Subgroups — C₂×#20241275 #26106417

Order 12032.

938 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂²⋊(C₄₇⋊C₈)	C₂	1	2	yes	maximal
2	#20241275	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₄₇×#407	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₄₇⋊C₈	—	2	2	no	—
5	C₂×C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
6	C₂×C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
7	C₂×C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
8	C₂×C₂²⋊(C₄₇⋊C₈)	C₄	1	4	yes	—
9	C₂×C₂²⋊(C₄₇⋊C₈)	C₄	1	4	yes	—
10	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
11	C₂⁴×C₄×C₄₇	C₂²	1	1	yes	characteristic
12	C₂²×M₁₆×C₄₇	C₄	1	2	yes	—
13	C₄₇×#407	—	2	2	no	—
14	C₄₇×#407	C₄	1	2	yes	—
15	C₄₇×#407	C₂²	1	2	yes	—
16	C₂×C₂²⋊C₈×C₄₇	—	2	2	no	—
17	C₂²×C₄₇⋊C₈	—	2	4	no	—
18	C₂²×C₄₇⋊C₈	—	2	4	no	—
19	C₂²×C₄₇⋊C₈	—	2	4	no	—
20	C₂²×C₄₇⋊C₈	—	4	8	no	—
21	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
22	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
23	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
24	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	—
25	C₂⁵×C₄₇	Q₈	1	1	yes	characteristic
26	C₂³×C₄×C₄₇	—	4	4	no	—
27	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
29	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
30	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
31	C₂³×C₄×C₄₇	D₄	1	2	yes	—
32	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
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	C₂³	1	1	yes	characteristic
37	C₂×M₁₆×C₄₇	—	2	2	no	—
38	C₂×M₁₆×C₄₇	—	4	4	no	—
39	C₂×M₁₆×C₄₇	C₂×C₄	1	4	yes	—
40	C₂²×C₈×C₄₇	—	2	2	no	—
41	C₂²⋊C₈×C₄₇	—	2	4	no	—
42	C₂²⋊C₈×C₄₇	—	2	4	no	—
43	C₂⁴×C₄₇	—	2	8	no	—
44	C₂⁴×C₄₇	—	4	2	no	—
45	C₂⁴×C₄₇	C₄⋊C₄	1	1	yes	characteristic
46	C₂⁴×C₄₇	C₄⋊C₄	1	2	yes	—
47	C₂⁴×C₄₇	C₄⋊C₄	1	1	yes	characteristic
48	C₂⁴×C₄₇	C₄⋊C₄	1	2	yes	—
49	C₂⁴×C₄₇	C₂×Q₈	1	1	yes	characteristic
50	C₂²×C₄×C₄₇	—	2	4	no	—
51	C₂²×C₄×C₄₇	—	2	8	no	—
52	C₂²×C₄×C₄₇	—	4	4	no	—
53	C₂²×C₄×C₄₇	—	4	2	no	—
54	C₂²×C₄×C₄₇	—	4	2	no	—
55	C₂²×C₄×C₄₇	—	4	4	no	—
56	C₂²×C₄×C₄₇	—	4	2	no	—
57	C₂²×C₄×C₄₇	—	4	2	no	—
58	C₂²×C₄×C₄₇	—	4	4	no	—
59	C₂²×C₄×C₄₇	—	4	2	no	—
60	C₂²×C₄×C₄₇	C₄²	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	C₄²	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
63	C₂²×C₄×C₄₇	C₂²⋊C₄	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	2	yes	—
69	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
71	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
73	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
78	M₁₆×C₄₇	—	4	4	no	—
79	M₁₆×C₄₇	—	4	4	no	—
80	C₂×C₈×C₄₇	—	2	4	no	—
81	C₂×C₈×C₄₇	—	4	4	no	—
82	C₂×C₄₇⋊C₈	—	2	8	no	—
83	C₂×C₄₇⋊C₈	—	4	8	no	—
84	C₂×C₄₇⋊C₈	—	4	8	no	—
85	C₂×C₄₇⋊C₈	—	4	8	no	—
86	C₂³×C₄₇	—	2	4	no	—
87	C₂³×C₄₇	—	2	4	no	—
88	C₂³×C₄₇	—	2	4	no	—
89	C₂³×C₄₇	—	2	4	no	—
90	C₂³×C₄₇	—	2	4	no	—
91	C₂³×C₄₇	—	2	2	no	—
92	C₂³×C₄₇	—	2	4	no	—
93	C₂³×C₄₇	—	2	4	no	—
94	C₂³×C₄₇	—	4	2	no	—
95	C₂³×C₄₇	—	4	1	no	—
96	C₂³×C₄₇	—	4	1	no	—
97	C₂³×C₄₇	—	4	16	no	—
98	C₂³×C₄₇	#95	1	1	yes	characteristic
99	C₂³×C₄₇	#95	1	2	yes	—
100	C₂³×C₄₇	#95	1	1	yes	characteristic
101	C₂³×C₄₇	#108	1	8	yes	—
102	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
103	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₄₇	—	2	4	no	—
105	C₂×C₄×C₄₇	—	2	2	no	—
106	C₂×C₄×C₄₇	—	2	4	no	—
107	C₂×C₄×C₄₇	—	2	2	no	—
108	C₂×C₄×C₄₇	—	2	4	no	—
109	C₂×C₄×C₄₇	—	2	2	no	—
110	C₂×C₄×C₄₇	—	2	4	no	—
111	C₂×C₄×C₄₇	—	2	2	no	—
112	C₂×C₄×C₄₇	—	2	2	no	—
113	C₂×C₄×C₄₇	—	2	4	no	—
114	C₂×C₄×C₄₇	—	4	2	no	—
115	C₂×C₄×C₄₇	—	4	2	no	—
116	C₂×C₄×C₄₇	—	4	2	no	—
117	C₂×C₄×C₄₇	—	4	2	no	—
118	C₂×C₄×C₄₇	—	4	1	no	—
119	C₂×C₄×C₄₇	—	4	2	no	—
120	C₂×C₄×C₄₇	—	4	1	no	—
121	C₂×C₄×C₄₇	#95	1	1	yes	characteristic
122	C₂×C₄×C₄₇	#95	1	2	yes	—
123	C₂×C₄×C₄₇	#95	1	1	yes	characteristic
124	C₂×C₄×C₄₇	#99	1	4	yes	—
125	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
126	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
127	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	C₈×C₄₇	—	4	4	no	—
129	C₄₇⋊C₈	—	4	8	no	—
130	C₂×#1298	—	47	1	no	maximal, Sylow
131	C₂²×C₄₇	—	2	16	no	—
132	C₂²×C₄₇	—	2	2	no	—
133	C₂²×C₄₇	—	2	2	no	—
134	C₂²×C₄₇	—	2	2	no	—
135	C₂²×C₄₇	—	2	2	no	—
136	C₂²×C₄₇	—	2	2	no	—
137	C₂²×C₄₇	—	2	2	no	—
138	C₂²×C₄₇	—	2	1	no	—
139	C₂²×C₄₇	—	2	1	no	—
140	C₂²×C₄₇	—	4	4	no	—
141	C₂²×C₄₇	—	4	8	no	—
142	C₂²×C₄₇	—	4	4	no	—
143	C₂²×C₄₇	—	4	4	no	—
144	C₂²×C₄₇	#341	1	4	yes	—
145	C₂²×C₄₇	#341	1	4	yes	—
146	C₂²×C₄₇	#342	1	2	yes	—
147	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
148	C₂²×C₄₇	C₂×#108	1	4	yes	—
149	C₄×C₄₇	—	2	2	no	—
150	C₄×C₄₇	—	2	2	no	—
151	C₄×C₄₇	—	2	1	no	—
152	C₄×C₄₇	—	2	1	no	—
153	C₄×C₄₇	#342	1	2	yes	—
154	C₄×C₄₇	C₂×#99	1	2	yes	—
155	C₂×#407	D₄₇	1	1	yes	characteristic
156	C₂²×C₂²⋊C₈	—	47	2	no	—
157	#1298	—	47	4	no	—
158	C₂×C₄₇	—	2	4	no	—
159	C₂×C₄₇	—	2	4	no	—
160	C₂×C₄₇	—	4	1	no	—
161	C₂×C₄₇	—	4	1	no	—
162	C₂×C₄₇	#1298	1	4	yes	complemented
163	C₂×C₄₇	C₂×#341	1	2	yes	—
164	C₂×C₄₇	C₂×#342	1	1	yes	characteristic
165	C₂⁴×C₄	C₂×D₄₇	1	1	yes	characteristic
166	C₂²×M₁₆	Dic₄₇	1	2	yes	—
167	C₂³×C₈	—	94	2	no	—
168	#407	—	2	2	no	—
169	#407	Dic₄₇	1	2	yes	—
170	#407	C₂×D₄₇	1	2	yes	—
171	C₂×C₂²⋊C₈	—	2	2	no	—
172	C₂×C₂²⋊C₈	—	47	4	no	—
173	C₂×C₂²⋊C₈	—	47	4	no	—
174	C₂×C₂²⋊C₈	—	47	4	no	—
175	C₂×C₂²⋊C₈	—	94	2	no	—
176	C₂×C₂²⋊C₈	—	94	2	no	—
177	C₂×C₂²⋊C₈	—	94	2	no	—
178	C₄₇	C₂×#1298	1	1	yes	characteristic, complemented, Sylow
179	C₂⁵	Dic₉₄	1	1	yes	characteristic
180	C₂³×C₄	—	4	4	no	—
181	C₂³×C₄	Dic₉₄	1	1	yes	characteristic
182	C₂³×C₄	C₄×D₄₇	1	2	yes	—
183	C₂³×C₄	C₄×D₄₇	1	2	yes	—
184	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
185	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
186	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
187	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
188	C₂³×C₄	C₂²⋊D₄₇	1	2	yes	—
189	C₂³×C₄	C₂²⋊D₄₇	1	2	yes	—
190	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
191	C₂×M₁₆	—	2	2	no	—
192	C₂×M₁₆	—	4	4	no	—
193	C₂×M₁₆	C₂×Dic₄₇	1	4	yes	—
194	C₂²×C₈	—	2	2	no	—
195	C₂²×C₈	—	94	4	no	—
196	C₂²×C₈	—	94	4	no	—
197	C₂²×C₈	—	94	4	no	—
198	C₂²×C₈	—	188	8	no	—
199	C₂²⋊C₈	—	2	4	no	—
200	C₂²⋊C₈	—	2	4	no	—
201	C₂²⋊C₈	—	47	8	no	—
202	C₂²⋊C₈	—	94	4	no	—
203	C₂²⋊C₈	—	94	4	no	—
204	C₂²⋊C₈	—	94	4	no	—
205	C₂⁴	—	2	8	no	—
206	C₂⁴	—	4	2	no	—
207	C₂⁴	Dic₄₇⋊C₄	1	2	yes	—
208	C₂⁴	Dic₄₇⋊C₄	1	2	yes	—
209	C₂⁴	C₄⋊Dic₄₇	1	1	yes	characteristic
210	C₂⁴	C₄⋊Dic₄₇	1	1	yes	characteristic
211	C₂⁴	C₂×Dic₉₄	1	1	yes	characteristic
212	C₂²×C₄	—	2	8	no	—
213	C₂²×C₄	—	2	4	no	—
214	C₂²×C₄	—	4	2	no	—
215	C₂²×C₄	—	4	4	no	—
216	C₂²×C₄	—	4	2	no	—
217	C₂²×C₄	—	4	2	no	—
218	C₂²×C₄	—	4	4	no	—
219	C₂²×C₄	—	4	2	no	—
220	C₂²×C₄	—	4	4	no	—
221	C₂²×C₄	—	4	2	no	—
222	C₂²×C₄	C₄×Dic₄₇	1	1	yes	characteristic
223	C₂²×C₄	C₄×Dic₄₇	1	1	yes	characteristic
224	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
225	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
226	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
227	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
228	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
229	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
230	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
231	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
232	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
233	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
234	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
235	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
236	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
237	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
238	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
239	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
240	M₁₆	—	4	4	no	—
241	M₁₆	—	4	4	no	—
242	C₂×C₈	—	2	4	no	—
243	C₂×C₈	—	4	4	no	—
244	C₂×C₈	—	94	8	no	—
245	C₂×C₈	—	188	8	no	—
246	C₂×C₈	—	188	8	no	—
247	C₂×C₈	—	188	8	no	—
248	C₂³	—	2	4	no	—
249	C₂³	—	2	4	no	—
250	C₂³	—	2	4	no	—
251	C₂³	—	2	4	no	—
252	C₂³	—	2	4	no	—
253	C₂³	—	2	2	no	—
254	C₂³	—	2	4	no	—
255	C₂³	—	2	4	no	—
256	C₂³	—	4	2	no	—
257	C₂³	—	4	1	no	—
258	C₂³	—	4	1	no	—
259	C₂³	—	4	16	no	—
260	C₂³	C₄₇⋊#95	1	1	yes	characteristic
261	C₂³	C₄₇⋊#95	1	2	yes	—
262	C₂³	C₄₇⋊#95	1	1	yes	characteristic
263	C₂³	#13093959	1	8	yes	—
264	C₂³	C₂×Dic₄₇⋊C₄	1	2	yes	—
265	C₂³	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	1	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	1	no	—
283	C₂×C₄	C₄₇⋊#95	1	1	yes	characteristic
284	C₂×C₄	C₄₇⋊#95	1	2	yes	—
285	C₂×C₄	C₄₇⋊#95	1	1	yes	characteristic
286	C₂×C₄	#13093939	1	4	yes	—
287	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
288	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	2	yes	—
289	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
290	C₈	—	4	4	no	—
291	C₈	—	188	8	no	—
292	C₂²	—	2	16	no	—
293	C₂²	—	2	2	no	—
294	C₂²	—	2	2	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	2	2	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	1	no	—
300	C₂²	—	2	1	no	—
301	C₂²	—	4	4	no	—
302	C₂²	—	4	8	no	—
303	C₂²	—	4	4	no	—
304	C₂²	—	4	4	no	—
305	C₂²	#15688887	1	4	yes	—
306	C₂²	#15688887	1	4	yes	—
307	C₂²	#15688890	1	2	yes	—
308	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
309	C₂²	C₂×#13093959	1	4	yes	—
310	C₄	—	2	2	no	—
311	C₄	—	2	2	no	—
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	#15688890	1	2	yes	—
315	C₄	C₂×#13093939	1	2	yes	—
316	C₂	—	2	4	no	—
317	C₂	—	2	4	no	—
318	C₂	—	4	1	no	—
319	C₂	—	4	1	no	—
320	C₂	#20241275	1	4	yes	direct product factor
321	C₂	C₂×#15688887	1	2	yes	—
322	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	Q₈
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂×#407	D₄₇
hypocenter	C₄₇	C₂×#1298
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. #20241275	C₂	direct (N × H)
162. C₂×C₄₇	#1298	semidirect
178. C₄₇	C₂×#1298	semidirect
320. C₂	#20241275	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₂²⋊(C₄₇⋊C₈)	2
#20241275	2
C₂×C₄₇×#407	2
C₂×#1298	47

Subgroups — C2×#20241275 #26106417

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20241275 #26106417