	yes	characteristic
343	C₂²	C₂×#13093949	1	1	yes	characteristic
344	C₂²	C₂×#13093950	1	1	yes	characteristic
345	C₂²	C₂×#13093952	1	1	yes	characteristic
346	C₄	—	2	2	no	—
347	C₄	—	2	1	no	—
348	C₄	—	2	1	no	—
349	C₄	—	2	1	no	—
350	C₄	—	2	2	no	—
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	2	2	no	—
354	C₄	—	2	1	no	—
355	C₄	—	376	2	no	—
356	C₄	—	188	4	no	—
357	C₂	#20241213	1	8	yes	direct product factor
358	C₂	C₂×#15688846	1	1	yes	characteristic
359	C₂	C₂×#15688853	1	1	yes	characteristic
360	C₂	C₂×#15688855	1	1	yes	characteristic
361	C₂	C₂×#15688859	1	1	yes	characteristic
362	C₂	C₂×#15688861	1	1	yes	characteristic
363	C₂	C₂×#15688862	1	1	yes	characteristic
364	C₂	C₂×#15688871	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	(C₂×D₄₇)⋊C₄
commutator	C₂×C₄×C₄₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₂×C₄₇×#336	C₂
socle, prime radical	C₂⁴×C₄₇	C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂×#336	D₄₇
hypocenter	C₄₇	C₂×#1272
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂×#336

Complements

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

N	H ≅ G/N	type
3. #20241213	C₂	direct (N × H)
24. C₂²×Dic₉₄	C₈	semidirect
86. C₂×Dic₉₄	C₂×C₈	semidirect
179. C₂×C₄₇	#1272	semidirect
202. C₄₇	C₂×#1272	semidirect
357. C₂	#20241213	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×#15689029	2
C₂×#15688878	2
#20241213	2
C₂×C₄₇×#336	2
C₂×#1272	47

Subgroups — C2×#20241213 #26104220

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20241213 #26104220