Subgroups — C₂×#20244337 #26220777

Order 12032.

1510 conjugacy classes of subgroups.

The subgroup lattice has 298 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×C₄×C₄⋊Dic₄₇	C₂	1	1	yes	maximal, characteristic
3	#20244337	C₂	1	8	yes	maximal, direct product factor
4	C₂×#15688885	C₂	1	4	yes	maximal
5	C₂²×C₄⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₄₇⋊C₈	C₂²	1	2	yes	—
7	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
8	C₂²×C₄×Dic₄₇	—	2	1	no	—
9	C₂×C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
10	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
11	C₄×C₄⋊Dic₄₇	—	2	2	no	—
12	C₄×C₄⋊Dic₄₇	C₂²	1	4	yes	—
13	C₂×C₄₇⋊#95	—	2	1	no	—
14	#15688885	C₂²	1	16	yes	—
15	C₂³×Q₈×C₄₇	—	2	1	no	—
16	C₂×C₄⋊Q₈×C₄₇	—	2	2	no	—
17	C₂×C₄⋊Q₈×C₄₇	—	2	4	no	—
18	C₂×C₄⋊Q₈×C₄₇	C₄	1	8	yes	complemented
19	C₂×C₄⋊Q₈×C₄₇	C₂²	1	4	yes	—
20	C₂²×C₄⋊C₄×C₄₇	—	2	1	no	—
21	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
22	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
23	C₂³×Dic₄₇	—	2	2	no	—
24	C₂²×C₄₇⋊C₈	—	2	4	no	—
25	C₂²×C₄₇⋊C₈	—	2	8	no	—
26	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
27	C₂×C₄⋊Dic₄₇	—	2	2	no	—
28	C₂×C₄⋊Dic₄₇	—	2	4	no	—
29	C₂×C₄⋊Dic₄₇	Q₈	1	8	yes	complemented
30	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
31	C₂×C₄×Dic₄₇	—	2	4	no	—
32	C₂×C₄×Dic₄₇	—	4	4	no	—
33	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
34	C₄₇⋊#95	—	2	4	no	—
35	C₂²×Q₈×C₄₇	—	2	8	no	—
36	C₂²×Q₈×C₄₇	—	2	4	no	—
37	C₂²×Q₈×C₄₇	—	4	8	no	—
38	C₂³×C₄×C₄₇	—	2	2	no	—
39	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
40	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
41	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
42	C₄⋊Q₈×C₄₇	—	2	16	no	—
43	C₄⋊Q₈×C₄₇	—	2	8	no	—
44	C₄⋊Q₈×C₄₇	C₂×C₄	1	16	yes	complemented
45	C₂×C₄⋊C₄×C₄₇	—	2	8	no	—
46	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
47	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
48	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
49	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	8	yes	—
50	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
51	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
52	C₂×C₄²×C₄₇	D₄	1	2	yes	—
53	C₂×C₄²×C₄₇	D₄	1	4	yes	—
54	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
55	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
56	C₂×Q₈×C₄₇	—	2	16	no	—
57	C₂×Q₈×C₄₇	—	4	16	no	—
58	C₂×Q₈×C₄₇	—	4	32	no	—
59	C₂²×C₄×C₄₇	—	2	4	no	—
60	C₂²×C₄×C₄₇	—	2	8	no	—
61	C₂²×C₄×C₄₇	—	2	4	no	—
62	C₂²×C₄×C₄₇	—	4	8	no	—
63	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
64	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
65	C₂²×C₄×C₄₇	SD₁₆	1	4	yes	—
66	C₂²×C₄×C₄₇	Q₁₆	1	4	yes	—
67	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
73	C₂²×C₄×C₄₇	#41	1	4	yes	—
74	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
75	C₂²×Dic₄₇	—	2	8	no	—
76	C₂²×Dic₄₇	—	2	4	no	—
77	C₂²×Dic₄₇	—	4	4	no	—
78	C₂²×Dic₄₇	—	4	4	no	—
79	C₄⋊C₄×C₄₇	—	2	16	no	—
80	C₄⋊C₄×C₄₇	—	2	8	no	—
81	C₄⋊C₄×C₄₇	—	2	4	no	—
82	C₄⋊C₄×C₄₇	C₂²×C₄	1	8	yes	—
83	C₄²×C₄₇	C₂²⋊C₄	1	8	yes	—
84	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
85	C₄²×C₄₇	C₂×D₄	1	4	yes	—
86	C₄⋊Dic₄₇	—	2	8	no	—
87	C₄⋊Dic₄₇	—	2	4	no	—
88	C₄⋊Dic₄₇	C₂×Q₈	1	8	yes	complemented
89	C₄×Dic₄₇	—	4	8	no	—
90	C₂×C₄₇⋊C₈	—	2	4	no	—
91	C₂×C₄₇⋊C₈	—	2	8	no	—
92	C₂×C₄₇⋊C₈	—	4	8	no	—
93	C₂³×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
94	C₂³×C₄₇	C₂²⋊Q₈	1	4	yes	—
95	C₂³×C₄₇	C₂×SD₁₆	1	2	yes	—
96	C₂³×C₄₇	C₂×Q₁₆	1	2	yes	—
97	C₂³×C₄₇	AGL(1,ℤ/8)	1	4	yes	—
98	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
99	Q₈×C₄₇	—	4	32	no	—
100	C₂×C₄×C₄₇	—	2	4	no	—
101	C₂×C₄×C₄₇	—	2	8	no	—
102	C₂×C₄×C₄₇	—	2	8	no	—
103	C₂×C₄×C₄₇	—	4	8	no	—
104	C₂×C₄×C₄₇	—	4	16	no	—
105	C₂×C₄×C₄₇	#103	1	16	yes	—
106	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
107	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
108	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
109	C₂×C₄×C₄₇	C₄×Q₈	1	8	yes	—
110	C₂×C₄×C₄₇	C₂²⋊Q₈	1	4	yes	—
111	C₂×C₄×C₄₇	#124	1	4	yes	—
112	C₂×C₄×C₄₇	C₄⋊Q₈	1	4	yes	—
113	C₂×C₄×C₄₇	C₂×SD₁₆	1	4	yes	—
114	C₂×C₄×C₄₇	C₂×Q₁₆	1	4	yes	—
115	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
116	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
118	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
119	C₂×Dic₄₇	—	2	8	no	—
120	C₂×Dic₄₇	—	4	4	no	—
121	C₂×Dic₄₇	—	4	4	no	—
122	C₂×Dic₄₇	—	4	16	no	—
123	C₄₇⋊C₈	—	4	4	no	—
124	C₂×#1987	—	47	1	no	maximal, Sylow
125	C₂²×C₄₇	#391	1	4	yes	—
126	C₂²×C₄₇	C₂×#103	1	4	yes	—
127	C₂²×C₄₇	#418	1	4	yes	—
128	C₂²×C₄₇	#476	1	8	yes	—
129	C₂²×C₄₇	#477	1	8	yes	—
130	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
131	C₂²×C₄₇	C₂×C₂²⋊Q₈	1	2	yes	—
132	C₂²×C₄₇	C₂²×SD₁₆	1	1	yes	characteristic
133	C₂²×C₄₇	C₂²×Q₁₆	1	1	yes	characteristic
134	C₂²×C₄₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
135	C₄×C₄₇	—	2	4	no	—
136	C₄×C₄₇	—	4	8	no	—
137	C₄×C₄₇	#391	1	4	yes	—
138	C₄×C₄₇	C₂×#103	1	8	yes	—
139	C₄×C₄₇	C₂×C₄×Q₈	1	2	yes	—
140	C₄×C₄₇	C₂×#124	1	1	yes	characteristic
141	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
142	Dic₄₇	—	4	8	no	—
143	C₂²×C₄⋊Q₈	D₄₇	1	1	yes	characteristic
144	C₂²×C₄⋊C₈	—	47	1	no	—
145	C₂×C₄×C₄⋊C₄	—	47	1	no	—
146	#1987	—	47	8	no	—
147	C₂×#340	—	47	4	no	—
148	C₂×C₄₇	#1987	1	8	yes	complemented
149	C₂×C₄₇	C₂×#391	1	1	yes	characteristic
150	C₂×C₄₇	C₂²×#103	1	1	yes	characteristic
151	C₂×C₄₇	C₂×#418	1	1	yes	characteristic
152	C₂×C₄₇	C₂×#476	1	2	yes	—
153	C₂×C₄₇	C₂×#477	1	2	yes	—
154	C₂³×Q₈	—	2	1	no	—
155	C₂³×C₈	—	47	2	no	—
156	C₂×C₄⋊Q₈	—	2	2	no	—
157	C₂×C₄⋊Q₈	—	2	4	no	—
158	C₂×C₄⋊Q₈	Dic₄₇	1	8	yes	complemented
159	C₂×C₄⋊Q₈	C₂×D₄₇	1	4	yes	—
160	C₂²×C₄⋊C₄	—	2	1	no	—
161	C₂²×C₄⋊C₄	—	47	2	no	—
162	C₂²×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
163	C₂²×C₄²	—	94	1	no	—
164	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
165	C₂×C₄⋊C₈	—	47	4	no	—
166	C₂×C₄⋊C₈	—	94	4	no	—
167	C₄×C₄⋊C₄	—	47	4	no	—
168	C₄×C₄⋊C₄	—	94	2	no	—
169	C₂×#95	—	94	1	no	—
170	#340	—	47	16	no	—
171	C₄₇	C₂×#1987	1	1	yes	characteristic, complemented, Sylow
172	C₂²×Q₈	—	2	8	no	—
173	C₂²×Q₈	—	2	4	no	—
174	C₂²×Q₈	—	4	8	no	—
175	C₂³×C₄	—	2	2	no	—
176	C₂³×C₄	—	94	2	no	—
177	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
178	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
179	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
180	C₂²×C₈	—	47	4	no	—
181	C₂²×C₈	—	94	8	no	—
182	C₂²×C₈	—	94	4	no	—
183	C₄⋊Q₈	—	2	16	no	—
184	C₄⋊Q₈	—	2	8	no	—
185	C₄⋊Q₈	C₂×Dic₄₇	1	16	yes	complemented
186	C₂×C₄⋊C₄	—	2	8	no	—
187	C₂×C₄⋊C₄	—	2	4	no	—
188	C₂×C₄⋊C₄	—	2	2	no	—
189	C₂×C₄⋊C₄	—	2	4	no	—
190	C₂×C₄⋊C₄	—	47	8	no	—
191	C₂×C₄⋊C₄	—	47	4	no	—
192	C₂×C₄⋊C₄	—	94	4	no	—
193	C₂×C₄⋊C₄	—	94	2	no	—
194	C₂×C₄⋊C₄	C₂×Dic₄₇	1	8	yes	—
195	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
196	C₂×C₄²	—	94	4	no	—
197	C₂×C₄²	—	188	4	no	—
198	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
199	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
200	C₂×C₄²	C₂²⋊D₄₇	1	4	yes	—
201	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
202	C₄⋊C₈	—	94	8	no	—
203	#95	—	94	4	no	—
204	C₂⁴	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
205	C₂×Q₈	—	2	16	no	—
206	C₂×Q₈	—	4	32	no	—
207	C₂×Q₈	—	4	16	no	—
208	C₂²×C₄	—	2	4	no	—
209	C₂²×C₄	—	2	8	no	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	4	8	no	—
212	C₂²×C₄	—	94	8	no	—
213	C₂²×C₄	—	94	4	no	—
214	C₂²×C₄	—	188	4	no	—
215	C₂²×C₄	—	188	4	no	—
216	C₂²×C₄	#10625336	1	4	yes	—
217	C₂²×C₄	#10625337	1	4	yes	—
218	C₂²×C₄	C₂²⋊Dic₄₇	1	4	yes	—
219	C₂²×C₄	C₂²⋊Dic₄₇	1	4	yes	—
220	C₂²×C₄	Q₈×D₄₇	1	4	yes	—
221	C₂²×C₄	#10625354	1	4	yes	—
222	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
223	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
224	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
225	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
226	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
227	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
228	C₂×C₈	—	94	4	no	—
229	C₂×C₈	—	94	8	no	—
230	C₂×C₈	—	188	8	no	—
231	C₄⋊C₄	—	2	16	no	—
232	C₄⋊C₄	—	2	8	no	—
233	C₄⋊C₄	—	2	4	no	—
234	C₄⋊C₄	—	47	8	no	—
235	C₄⋊C₄	—	94	8	no	—
236	C₄⋊C₄	—	94	4	no	—
237	C₄⋊C₄	C₂²×Dic₄₇	1	8	yes	—
238	C₄²	—	188	8	no	—
239	C₄²	C₂²⋊Dic₄₇	1	8	yes	—
240	C₄²	C₂²×Dic₄₇	1	4	yes	—
241	C₄²	C₂×C₂²⋊D₄₇	1	4	yes	—
242	C₂³	C₂×C₂²⋊Dic₄₇	1	2	yes	—
243	C₂³	#13094000	1	4	yes	—
244	C₂³	C₂×#10625336	1	2	yes	—
245	C₂³	C₂×#10625337	1	2	yes	—
246	C₂³	#13094050	1	4	yes	—
247	C₂³	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
248	Q₈	—	4	32	no	—
249	C₂×C₄	—	2	8	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	2	8	no	—
252	C₂×C₄	—	4	16	no	—
253	C₂×C₄	—	4	8	no	—
254	C₂×C₄	—	94	8	no	—
255	C₂×C₄	—	188	4	no	—
256	C₂×C₄	—	188	4	no	—
257	C₂×C₄	—	188	16	no	—
258	C₂×C₄	#13093948	1	16	yes	—
259	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	4	yes	—
260	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
261	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	4	yes	—
262	C₂×C₄	Q₈×Dic₄₇	1	8	yes	—
263	C₂×C₄	#13094000	1	4	yes	—
264	C₂×C₄	#13094013	1	4	yes	—
265	C₂×C₄	#13094022	1	4	yes	—
266	C₂×C₄	C₂×#10625336	1	4	yes	—
267	C₂×C₄	C₂×#10625337	1	4	yes	—
268	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
269	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
270	C₂×C₄	C₂×Q₈×D₄₇	1	2	yes	—
271	C₂×C₄	C₂×#10625354	1	2	yes	—
272	C₈	—	188	4	no	—
273	C₂²	#15689030	1	4	yes	—
274	C₂²	C₂×#13093948	1	4	yes	—
275	C₂²	#15689128	1	4	yes	—
276	C₂²	#15689410	1	8	yes	—
277	C₂²	#15689415	1	8	yes	—
278	C₂²	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
279	C₂²	C₂×#13094000	1	2	yes	—
280	C₂²	C₂²×#10625336	1	1	yes	characteristic
281	C₂²	C₂²×#10625337	1	1	yes	characteristic
282	C₂²	C₂×#13094050	1	2	yes	—
283	C₄	—	2	4	no	—
284	C₄	—	4	8	no	—
285	C₄	—	188	8	no	—
286	C₄	#15689030	1	4	yes	—
287	C₄	C₂×#13093948	1	8	yes	—
288	C₄	C₂×Q₈×Dic₄₇	1	2	yes	—
289	C₄	C₂×#13094013	1	1	yes	characteristic
290	C₄	C₂×#13094022	1	1	yes	characteristic
291	C₂	#20244337	1	8	yes	direct product factor
292	C₂	C₂×#15689030	1	1	yes	characteristic
293	C₂	C₂²×#13093948	1	1	yes	characteristic
294	C₂	C₂×#15689128	1	1	yes	characteristic
295	C₂	C₂×#15689410	1	2	yes	—
296	C₂	C₂×#15689415	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂²×C₄⋊Q₈×C₄₇	C₂
socle, prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂²×C₄⋊Q₈	D₄₇
hypocenter	C₄₇	C₂×#1987
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
3. #20244337	C₂	direct (N × H)
18. C₂×C₄⋊Q₈×C₄₇	C₄	semidirect
29. C₂×C₄⋊Dic₄₇	Q₈	semidirect
44. C₄⋊Q₈×C₄₇	C₂×C₄	semidirect
88. C₄⋊Dic₄₇	C₂×Q₈	semidirect
148. C₂×C₄₇	#1987	semidirect
158. C₂×C₄⋊Q₈	Dic₄₇	semidirect
171. C₄₇	C₂×#1987	semidirect
185. C₄⋊Q₈	C₂×Dic₄₇	semidirect
291. C₂	#20244337	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×C₄×C₄⋊Dic₄₇	2
#20244337	2
C₂×#15688885	2
C₂²×C₄⋊Q₈×C₄₇	2
C₂×#1987	47

Subgroups — C2×#20244337 #26220777

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20244337 #26220777