Subgroups — C₂×#20244250 #26220089

Order 12032.

1614 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093958	C₂	1	1	yes	maximal, characteristic
2	C₂²×#13093948	C₂	1	2	yes	maximal
3	C₂²×C₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#15689030	C₂	1	2	yes	maximal
5	#20244250	C₂	1	8	yes	maximal, direct product factor
6	C₂³×Q₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
9	C₂²×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
10	C₂×#13093958	—	2	4	no	—
11	C₂×#13093958	C₂²	1	4	yes	—
12	C₂×#13093948	—	2	8	no	—
13	C₂×#13093948	C₂²	1	8	yes	—
14	C₂×C₈×Dic₄₇	—	2	4	no	—
15	C₂×C₈×Dic₄₇	C₂²	1	4	yes	—
16	#15689030	—	2	4	no	—
17	#15689030	C₂²	1	8	yes	—
18	C₂×C₄₇⋊#95	—	2	4	no	—
19	C₂³×Q₈×C₄₇	C₂²	1	2	yes	—
20	C₂²×Q₁₆×C₄₇	—	2	4	no	—
21	C₂²×Q₁₆×C₄₇	—	2	4	no	—
22	C₂²×Q₁₆×C₄₇	C₄	1	8	yes	complemented
23	C₂²×Q₁₆×C₄₇	C₂²	1	4	yes	—
24	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
25	C₂³×Dic₄₇	—	2	2	no	—
26	C₂³×Dic₄₇	D₄	1	2	yes	—
27	C₂²×C₄₇⋊C₈	—	2	2	no	—
28	C₂²×C₄₇⋊C₈	—	2	2	no	—
29	C₂²×C₄₇⋊C₈	D₄	1	4	yes	—
30	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
31	C₂×C₄⋊Dic₄₇	—	2	4	no	—
32	C₂×C₄⋊Dic₄₇	—	2	4	no	—
33	C₂×C₄⋊Dic₄₇	—	2	2	no	—
34	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
35	C₂×C₄×Dic₄₇	—	2	2	no	—
36	C₂×C₄×Dic₄₇	—	2	2	no	—
37	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
38	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
39	#13093958	—	2	8	no	—
40	#13093948	—	2	16	no	—
41	C₈×Dic₄₇	—	2	8	no	—
42	C₄₇⋊#95	—	2	16	no	—
43	C₂²×Q₈×C₄₇	—	2	2	no	—
44	C₂²×Q₈×C₄₇	—	2	4	no	—
45	C₂²×Q₈×C₄₇	—	2	4	no	—
46	C₂²×Q₈×C₄₇	—	2	4	no	—
47	C₂²×Q₈×C₄₇	—	2	4	no	—
48	C₂²×Q₈×C₄₇	—	2	4	no	—
49	C₂²×Q₈×C₄₇	C₂×C₄	1	8	yes	—
50	C₂²×Q₈×C₄₇	C₂³	1	4	yes	—
51	C₂³×C₄×C₄₇	—	2	2	no	—
52	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
53	C₂×Q₁₆×C₄₇	—	2	32	no	—
54	C₂×Q₁₆×C₄₇	—	2	8	no	—
55	C₂×Q₁₆×C₄₇	—	2	8	no	—
56	C₂×Q₁₆×C₄₇	C₂×C₄	1	16	yes	complemented
57	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
58	C₂²×C₈×C₄₇	D₄	1	4	yes	—
59	C₂²×C₈×C₄₇	D₄	1	4	yes	—
60	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
61	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
62	C₂×Q₈×C₄₇	—	2	8	no	—
63	C₂×Q₈×C₄₇	—	2	4	no	—
64	C₂×Q₈×C₄₇	—	2	16	no	—
65	C₂×Q₈×C₄₇	—	2	16	no	—
66	C₂×Q₈×C₄₇	—	2	4	no	—
67	C₂×Q₈×C₄₇	—	2	4	no	—
68	C₂×Q₈×C₄₇	—	2	4	no	—
69	C₂×Q₈×C₄₇	—	2	4	no	—
70	C₂×Q₈×C₄₇	—	4	4	no	—
71	C₂×Q₈×C₄₇	—	4	8	no	—
72	C₂×Q₈×C₄₇	—	4	8	no	—
73	C₂×Q₈×C₄₇	—	4	4	no	—
74	C₂×Q₈×C₄₇	C₂²×C₄	1	8	yes	—
75	C₂²×C₄×C₄₇	—	2	8	no	—
76	C₂²×C₄×C₄₇	—	2	4	no	—
77	C₂²×C₄×C₄₇	—	4	4	no	—
78	C₂²×C₄×C₄₇	—	4	4	no	—
79	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
82	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
84	C₂²×C₄×C₄₇	#41	1	2	yes	—
85	C₂²×C₄×C₄₇	#41	1	2	yes	—
86	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
87	C₂²×Dic₄₇	—	2	4	no	—
88	C₂²×Dic₄₇	—	2	4	no	—
89	C₂²×Dic₄₇	—	2	4	no	—
90	C₂²×Dic₄₇	—	4	4	no	—
91	C₂²×Dic₄₇	—	4	8	no	—
92	C₂²×Dic₄₇	Q₁₆	1	8	yes	complemented
93	C₂²×Dic₄₇	C₂×D₄	1	4	yes	—
94	Q₁₆×C₄₇	—	2	32	no	—
95	C₂×C₈×C₄₇	C₂²⋊C₄	1	16	yes	—
96	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
97	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
98	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
99	C₄⋊Dic₄₇	—	2	4	no	—
100	C₄⋊Dic₄₇	—	2	4	no	—
101	C₄⋊Dic₄₇	—	4	4	no	—
102	C₄×Dic₄₇	—	2	4	no	—
103	C₄×Dic₄₇	—	2	2	no	—
104	C₄×Dic₄₇	—	2	2	no	—
105	C₂×C₄₇⋊C₈	—	2	8	no	—
106	C₂×C₄₇⋊C₈	—	2	2	no	—
107	C₂×C₄₇⋊C₈	—	2	2	no	—
108	C₂×C₄₇⋊C₈	C₂×D₄	1	4	yes	—
109	C₂³×C₄₇	C₄×D₄	1	2	yes	—
110	C₂³×C₄₇	#121	1	2	yes	—
111	C₂³×C₄₇	#127	1	2	yes	—
112	C₂³×C₄₇	C₂×Q₁₆	1	4	yes	—
113	C₂³×C₄₇	#135	1	4	yes	—
114	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
115	Q₈×C₄₇	—	2	8	no	—
116	Q₈×C₄₇	—	2	8	no	—
117	Q₈×C₄₇	—	4	4	no	—
118	Q₈×C₄₇	—	4	4	no	—
119	Q₈×C₄₇	—	4	16	no	—
120	C₂×C₄×C₄₇	—	2	8	no	—
121	C₂×C₄×C₄₇	—	4	4	no	—
122	C₂×C₄×C₄₇	—	4	4	no	—
123	C₂×C₄×C₄₇	—	4	16	no	—
124	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
125	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
126	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
127	C₂×C₄×C₄₇	#121	1	4	yes	—
128	C₂×C₄×C₄₇	#121	1	2	yes	—
129	C₂×C₄×C₄₇	#124	1	4	yes	—
130	C₂×C₄×C₄₇	#127	1	2	yes	—
131	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
132	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
133	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
135	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
136	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
137	C₂×Dic₄₇	—	2	4	no	—
138	C₂×Dic₄₇	—	2	4	no	—
139	C₂×Dic₄₇	—	2	16	no	—
140	C₂×Dic₄₇	—	4	4	no	—
141	C₂×Dic₄₇	—	4	8	no	—
142	C₂×Dic₄₇	—	8	8	no	—
143	C₂×Dic₄₇	C₂×Q₁₆	1	8	yes	complemented
144	C₈×C₄₇	C₂×C₂²⋊C₄	1	8	yes	—
145	C₄₇⋊C₈	—	2	4	no	—
146	C₂×#1969	—	47	1	no	maximal, Sylow
147	C₂²×C₄₇	#390	1	4	yes	—
148	C₂²×C₄₇	C₄×Q₁₆	1	8	yes	—
149	C₂²×C₄₇	#467	1	8	yes	—
150	C₂²×C₄₇	#494	1	4	yes	—
151	C₂²×C₄₇	#495	1	4	yes	—
152	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
153	C₂²×C₄₇	C₂×#121	1	1	yes	characteristic
154	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
155	C₂²×C₄₇	C₂²×Q₁₆	1	2	yes	—
156	C₂²×C₄₇	C₂×#135	1	2	yes	—
157	C₄×C₄₇	—	4	8	no	—
158	C₄×C₄₇	#390	1	4	yes	—
159	C₄×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
160	C₄×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
161	C₄×C₄₇	C₂×#121	1	1	yes	characteristic
162	C₄×C₄₇	C₂×#124	1	1	yes	characteristic
163	Dic₄₇	—	2	8	no	—
164	Dic₄₇	—	8	4	no	—
165	C₂³×Q₁₆	D₄₇	1	1	yes	characteristic
166	C₂²×#107	—	47	1	no	—
167	C₂²×#103	—	47	2	no	—
168	C₂²×C₄×C₈	—	47	1	no	—
169	C₂×#391	—	47	2	no	—
170	#1969	—	47	8	no	—
171	C₂×C₄₇	#1969	1	8	yes	complemented
172	C₂×C₄₇	C₂×#390	1	1	yes	characteristic
173	C₂×C₄₇	C₂×C₄×Q₁₆	1	2	yes	—
174	C₂×C₄₇	C₂×#467	1	2	yes	—
175	C₂×C₄₇	C₂×#494	1	1	yes	characteristic
176	C₂×C₄₇	C₂×#495	1	1	yes	characteristic
177	C₂³×Q₈	C₂×D₄₇	1	2	yes	—
178	C₂²×Q₁₆	—	2	4	no	—
179	C₂²×Q₁₆	—	2	4	no	—
180	C₂²×Q₁₆	Dic₄₇	1	8	yes	complemented
181	C₂²×Q₁₆	C₂×D₄₇	1	4	yes	—
182	C₂³×C₈	—	47	1	no	—
183	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
184	C₂²×C₄⋊C₄	—	47	2	no	—
185	C₂²×C₄²	—	47	1	no	—
186	C₂×#107	—	47	4	no	—
187	C₂×#107	—	94	4	no	—
188	C₂×#103	—	47	8	no	—
189	C₂×#103	—	94	8	no	—
190	C₂×C₄×C₈	—	47	4	no	—
191	C₂×C₄×C₈	—	94	4	no	—
192	#391	—	47	8	no	—
193	#391	—	94	4	no	—
194	C₂×#95	—	94	4	no	—
195	C₄₇	C₂×#1969	1	1	yes	characteristic, complemented, Sylow
196	C₂²×Q₈	—	2	2	no	—
197	C₂²×Q₈	—	2	4	no	—
198	C₂²×Q₈	—	2	4	no	—
199	C₂²×Q₈	—	2	4	no	—
200	C₂²×Q₈	—	2	4	no	—
201	C₂²×Q₈	—	2	4	no	—
202	C₂²×Q₈	C₂×Dic₄₇	1	8	yes	—
203	C₂²×Q₈	C₂²×D₄₇	1	4	yes	—
204	C₂³×C₄	—	2	2	no	—
205	C₂³×C₄	—	47	2	no	—
206	C₂³×C₄	—	94	2	no	—
207	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
208	C₂×Q₁₆	—	2	32	no	—
209	C₂×Q₁₆	—	2	8	no	—
210	C₂×Q₁₆	—	2	8	no	—
211	C₂×Q₁₆	C₂×Dic₄₇	1	16	yes	complemented
212	C₂²×C₈	—	47	4	no	—
213	C₂²×C₈	—	47	2	no	—
214	C₂²×C₈	—	94	2	no	—
215	C₂²×C₈	—	94	2	no	—
216	C₂²×C₈	C₂×Dic₄₇	1	4	yes	—
217	C₂²×C₈	C₂²⋊D₄₇	1	4	yes	—
218	C₂²×C₈	C₂²⋊D₄₇	1	4	yes	—
219	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
220	C₂×C₄⋊C₄	—	47	4	no	—
221	C₂×C₄⋊C₄	—	94	4	no	—
222	C₂×C₄⋊C₄	—	94	4	no	—
223	C₂×C₄⋊C₄	—	94	2	no	—
224	C₂×C₄²	—	47	2	no	—
225	C₂×C₄²	—	47	2	no	—
226	C₂×C₄²	—	94	2	no	—
227	C₂×C₄²	—	94	2	no	—
228	#107	—	94	8	no	—
229	#103	—	94	16	no	—
230	C₄×C₈	—	94	8	no	—
231	#95	—	94	16	no	—
232	C₂⁴	D₄×D₄₇	1	1	yes	characteristic
233	C₂×Q₈	—	2	8	no	—
234	C₂×Q₈	—	2	4	no	—
235	C₂×Q₈	—	2	16	no	—
236	C₂×Q₈	—	2	16	no	—
237	C₂×Q₈	—	2	4	no	—
238	C₂×Q₈	—	2	4	no	—
239	C₂×Q₈	—	2	4	no	—
240	C₂×Q₈	—	2	4	no	—
241	C₂×Q₈	—	4	8	no	—
242	C₂×Q₈	—	4	4	no	—
243	C₂×Q₈	—	4	8	no	—
244	C₂×Q₈	—	4	4	no	—
245	C₂×Q₈	C₂²×Dic₄₇	1	8	yes	—
246	C₂²×C₄	—	2	8	no	—
247	C₂²×C₄	—	2	4	no	—
248	C₂²×C₄	—	4	4	no	—
249	C₂²×C₄	—	4	4	no	—
250	C₂²×C₄	—	47	8	no	—
251	C₂²×C₄	—	47	4	no	—
252	C₂²×C₄	—	94	4	no	—
253	C₂²×C₄	—	94	4	no	—
254	C₂²×C₄	—	94	4	no	—
255	C₂²×C₄	—	188	4	no	—
256	C₂²×C₄	—	188	8	no	—
257	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
258	C₂²×C₄	D₄×D₄₇	1	2	yes	—
259	C₂²×C₄	#10625352	1	2	yes	—
260	C₂²×C₄	#10625352	1	2	yes	—
261	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
262	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
263	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
264	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
265	Q₁₆	—	2	32	no	—
266	C₂×C₈	—	47	4	no	—
267	C₂×C₈	—	94	8	no	—
268	C₂×C₈	—	94	2	no	—
269	C₂×C₈	—	94	2	no	—
270	C₂×C₈	C₂²⋊Dic₄₇	1	16	yes	—
271	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
272	C₂×C₈	C₂×C₂²⋊D₄₇	1	4	yes	—
273	C₂×C₈	C₂×C₂²⋊D₄₇	1	4	yes	—
274	C₄⋊C₄	—	94	4	no	—
275	C₄⋊C₄	—	94	4	no	—
276	C₄⋊C₄	—	188	4	no	—
277	C₄²	—	94	4	no	—
278	C₄²	—	94	2	no	—
279	C₄²	—	94	2	no	—
280	C₂³	D₄×Dic₄₇	1	2	yes	—
281	C₂³	#13093995	1	2	yes	—
282	C₂³	#13094020	1	2	yes	—
283	C₂³	Q₁₆×D₄₇	1	4	yes	—
284	C₂³	#13094047	1	4	yes	—
285	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
286	Q₈	—	2	8	no	—
287	Q₈	—	2	8	no	—
288	Q₈	—	4	16	no	—
289	Q₈	—	4	4	no	—
290	Q₈	—	4	4	no	—
291	C₂×C₄	—	2	8	no	—
292	C₂×C₄	—	4	16	no	—
293	C₂×C₄	—	4	4	no	—
294	C₂×C₄	—	4	4	no	—
295	C₂×C₄	—	47	8	no	—
296	C₂×C₄	—	94	4	no	—
297	C₂×C₄	—	94	16	no	—
298	C₂×C₄	—	94	4	no	—
299	C₂×C₄	—	376	8	no	—
300	C₂×C₄	—	188	8	no	—
301	C₂×C₄	—	188	4	no	—
302	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	4	yes	—
303	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
304	C₂×C₄	D₄×Dic₄₇	1	4	yes	—
305	C₂×C₄	#13093995	1	2	yes	—
306	C₂×C₄	#13093995	1	4	yes	—
307	C₂×C₄	#13094011	1	4	yes	—
308	C₂×C₄	#13094020	1	2	yes	—
309	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
310	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
311	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
312	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
313	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
314	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
315	C₈	—	94	4	no	—
316	C₈	C₂×C₂²⋊Dic₄₇	1	8	yes	—
317	C₂²	#15689026	1	4	yes	—
318	C₂²	Q₁₆×Dic₄₇	1	8	yes	—
319	C₂²	#15689361	1	8	yes	—
320	C₂²	#15689501	1	4	yes	—
321	C₂²	#15689508	1	4	yes	—
322	C₂²	C₂×D₄×Dic₄₇	1	1	yes	characteristic
323	C₂²	C₂×#13093995	1	1	yes	characteristic
324	C₂²	C₂×#13094020	1	1	yes	characteristic
325	C₂²	C₂×Q₁₆×D₄₇	1	2	yes	—
326	C₂²	C₂×#13094047	1	2	yes	—
327	C₄	—	4	8	no	—
328	C₄	—	94	8	no	—
329	C₄	—	376	4	no	—
330	C₄	#15689026	1	4	yes	—
331	C₄	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
332	C₄	C₂×D₄×Dic₄₇	1	1	yes	characteristic
333	C₄	C₂×#13093995	1	1	yes	characteristic
334	C₄	C₂×#13094011	1	1	yes	characteristic
335	C₂	#20244250	1	8	yes	direct product factor
336	C₂	C₂×#15689026	1	1	yes	characteristic
337	C₂	C₂×Q₁₆×Dic₄₇	1	2	yes	—
338	C₂	C₂×#15689361	1	2	yes	—
339	C₂	C₂×#15689501	1	1	yes	characteristic
340	C₂	C₂×#15689508	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂³×Q₁₆×C₄₇	C₂
socle, prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂³×Q₁₆	D₄₇
hypocenter	C₄₇	C₂×#1969
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×Q₁₆

Complements

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

N	H ≅ G/N	type
5. #20244250	C₂	direct (N × H)
22. C₂²×Q₁₆×C₄₇	C₄	semidirect
56. C₂×Q₁₆×C₄₇	C₂×C₄	semidirect
92. C₂²×Dic₄₇	Q₁₆	semidirect
143. C₂×Dic₄₇	C₂×Q₁₆	semidirect
171. C₂×C₄₇	#1969	semidirect
180. C₂²×Q₁₆	Dic₄₇	semidirect
195. C₄₇	C₂×#1969	semidirect
211. C₂×Q₁₆	C₂×Dic₄₇	semidirect
335. C₂	#20244250	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×#13093958	2
C₂²×#13093948	2
C₂²×C₈×Dic₄₇	2
C₂×#15689030	2
#20244250	2
C₂³×Q₁₆×C₄₇	2
C₂×#1969	47

Subgroups — C2×#20244250 #26220089

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20244250 #26220089