Subgroups — C₂×#20243456 #26215979

Order 12032.

2110 conjugacy classes of subgroups.

The subgroup lattice has 300 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	#20243456	C₂	1	8	yes	maximal, direct product factor
3	C₂×#15688878	C₂	1	4	yes	maximal, complemented
4	C₂³×C₄²×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₄₇⋊C₈	C₂²	1	4	yes	—
6	C₂×C₂²⋊(C₄₇⋊C₈)	—	2	8	no	—
7	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	8	yes	—
8	#15688878	C₂²	1	16	yes	complemented
9	C₂⁴×C₄×C₄₇	C₄	1	1	yes	characteristic
10	C₂⁴×C₄×C₄₇	C₄	1	1	yes	characteristic
11	C₂⁴×C₄×C₄₇	C₂²	1	1	yes	characteristic
12	C₂²×C₄²×C₄₇	—	2	8	no	—
13	C₂²×C₄²×C₄₇	C₄	1	4	yes	—
14	C₂²×C₄²×C₄₇	C₄	1	2	yes	—
15	C₂²×C₄²×C₄₇	C₂²	1	4	yes	—
16	C₂²×C₄²×C₄₇	C₂²	1	2	yes	—
17	C₂²×C₄₇⋊C₈	—	2	8	no	—
18	C₂²×C₄₇⋊C₈	—	2	16	no	—
19	C₂²×C₄₇⋊C₈	C₂³	1	8	yes	—
20	C₂²⋊(C₄₇⋊C₈)	—	2	16	no	—
21	C₂⁵×C₄₇	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₄₇	—	2	4	no	—
23	C₂³×C₄×C₄₇	—	2	4	no	—
24	C₂³×C₄×C₄₇	—	2	4	no	—
25	C₂³×C₄×C₄₇	—	2	8	no	—
26	C₂³×C₄×C₄₇	—	2	8	no	—
27	C₂³×C₄×C₄₇	C₈	1	8	yes	complemented
28	C₂³×C₄×C₄₇	C₂×C₄	1	4	yes	—
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	C₂×C₄	1	4	yes	—
33	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
34	C₂³×C₄×C₄₇	D₄	1	8	yes	—
35	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
36	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
37	C₂×C₄²×C₄₇	—	2	8	no	—
38	C₂×C₄²×C₄₇	—	2	16	no	—
39	C₂×C₄²×C₄₇	—	2	16	no	—
40	C₂×C₄²×C₄₇	—	2	8	no	—
41	C₂×C₄²×C₄₇	C₂×C₄	1	8	yes	—
42	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
43	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
44	C₂⁴×C₄₇	—	2	4	no	—
45	C₂⁴×C₄₇	—	2	4	no	—
46	C₂⁴×C₄₇	C₂²⋊C₄	1	4	yes	—
47	C₂⁴×C₄₇	C₂×C₈	1	4	yes	—
48	C₂⁴×C₄₇	M₁₆	1	4	yes	—
49	C₂⁴×C₄₇	C₂²×C₄	1	2	yes	—
50	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₄₇	—	2	32	no	—
52	C₂²×C₄×C₄₇	—	2	32	no	—
53	C₂²×C₄×C₄₇	—	2	8	no	—
54	C₂²×C₄×C₄₇	—	2	4	no	—
55	C₂²×C₄×C₄₇	—	2	8	no	—
56	C₂²×C₄×C₄₇	—	2	4	no	—
57	C₂²×C₄×C₄₇	—	2	4	no	—
58	C₂²×C₄×C₄₇	—	2	4	no	—
59	C₂²×C₄×C₄₇	—	2	8	no	—
60	C₂²×C₄×C₄₇	—	2	4	no	—
61	C₂²×C₄×C₄₇	—	2	8	no	—
62	C₂²×C₄×C₄₇	—	2	32	no	—
63	C₂²×C₄×C₄₇	—	2	16	no	—
64	C₂²×C₄×C₄₇	—	2	8	no	—
65	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
66	C₂²×C₄×C₄₇	C₂²⋊C₄	1	8	yes	—
67	C₂²×C₄×C₄₇	C₂×C₈	1	8	yes	—
68	C₂²×C₄×C₄₇	C₂×C₈	1	16	yes	complemented
69	C₂²×C₄×C₄₇	M₁₆	1	8	yes	—
70	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
71	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
72	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
73	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₇	C₂×D₄	1	8	yes	—
75	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
76	C₂²×C₄×C₄₇	#41	1	8	yes	—
77	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
78	C₄²×C₄₇	—	2	16	no	—
79	C₄²×C₄₇	—	2	8	no	—
80	C₄²×C₄₇	—	2	8	no	—
81	C₂×C₄₇⋊C₈	—	2	8	no	—
82	C₂×C₄₇⋊C₈	—	2	16	no	—
83	C₂×C₄₇⋊C₈	—	4	16	no	—
84	C₂³×C₄₇	—	2	4	no	—
85	C₂³×C₄₇	—	2	4	no	—
86	C₂³×C₄₇	—	2	8	no	—
87	C₂³×C₄₇	—	2	8	no	—
88	C₂³×C₄₇	—	2	16	no	—
89	C₂³×C₄₇	—	2	16	no	—
90	C₂³×C₄₇	C₂²⋊C₈	1	16	yes	—
91	C₂³×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
92	C₂³×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
93	C₂³×C₄₇	#117	1	4	yes	—
94	C₂³×C₄₇	C₂²×C₈	1	4	yes	—
95	C₂³×C₄₇	C₂²×C₈	1	2	yes	—
96	C₂³×C₄₇	C₂×M₁₆	1	4	yes	—
97	C₂³×C₄₇	C₂×M₁₆	1	2	yes	—
98	C₂³×C₄₇	C₂×M₁₆	1	4	yes	—
99	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
100	C₂×C₄×C₄₇	—	2	32	no	—
101	C₂×C₄×C₄₇	—	2	4	no	—
102	C₂×C₄×C₄₇	—	2	4	no	—
103	C₂×C₄×C₄₇	—	2	4	no	—
104	C₂×C₄×C₄₇	—	2	4	no	—
105	C₂×C₄×C₄₇	—	2	4	no	—
106	C₂×C₄×C₄₇	—	2	8	no	—
107	C₂×C₄×C₄₇	—	2	16	no	—
108	C₂×C₄×C₄₇	—	2	32	no	—
109	C₂×C₄×C₄₇	—	2	16	no	—
110	C₂×C₄×C₄₇	—	2	16	no	—
111	C₂×C₄×C₄₇	—	2	16	no	—
112	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
113	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
114	C₂×C₄×C₄₇	#117	1	4	yes	—
115	C₂×C₄×C₄₇	#123	1	16	yes	—
116	C₂×C₄×C₄₇	C₂²×C₈	1	8	yes	—
117	C₂×C₄×C₄₇	C₂×M₁₆	1	8	yes	—
118	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
119	C₂×C₄×C₄₇	C₂×#41	1	4	yes	—
120	C₄₇⋊C₈	—	4	8	no	—
121	C₂×#1817	—	47	1	no	maximal, Sylow
122	C₂²×C₄₇	—	2	32	no	—
123	C₂²×C₄₇	—	2	4	no	—
124	C₂²×C₄₇	—	2	8	no	—
125	C₂²×C₄₇	—	2	8	no	—
126	C₂²×C₄₇	—	2	4	no	—
127	C₂²×C₄₇	#381	1	4	yes	—
128	C₂²×C₄₇	C₂×C₂²⋊C₈	1	8	yes	—
129	C₂²×C₄₇	C₂×C₂²⋊C₈	1	4	yes	—
130	C₂²×C₄₇	#407	1	4	yes	—
131	C₂²×C₄₇	#431	1	8	yes	—
132	C₂²×C₄₇	#432	1	8	yes	—
133	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
134	C₂²×C₄₇	C₂×#117	1	2	yes	—
135	C₂²×C₄₇	C₂³×C₈	1	1	yes	characteristic
136	C₂²×C₄₇	C₂²×M₁₆	1	1	yes	characteristic
137	C₂²×C₄₇	C₂²×M₁₆	1	2	yes	—
138	C₄×C₄₇	—	2	8	no	—
139	C₄×C₄₇	—	2	8	no	—
140	C₄×C₄₇	—	2	4	no	—
141	C₄×C₄₇	#381	1	4	yes	—
142	C₄×C₄₇	C₂×#123	1	4	yes	—
143	C₂³×C₄²	D₄₇	1	1	yes	characteristic
144	C₂²×C₂²⋊C₈	—	47	2	no	—
145	#1817	—	47	8	no	—
146	C₂×#336	—	47	4	no	—
147	C₂×C₄₇	—	2	8	no	—
148	C₂×C₄₇	#1817	1	8	yes	complemented
149	C₂×C₄₇	C₂×#381	1	1	yes	characteristic
150	C₂×C₄₇	C₂²×C₂²⋊C₈	1	1	yes	characteristic
151	C₂×C₄₇	C₂×#407	1	1	yes	characteristic
152	C₂×C₄₇	C₂×#431	1	2	yes	—
153	C₂×C₄₇	C₂×#432	1	2	yes	—
154	C₂⁴×C₄	Dic₄₇	1	1	yes	characteristic
155	C₂⁴×C₄	Dic₄₇	1	1	yes	characteristic
156	C₂⁴×C₄	C₂×D₄₇	1	1	yes	characteristic
157	C₂³×C₈	—	47	4	no	—
158	C₂²×C₄²	—	2	8	no	—
159	C₂²×C₄²	Dic₄₇	1	4	yes	—
160	C₂²×C₄²	Dic₄₇	1	2	yes	—
161	C₂²×C₄²	C₂×D₄₇	1	4	yes	—
162	C₂²×C₄²	C₂×D₄₇	1	2	yes	—
163	C₂×C₂²⋊C₈	—	47	8	no	—
164	C₂×C₂²⋊C₈	—	94	8	no	—
165	#336	—	47	16	no	—
166	C₄₇	C₂×#1817	1	1	yes	characteristic, complemented, Sylow
167	C₂⁵	C₂×Dic₄₇	1	1	yes	characteristic
168	C₂³×C₄	—	2	4	no	—
169	C₂³×C₄	—	2	4	no	—
170	C₂³×C₄	—	2	4	no	—
171	C₂³×C₄	—	2	8	no	—
172	C₂³×C₄	—	2	8	no	—
173	C₂³×C₄	C₄₇⋊C₈	1	8	yes	complemented
174	C₂³×C₄	C₂×Dic₄₇	1	4	yes	—
175	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
176	C₂³×C₄	C₂×Dic₄₇	1	4	yes	—
177	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
178	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
179	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
180	C₂³×C₄	C₂²⋊D₄₇	1	8	yes	—
181	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
182	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
183	C₂²×C₈	—	47	8	no	—
184	C₂²×C₈	—	94	16	no	—
185	C₂²×C₈	—	94	8	no	—
186	C₂×C₄²	—	2	8	no	—
187	C₂×C₄²	—	2	16	no	—
188	C₂×C₄²	—	2	16	no	—
189	C₂×C₄²	—	2	8	no	—
190	C₂×C₄²	C₂×Dic₄₇	1	8	yes	—
191	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
192	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
193	C₂²⋊C₈	—	94	16	no	—
194	C₂⁴	—	2	4	no	—
195	C₂⁴	—	2	4	no	—
196	C₂⁴	C₂×C₄₇⋊C₈	1	4	yes	—
197	C₂⁴	#10625329	1	4	yes	—
198	C₂⁴	C₂²⋊Dic₄₇	1	4	yes	—
199	C₂⁴	C₂²×Dic₄₇	1	2	yes	—
200	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
201	C₂²×C₄	—	2	32	no	—
202	C₂²×C₄	—	2	32	no	—
203	C₂²×C₄	—	2	8	no	—
204	C₂²×C₄	—	2	4	no	—
205	C₂²×C₄	—	2	8	no	—
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	—	2	4	no	—
209	C₂²×C₄	—	2	8	no	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	2	8	no	—
212	C₂²×C₄	—	2	32	no	—
213	C₂²×C₄	—	2	16	no	—
214	C₂²×C₄	—	2	8	no	—
215	C₂²×C₄	C₂×C₄₇⋊C₈	1	8	yes	—
216	C₂²×C₄	C₂×C₄₇⋊C₈	1	16	yes	complemented
217	C₂²×C₄	#10625329	1	8	yes	—
218	C₂²×C₄	C₂²⋊Dic₄₇	1	4	yes	—
219	C₂²×C₄	C₂²⋊Dic₄₇	1	8	yes	—
220	C₂²×C₄	#10625350	1	8	yes	—
221	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
222	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
223	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
224	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
225	C₂²×C₄	C₂×C₂²⋊D₄₇	1	8	yes	—
226	C₂²×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
227	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
228	C₂×C₈	—	94	8	no	—
229	C₂×C₈	—	94	16	no	—
230	C₂×C₈	—	188	16	no	—
231	C₄²	—	2	16	no	—
232	C₄²	—	2	8	no	—
233	C₄²	—	2	8	no	—
234	C₂³	—	2	8	no	—
235	C₂³	—	2	4	no	—
236	C₂³	—	2	8	no	—
237	C₂³	—	2	4	no	—
238	C₂³	—	2	16	no	—
239	C₂³	—	2	16	no	—
240	C₂³	C₂²⋊(C₄₇⋊C₈)	1	16	yes	—
241	C₂³	C₂×C₂²⋊Dic₄₇	1	4	yes	—
242	C₂³	C₂×C₂²⋊Dic₄₇	1	2	yes	—
243	C₂³	#13093983	1	4	yes	—
244	C₂³	C₂²×C₄₇⋊C₈	1	4	yes	—
245	C₂³	C₂²×C₄₇⋊C₈	1	2	yes	—
246	C₂³	C₂×#10625329	1	4	yes	—
247	C₂³	C₂×#10625329	1	2	yes	—
248	C₂³	C₂×#10625329	1	4	yes	—
249	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
250	C₂×C₄	—	2	32	no	—
251	C₂×C₄	—	2	8	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	16	no	—
256	C₂×C₄	—	2	32	no	—
257	C₂×C₄	—	2	16	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	2	16	no	—
261	C₂×C₄	—	2	16	no	—
262	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	4	yes	—
263	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
264	C₂×C₄	#13093983	1	4	yes	—
265	C₂×C₄	#13094007	1	16	yes	—
266	C₂×C₄	C₂²×C₄₇⋊C₈	1	8	yes	—
267	C₂×C₄	C₂×#10625329	1	8	yes	—
268	C₂×C₄	C₂²×C₂²⋊D₄₇	1	2	yes	—
269	C₂×C₄	C₂×#10625350	1	4	yes	—
270	C₈	—	188	8	no	—
271	C₂²	—	2	32	no	—
272	C₂²	—	2	4	no	—
273	C₂²	—	2	8	no	—
274	C₂²	—	2	8	no	—
275	C₂²	—	2	4	no	—
276	C₂²	#15688986	1	4	yes	—
277	C₂²	C₂×C₂²⋊(C₄₇⋊C₈)	1	8	yes	—
278	C₂²	C₂×C₂²⋊(C₄₇⋊C₈)	1	4	yes	—
279	C₂²	#15689087	1	4	yes	—
280	C₂²	#15689176	1	8	yes	—
281	C₂²	#15689180	1	8	yes	—
282	C₂²	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
283	C₂²	C₂×#13093983	1	2	yes	—
284	C₂²	C₂³×C₄₇⋊C₈	1	1	yes	characteristic
285	C₂²	C₂²×#10625329	1	2	yes	—
286	C₂²	C₂²×#10625329	1	1	yes	characteristic
287	C₄	—	2	4	no	—
288	C₄	—	2	8	no	—
289	C₄	—	2	8	no	—
290	C₄	#15688986	1	4	yes	—
291	C₄	C₂×#13094007	1	4	yes	—
292	C₂	—	2	8	no	—
293	C₂	#20243456	1	8	yes	direct product factor
294	C₂	C₂×#15688986	1	1	yes	characteristic
295	C₂	C₂²×C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
296	C₂	C₂×#15689087	1	1	yes	characteristic
297	C₂	C₂×#15689176	1	2	yes	—
298	C₂	C₂×#15689180	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₄₇
commutator	C₂²×C₄₇	C₂³×C₈
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂³×C₄²×C₄₇	C₂
socle	C₂⁴×C₄₇	C₂²×C₄
prime radical	C₂⁵×C₄₇	C₂×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂³×C₄²	D₄₇
hypocenter	C₄₇	C₂×#1817
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³×C₄ ◁ C₂³×C₄²

Complements

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

N	H ≅ G/N	type
2. #20243456	C₂	direct (N × H)
3. C₂×#15688878	C₂	semidirect
8. #15688878	C₂²	semidirect
27. C₂³×C₄×C₄₇	C₈	semidirect
68. C₂²×C₄×C₄₇	C₂×C₈	semidirect
148. C₂×C₄₇	#1817	semidirect
166. C₄₇	C₂×#1817	semidirect
173. C₂³×C₄	C₄₇⋊C₈	semidirect
216. C₂²×C₄	C₂×C₄₇⋊C₈	semidirect
293. C₂	#20243456	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
#20243456	2
C₂×#15688878	2
C₂³×C₄²×C₄₇	2
C₂×#1817	47

Subgroups — C2×#20243456 #26215979

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20243456 #26215979