Subgroups — C₂×#20241273 #26106397

Order 12032.

1226 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₂²⋊Dic₄₇	C₂	1	2	yes	maximal
2	#20241273	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₄₇×#407	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₂²⋊Dic₄₇	—	2	2	no	—
5	C₂²×C₄×Dic₄₇	—	2	2	no	—
6	C₄×C₂²⋊Dic₄₇	—	2	2	no	—
7	C₄×C₂²⋊Dic₄₇	—	2	2	no	—
8	C₄×C₂²⋊Dic₄₇	C₄	1	4	yes	complemented
9	C₄×C₂²⋊Dic₄₇	C₂²	1	4	yes	—
10	C₂×C₄₇⋊#95	C₄	1	4	yes	complemented
11	C₂⁴×C₄×C₄₇	C₂²	1	1	yes	characteristic
12	C₂²×M₁₆×C₄₇	C₄	1	2	yes	complemented
13	C₄₇×#407	—	2	2	no	—
14	C₄₇×#407	C₄	1	2	yes	complemented
15	C₄₇×#407	C₂²	1	2	yes	—
16	C₂×C₂²⋊C₈×C₄₇	—	2	2	no	—
17	C₂³×Dic₄₇	—	2	4	no	—
18	C₂×C₂²⋊Dic₄₇	—	2	16	no	—
19	C₂×C₂²⋊Dic₄₇	—	2	4	no	—
20	C₂×C₂²⋊Dic₄₇	—	2	4	no	—
21	C₂×C₄×Dic₄₇	—	2	4	no	—
22	C₂×C₄×Dic₄₇	—	2	4	no	—
23	C₂×C₄×Dic₄₇	—	4	8	no	—
24	C₄₇⋊#95	—	2	4	no	—
25	C₄₇⋊#95	C₂×C₄	1	8	yes	complemented
26	C₂⁵×C₄₇	D₄	1	1	yes	characteristic
27	C₂³×C₄×C₄₇	—	4	4	no	—
28	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
31	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
32	C₂³×C₄×C₄₇	D₄	1	2	yes	—
33	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
34	C₂³×C₄×C₄₇	D₄	1	2	yes	—
35	C₂³×C₄×C₄₇	Q₈	1	1	yes	characteristic
36	C₂³×C₄×C₄₇	Q₈	1	1	yes	characteristic
37	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
38	C₂×M₁₆×C₄₇	—	2	2	no	—
39	C₂×M₁₆×C₄₇	—	4	4	no	—
40	C₂×M₁₆×C₄₇	C₂×C₄	1	4	yes	complemented
41	C₂²×C₈×C₄₇	—	2	2	no	—
42	C₂²⋊C₈×C₄₇	—	2	4	no	—
43	C₂²⋊C₈×C₄₇	—	2	4	no	—
44	C₂⁴×C₄₇	—	2	8	no	—
45	C₂⁴×C₄₇	—	4	2	no	—
46	C₂⁴×C₄₇	C₂²⋊C₄	1	2	yes	—
47	C₂⁴×C₄₇	C₂²⋊C₄	1	2	yes	—
48	C₂⁴×C₄₇	C₄⋊C₄	1	1	yes	characteristic
49	C₂⁴×C₄₇	C₄⋊C₄	1	1	yes	characteristic
50	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₄₇	—	2	4	no	—
52	C₂²×C₄×C₄₇	—	2	8	no	—
53	C₂²×C₄×C₄₇	—	4	4	no	—
54	C₂²×C₄×C₄₇	—	4	2	no	—
55	C₂²×C₄×C₄₇	—	4	2	no	—
56	C₂²×C₄×C₄₇	—	4	4	no	—
57	C₂²×C₄×C₄₇	—	4	2	no	—
58	C₂²×C₄×C₄₇	—	4	2	no	—
59	C₂²×C₄×C₄₇	—	4	4	no	—
60	C₂²×C₄×C₄₇	—	4	2	no	—
61	C₂²×C₄×C₄₇	C₄²	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₄²	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
64	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
65	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
67	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
68	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
71	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
72	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
74	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
75	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
77	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
79	C₂²×Dic₄₇	—	2	16	no	—
80	C₂²×Dic₄₇	—	2	8	no	—
81	C₂²×Dic₄₇	—	4	8	no	—
82	C₂²×Dic₄₇	—	4	8	no	—
83	M₁₆×C₄₇	—	4	4	no	—
84	M₁₆×C₄₇	—	4	4	no	—
85	C₂×C₈×C₄₇	—	2	4	no	—
86	C₂×C₈×C₄₇	—	4	4	no	—
87	C₂²⋊Dic₄₇	—	2	16	no	—
88	C₂²⋊Dic₄₇	—	2	16	no	—
89	C₄×Dic₄₇	—	4	8	no	—
90	C₄×Dic₄₇	—	4	8	no	—
91	C₂³×C₄₇	—	2	4	no	—
92	C₂³×C₄₇	—	2	4	no	—
93	C₂³×C₄₇	—	2	4	no	—
94	C₂³×C₄₇	—	2	4	no	—
95	C₂³×C₄₇	—	2	4	no	—
96	C₂³×C₄₇	—	2	2	no	—
97	C₂³×C₄₇	—	2	4	no	—
98	C₂³×C₄₇	—	2	4	no	—
99	C₂³×C₄₇	—	4	2	no	—
100	C₂³×C₄₇	—	4	1	no	—
101	C₂³×C₄₇	—	4	1	no	—
102	C₂³×C₄₇	—	4	16	no	—
103	C₂³×C₄₇	#95	1	1	yes	characteristic
104	C₂³×C₄₇	#95	1	2	yes	—
105	C₂³×C₄₇	#95	1	1	yes	characteristic
106	C₂³×C₄₇	#104	1	8	yes	—
107	C₂³×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
108	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₄₇	—	2	2	no	—
110	C₂×C₄×C₄₇	—	2	4	no	—
111	C₂×C₄×C₄₇	—	2	4	no	—
112	C₂×C₄×C₄₇	—	2	2	no	—
113	C₂×C₄×C₄₇	—	2	4	no	—
114	C₂×C₄×C₄₇	—	2	2	no	—
115	C₂×C₄×C₄₇	—	2	4	no	—
116	C₂×C₄×C₄₇	—	2	2	no	—
117	C₂×C₄×C₄₇	—	2	2	no	—
118	C₂×C₄×C₄₇	—	2	4	no	—
119	C₂×C₄×C₄₇	—	4	2	no	—
120	C₂×C₄×C₄₇	—	4	2	no	—
121	C₂×C₄×C₄₇	—	4	2	no	—
122	C₂×C₄×C₄₇	—	4	2	no	—
123	C₂×C₄×C₄₇	—	4	1	no	—
124	C₂×C₄×C₄₇	—	4	2	no	—
125	C₂×C₄×C₄₇	—	4	1	no	—
126	C₂×C₄×C₄₇	#95	1	1	yes	characteristic
127	C₂×C₄×C₄₇	#95	1	2	yes	—
128	C₂×C₄×C₄₇	#95	1	1	yes	characteristic
129	C₂×C₄×C₄₇	#99	1	4	yes	—
130	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
131	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
133	C₂×Dic₄₇	—	2	16	no	—
134	C₂×Dic₄₇	—	4	8	no	—
135	C₂×Dic₄₇	—	4	32	no	—
136	C₂×Dic₄₇	—	4	8	no	—
137	C₈×C₄₇	—	4	4	no	—
138	C₂×#1297	—	47	1	no	maximal, Sylow
139	C₂²×C₄₇	—	2	16	no	—
140	C₂²×C₄₇	—	2	2	no	—
141	C₂²×C₄₇	—	2	2	no	—
142	C₂²×C₄₇	—	2	2	no	—
143	C₂²×C₄₇	—	2	2	no	—
144	C₂²×C₄₇	—	2	2	no	—
145	C₂²×C₄₇	—	2	2	no	—
146	C₂²×C₄₇	—	2	1	no	—
147	C₂²×C₄₇	—	2	1	no	—
148	C₂²×C₄₇	—	4	4	no	—
149	C₂²×C₄₇	—	4	8	no	—
150	C₂²×C₄₇	—	4	4	no	—
151	C₂²×C₄₇	—	4	4	no	—
152	C₂²×C₄₇	#339	1	4	yes	—
153	C₂²×C₄₇	#339	1	4	yes	—
154	C₂²×C₄₇	#342	1	2	yes	—
155	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
156	C₂²×C₄₇	C₂×#104	1	4	yes	—
157	C₄×C₄₇	—	2	1	no	—
158	C₄×C₄₇	—	2	2	no	—
159	C₄×C₄₇	—	2	1	no	—
160	C₄×C₄₇	—	2	2	no	—
161	C₄×C₄₇	#342	1	2	yes	—
162	C₄×C₄₇	C₂×#99	1	2	yes	—
163	Dic₄₇	—	4	16	no	—
164	C₂×#407	D₄₇	1	1	yes	characteristic
165	C₂×C₄×C₂²⋊C₄	—	47	2	no	—
166	#1297	—	47	4	no	—
167	C₂×C₄₇	—	2	4	no	—
168	C₂×C₄₇	—	2	4	no	—
169	C₂×C₄₇	—	4	1	no	—
170	C₂×C₄₇	—	4	1	no	—
171	C₂×C₄₇	#1297	1	4	yes	complemented
172	C₂×C₄₇	C₂×#339	1	2	yes	—
173	C₂×C₄₇	C₂×#342	1	1	yes	characteristic
174	C₂⁴×C₄	C₂×D₄₇	1	1	yes	characteristic
175	C₂²×M₁₆	Dic₄₇	1	2	yes	complemented
176	C₂²×C₂²⋊C₄	—	94	2	no	—
177	C₂²×C₄²	—	94	2	no	—
178	#407	—	2	2	no	—
179	#407	Dic₄₇	1	2	yes	complemented
180	#407	C₂×D₄₇	1	2	yes	—
181	C₂×C₂²⋊C₈	—	2	2	no	—
182	C₄×C₂²⋊C₄	—	47	4	no	—
183	C₄×C₂²⋊C₄	—	47	4	no	—
184	C₄×C₂²⋊C₄	—	94	2	no	—
185	C₄×C₂²⋊C₄	—	94	2	no	—
186	C₂×#95	—	47	4	no	—
187	C₄₇	C₂×#1297	1	1	yes	characteristic, complemented, Sylow
188	C₂⁵	D₁₈₈	1	1	yes	characteristic
189	C₂³×C₄	—	4	4	no	—
190	C₂³×C₄	—	94	4	no	—
191	C₂³×C₄	Dic₉₄	1	1	yes	characteristic
192	C₂³×C₄	Dic₉₄	1	1	yes	characteristic
193	C₂³×C₄	C₄×D₄₇	1	2	yes	—
194	C₂³×C₄	C₄×D₄₇	1	2	yes	—
195	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
196	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
197	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
198	C₂³×C₄	C₂²⋊D₄₇	1	2	yes	—
199	C₂³×C₄	C₂²⋊D₄₇	1	2	yes	—
200	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
201	C₂×M₁₆	—	2	2	no	—
202	C₂×M₁₆	—	4	4	no	—
203	C₂×M₁₆	C₂×Dic₄₇	1	4	yes	complemented
204	C₂²×C₈	—	2	2	no	—
205	C₂×C₂²⋊C₄	—	94	16	no	—
206	C₂×C₂²⋊C₄	—	94	4	no	—
207	C₂×C₂²⋊C₄	—	94	4	no	—
208	C₂×C₄²	—	94	4	no	—
209	C₂×C₄²	—	94	4	no	—
210	C₂×C₄²	—	188	8	no	—
211	C₂²⋊C₈	—	2	4	no	—
212	C₂²⋊C₈	—	2	4	no	—
213	#95	—	47	8	no	—
214	#95	—	94	4	no	—
215	C₂⁴	—	2	8	no	—
216	C₂⁴	—	4	2	no	—
217	C₂⁴	C₄⋊Dic₄₇	1	1	yes	characteristic
218	C₂⁴	C₄⋊Dic₄₇	1	1	yes	characteristic
219	C₂⁴	(C₂×D₄₇)⋊C₄	1	2	yes	—
220	C₂⁴	(C₂×D₄₇)⋊C₄	1	2	yes	—
221	C₂⁴	C₂×D₁₈₈	1	1	yes	characteristic
222	C₂²×C₄	—	2	8	no	—
223	C₂²×C₄	—	2	4	no	—
224	C₂²×C₄	—	4	2	no	—
225	C₂²×C₄	—	4	4	no	—
226	C₂²×C₄	—	4	2	no	—
227	C₂²×C₄	—	4	2	no	—
228	C₂²×C₄	—	4	4	no	—
229	C₂²×C₄	—	4	2	no	—
230	C₂²×C₄	—	4	2	no	—
231	C₂²×C₄	—	4	4	no	—
232	C₂²×C₄	—	94	16	no	—
233	C₂²×C₄	—	94	8	no	—
234	C₂²×C₄	—	188	8	no	—
235	C₂²×C₄	—	188	8	no	—
236	C₂²×C₄	C₄×Dic₄₇	1	1	yes	characteristic
237	C₂²×C₄	C₄×Dic₄₇	1	1	yes	characteristic
238	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
239	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
240	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
241	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
242	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
243	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
244	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
245	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
246	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
247	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
248	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
249	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
250	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
251	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
252	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
253	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
254	M₁₆	—	4	4	no	—
255	M₁₆	—	4	4	no	—
256	C₂×C₈	—	2	4	no	—
257	C₂×C₈	—	4	4	no	—
258	C₂²⋊C₄	—	94	16	no	—
259	C₂²⋊C₄	—	94	16	no	—
260	C₄²	—	188	8	no	—
261	C₄²	—	188	8	no	—
262	C₂³	—	2	4	no	—
263	C₂³	—	2	4	no	—
264	C₂³	—	2	4	no	—
265	C₂³	—	2	4	no	—
266	C₂³	—	2	4	no	—
267	C₂³	—	2	4	no	—
268	C₂³	—	2	4	no	—
269	C₂³	—	2	2	no	—
270	C₂³	—	4	1	no	—
271	C₂³	—	4	2	no	—
272	C₂³	—	4	1	no	—
273	C₂³	—	4	16	no	—
274	C₂³	C₄₇⋊#95	1	1	yes	characteristic
275	C₂³	C₄₇⋊#95	1	2	yes	—
276	C₂³	C₄₇⋊#95	1	1	yes	characteristic
277	C₂³	#13093950	1	8	yes	—
278	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	2	yes	—
279	C₂³	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	2	4	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	1	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	94	16	no	—
298	C₂×C₄	—	188	8	no	—
299	C₂×C₄	—	188	32	no	—
300	C₂×C₄	—	188	8	no	—
301	C₂×C₄	C₄₇⋊#95	1	1	yes	characteristic
302	C₂×C₄	C₄₇⋊#95	1	2	yes	—
303	C₂×C₄	C₄₇⋊#95	1	1	yes	characteristic
304	C₂×C₄	#13093939	1	4	yes	—
305	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
306	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
307	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
308	C₈	—	4	4	no	—
309	C₂²	—	2	16	no	—
310	C₂²	—	2	2	no	—
311	C₂²	—	2	2	no	—
312	C₂²	—	2	2	no	—
313	C₂²	—	2	2	no	—
314	C₂²	—	2	2	no	—
315	C₂²	—	2	2	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	4	4	no	—
319	C₂²	—	4	8	no	—
320	C₂²	—	4	4	no	—
321	C₂²	—	4	4	no	—
322	C₂²	#15688884	1	4	yes	—
323	C₂²	#15688884	1	4	yes	—
324	C₂²	#15688890	1	2	yes	—
325	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
326	C₂²	C₂×#13093950	1	4	yes	—
327	C₄	—	2	1	no	—
328	C₄	—	2	2	no	—
329	C₄	—	2	1	no	—
330	C₄	—	2	2	no	—
331	C₄	—	188	16	no	—
332	C₄	#15688890	1	2	yes	—
333	C₄	C₂×#13093939	1	2	yes	—
334	C₂	—	2	4	no	—
335	C₂	—	2	4	no	—
336	C₂	—	4	1	no	—
337	C₂	—	4	1	no	—
338	C₂	#20241273	1	4	yes	direct product factor
339	C₂	C₂×#15688884	1	2	yes	—
340	C₂	C₂×#15688890	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²⋊Dic₄₇
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₂×C₄₇×#407	C₂
socle	C₂³×C₄₇	#95
prime radical	C₂⁵×C₄₇	D₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂×#407	D₄₇
hypocenter	C₄₇	C₂×#1297
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄ ◁ C₂×#407

Complements

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

N	H ≅ G/N	type
2. #20241273	C₂	direct (N × H)
8. C₄×C₂²⋊Dic₄₇	C₄	semidirect
10. C₂×C₄₇⋊#95	C₄	semidirect
12. C₂²×M₁₆×C₄₇	C₄	semidirect
14. C₄₇×#407	C₄	semidirect
25. C₄₇⋊#95	C₂×C₄	semidirect
40. C₂×M₁₆×C₄₇	C₂×C₄	semidirect
171. C₂×C₄₇	#1297	semidirect
175. C₂²×M₁₆	Dic₄₇	semidirect
179. #407	Dic₄₇	semidirect
187. C₄₇	C₂×#1297	semidirect
203. C₂×M₁₆	C₂×Dic₄₇	semidirect
338. C₂	#20241273	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×C₄×C₂²⋊Dic₄₇	2
#20241273	2
C₂×C₄₇×#407	2
C₂×#1297	47

Subgroups — C2×#20241273 #26106397

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20241273 #26106397