Subgroups — C₂×#20244366 #26220896

Order 12032.

1422 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	C₂×#15689147	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄²×Dic₄₇	C₂	1	1	yes	maximal, characteristic
3	#20244366	C₂	1	8	yes	maximal, direct product factor
4	C₂×#15688882	C₂	1	4	yes	maximal
5	C₂×C₄₇×#533	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10625329	C₂²	1	2	yes	—
7	C₂²×C₄×Dic₄₇	—	2	2	no	—
8	C₂²×C₄×Dic₄₇	C₂²	1	2	yes	—
9	#15689147	—	2	2	no	—
10	#15689147	C₂²	1	4	yes	—
11	C₂×C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
12	C₄²×Dic₄₇	—	2	2	no	—
13	C₄²×Dic₄₇	C₂²	1	4	yes	—
14	#15688882	C₂²	1	16	yes	—
15	C₄₇×#533	—	2	2	no	—
16	C₄₇×#533	—	2	2	no	—
17	C₄₇×#533	C₄	1	4	yes	complemented
18	C₄₇×#533	C₂²	1	4	yes	—
19	C₂×C₄⋊Q₈×C₄₇	C₄	1	2	yes	complemented
20	C₂×#125×C₄₇	C₄	1	2	yes	complemented
21	C₂×C₂²⋊Q₈×C₄₇	—	2	2	no	—
22	C₂×C₄×Q₈×C₄₇	—	2	2	no	—
23	C₂×#117×C₄₇	C₂²	1	2	yes	—
24	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
25	C₂³×Dic₄₇	—	2	2	no	—
26	C₂×#10625329	—	2	2	no	—
27	C₂×#10625329	—	2	8	no	—
28	C₂×#10625329	C₂³	1	4	yes	—
29	C₂²×C₄₇⋊C₈	—	2	2	no	—
30	C₂×C₄×Dic₄₇	—	2	16	no	—
31	C₂×C₄×Dic₄₇	—	2	2	no	—
32	C₂×C₄×Dic₄₇	—	2	8	no	—
33	C₂×C₄×Dic₄₇	—	2	4	no	—
34	C₂×C₄×Dic₄₇	Q₈	1	8	yes	complemented
35	C₂×C₄×Dic₄₇	C₂³	1	4	yes	—
36	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
37	C₂²×Q₈×C₄₇	—	2	2	no	—
38	C₂³×C₄×C₄₇	D₄	1	2	yes	—
39	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
40	C₄⋊Q₈×C₄₇	—	2	4	no	—
41	C₄⋊Q₈×C₄₇	C₂×C₄	1	8	yes	complemented
42	#125×C₄₇	—	2	4	no	—
43	#125×C₄₇	C₂×C₄	1	8	yes	complemented
44	C₂²⋊Q₈×C₄₇	—	2	8	no	—
45	C₂²⋊Q₈×C₄₇	—	2	8	no	—
46	C₄×Q₈×C₄₇	—	2	8	no	—
47	C₄×Q₈×C₄₇	—	2	8	no	—
48	#117×C₄₇	—	2	2	no	—
49	#117×C₄₇	—	2	2	no	—
50	#117×C₄₇	C₂×C₄	1	4	yes	—
51	#117×C₄₇	C₂³	1	4	yes	—
52	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
53	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
54	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
55	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
56	C₂×C₄²×C₄₇	—	2	2	no	—
57	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
60	C₂×C₄²×C₄₇	D₄	1	2	yes	—
61	C₂×C₄²×C₄₇	D₄	1	2	yes	—
62	C₂×C₄²×C₄₇	D₄	1	2	yes	—
63	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
64	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
65	C₂×Q₈×C₄₇	—	2	8	no	—
66	C₂×Q₈×C₄₇	—	4	8	no	—
67	C₂²×C₄×C₄₇	—	2	8	no	—
68	C₂²×C₄×C₄₇	—	2	4	no	—
69	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
70	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
71	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
72	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
75	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
79	C₂²×C₄×C₄₇	#41	1	4	yes	—
80	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
81	C₂²×Dic₄₇	—	2	16	no	—
82	C₂²×Dic₄₇	—	2	8	no	—
83	C₂²×Dic₄₇	—	2	4	no	—
84	C₄⋊C₄×C₄₇	—	2	16	no	—
85	C₄⋊C₄×C₄₇	—	2	8	no	—
86	C₄⋊C₄×C₄₇	—	2	4	no	—
87	C₄⋊C₄×C₄₇	C₂²×C₄	1	8	yes	—
88	C₂²⋊C₄×C₄₇	—	2	4	no	—
89	C₂²⋊C₄×C₄₇	—	2	4	no	—
90	C₄²×C₄₇	—	2	4	no	—
91	C₄²×C₄₇	—	2	4	no	—
92	C₄²×C₄₇	C₂²⋊C₄	1	4	yes	—
93	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
94	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
95	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
96	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
97	C₄²×C₄₇	C₂×D₄	1	4	yes	—
98	C₄×Dic₄₇	—	2	32	no	—
99	C₄×Dic₄₇	—	2	8	no	—
100	C₄×Dic₄₇	—	2	4	no	—
101	C₄×Dic₄₇	C₂×Q₈	1	8	yes	complemented
102	#10625329	—	2	8	no	—
103	#10625329	—	4	4	no	—
104	C₂×C₄₇⋊C₈	—	2	4	no	—
105	C₂×C₄₇⋊C₈	—	4	4	no	—
106	C₂³×C₄₇	—	2	4	no	—
107	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂³×C₄₇	C₂²⋊Q₈	1	4	yes	—
110	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
111	Q₈×C₄₇	—	4	16	no	—
112	C₂×C₄×C₄₇	—	2	4	no	—
113	C₂×C₄×C₄₇	—	2	8	no	—
114	C₂×C₄×C₄₇	—	2	8	no	—
115	C₂×C₄×C₄₇	—	2	8	no	—
116	C₂×C₄×C₄₇	—	4	8	no	—
117	C₂×C₄×C₄₇	#104	1	8	yes	—
118	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
119	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
122	C₂×C₄×C₄₇	C₄×Q₈	1	4	yes	—
123	C₂×C₄×C₄₇	C₄×Q₈	1	4	yes	—
124	C₂×C₄×C₄₇	C₂²⋊Q₈	1	4	yes	—
125	C₂×C₄×C₄₇	#124	1	2	yes	—
126	C₂×C₄×C₄₇	#124	1	2	yes	—
127	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
128	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
129	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
130	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
132	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
133	C₂×Dic₄₇	—	2	16	no	—
134	C₂×Dic₄₇	—	2	32	no	—
135	C₂×Dic₄₇	—	2	8	no	—
136	C₄₇⋊C₈	—	4	4	no	—
137	C₂×#1994	—	47	1	no	maximal, Sylow
138	C₂²×C₄₇	—	2	2	no	—
139	C₂²×C₄₇	—	2	2	no	—
140	C₂²×C₄₇	—	2	8	no	—
141	C₂²×C₄₇	#391	1	2	yes	—
142	C₂²×C₄₇	#391	1	2	yes	—
143	C₂²×C₄₇	C₂×#104	1	4	yes	—
144	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
145	C₂²×C₄₇	C₂×C₂²⋊Q₈	1	2	yes	—
146	C₄×C₄₇	—	2	4	no	—
147	C₄×C₄₇	—	4	8	no	—
148	C₄×C₄₇	#391	1	2	yes	—
149	C₄×C₄₇	#391	1	2	yes	—
150	C₄×C₄₇	C₂×#104	1	8	yes	—
151	C₄×C₄₇	C₂×C₄×Q₈	1	2	yes	—
152	C₄×C₄₇	C₂×#124	1	1	yes	characteristic
153	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
154	Dic₄₇	—	2	16	no	—
155	C₂×#533	D₄₇	1	1	yes	characteristic
156	C₂×#423	—	47	1	no	—
157	C₂×C₄³	—	47	1	no	—
158	#1994	—	47	8	no	—
159	C₂×#339	—	47	4	no	—
160	C₂×C₄₇	—	2	2	no	—
161	C₂×C₄₇	—	2	2	no	—
162	C₂×C₄₇	#1994	1	4	yes	complemented
163	C₂×C₄₇	C₂×#391	1	1	yes	characteristic
164	C₂×C₄₇	C₂²×#104	1	2	yes	—
165	C₂²×M₁₆	—	47	2	no	—
166	#533	—	2	2	no	—
167	#533	—	2	2	no	—
168	#533	Dic₄₇	1	4	yes	complemented
169	#533	C₂×D₄₇	1	4	yes	—
170	C₂×C₄⋊Q₈	Dic₄₇	1	2	yes	complemented
171	C₂×#125	Dic₄₇	1	2	yes	complemented
172	C₂×C₂²⋊Q₈	—	2	2	no	—
173	C₂×C₄×Q₈	—	2	2	no	—
174	C₂×#117	C₂×D₄₇	1	2	yes	—
175	C₂²×C₄²	—	47	2	no	—
176	C₂²×C₄²	—	94	2	no	—
177	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
178	#423	—	47	4	no	—
179	#423	—	94	2	no	—
180	C₂×C₄⋊C₈	—	94	2	no	—
181	C₄³	—	47	4	no	—
182	C₄³	—	94	2	no	—
183	#339	—	47	16	no	—
184	C₄₇	C₂×#1994	1	1	yes	characteristic, complemented, Sylow
185	C₂²×Q₈	—	2	2	no	—
186	C₂³×C₄	—	94	2	no	—
187	C₂³×C₄	C₂²⋊D₄₇	1	2	yes	—
188	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
189	C₂×M₁₆	—	47	4	no	—
190	C₂×M₁₆	—	94	2	no	—
191	C₂×M₁₆	—	94	8	no	—
192	C₂²×C₈	—	94	2	no	—
193	C₄⋊Q₈	—	2	4	no	—
194	C₄⋊Q₈	C₂×Dic₄₇	1	8	yes	complemented
195	#125	—	2	4	no	—
196	#125	C₂×Dic₄₇	1	8	yes	complemented
197	C₂²⋊Q₈	—	2	8	no	—
198	C₂²⋊Q₈	—	2	8	no	—
199	C₄×Q₈	—	2	8	no	—
200	C₄×Q₈	—	2	8	no	—
201	#117	—	2	2	no	—
202	#117	—	2	2	no	—
203	#117	C₂×Dic₄₇	1	4	yes	—
204	#117	C₂²×D₄₇	1	4	yes	—
205	C₂×C₄⋊C₄	—	2	4	no	—
206	C₂×C₄⋊C₄	—	2	2	no	—
207	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
208	C₂×C₂²⋊C₄	—	2	2	no	—
209	C₂×C₄²	—	2	2	no	—
210	C₂×C₄²	—	47	8	no	—
211	C₂×C₄²	—	47	4	no	—
212	C₂×C₄²	—	94	16	no	—
213	C₂×C₄²	—	94	4	no	—
214	C₂×C₄²	—	94	8	no	—
215	C₂×C₄²	—	94	2	no	—
216	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
217	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
218	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
219	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
220	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
221	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
222	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
223	C₄⋊C₈	—	94	8	no	—
224	C₂⁴	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
225	C₂×Q₈	—	2	8	no	—
226	C₂×Q₈	—	4	8	no	—
227	C₂²×C₄	—	2	8	no	—
228	C₂²×C₄	—	2	4	no	—
229	C₂²×C₄	—	94	16	no	—
230	C₂²×C₄	—	94	8	no	—
231	C₂²×C₄	—	94	4	no	—
232	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
233	C₂²×C₄	C₂²⋊Dic₄₇	1	4	yes	—
234	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
235	C₂²×C₄	Q₈×D₄₇	1	4	yes	—
236	C₂²×C₄	#10625354	1	4	yes	—
237	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
238	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
239	C₂²×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
240	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
241	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
242	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
243	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
244	M₁₆	—	94	8	no	—
245	M₁₆	—	188	4	no	—
246	C₂×C₈	—	94	4	no	—
247	C₂×C₈	—	188	4	no	—
248	C₄⋊C₄	—	2	16	no	—
249	C₄⋊C₄	—	2	8	no	—
250	C₄⋊C₄	—	2	4	no	—
251	C₄⋊C₄	C₂²×Dic₄₇	1	8	yes	—
252	C₂²⋊C₄	—	2	4	no	—
253	C₂²⋊C₄	—	2	4	no	—
254	C₄²	—	2	4	no	—
255	C₄²	—	2	4	no	—
256	C₄²	—	47	8	no	—
257	C₄²	—	94	32	no	—
258	C₄²	—	94	8	no	—
259	C₄²	—	94	4	no	—
260	C₄²	C₂²⋊Dic₄₇	1	4	yes	—
261	C₄²	C₂²⋊Dic₄₇	1	2	yes	—
262	C₄²	C₂²⋊Dic₄₇	1	2	yes	—
263	C₄²	C₂²×Dic₄₇	1	2	yes	—
264	C₄²	C₂²×Dic₄₇	1	2	yes	—
265	C₄²	C₂×C₂²⋊D₄₇	1	4	yes	—
266	C₂³	—	2	4	no	—
267	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
268	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
269	C₂³	#13094000	1	4	yes	—
270	C₂³	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
271	Q₈	—	4	16	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	8	no	—
274	C₂×C₄	—	2	8	no	—
275	C₂×C₄	—	2	8	no	—
276	C₂×C₄	—	4	8	no	—
277	C₂×C₄	—	94	16	no	—
278	C₂×C₄	—	94	32	no	—
279	C₂×C₄	—	94	8	no	—
280	C₂×C₄	#13093951	1	8	yes	—
281	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	4	yes	—
282	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
283	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
284	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	4	yes	—
285	C₂×C₄	Q₈×Dic₄₇	1	4	yes	—
286	C₂×C₄	Q₈×Dic₄₇	1	4	yes	—
287	C₂×C₄	#13094000	1	4	yes	—
288	C₂×C₄	#13094013	1	2	yes	—
289	C₂×C₄	#13094013	1	2	yes	—
290	C₂×C₄	#13094022	1	2	yes	—
291	C₂×C₄	#13094022	1	2	yes	—
292	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
293	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
294	C₂×C₄	C₂×Q₈×D₄₇	1	2	yes	—
295	C₂×C₄	C₂×#10625354	1	2	yes	—
296	C₈	—	188	4	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	8	no	—
300	C₂²	#15689030	1	2	yes	—
301	C₂²	#15689030	1	2	yes	—
302	C₂²	C₂×#13093951	1	4	yes	—
303	C₂²	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
304	C₂²	C₂×#13094000	1	2	yes	—
305	C₄	—	2	4	no	—
306	C₄	—	4	8	no	—
307	C₄	—	94	16	no	—
308	C₄	#15689030	1	2	yes	—
309	C₄	#15689030	1	2	yes	—
310	C₄	C₂×#13093951	1	8	yes	—
311	C₄	C₂×Q₈×Dic₄₇	1	2	yes	—
312	C₄	C₂×#13094013	1	1	yes	characteristic
313	C₄	C₂×#13094022	1	1	yes	characteristic
314	C₂	—	2	2	no	—
315	C₂	—	2	2	no	—
316	C₂	#20244366	1	4	yes	direct product factor
317	C₂	C₂×#15689030	1	1	yes	characteristic
318	C₂	C₂²×#13093951	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×#533	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#533	D₄₇
hypocenter	C₄₇	C₂×#1994
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#533

Complements

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

N	H ≅ G/N	type
3. #20244366	C₂	direct (N × H)
17. C₄₇×#533	C₄	semidirect
19. C₂×C₄⋊Q₈×C₄₇	C₄	semidirect
20. C₂×#125×C₄₇	C₄	semidirect
34. C₂×C₄×Dic₄₇	Q₈	semidirect
41. C₄⋊Q₈×C₄₇	C₂×C₄	semidirect
43. #125×C₄₇	C₂×C₄	semidirect
101. C₄×Dic₄₇	C₂×Q₈	semidirect
162. C₂×C₄₇	#1994	semidirect
168. #533	Dic₄₇	semidirect
170. C₂×C₄⋊Q₈	Dic₄₇	semidirect
171. C₂×#125	Dic₄₇	semidirect
184. C₄₇	C₂×#1994	semidirect
194. C₄⋊Q₈	C₂×Dic₄₇	semidirect
196. #125	C₂×Dic₄₇	semidirect
316. C₂	#20244366	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
C₂×C₄²×Dic₄₇	2
#20244366	2
C₂×#15688882	2
C₂×C₄₇×#533	2
C₂×#1994	47

Subgroups — C2×#20244366 #26220896

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20244366 #26220896