Subgroups — C₂×#20244946 #26246172

Order 12032.

998 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15688993	C₂	1	1	yes	maximal, characteristic
2	#20244946	C₂	1	8	yes	maximal, direct product factor
3	C₂×#15688885	C₂	1	2	yes	maximal
4	C₂×#15688885	C₂	1	2	yes	maximal
5	C₂×#15688878	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄₇×#389	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₄₇⋊C₈	C₂²	1	2	yes	—
8	C₂²×C₄⋊Dic₄₇	—	2	1	no	—
9	C₂²×C₄⋊Dic₄₇	—	2	1	no	—
10	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
11	#15688993	—	2	2	no	—
12	#15688993	C₂²	1	4	yes	—
13	#15688885	C₂²	1	8	yes	—
14	#15688885	C₂²	1	8	yes	—
15	#15688878	C₂²	1	4	yes	—
16	C₂²×C₄⋊C₄×C₄₇	—	2	1	no	—
17	C₂²×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
18	C₂²×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
19	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
20	C₄₇×#389	—	2	2	no	—
21	C₄₇×#389	C₂²	1	4	yes	—
22	C₂×#95×C₄₇	—	2	1	no	—
23	C₂³×Dic₄₇	—	2	2	no	—
24	C₂²×C₄₇⋊C₈	—	2	4	no	—
25	C₂²×C₄₇⋊C₈	—	2	4	no	—
26	C₂²×C₄₇⋊C₈	—	2	4	no	—
27	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
28	C₂×C₄⋊Dic₄₇	—	2	8	no	—
29	C₂×C₄⋊Dic₄₇	—	2	2	no	—
30	C₂×C₄⋊Dic₄₇	—	2	4	no	—
31	C₂×C₄⋊Dic₄₇	—	2	4	no	—
32	C₂×C₄⋊Dic₄₇	—	4	4	no	—
33	C₂×C₄⋊Dic₄₇	—	4	4	no	—
34	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
35	C₂³×C₄×C₄₇	—	2	1	no	—
36	C₂³×C₄×C₄₇	—	2	1	no	—
37	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
38	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
39	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
40	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
41	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
42	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
43	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
44	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
45	C₂×C₄⋊C₄×C₄₇	—	4	4	no	—
46	C₂×C₄⋊C₄×C₄₇	Q₈	1	4	yes	—
47	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
48	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
49	C₂×C₄²×C₄₇	—	2	4	no	—
50	C₂×C₄²×C₄₇	D₄	1	2	yes	—
51	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
52	#95×C₄₇	—	2	4	no	—
53	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	—	2	2	no	—
55	C₂²×C₄×C₄₇	—	2	2	no	—
56	C₂²×C₄×C₄₇	—	2	2	no	—
57	C₂²×C₄×C₄₇	—	2	2	no	—
58	C₂²×C₄×C₄₇	—	2	4	no	—
59	C₂²×C₄×C₄₇	—	2	2	no	—
60	C₂²×C₄×C₄₇	—	2	2	no	—
61	C₂²×C₄×C₄₇	—	4	2	no	—
62	C₂²×C₄×C₄₇	—	4	2	no	—
63	C₂²×C₄×C₄₇	—	4	2	no	—
64	C₂²×C₄×C₄₇	—	4	2	no	—
65	C₂²×C₄×C₄₇	—	4	2	no	—
66	C₂²×C₄×C₄₇	D₈	1	4	yes	—
67	C₂²×C₄×C₄₇	SD₁₆	1	4	yes	—
68	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
72	C₂²×C₄×C₄₇	#41	1	2	yes	—
73	C₂²×C₄×C₄₇	#41	1	4	yes	—
74	C₂²×C₄×C₄₇	#41	1	4	yes	—
75	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
76	C₂²×Dic₄₇	—	2	4	no	—
77	C₂²×Dic₄₇	—	4	4	no	—
78	C₂²×Dic₄₇	—	4	4	no	—
79	C₂²×Dic₄₇	—	4	4	no	—
80	C₄⋊C₄×C₄₇	—	2	4	no	—
81	C₄⋊C₄×C₄₇	—	2	2	no	—
82	C₄⋊C₄×C₄₇	—	2	4	no	—
83	C₄⋊C₄×C₄₇	—	4	2	no	—
84	C₄⋊C₄×C₄₇	—	4	8	no	—
85	C₄⋊C₄×C₄₇	C₂×Q₈	1	4	yes	—
86	C₄²×C₄₇	—	2	4	no	—
87	C₄²×C₄₇	—	2	4	no	—
88	C₄⋊Dic₄₇	—	2	8	no	—
89	C₄⋊Dic₄₇	—	4	4	no	—
90	C₄⋊Dic₄₇	—	4	8	no	—
91	C₄⋊Dic₄₇	—	4	8	no	—
92	C₂×C₄₇⋊C₈	—	2	4	no	—
93	C₂×C₄₇⋊C₈	—	2	4	no	—
94	C₂×C₄₇⋊C₈	—	2	4	no	—
95	C₂×C₄₇⋊C₈	—	4	8	no	—
96	C₂³×C₄₇	C₂²⋊Q₈	1	2	yes	—
97	C₂³×C₄₇	#123	1	2	yes	—
98	C₂³×C₄₇	#124	1	2	yes	—
99	C₂³×C₄₇	C₂×D₈	1	2	yes	—
100	C₂³×C₄₇	C₂×SD₁₆	1	2	yes	—
101	C₂³×C₄₇	#137	1	2	yes	—
102	C₂³×C₄₇	#137	1	2	yes	—
103	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
104	C₂×C₄×C₄₇	—	2	2	no	—
105	C₂×C₄×C₄₇	—	2	2	no	—
106	C₂×C₄×C₄₇	—	2	2	no	—
107	C₂×C₄×C₄₇	—	2	2	no	—
108	C₂×C₄×C₄₇	—	2	8	no	—
109	C₂×C₄×C₄₇	—	2	8	no	—
110	C₂×C₄×C₄₇	—	2	4	no	—
111	C₂×C₄×C₄₇	—	4	2	no	—
112	C₂×C₄×C₄₇	—	4	2	no	—
113	C₂×C₄×C₄₇	—	4	8	no	—
114	C₂×C₄×C₄₇	—	4	2	no	—
115	C₂×C₄×C₄₇	—	4	2	no	—
116	C₂×C₄×C₄₇	—	4	2	no	—
117	C₂×C₄×C₄₇	—	8	4	no	—
118	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
119	C₂×C₄×C₄₇	#123	1	2	yes	—
120	C₂×C₄×C₄₇	#124	1	2	yes	—
121	C₂×C₄×C₄₇	#125	1	8	yes	—
122	C₂×C₄×C₄₇	#126	1	8	yes	—
123	C₂×C₄×C₄₇	C₂×D₈	1	4	yes	—
124	C₂×C₄×C₄₇	C₂×SD₁₆	1	4	yes	—
125	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
126	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
127	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
128	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
129	C₂×Dic₄₇	—	4	4	no	—
130	C₂×Dic₄₇	—	4	4	no	—
131	C₂×Dic₄₇	—	4	4	no	—
132	C₂×Dic₄₇	—	8	8	no	—
133	C₄₇⋊C₈	—	4	4	no	—
134	C₂×#2099	—	47	1	no	maximal, Sylow
135	C₂²×C₄₇	#400	1	4	yes	—
136	C₂²×C₄₇	#474	1	4	yes	—
137	C₂²×C₄₇	#475	1	4	yes	—
138	C₂²×C₄₇	#480	1	4	yes	—
139	C₂²×C₄₇	#483	1	4	yes	—
140	C₂²×C₄₇	#486	1	4	yes	—
141	C₂²×C₄₇	#491	1	4	yes	—
142	C₂²×C₄₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
143	C₂²×C₄₇	C₂×#123	1	1	yes	characteristic
144	C₂²×C₄₇	C₂×#124	1	1	yes	characteristic
145	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
146	C₂²×C₄₇	C₂²×SD₁₆	1	1	yes	characteristic
147	C₂²×C₄₇	C₂×#137	1	1	yes	characteristic
148	C₂²×C₄₇	C₂×#137	1	1	yes	characteristic
149	C₄×C₄₇	—	2	4	no	—
150	C₄×C₄₇	—	2	4	no	—
151	C₄×C₄₇	—	4	4	no	—
152	C₄×C₄₇	—	8	2	no	—
153	C₄×C₄₇	#400	1	4	yes	—
154	C₄×C₄₇	C₂×#125	1	2	yes	—
155	C₄×C₄₇	C₂×#126	1	2	yes	—
156	Dic₄₇	—	8	4	no	—
157	C₂×#389	D₄₇	1	1	yes	characteristic
158	C₂×#384	—	47	1	no	—
159	#2099	—	47	8	no	—
160	C₂×#340	—	47	2	no	—
161	C₂×#340	—	47	2	no	—
162	C₂×#336	—	47	1	no	—
163	C₂×C₄₇	#2099	1	8	yes	complemented
164	C₂×C₄₇	C₂×#400	1	1	yes	characteristic
165	C₂×C₄₇	C₂×#474	1	1	yes	characteristic
166	C₂×C₄₇	C₂×#475	1	1	yes	characteristic
167	C₂×C₄₇	C₂×#480	1	1	yes	characteristic
168	C₂×C₄₇	C₂×#483	1	1	yes	characteristic
169	C₂×C₄₇	C₂×#486	1	1	yes	characteristic
170	C₂×C₄₇	C₂×#491	1	1	yes	characteristic
171	C₂³×C₈	—	47	2	no	—
172	C₂²×C₄⋊C₄	—	2	1	no	—
173	C₂²×C₄⋊C₄	—	47	2	no	—
174	C₂²×C₄⋊C₄	—	94	1	no	—
175	C₂²×C₄⋊C₄	—	94	1	no	—
176	C₂²×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
177	C₂²×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
178	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
179	#389	—	2	2	no	—
180	#389	C₂×D₄₇	1	4	yes	—
181	#384	—	47	4	no	—
182	#384	—	94	2	no	—
183	C₂×#95	—	2	1	no	—
184	#340	—	47	8	no	—
185	#340	—	47	8	no	—
186	#336	—	47	4	no	—
187	C₄₇	C₂×#2099	1	1	yes	characteristic, complemented, Sylow
188	C₂³×C₄	—	2	1	no	—
189	C₂³×C₄	—	2	1	no	—
190	C₂³×C₄	—	94	2	no	—
191	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
192	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
193	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
194	C₂²×C₈	—	47	4	no	—
195	C₂²×C₈	—	94	4	no	—
196	C₂²×C₈	—	94	4	no	—
197	C₂²×C₈	—	94	4	no	—
198	C₂×C₄⋊C₄	—	2	2	no	—
199	C₂×C₄⋊C₄	—	2	1	no	—
200	C₂×C₄⋊C₄	—	2	4	no	—
201	C₂×C₄⋊C₄	—	2	4	no	—
202	C₂×C₄⋊C₄	—	2	1	no	—
203	C₂×C₄⋊C₄	—	4	4	no	—
204	C₂×C₄⋊C₄	—	47	4	no	—
205	C₂×C₄⋊C₄	—	94	8	no	—
206	C₂×C₄⋊C₄	—	94	2	no	—
207	C₂×C₄⋊C₄	—	94	4	no	—
208	C₂×C₄⋊C₄	—	94	4	no	—
209	C₂×C₄⋊C₄	—	188	4	no	—
210	C₂×C₄⋊C₄	—	188	4	no	—
211	C₂×C₄⋊C₄	Dic₉₄	1	4	yes	—
212	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
213	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
214	C₂×C₄²	—	2	4	no	—
215	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
216	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
217	#95	—	2	4	no	—
218	C₂⁴	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	4	no	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	4	2	no	—
227	C₂²×C₄	—	4	2	no	—
228	C₂²×C₄	—	4	2	no	—
229	C₂²×C₄	—	4	2	no	—
230	C₂²×C₄	—	4	2	no	—
231	C₂²×C₄	—	94	4	no	—
232	C₂²×C₄	—	188	4	no	—
233	C₂²×C₄	—	188	4	no	—
234	C₂²×C₄	—	188	4	no	—
235	C₂²×C₄	#10625334	1	4	yes	—
236	C₂²×C₄	#10625336	1	4	yes	—
237	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
238	C₂²×C₄	#10625350	1	2	yes	—
239	C₂²×C₄	#10625352	1	4	yes	—
240	C₂²×C₄	#10625354	1	4	yes	—
241	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
242	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
243	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
244	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
245	C₂×C₈	—	94	4	no	—
246	C₂×C₈	—	94	4	no	—
247	C₂×C₈	—	94	4	no	—
248	C₂×C₈	—	188	8	no	—
249	C₄⋊C₄	—	2	4	no	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	4	no	—
252	C₄⋊C₄	—	4	2	no	—
253	C₄⋊C₄	—	4	8	no	—
254	C₄⋊C₄	—	94	8	no	—
255	C₄⋊C₄	—	188	4	no	—
256	C₄⋊C₄	—	188	8	no	—
257	C₄⋊C₄	—	188	8	no	—
258	C₄⋊C₄	C₂×Dic₉₄	1	4	yes	—
259	C₄²	—	2	4	no	—
260	C₄²	—	2	4	no	—
261	C₂³	#13094003	1	2	yes	—
262	C₂³	#13094005	1	2	yes	—
263	C₂³	#13094013	1	2	yes	—
264	C₂³	C₂×#10625334	1	2	yes	—
265	C₂³	C₂×#10625336	1	2	yes	—
266	C₂³	#13094054	1	2	yes	—
267	C₂³	#13094054	1	2	yes	—
268	C₂³	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	8	no	—
274	C₂×C₄	—	2	8	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	8	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	8	4	no	—
283	C₂×C₄	—	376	8	no	—
284	C₂×C₄	—	188	4	no	—
285	C₂×C₄	—	188	4	no	—
286	C₂×C₄	—	188	4	no	—
287	C₂×C₄	#13094003	1	2	yes	—
288	C₂×C₄	#13094005	1	2	yes	—
289	C₂×C₄	#13094013	1	2	yes	—
290	C₂×C₄	#13094015	1	8	yes	—
291	C₂×C₄	#13094019	1	8	yes	—
292	C₂×C₄	C₂×#10625334	1	4	yes	—
293	C₂×C₄	C₂×#10625336	1	4	yes	—
294	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
295	C₂×C₄	C₂×#10625352	1	2	yes	—
296	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
297	C₂×C₄	C₂×#10625354	1	2	yes	—
298	C₈	—	188	4	no	—
299	C₂²	#15689062	1	4	yes	—
300	C₂²	#15689403	1	4	yes	—
301	C₂²	#15689408	1	4	yes	—
302	C₂²	#15689434	1	4	yes	—
303	C₂²	#15689451	1	4	yes	—
304	C₂²	#15689471	1	4	yes	—
305	C₂²	#15689493	1	4	yes	—
306	C₂²	C₂×#13094003	1	1	yes	characteristic
307	C₂²	C₂×#13094005	1	1	yes	characteristic
308	C₂²	C₂×#13094013	1	1	yes	characteristic
309	C₂²	C₂²×#10625334	1	1	yes	characteristic
310	C₂²	C₂²×#10625336	1	1	yes	characteristic
311	C₂²	C₂×#13094054	1	1	yes	characteristic
312	C₂²	C₂×#13094054	1	1	yes	characteristic
313	C₄	—	2	4	no	—
314	C₄	—	2	4	no	—
315	C₄	—	4	4	no	—
316	C₄	—	8	2	no	—
317	C₄	—	376	4	no	—
318	C₄	#15689062	1	4	yes	—
319	C₄	C₂×#13094015	1	2	yes	—
320	C₄	C₂×#13094019	1	2	yes	—
321	C₂	#20244946	1	8	yes	direct product factor
322	C₂	C₂×#15689062	1	1	yes	characteristic
323	C₂	C₂×#15689403	1	1	yes	characteristic
324	C₂	C₂×#15689408	1	1	yes	characteristic
325	C₂	C₂×#15689434	1	1	yes	characteristic
326	C₂	C₂×#15689451	1	1	yes	characteristic
327	C₂	C₂×#15689471	1	1	yes	characteristic
328	C₂	C₂×#15689493	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂×C₂²⋊D₄₇
commutator, cosocle	C₂²×C₄×C₄₇	C₂⁴
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×#389	C₂
socle, prime radical	C₂⁴×C₄₇	C₂×D₄
hypercenter	C₂×#389	D₄₇
hypocenter	C₄₇	C₂×#2099
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂×#389

Complements

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

N	H ≅ G/N	type
2. #20244946	C₂	direct (N × H)
163. C₂×C₄₇	#2099	semidirect
187. C₄₇	C₂×#2099	semidirect
321. C₂	#20244946	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×#15688993	2
#20244946	2
C₂×#15688885	2
C₂×#15688878	2
C₂×C₄₇×#389	2
C₂×#2099	47

Subgroups — C2×#20244946 #26246172

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20244946 #26246172