Subgroups — C₂×#20244233 #26220048

Order 12032.

1518 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689147	C₂	1	1	yes	maximal, characteristic
2	C₂²×#13093943	C₂	1	4	yes	maximal
3	C₂×C₄×#10625329	C₂	1	1	yes	maximal, characteristic
4	#20244233	C₂	1	8	yes	maximal, direct product factor
5	C₂²×C₄⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10625329	C₂²	1	2	yes	—
7	C₂²×#10625329	C₂²	1	2	yes	—
8	#15689147	—	2	2	no	—
9	#15689147	C₂²	1	4	yes	—
10	C₂×C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
11	C₂×#13093943	—	2	16	no	—
12	C₂×#13093943	C₂²	1	16	yes	—
13	C₄×#10625329	—	2	2	no	—
14	C₄×#10625329	C₂²	1	4	yes	—
15	C₂×#13093935	—	2	1	no	—
16	C₂×C₄×C₄₇⋊C₈	—	2	1	no	—
17	C₂³×Q₈×C₄₇	C₂²	1	2	yes	—
18	C₂×C₄⋊Q₈×C₄₇	—	2	1	no	—
19	C₂×C₄⋊Q₈×C₄₇	—	2	1	no	—
20	C₂×C₄⋊Q₈×C₄₇	—	2	2	no	—
21	C₂×C₄⋊Q₈×C₄₇	—	2	2	no	—
22	C₂×C₄⋊Q₈×C₄₇	—	2	2	no	—
23	C₂×C₄⋊Q₈×C₄₇	C₄	1	4	yes	—
24	C₂×C₄⋊Q₈×C₄₇	C₂²	1	4	yes	—
25	C₂²×C₄⋊C₄×C₄₇	C₄	1	4	yes	—
26	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
27	C₂×#10625329	—	2	2	no	—
28	C₂×#10625329	—	2	4	no	—
29	C₂×#10625329	—	2	2	no	—
30	C₂×#10625329	—	2	8	no	—
31	C₂×#10625329	D₄	1	8	yes	—
32	C₂×#10625329	C₂³	1	4	yes	—
33	C₂×#10625329	C₂³	1	4	yes	—
34	C₂²×C₄₇⋊C₈	—	2	2	no	—
35	C₂²×C₄₇⋊C₈	—	2	2	no	—
36	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
37	#13093943	—	2	32	no	—
38	#13093935	—	2	4	no	—
39	C₄×C₄₇⋊C₈	—	2	4	no	—
40	C₂²×Q₈×C₄₇	—	2	2	no	—
41	C₂²×Q₈×C₄₇	—	2	2	no	—
42	C₂²×Q₈×C₄₇	—	2	2	no	—
43	C₂²×Q₈×C₄₇	—	2	2	no	—
44	C₂²×Q₈×C₄₇	—	2	8	no	—
45	C₂²×Q₈×C₄₇	—	2	8	no	—
46	C₂²×Q₈×C₄₇	C₂×C₄	1	4	yes	—
47	C₂²×Q₈×C₄₇	C₂³	1	4	yes	—
48	C₂³×C₄×C₄₇	C₂×C₄	1	4	yes	—
49	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
50	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
51	C₄⋊Q₈×C₄₇	—	2	4	no	—
52	C₄⋊Q₈×C₄₇	—	2	8	no	—
53	C₄⋊Q₈×C₄₇	—	2	4	no	—
54	C₄⋊Q₈×C₄₇	—	2	4	no	—
55	C₄⋊Q₈×C₄₇	—	2	4	no	—
56	C₄⋊Q₈×C₄₇	—	2	8	no	—
57	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
58	C₂×C₄⋊C₄×C₄₇	—	2	8	no	—
59	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
60	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	16	yes	—
61	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
62	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
63	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
64	C₂×C₄²×C₄₇	D₄	1	2	yes	—
65	C₂×C₄²×C₄₇	D₄	1	2	yes	—
66	C₂×C₄²×C₄₇	D₄	1	2	yes	—
67	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
68	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
69	C₂×Q₈×C₄₇	—	2	4	no	—
70	C₂×Q₈×C₄₇	—	2	4	no	—
71	C₂×Q₈×C₄₇	—	2	16	no	—
72	C₂×Q₈×C₄₇	—	2	4	no	—
73	C₂×Q₈×C₄₇	—	2	4	no	—
74	C₂×Q₈×C₄₇	—	2	8	no	—
75	C₂×Q₈×C₄₇	—	2	8	no	—
76	C₂×Q₈×C₄₇	—	4	8	no	—
77	C₂×Q₈×C₄₇	—	4	8	no	—
78	C₂×Q₈×C₄₇	—	4	4	no	—
79	C₂×Q₈×C₄₇	—	4	4	no	—
80	C₂×Q₈×C₄₇	—	4	4	no	—
81	C₂×Q₈×C₄₇	—	4	4	no	—
82	C₂²×C₄×C₄₇	—	2	8	no	—
83	C₂²×C₄×C₄₇	—	2	4	no	—
84	C₂²×C₄×C₄₇	—	2	4	no	—
85	C₂²×C₄×C₄₇	—	2	16	no	—
86	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
87	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
88	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
89	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
90	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
91	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
97	C₂²×C₄×C₄₇	#41	1	4	yes	—
98	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
99	C₄⋊C₄×C₄₇	—	2	16	no	—
100	C₄⋊C₄×C₄₇	—	2	16	no	—
101	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
102	C₄²×C₄₇	C₂²⋊C₄	1	4	yes	—
103	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
104	C₄²×C₄₇	C₂×D₄	1	2	yes	—
105	C₄²×C₄₇	C₂×D₄	1	2	yes	—
106	C₄²×C₄₇	C₂×D₄	1	4	yes	—
107	#10625329	—	2	8	no	—
108	#10625329	—	2	4	no	—
109	#10625329	—	2	8	no	—
110	#10625329	—	4	4	no	—
111	#10625329	C₂×D₄	1	8	yes	—
112	C₂×C₄₇⋊C₈	—	2	8	no	—
113	C₂×C₄₇⋊C₈	—	2	4	no	—
114	C₂×C₄₇⋊C₈	—	2	4	no	—
115	C₂×C₄₇⋊C₈	—	4	4	no	—
116	C₂³×C₄₇	—	2	4	no	—
117	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂³×C₄₇	C₄×D₄	1	4	yes	—
120	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
121	Q₈×C₄₇	—	4	8	no	—
122	Q₈×C₄₇	—	4	4	no	—
123	Q₈×C₄₇	—	4	4	no	—
124	Q₈×C₄₇	—	4	8	no	—
125	Q₈×C₄₇	—	4	8	no	—
126	C₂×C₄×C₄₇	—	2	4	no	—
127	C₂×C₄×C₄₇	—	2	8	no	—
128	C₂×C₄×C₄₇	—	2	8	no	—
129	C₂×C₄×C₄₇	—	2	8	no	—
130	C₂×C₄×C₄₇	—	2	16	no	—
131	C₂×C₄×C₄₇	—	4	8	no	—
132	C₂×C₄×C₄₇	—	4	8	no	—
133	C₂×C₄×C₄₇	#101	1	8	yes	complemented
134	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
135	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
136	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
137	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
138	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
139	C₂×C₄×C₄₇	#121	1	4	yes	—
140	C₂×C₄×C₄₇	#121	1	4	yes	—
141	C₂×C₄×C₄₇	#124	1	2	yes	—
142	C₂×C₄×C₄₇	#124	1	2	yes	—
143	C₂×C₄×C₄₇	#127	1	2	yes	—
144	C₂×C₄×C₄₇	#127	1	2	yes	—
145	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
146	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
148	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
149	C₄₇⋊C₈	—	2	8	no	—
150	C₄₇⋊C₈	—	4	4	no	—
151	C₂×#1966	—	47	1	no	maximal, Sylow
152	C₂²×C₄₇	—	2	2	no	—
153	C₂²×C₄₇	—	2	2	no	—
154	C₂²×C₄₇	—	2	8	no	—
155	C₂²×C₄₇	#390	1	2	yes	—
156	C₂²×C₄₇	#390	1	2	yes	—
157	C₂²×C₄₇	C₂×#101	1	4	yes	—
158	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
159	C₂²×C₄₇	C₂×C₄×D₄	1	2	yes	—
160	C₄×C₄₇	—	2	4	no	—
161	C₄×C₄₇	—	4	8	no	—
162	C₄×C₄₇	#390	1	2	yes	—
163	C₄×C₄₇	#390	1	2	yes	—
164	C₄×C₄₇	C₂×#101	1	8	yes	complemented
165	C₄×C₄₇	C₂×#121	1	2	yes	—
166	C₄×C₄₇	C₂×#124	1	1	yes	characteristic
167	C₄×C₄₇	C₂×#127	1	1	yes	characteristic
168	C₂²×C₄⋊Q₈	D₄₇	1	1	yes	characteristic
169	C₂×#423	—	47	1	no	—
170	C₂²×#101	—	47	4	no	—
171	C₂×C₄×M₁₆	—	47	1	no	—
172	#1966	—	47	8	no	—
173	C₂×C₄₇	—	2	2	no	—
174	C₂×C₄₇	—	2	2	no	—
175	C₂×C₄₇	#1966	1	4	yes	complemented
176	C₂×C₄₇	C₂×#390	1	1	yes	characteristic
177	C₂×C₄₇	C₂²×#101	1	2	yes	—
178	C₂³×Q₈	C₂×D₄₇	1	2	yes	—
179	C₂²×M₁₆	—	47	2	no	—
180	C₂²×M₁₆	—	47	2	no	—
181	C₂×C₄⋊Q₈	—	2	1	no	—
182	C₂×C₄⋊Q₈	—	2	1	no	—
183	C₂×C₄⋊Q₈	—	2	2	no	—
184	C₂×C₄⋊Q₈	—	2	2	no	—
185	C₂×C₄⋊Q₈	—	2	2	no	—
186	C₂×C₄⋊Q₈	Dic₄₇	1	4	yes	—
187	C₂×C₄⋊Q₈	C₂×D₄₇	1	4	yes	—
188	C₂²×C₄⋊C₄	Dic₄₇	1	4	yes	—
189	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
190	#423	—	47	4	no	—
191	#423	—	94	2	no	—
192	C₂×C₄⋊C₈	—	94	2	no	—
193	C₂×#101	—	47	16	no	—
194	C₂×#101	—	94	16	no	—
195	C₄×M₁₆	—	47	4	no	—
196	C₄×M₁₆	—	94	2	no	—
197	C₂×#97	—	94	1	no	—
198	C₂×C₄×C₈	—	94	1	no	—
199	C₄₇	C₂×#1966	1	1	yes	characteristic, complemented, Sylow
200	C₂²×Q₈	—	2	8	no	—
201	C₂²×Q₈	—	2	8	no	—
202	C₂²×Q₈	—	2	2	no	—
203	C₂²×Q₈	—	2	2	no	—
204	C₂²×Q₈	—	2	2	no	—
205	C₂²×Q₈	—	2	2	no	—
206	C₂²×Q₈	C₂×Dic₄₇	1	4	yes	—
207	C₂²×Q₈	C₂²×D₄₇	1	4	yes	—
208	C₂³×C₄	C₂×Dic₄₇	1	4	yes	—
209	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
210	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
211	C₂×M₁₆	—	47	8	no	—
212	C₂×M₁₆	—	47	4	no	—
213	C₂×M₁₆	—	47	4	no	—
214	C₂×M₁₆	—	94	4	no	—
215	C₂×M₁₆	—	94	2	no	—
216	C₂×M₁₆	—	94	2	no	—
217	C₂×M₁₆	—	94	8	no	—
218	C₂²×C₈	—	94	2	no	—
219	C₂²×C₈	—	94	2	no	—
220	C₄⋊Q₈	—	2	4	no	—
221	C₄⋊Q₈	—	2	8	no	—
222	C₄⋊Q₈	—	2	4	no	—
223	C₄⋊Q₈	—	2	4	no	—
224	C₄⋊Q₈	—	2	4	no	—
225	C₄⋊Q₈	—	2	8	no	—
226	C₂×C₄⋊C₄	—	2	8	no	—
227	C₂×C₄⋊C₄	—	2	4	no	—
228	C₂×C₄⋊C₄	—	2	4	no	—
229	C₂×C₄⋊C₄	C₂×Dic₄₇	1	16	yes	—
230	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
231	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
232	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
233	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
234	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
235	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
236	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
237	C₄⋊C₈	—	94	8	no	—
238	#101	—	94	32	no	—
239	#97	—	94	4	no	—
240	C₄×C₈	—	94	4	no	—
241	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
242	C₂×Q₈	—	2	4	no	—
243	C₂×Q₈	—	2	4	no	—
244	C₂×Q₈	—	2	16	no	—
245	C₂×Q₈	—	2	8	no	—
246	C₂×Q₈	—	2	4	no	—
247	C₂×Q₈	—	2	4	no	—
248	C₂×Q₈	—	2	8	no	—
249	C₂×Q₈	—	4	8	no	—
250	C₂×Q₈	—	4	8	no	—
251	C₂×Q₈	—	4	4	no	—
252	C₂×Q₈	—	4	4	no	—
253	C₂×Q₈	—	4	4	no	—
254	C₂×Q₈	—	4	4	no	—
255	C₂²×C₄	—	2	8	no	—
256	C₂²×C₄	—	2	4	no	—
257	C₂²×C₄	—	2	4	no	—
258	C₂²×C₄	—	2	16	no	—
259	C₂²×C₄	C₂²⋊Dic₄₇	1	4	yes	—
260	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
261	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
262	C₂²×C₄	D₄×D₄₇	1	4	yes	—
263	C₂²×C₄	#10625352	1	4	yes	—
264	C₂²×C₄	C₂²×Dic₄₇	1	8	yes	—
265	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
266	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
267	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
268	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
269	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
270	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
271	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
272	M₁₆	—	47	8	no	—
273	M₁₆	—	94	8	no	—
274	M₁₆	—	94	4	no	—
275	M₁₆	—	94	8	no	—
276	M₁₆	—	188	4	no	—
277	C₂×C₈	—	94	8	no	—
278	C₂×C₈	—	94	4	no	—
279	C₂×C₈	—	94	4	no	—
280	C₂×C₈	—	188	4	no	—
281	C₄⋊C₄	—	2	16	no	—
282	C₄⋊C₄	—	2	16	no	—
283	C₄²	C₂²⋊Dic₄₇	1	2	yes	—
284	C₄²	C₂²⋊Dic₄₇	1	4	yes	—
285	C₄²	C₂²⋊Dic₄₇	1	2	yes	—
286	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
287	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
288	C₄²	C₂×C₂²⋊D₄₇	1	4	yes	—
289	C₂³	—	2	4	no	—
290	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
291	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
292	C₂³	D₄×Dic₄₇	1	4	yes	—
293	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
294	Q₈	—	4	8	no	—
295	Q₈	—	4	8	no	—
296	Q₈	—	4	8	no	—
297	Q₈	—	4	4	no	—
298	Q₈	—	4	4	no	—
299	C₂×C₄	—	2	4	no	—
300	C₂×C₄	—	2	8	no	—
301	C₂×C₄	—	2	8	no	—
302	C₂×C₄	—	2	8	no	—
303	C₂×C₄	—	2	16	no	—
304	C₂×C₄	—	4	8	no	—
305	C₂×C₄	—	4	8	no	—
306	C₂×C₄	#13093943	1	8	yes	complemented
307	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	4	yes	—
308	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
309	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
310	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	4	yes	—
311	C₂×C₄	D₄×Dic₄₇	1	4	yes	—
312	C₂×C₄	#13093995	1	4	yes	—
313	C₂×C₄	#13093995	1	4	yes	—
314	C₂×C₄	#13094011	1	2	yes	—
315	C₂×C₄	#13094011	1	2	yes	—
316	C₂×C₄	#13094020	1	2	yes	—
317	C₂×C₄	#13094020	1	2	yes	—
318	C₂×C₄	C₂×D₄×D₄₇	1	2	yes	—
319	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
320	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
321	C₂×C₄	C₂×#10625352	1	2	yes	—
322	C₈	—	94	8	no	—
323	C₈	—	188	4	no	—
324	C₂²	—	2	8	no	—
325	C₂²	—	2	2	no	—
326	C₂²	—	2	2	no	—
327	C₂²	#15689026	1	2	yes	—
328	C₂²	#15689026	1	2	yes	—
329	C₂²	C₂×#13093943	1	4	yes	—
330	C₂²	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
331	C₂²	C₂×D₄×Dic₄₇	1	2	yes	—
332	C₄	—	2	4	no	—
333	C₄	—	4	8	no	—
334	C₄	#15689026	1	2	yes	—
335	C₄	#15689026	1	2	yes	—
336	C₄	C₂×#13093943	1	8	yes	complemented
337	C₄	C₂×#13093995	1	2	yes	—
338	C₄	C₂×#13094011	1	1	yes	characteristic
339	C₄	C₂×#13094020	1	1	yes	characteristic
340	C₂	—	2	2	no	—
341	C₂	—	2	2	no	—
342	C₂	#20244233	1	4	yes	direct product factor
343	C₂	C₂×#15689026	1	1	yes	characteristic
344	C₂	C₂²×#13093943	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₄₇
commutator	C₂³×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂²×C₄⋊Q₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂²×C₄⋊Q₈	D₄₇
hypocenter	C₄₇	C₂×#1966
LCS	G ▷ C₂³×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
4. #20244233	C₂	direct (N × H)
133. C₂×C₄×C₄₇	#101	semidirect
164. C₄×C₄₇	C₂×#101	semidirect
175. C₂×C₄₇	#1966	semidirect
199. C₄₇	C₂×#1966	semidirect
306. C₂×C₄	#13093943	semidirect
336. C₄	C₂×#13093943	semidirect
342. C₂	#20244233	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#15689147	2
C₂²×#13093943	2
C₂×C₄×#10625329	2
#20244233	2
C₂²×C₄⋊Q₈×C₄₇	2
C₂×#1966	47

Subgroups — C2×#20244233 #26220048

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20244233 #26220048