Subgroups — C₂×#20243515 #26216461

Order 12032.

1182 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688993	C₂	1	1	yes	maximal, characteristic
3	#20243515	C₂	1	8	yes	maximal, direct product factor
4	C₂×#15688885	C₂	1	4	yes	maximal
5	C₂×C₄₇×#384	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₄₇⋊C₈	C₂²	1	2	yes	—
7	C₂²×C₄⋊Dic₄₇	—	2	1	no	—
8	C₂²×C₄⋊Dic₄₇	—	2	1	no	—
9	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
10	C₂×C₄×C₄₇⋊C₈	C₄	1	8	yes	complemented
11	C₂×C₄×C₄₇⋊C₈	C₂²	1	4	yes	—
12	#15688993	—	2	2	no	—
13	#15688993	C₂²	1	4	yes	—
14	#15688885	C₂²	1	16	yes	—
15	C₂²×C₄⋊C₄×C₄₇	—	2	1	no	—
16	C₂²×C₄⋊C₄×C₄₇	—	2	1	no	—
17	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
18	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
19	C₄₇×#384	—	2	2	no	—
20	C₄₇×#384	C₂²	1	4	yes	—
21	C₂³×Dic₄₇	—	2	2	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	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
26	C₂×C₄⋊Dic₄₇	—	2	8	no	—
27	C₂×C₄⋊Dic₄₇	—	2	2	no	—
28	C₂×C₄⋊Dic₄₇	—	2	4	no	—
29	C₂×C₄⋊Dic₄₇	—	2	4	no	—
30	C₂×C₄⋊Dic₄₇	—	4	4	no	—
31	C₂×C₄⋊Dic₄₇	—	4	4	no	—
32	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
33	C₄×C₄₇⋊C₈	C₂×C₄	1	16	yes	complemented
34	C₂³×C₄×C₄₇	—	2	2	no	—
35	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
36	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
37	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
38	C₂×C₄⋊C₄×C₄₇	—	2	8	no	—
39	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
40	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
41	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
42	C₂×C₄⋊C₄×C₄₇	—	4	4	no	—
43	C₂×C₄⋊C₄×C₄₇	—	4	4	no	—
44	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
45	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
46	C₂×C₄²×C₄₇	D₄	1	2	yes	—
47	C₂×C₄²×C₄₇	Q₈	1	4	yes	—
48	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
49	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₄₇	—	2	4	no	—
51	C₂²×C₄×C₄₇	—	4	4	no	—
52	C₂²×C₄×C₄₇	—	4	4	no	—
53	C₂²×C₄×C₄₇	—	4	4	no	—
54	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
55	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
56	C₂²×C₄×C₄₇	D₈	1	4	yes	—
57	C₂²×C₄×C₄₇	SD₁₆	1	4	yes	—
58	C₂²×C₄×C₄₇	SD₁₆	1	4	yes	—
59	C₂²×C₄×C₄₇	Q₁₆	1	4	yes	—
60	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
65	C₂²×C₄×C₄₇	#41	1	4	yes	—
66	C₂²×C₄×C₄₇	#41	1	4	yes	—
67	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
68	C₂²×Dic₄₇	—	2	4	no	—
69	C₂²×Dic₄₇	—	4	4	no	—
70	C₂²×Dic₄₇	—	4	4	no	—
71	C₂²×Dic₄₇	—	4	4	no	—
72	C₄⋊C₄×C₄₇	—	2	8	no	—
73	C₄⋊C₄×C₄₇	—	4	4	no	—
74	C₄⋊C₄×C₄₇	—	4	8	no	—
75	C₄⋊C₄×C₄₇	—	4	8	no	—
76	C₄²×C₄₇	C₄⋊C₄	1	8	yes	—
77	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
78	C₄²×C₄₇	C₂×Q₈	1	4	yes	—
79	C₄⋊Dic₄₇	—	2	8	no	—
80	C₄⋊Dic₄₇	—	4	4	no	—
81	C₄⋊Dic₄₇	—	4	8	no	—
82	C₄⋊Dic₄₇	—	4	8	no	—
83	C₂×C₄₇⋊C₈	—	2	16	no	—
84	C₂×C₄₇⋊C₈	—	2	4	no	—
85	C₂×C₄₇⋊C₈	—	2	4	no	—
86	C₂×C₄₇⋊C₈	C₂²×C₄	1	8	yes	—
87	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
88	C₂³×C₄₇	#124	1	2	yes	—
89	C₂³×C₄₇	#124	1	2	yes	—
90	C₂³×C₄₇	C₂×D₈	1	2	yes	—
91	C₂³×C₄₇	C₂×SD₁₆	1	2	yes	—
92	C₂³×C₄₇	C₂×SD₁₆	1	2	yes	—
93	C₂³×C₄₇	C₂×Q₁₆	1	2	yes	—
94	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
95	C₂×C₄×C₄₇	—	4	4	no	—
96	C₂×C₄×C₄₇	—	4	4	no	—
97	C₂×C₄×C₄₇	—	4	4	no	—
98	C₂×C₄×C₄₇	—	8	8	no	—
99	C₂×C₄×C₄₇	#106	1	16	yes	—
100	C₂×C₄×C₄₇	#107	1	16	yes	—
101	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
102	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
103	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
104	C₂×C₄×C₄₇	#117	1	8	yes	—
105	C₂×C₄×C₄₇	#124	1	2	yes	—
106	C₂×C₄×C₄₇	#124	1	2	yes	—
107	C₂×C₄×C₄₇	#125	1	8	yes	—
108	C₂×C₄×C₄₇	C₂×D₈	1	4	yes	—
109	C₂×C₄×C₄₇	C₂×SD₁₆	1	4	yes	—
110	C₂×C₄×C₄₇	C₂×SD₁₆	1	4	yes	—
111	C₂×C₄×C₄₇	C₂×Q₁₆	1	4	yes	—
112	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
113	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
114	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
115	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
116	C₂×Dic₄₇	—	4	4	no	—
117	C₂×Dic₄₇	—	4	4	no	—
118	C₂×Dic₄₇	—	4	4	no	—
119	C₂×Dic₄₇	—	8	8	no	—
120	C₄₇⋊C₈	—	2	8	no	—
121	C₂×#1834	—	47	1	no	maximal, Sylow
122	C₂²×C₄₇	#382	1	4	yes	—
123	C₂²×C₄₇	C₂×#106	1	4	yes	—
124	C₂²×C₄₇	C₂×#107	1	4	yes	—
125	C₂²×C₄₇	#486	1	4	yes	—
126	C₂²×C₄₇	#486	1	4	yes	—
127	C₂²×C₄₇	#487	1	4	yes	—
128	C₂²×C₄₇	#487	1	4	yes	—
129	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
130	C₂²×C₄₇	C₂×#124	1	1	yes	characteristic
131	C₂²×C₄₇	C₂×#124	1	1	yes	characteristic
132	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
133	C₂²×C₄₇	C₂²×SD₁₆	1	1	yes	characteristic
134	C₂²×C₄₇	C₂²×SD₁₆	1	1	yes	characteristic
135	C₂²×C₄₇	C₂²×Q₁₆	1	1	yes	characteristic
136	C₄×C₄₇	—	8	4	no	—
137	C₄×C₄₇	#382	1	4	yes	—
138	C₄×C₄₇	C₂×#106	1	8	yes	—
139	C₄×C₄₇	C₂×#107	1	8	yes	—
140	C₄×C₄₇	C₂×#117	1	2	yes	—
141	C₄×C₄₇	C₂×#125	1	2	yes	—
142	Dic₄₇	—	8	4	no	—
143	C₂²×C₄×C₈	—	47	1	no	—
144	C₂×#384	—	47	1	no	—
145	C₂×#384	D₄₇	1	1	yes	characteristic
146	#1834	—	47	8	no	—
147	C₂×#340	—	47	4	no	—
148	C₂×C₄₇	#1834	1	8	yes	complemented
149	C₂×C₄₇	C₂×#382	1	1	yes	characteristic
150	C₂×C₄₇	C₂²×#106	1	1	yes	characteristic
151	C₂×C₄₇	C₂²×#107	1	1	yes	characteristic
152	C₂×C₄₇	C₂×#486	1	1	yes	characteristic
153	C₂×C₄₇	C₂×#486	1	1	yes	characteristic
154	C₂×C₄₇	C₂×#487	1	1	yes	characteristic
155	C₂×C₄₇	C₂×#487	1	1	yes	characteristic
156	C₂³×C₈	—	47	2	no	—
157	C₂²×C₄⋊C₄	—	2	1	no	—
158	C₂²×C₄⋊C₄	—	2	1	no	—
159	C₂²×C₄⋊C₄	—	47	2	no	—
160	C₂²×C₄⋊C₄	—	94	1	no	—
161	C₂²×C₄⋊C₄	—	94	1	no	—
162	C₂²×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
163	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
164	C₂×C₄×C₈	—	47	8	no	—
165	C₂×C₄×C₈	—	47	4	no	—
166	#384	—	2	2	no	—
167	#384	—	47	4	no	—
168	#384	—	94	2	no	—
169	#384	C₂×D₄₇	1	4	yes	—
170	#340	—	47	16	no	—
171	C₄₇	C₂×#1834	1	1	yes	characteristic, complemented, Sylow
172	C₂³×C₄	—	2	2	no	—
173	C₂³×C₄	—	94	2	no	—
174	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
175	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
176	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
177	C₂²×C₈	—	47	8	no	—
178	C₂²×C₈	—	47	4	no	—
179	C₂²×C₈	—	94	4	no	—
180	C₂²×C₈	—	94	4	no	—
181	C₂×C₄⋊C₄	—	2	8	no	—
182	C₂×C₄⋊C₄	—	2	4	no	—
183	C₂×C₄⋊C₄	—	2	4	no	—
184	C₂×C₄⋊C₄	—	2	2	no	—
185	C₂×C₄⋊C₄	—	4	4	no	—
186	C₂×C₄⋊C₄	—	4	4	no	—
187	C₂×C₄⋊C₄	—	47	4	no	—
188	C₂×C₄⋊C₄	—	94	8	no	—
189	C₂×C₄⋊C₄	—	94	2	no	—
190	C₂×C₄⋊C₄	—	94	4	no	—
191	C₂×C₄⋊C₄	—	94	4	no	—
192	C₂×C₄⋊C₄	—	188	4	no	—
193	C₂×C₄⋊C₄	—	188	4	no	—
194	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
195	C₂×C₄²	Dic₉₄	1	4	yes	—
196	C₂×C₄²	C₄×D₄₇	1	4	yes	—
197	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
198	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
199	C₄×C₈	—	47	16	no	—
200	C₂⁴	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
201	C₂²×C₄	—	2	4	no	—
202	C₂²×C₄	—	4	4	no	—
203	C₂²×C₄	—	4	4	no	—
204	C₂²×C₄	—	4	4	no	—
205	C₂²×C₄	—	94	4	no	—
206	C₂²×C₄	—	188	4	no	—
207	C₂²×C₄	—	188	4	no	—
208	C₂²×C₄	—	188	4	no	—
209	C₂²×C₄	Dic₄₇⋊C₄	1	4	yes	—
210	C₂²×C₄	Dic₄₇⋊C₄	1	4	yes	—
211	C₂²×C₄	#10625334	1	4	yes	—
212	C₂²×C₄	#10625335	1	4	yes	—
213	C₂²×C₄	#10625336	1	4	yes	—
214	C₂²×C₄	#10625337	1	4	yes	—
215	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
216	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
217	C₂²×C₄	#10625352	1	4	yes	—
218	C₂²×C₄	#10625354	1	4	yes	—
219	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
220	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
221	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
222	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
223	C₂×C₈	—	47	8	no	—
224	C₂×C₈	—	94	16	no	—
225	C₂×C₈	—	94	4	no	—
226	C₂×C₈	—	94	4	no	—
227	C₄⋊C₄	—	2	8	no	—
228	C₄⋊C₄	—	4	4	no	—
229	C₄⋊C₄	—	4	8	no	—
230	C₄⋊C₄	—	4	8	no	—
231	C₄⋊C₄	—	94	8	no	—
232	C₄⋊C₄	—	188	4	no	—
233	C₄⋊C₄	—	188	8	no	—
234	C₄⋊C₄	—	188	8	no	—
235	C₄²	Dic₄₇⋊C₄	1	8	yes	—
236	C₄²	C₂×Dic₉₄	1	4	yes	—
237	C₄²	C₂×C₄×D₄₇	1	4	yes	—
238	C₂³	C₂×Dic₄₇⋊C₄	1	2	yes	—
239	C₂³	#13094011	1	2	yes	—
240	C₂³	#13094013	1	2	yes	—
241	C₂³	C₂×#10625334	1	2	yes	—
242	C₂³	C₂×#10625336	1	2	yes	—
243	C₂³	C₂×#10625335	1	2	yes	—
244	C₂³	C₂×#10625337	1	2	yes	—
245	C₂³	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
246	C₂×C₄	—	4	4	no	—
247	C₂×C₄	—	4	4	no	—
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	—	8	8	no	—
250	C₂×C₄	—	376	8	no	—
251	C₂×C₄	—	188	4	no	—
252	C₂×C₄	—	188	4	no	—
253	C₂×C₄	—	188	4	no	—
254	C₂×C₄	#13093955	1	16	yes	—
255	C₂×C₄	#13093957	1	16	yes	—
256	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
257	C₂×C₄	C₂×Dic₄₇⋊C₄	1	4	yes	—
258	C₂×C₄	C₂×Dic₄₇⋊C₄	1	4	yes	—
259	C₂×C₄	#13093982	1	8	yes	—
260	C₂×C₄	#13094011	1	2	yes	—
261	C₂×C₄	#13094013	1	2	yes	—
262	C₂×C₄	#13094015	1	8	yes	—
263	C₂×C₄	C₂×#10625334	1	4	yes	—
264	C₂×C₄	C₂×#10625336	1	4	yes	—
265	C₂×C₄	C₂×#10625335	1	4	yes	—
266	C₂×C₄	C₂×#10625337	1	4	yes	—
267	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
268	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
269	C₂×C₄	C₂×#10625352	1	2	yes	—
270	C₂×C₄	C₂×#10625354	1	2	yes	—
271	C₈	—	94	8	no	—
272	C₂²	#15688988	1	4	yes	—
273	C₂²	C₂×#13093955	1	4	yes	—
274	C₂²	C₂×#13093957	1	4	yes	—
275	C₂²	#15689469	1	4	yes	—
276	C₂²	#15689471	1	4	yes	—
277	C₂²	#15689473	1	4	yes	—
278	C₂²	#15689475	1	4	yes	—
279	C₂²	C₂²×Dic₄₇⋊C₄	1	1	yes	characteristic
280	C₂²	C₂×#13094011	1	1	yes	characteristic
281	C₂²	C₂×#13094013	1	1	yes	characteristic
282	C₂²	C₂²×#10625334	1	1	yes	characteristic
283	C₂²	C₂²×#10625336	1	1	yes	characteristic
284	C₂²	C₂²×#10625335	1	1	yes	characteristic
285	C₂²	C₂²×#10625337	1	1	yes	characteristic
286	C₄	—	8	4	no	—
287	C₄	—	376	4	no	—
288	C₄	#15688988	1	4	yes	—
289	C₄	C₂×#13093955	1	8	yes	—
290	C₄	C₂×#13093957	1	8	yes	—
291	C₄	C₂×#13093982	1	2	yes	—
292	C₄	C₂×#13094015	1	2	yes	—
293	C₂	#20243515	1	8	yes	direct product factor
294	C₂	C₂×#15688988	1	1	yes	characteristic
295	C₂	C₂²×#13093955	1	1	yes	characteristic
296	C₂	C₂²×#13093957	1	1	yes	characteristic
297	C₂	C₂×#15689469	1	1	yes	characteristic
298	C₂	C₂×#15689471	1	1	yes	characteristic
299	C₂	C₂×#15689473	1	1	yes	characteristic
300	C₂	C₂×#15689475	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×#384	C₂
socle, prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#384	D₄₇
hypocenter	C₄₇	C₂×#1834
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂×#384

Complements

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

N	H ≅ G/N	type
3. #20243515	C₂	direct (N × H)
10. C₂×C₄×C₄₇⋊C₈	C₄	semidirect
33. C₄×C₄₇⋊C₈	C₂×C₄	semidirect
148. C₂×C₄₇	#1834	semidirect
171. C₄₇	C₂×#1834	semidirect
293. C₂	#20243515	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×C₄×C₄₇⋊C₈	2
C₂×#15688993	2
#20243515	2
C₂×#15688885	2
C₂×C₄₇×#384	2
C₂×#1834	47

Subgroups — C2×#20243515 #26216461

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20243515 #26216461