Subgroups — C₂×#20244109 #26219547

Order 12032.

1182 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₄₇⋊C₈	C₂	1	2	yes	maximal
2	C₂²×#13093935	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₄⋊Dic₄₇	C₂	1	1	yes	maximal, characteristic
4	#20244109	C₂	1	8	yes	maximal, direct product factor
5	C₂×#15688877	C₂	1	2	yes	maximal
6	C₂²×C₄⋊C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
8	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
9	C₂²×C₄⋊Dic₄₇	C₄	1	2	yes	—
10	C₂²×C₄×Dic₄₇	C₂²	1	2	yes	—
11	C₂×Dic₄₇⋊C₈	—	2	8	no	—
12	C₂×Dic₄₇⋊C₈	C₂²	1	8	yes	—
13	C₂×#13093935	—	2	4	no	—
14	C₂×#13093935	C₂²	1	4	yes	—
15	C₄×C₄⋊Dic₄₇	C₄	1	4	yes	—
16	C₄×C₄⋊Dic₄₇	C₂²	1	4	yes	—
17	C₂×C₄₇⋊#95	C₄	1	1	yes	characteristic
18	C₂×C₄₇⋊#95	C₄	1	1	yes	characteristic
19	#15688877	C₂²	1	8	yes	—
20	C₂³×C₈×C₄₇	C₂²	1	2	yes	—
21	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₈×C₄₇	—	2	4	no	—
23	C₂×C₄⋊C₈×C₄₇	C₂²	1	4	yes	—
24	C₂³×Dic₄₇	C₂×C₄	1	2	yes	—
25	C₂³×Dic₄₇	C₂×C₄	1	2	yes	—
26	C₂²×C₄₇⋊C₈	—	2	2	no	—
27	C₂²×C₄₇⋊C₈	—	2	2	no	—
28	C₂²×C₄₇⋊C₈	—	2	2	no	—
29	C₂²×C₄₇⋊C₈	—	2	2	no	—
30	C₂²×C₄₇⋊C₈	D₄	1	4	yes	—
31	C₂²×C₄₇⋊C₈	Q₈	1	4	yes	—
32	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
33	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
34	C₂×C₄⋊Dic₄₇	—	2	8	no	—
35	C₂×C₄⋊Dic₄₇	C₂×C₄	1	8	yes	—
36	C₂×C₄×Dic₄₇	—	2	4	no	—
37	C₂×C₄×Dic₄₇	—	2	4	no	—
38	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
39	C₂×C₄×Dic₄₇	C₂³	1	4	yes	—
40	Dic₄₇⋊C₈	—	2	16	no	—
41	#13093935	—	2	8	no	—
42	C₄₇⋊#95	C₂×C₄	1	4	yes	—
43	C₄₇⋊#95	C₂×C₄	1	4	yes	—
44	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
45	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
46	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₄₇	—	2	4	no	—
48	C₂²×C₈×C₄₇	—	2	4	no	—
49	C₂²×C₈×C₄₇	—	2	4	no	—
50	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
51	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₄₇	D₄	1	4	yes	—
53	C₂×C₄²×C₄₇	Q₈	1	4	yes	—
54	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
55	C₄⋊C₈×C₄₇	—	2	8	no	—
56	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	—	2	4	no	—
58	C₂²×C₄×C₄₇	—	2	4	no	—
59	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
60	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
61	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
62	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
67	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
68	C₂²×C₄×C₄₇	#41	1	2	yes	—
69	C₂²×C₄×C₄₇	#41	1	2	yes	—
70	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
71	C₂²×Dic₄₇	—	2	4	no	—
72	C₂²×Dic₄₇	—	2	4	no	—
73	C₂²×Dic₄₇	—	2	4	no	—
74	C₂²×Dic₄₇	—	2	4	no	—
75	C₂²×Dic₄₇	—	2	4	no	—
76	C₂²×Dic₄₇	M₁₆	1	8	yes	—
77	C₂²×Dic₄₇	C₂²×C₄	1	4	yes	—
78	C₂²×Dic₄₇	C₂²×C₄	1	4	yes	—
79	C₂×C₈×C₄₇	—	2	4	no	—
80	C₂×C₈×C₄₇	—	2	4	no	—
81	C₂×C₈×C₄₇	—	2	4	no	—
82	C₂×C₈×C₄₇	—	4	8	no	—
83	C₄²×C₄₇	C₄⋊C₄	1	8	yes	—
84	C₄²×C₄₇	C₂×D₄	1	4	yes	—
85	C₄²×C₄₇	C₂×Q₈	1	4	yes	—
86	C₄⋊Dic₄₇	—	2	16	no	—
87	C₄×Dic₄₇	—	2	8	no	—
88	C₄×Dic₄₇	—	2	4	no	—
89	C₄×Dic₄₇	—	2	4	no	—
90	C₂×C₄₇⋊C₈	—	2	8	no	—
91	C₂×C₄₇⋊C₈	—	2	8	no	—
92	C₂×C₄₇⋊C₈	—	2	2	no	—
93	C₂×C₄₇⋊C₈	—	2	2	no	—
94	C₂×C₄₇⋊C₈	—	2	2	no	—
95	C₂×C₄₇⋊C₈	—	2	2	no	—
96	C₂×C₄₇⋊C₈	C₂×D₄	1	4	yes	—
97	C₂×C₄₇⋊C₈	C₂×Q₈	1	4	yes	—
98	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
99	C₂³×C₄₇	C₄×D₄	1	2	yes	—
100	C₂³×C₄₇	C₄×Q₈	1	2	yes	—
101	C₂³×C₄₇	C₂×M₁₆	1	4	yes	—
102	C₂³×C₄₇	#131	1	4	yes	—
103	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
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	8	no	—
108	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
109	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
110	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
111	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
112	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
113	C₂×C₄×C₄₇	#121	1	4	yes	—
114	C₂×C₄×C₄₇	C₂²⋊Q₈	1	4	yes	—
115	C₂×C₄×C₄₇	#125	1	4	yes	—
116	C₂×C₄×C₄₇	C₄⋊Q₈	1	4	yes	—
117	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
119	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
120	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
121	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
122	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
123	C₂×Dic₄₇	—	2	4	no	—
124	C₂×Dic₄₇	—	2	4	no	—
125	C₂×Dic₄₇	—	2	16	no	—
126	C₂×Dic₄₇	—	2	4	no	—
127	C₂×Dic₄₇	—	2	4	no	—
128	C₂×Dic₄₇	—	2	4	no	—
129	C₂×Dic₄₇	—	4	8	no	—
130	C₂×Dic₄₇	C₂×M₁₆	1	8	yes	—
131	C₈×C₄₇	—	4	4	no	—
132	C₄₇⋊C₈	—	2	4	no	—
133	C₄₇⋊C₈	—	2	4	no	—
134	C₂×#1940	—	47	1	no	maximal, Sylow
135	C₂²×C₄₇	#389	1	4	yes	—
136	C₂²×C₄₇	#423	1	4	yes	—
137	C₂²×C₄₇	#424	1	4	yes	—
138	C₂²×C₄₇	#435	1	8	yes	—
139	C₂²×C₄₇	#446	1	8	yes	—
140	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
141	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
142	C₂²×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
143	C₂²×C₄₇	C₂²×M₁₆	1	2	yes	—
144	C₂²×C₄₇	C₂×#131	1	2	yes	—
145	C₄×C₄₇	—	2	4	no	—
146	C₄×C₄₇	—	2	4	no	—
147	C₄×C₄₇	#389	1	4	yes	—
148	C₄×C₄₇	C₂×#121	1	1	yes	characteristic
149	C₄×C₄₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
150	C₄×C₄₇	C₂×#125	1	1	yes	characteristic
151	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
152	Dic₄₇	—	2	8	no	—
153	Dic₄₇	—	4	4	no	—
154	C₂²×C₄⋊C₈	—	47	2	no	—
155	C₂²×C₄⋊C₈	D₄₇	1	1	yes	characteristic
156	C₂²×#97	—	47	1	no	—
157	C₂×C₄×C₄⋊C₄	—	47	1	no	—
158	#1940	—	47	8	no	—
159	C₂×#336	—	47	2	no	—
160	C₂×C₄₇	#1940	1	8	yes	complemented
161	C₂×C₄₇	C₂×#389	1	1	yes	characteristic
162	C₂×C₄₇	C₂×#423	1	1	yes	characteristic
163	C₂×C₄₇	C₂×#424	1	1	yes	characteristic
164	C₂×C₄₇	C₂×#435	1	2	yes	—
165	C₂×C₄₇	C₂×#446	1	2	yes	—
166	C₂³×C₈	—	47	1	no	—
167	C₂³×C₈	—	47	1	no	—
168	C₂³×C₈	C₂×D₄₇	1	2	yes	—
169	C₂²×C₄⋊C₄	—	47	2	no	—
170	C₂²×C₄²	—	47	2	no	—
171	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
172	C₂×C₄⋊C₈	—	2	4	no	—
173	C₂×C₄⋊C₈	—	47	8	no	—
174	C₂×C₄⋊C₈	—	94	8	no	—
175	C₂×C₄⋊C₈	C₂×D₄₇	1	4	yes	—
176	C₂×#97	—	47	4	no	—
177	C₂×#97	—	94	4	no	—
178	C₄×C₄⋊C₄	—	47	4	no	—
179	C₄×C₄⋊C₄	—	47	4	no	—
180	C₂×#95	—	47	1	no	—
181	C₂×#95	—	47	1	no	—
182	#336	—	47	8	no	—
183	C₄₇	C₂×#1940	1	1	yes	characteristic, complemented, Sylow
184	C₂³×C₄	—	47	2	no	—
185	C₂³×C₄	—	47	2	no	—
186	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
187	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
188	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
189	C₂²×C₈	—	2	4	no	—
190	C₂²×C₈	—	2	4	no	—
191	C₂²×C₈	—	2	4	no	—
192	C₂²×C₈	—	47	4	no	—
193	C₂²×C₈	—	47	4	no	—
194	C₂²×C₈	—	47	2	no	—
195	C₂²×C₈	—	47	2	no	—
196	C₂²×C₈	—	94	2	no	—
197	C₂²×C₈	—	94	2	no	—
198	C₂²×C₈	—	94	2	no	—
199	C₂²×C₈	—	94	2	no	—
200	C₂²×C₈	C₂²×D₄₇	1	4	yes	—
201	C₂×C₄⋊C₄	—	47	8	no	—
202	C₂×C₄⋊C₄	—	94	8	no	—
203	C₂×C₄²	—	47	4	no	—
204	C₂×C₄²	—	47	4	no	—
205	C₂×C₄²	—	94	4	no	—
206	C₂×C₄²	—	94	4	no	—
207	C₂×C₄²	Dic₉₄	1	4	yes	—
208	C₂×C₄²	C₄×D₄₇	1	2	yes	—
209	C₂×C₄²	C₂²⋊D₄₇	1	4	yes	—
210	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
211	C₄⋊C₈	—	2	8	no	—
212	C₄⋊C₈	—	94	16	no	—
213	#97	—	94	8	no	—
214	#95	—	47	4	no	—
215	#95	—	47	4	no	—
216	C₂⁴	C₂×C₄×D₄₇	1	1	yes	characteristic
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	47	8	no	—
220	C₂²×C₄	—	47	4	no	—
221	C₂²×C₄	—	47	4	no	—
222	C₂²×C₄	—	94	4	no	—
223	C₂²×C₄	—	94	4	no	—
224	C₂²×C₄	—	94	4	no	—
225	C₂²×C₄	—	94	4	no	—
226	C₂²×C₄	—	94	4	no	—
227	C₂²×C₄	Dic₄₇⋊C₄	1	4	yes	—
228	C₂²×C₄	Dic₄₇⋊C₄	1	4	yes	—
229	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
230	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
231	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
232	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
233	C₂²×C₄	D₄×D₄₇	1	2	yes	—
234	C₂²×C₄	#10625352	1	2	yes	—
235	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
236	C₂²×C₄	#10625354	1	2	yes	—
237	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
238	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
239	C₂×C₈	—	2	4	no	—
240	C₂×C₈	—	2	4	no	—
241	C₂×C₈	—	2	4	no	—
242	C₂×C₈	—	4	8	no	—
243	C₂×C₈	—	47	4	no	—
244	C₂×C₈	—	47	4	no	—
245	C₂×C₈	—	94	8	no	—
246	C₂×C₈	—	94	8	no	—
247	C₂×C₈	—	94	2	no	—
248	C₂×C₈	—	94	2	no	—
249	C₂×C₈	—	94	2	no	—
250	C₂×C₈	—	94	2	no	—
251	C₄⋊C₄	—	94	16	no	—
252	C₄²	—	94	8	no	—
253	C₄²	—	94	4	no	—
254	C₄²	—	94	4	no	—
255	C₄²	Dic₄₇⋊C₄	1	8	yes	—
256	C₄²	C₂×Dic₉₄	1	4	yes	—
257	C₄²	C₂×C₂²⋊D₄₇	1	4	yes	—
258	C₂³	C₂×Dic₄₇⋊C₄	1	2	yes	—
259	C₂³	#13093986	1	2	yes	—
260	C₂³	#13093989	1	2	yes	—
261	C₂³	M₁₆×D₄₇	1	4	yes	—
262	C₂³	#13094030	1	4	yes	—
263	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	8	no	—
267	C₂×C₄	—	2	8	no	—
268	C₂×C₄	—	47	8	no	—
269	C₂×C₄	—	94	4	no	—
270	C₂×C₄	—	94	4	no	—
271	C₂×C₄	—	94	16	no	—
272	C₂×C₄	—	94	4	no	—
273	C₂×C₄	—	94	4	no	—
274	C₂×C₄	—	94	4	no	—
275	C₂×C₄	—	188	8	no	—
276	C₂×C₄	C₂×Dic₄₇⋊C₄	1	4	yes	—
277	C₂×C₄	C₂×Dic₄₇⋊C₄	1	4	yes	—
278	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
279	C₂×C₄	#13093986	1	2	yes	—
280	C₂×C₄	#13093989	1	2	yes	—
281	C₂×C₄	#13093995	1	4	yes	—
282	C₂×C₄	#13094000	1	4	yes	—
283	C₂×C₄	#13094015	1	4	yes	—
284	C₂×C₄	#13094023	1	4	yes	—
285	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
286	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
287	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
288	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
289	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
290	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
291	C₈	—	4	4	no	—
292	C₈	—	94	4	no	—
293	C₈	—	94	4	no	—
294	C₂²	#15689019	1	4	yes	—
295	C₂²	#15689146	1	4	yes	—
296	C₂²	#15689149	1	4	yes	—
297	C₂²	#15689193	1	8	yes	—
298	C₂²	#15689252	1	8	yes	—
299	C₂²	C₂²×Dic₄₇⋊C₄	1	1	yes	characteristic
300	C₂²	C₂×#13093986	1	1	yes	characteristic
301	C₂²	C₂×#13093989	1	1	yes	characteristic
302	C₂²	C₂×M₁₆×D₄₇	1	2	yes	—
303	C₂²	C₂×#13094030	1	2	yes	—
304	C₄	—	2	4	no	—
305	C₄	—	2	4	no	—
306	C₄	—	94	8	no	—
307	C₄	—	188	4	no	—
308	C₄	#15689019	1	4	yes	—
309	C₄	C₂×#13093995	1	1	yes	characteristic
310	C₄	C₂×#13094000	1	1	yes	characteristic
311	C₄	C₂×#13094015	1	1	yes	characteristic
312	C₄	C₂×#13094023	1	1	yes	characteristic
313	C₂	#20244109	1	8	yes	direct product factor
314	C₂	C₂×#15689019	1	1	yes	characteristic
315	C₂	C₂×#15689146	1	1	yes	characteristic
316	C₂	C₂×#15689149	1	1	yes	characteristic
317	C₂	C₂×#15689193	1	2	yes	—
318	C₂	C₂×#15689252	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₄₇
commutator	C₂³×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂²×C₄⋊C₈×C₄₇	C₂
socle, prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂²×C₄⋊C₈	D₄₇
hypocenter	C₄₇	C₂×#1940
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
4. #20244109	C₂	direct (N × H)
160. C₂×C₄₇	#1940	semidirect
183. C₄₇	C₂×#1940	semidirect
313. C₂	#20244109	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×Dic₄₇⋊C₈	2
C₂²×#13093935	2
C₂×C₄×C₄⋊Dic₄₇	2
#20244109	2
C₂×#15688877	2
C₂²×C₄⋊C₈×C₄₇	2
C₂×#1940	47

Subgroups — C2×#20244109 #26219547

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20244109 #26219547