Subgroups — C₂×#20244866 #26245253

Order 12032.

1086 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093956	C₂	1	2	yes	maximal
2	C₂×#15689021	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688993	C₂	1	1	yes	maximal, characteristic
4	#20244866	C₂	1	8	yes	maximal, direct product factor
5	C₂×#15688886	C₂	1	2	yes	maximal
6	C₂×C₄₇×#336	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₄⋊Dic₄₇	—	2	1	no	—
8	C₂²×C₄⋊Dic₄₇	—	2	1	no	—
9	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
10	C₂²×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
11	C₂²×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
12	C₂²×Dic₄₇⋊C₄	—	2	1	no	—
13	C₂×#13093956	—	2	8	no	—
14	C₂×#13093956	C₂²	1	8	yes	—
15	#15689021	—	2	2	no	—
16	#15689021	C₂²	1	4	yes	—
17	#15688993	—	2	2	no	—
18	#15688993	C₂²	1	4	yes	—
19	C₂×C₄₇⋊#95	—	2	1	no	—
20	#15688886	C₂²	1	8	yes	—
21	C₂³×C₈×C₄₇	C₂²	1	2	yes	—
22	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
23	C₄₇×#336	C₂²	1	4	yes	—
24	C₂³×Dic₄₇	—	2	1	no	—
25	C₂³×Dic₄₇	—	2	1	no	—
26	C₂³×Dic₄₇	—	2	2	no	—
27	C₂×C₄⋊Dic₄₇	—	2	1	no	—
28	C₂×C₄⋊Dic₄₇	—	2	2	no	—
29	C₂×C₄⋊Dic₄₇	—	2	4	no	—
30	C₂×C₄⋊Dic₄₇	—	2	4	no	—
31	C₂×C₄⋊Dic₄₇	—	2	4	no	—
32	C₂×C₄⋊Dic₄₇	—	2	2	no	—
33	C₂×C₄⋊Dic₄₇	—	2	4	no	—
34	C₂×C₄⋊Dic₄₇	—	2	4	no	—
35	C₂×C₄⋊Dic₄₇	—	2	1	no	—
36	C₂×C₄⋊Dic₄₇	—	4	4	no	—
37	C₂×C₄⋊Dic₄₇	—	4	4	no	—
38	C₂×C₄⋊Dic₄₇	D₄	1	4	yes	—
39	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
40	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
41	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
42	C₂×Dic₄₇⋊C₄	—	2	4	no	—
43	C₂×Dic₄₇⋊C₄	—	4	4	no	—
44	#13093956	—	2	16	no	—
45	C₄₇⋊#95	—	2	4	no	—
46	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
47	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
48	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₄₇	—	2	4	no	—
50	C₂²×C₈×C₄₇	—	2	4	no	—
51	C₂²×C₈×C₄₇	Q₈	1	8	yes	—
52	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
53	C₂×C₄²×C₄₇	—	2	4	no	—
54	C₂×C₄²×C₄₇	D₄	1	2	yes	—
55	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
56	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	—	2	2	no	—
58	C₂²×C₄×C₄₇	—	2	2	no	—
59	C₂²×C₄×C₄₇	—	2	2	no	—
60	C₂²×C₄×C₄₇	—	2	2	no	—
61	C₂²×C₄×C₄₇	SD₁₆	1	4	yes	—
62	C₂²×C₄×C₄₇	SD₁₆	1	4	yes	—
63	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
68	C₂²×C₄×C₄₇	#41	1	2	yes	—
69	C₂²×C₄×C₄₇	#41	1	4	yes	—
70	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
71	C₂²×Dic₄₇	—	2	4	no	—
72	C₂²×Dic₄₇	—	2	2	no	—
73	C₂²×Dic₄₇	—	2	2	no	—
74	C₂²×Dic₄₇	—	2	4	no	—
75	C₂²×Dic₄₇	—	4	2	no	—
76	C₂²×Dic₄₇	—	4	2	no	—
77	C₂²×Dic₄₇	—	4	4	no	—
78	C₂²×Dic₄₇	—	4	2	no	—
79	C₂²×Dic₄₇	—	4	2	no	—
80	C₂²×Dic₄₇	—	4	4	no	—
81	C₂²×Dic₄₇	—	4	4	no	—
82	C₂²×Dic₄₇	—	4	2	no	—
83	C₂×C₈×C₄₇	—	2	16	no	—
84	C₂×C₈×C₄₇	—	2	4	no	—
85	C₂×C₈×C₄₇	—	2	4	no	—
86	C₂×C₈×C₄₇	C₂×Q₈	1	8	yes	—
87	C₄²×C₄₇	—	2	4	no	—
88	C₄²×C₄₇	—	2	4	no	—
89	C₄⋊Dic₄₇	—	2	4	no	—
90	C₄⋊Dic₄₇	—	2	2	no	—
91	C₄⋊Dic₄₇	—	2	4	no	—
92	C₄⋊Dic₄₇	—	2	4	no	—
93	C₄⋊Dic₄₇	—	2	4	no	—
94	C₄⋊Dic₄₇	—	4	8	no	—
95	C₄⋊Dic₄₇	—	4	4	no	—
96	C₄⋊Dic₄₇	—	4	2	no	—
97	C₄⋊Dic₄₇	—	4	8	no	—
98	C₄⋊Dic₄₇	C₂×D₄	1	4	yes	—
99	Dic₄₇⋊C₄	—	4	8	no	—
100	C₂³×C₄₇	#121	1	2	yes	—
101	C₂³×C₄₇	#123	1	2	yes	—
102	C₂³×C₄₇	C₄⋊Q₈	1	2	yes	—
103	C₂³×C₄₇	C₂×SD₁₆	1	2	yes	—
104	C₂³×C₄₇	C₂×SD₁₆	1	2	yes	—
105	C₂³×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
106	C₂³×C₄₇	#137	1	2	yes	—
107	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
108	C₂×C₄×C₄₇	—	2	2	no	—
109	C₂×C₄×C₄₇	—	2	2	no	—
110	C₂×C₄×C₄₇	—	2	2	no	—
111	C₂×C₄×C₄₇	—	2	2	no	—
112	C₂×C₄×C₄₇	—	2	8	no	—
113	C₂×C₄×C₄₇	—	2	8	no	—
114	C₂×C₄×C₄₇	#121	1	2	yes	—
115	C₂×C₄×C₄₇	C₂²⋊Q₈	1	8	yes	—
116	C₂×C₄×C₄₇	#123	1	2	yes	—
117	C₂×C₄×C₄₇	#125	1	8	yes	—
118	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
119	C₂×C₄×C₄₇	C₂×SD₁₆	1	4	yes	—
120	C₂×C₄×C₄₇	C₂×SD₁₆	1	4	yes	—
121	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
123	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
124	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
125	C₂×Dic₄₇	—	2	4	no	—
126	C₂×Dic₄₇	—	4	2	no	—
127	C₂×Dic₄₇	—	4	2	no	—
128	C₂×Dic₄₇	—	4	8	no	—
129	C₂×Dic₄₇	—	4	4	no	—
130	C₂×Dic₄₇	—	4	2	no	—
131	C₂×Dic₄₇	—	4	2	no	—
132	C₂×Dic₄₇	—	4	4	no	—
133	C₂×Dic₄₇	—	4	4	no	—
134	C₂×Dic₄₇	—	4	2	no	—
135	C₂×Dic₄₇	—	8	8	no	—
136	C₂×Dic₄₇	—	8	4	no	—
137	C₈×C₄₇	—	2	8	no	—
138	C₂×#2084	—	47	1	no	maximal, Sylow
139	C₂²×C₄₇	#398	1	4	yes	—
140	C₂²×C₄₇	#460	1	4	yes	—
141	C₂²×C₄₇	#461	1	4	yes	—
142	C₂²×C₄₇	#481	1	4	yes	—
143	C₂²×C₄₇	#483	1	4	yes	—
144	C₂²×C₄₇	#498	1	4	yes	—
145	C₂²×C₄₇	#501	1	4	yes	—
146	C₂²×C₄₇	C₂×#121	1	1	yes	characteristic
147	C₂²×C₄₇	C₂×#123	1	1	yes	characteristic
148	C₂²×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
149	C₂²×C₄₇	C₂²×SD₁₆	1	1	yes	characteristic
150	C₂²×C₄₇	C₂²×SD₁₆	1	1	yes	characteristic
151	C₂²×C₄₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
152	C₂²×C₄₇	C₂×#137	1	1	yes	characteristic
153	C₄×C₄₇	—	2	4	no	—
154	C₄×C₄₇	—	2	4	no	—
155	C₄×C₄₇	#398	1	4	yes	—
156	C₄×C₄₇	C₂×C₂²⋊Q₈	1	2	yes	—
157	C₄×C₄₇	C₂×#125	1	2	yes	—
158	Dic₄₇	—	4	4	no	—
159	Dic₄₇	—	8	2	no	—
160	Dic₄₇	—	8	4	no	—
161	C₂²×#106	—	47	2	no	—
162	C₂×#389	—	47	1	no	—
163	C₂×#384	—	47	1	no	—
164	#2084	—	47	8	no	—
165	C₂×#340	—	47	2	no	—
166	C₂×#336	D₄₇	1	1	yes	characteristic
167	C₂×C₄₇	#2084	1	8	yes	complemented
168	C₂×C₄₇	C₂×#398	1	1	yes	characteristic
169	C₂×C₄₇	C₂×#461	1	1	yes	characteristic
170	C₂×C₄₇	C₂×#460	1	1	yes	characteristic
171	C₂×C₄₇	C₂×#483	1	1	yes	characteristic
172	C₂×C₄₇	C₂×#481	1	1	yes	characteristic
173	C₂×C₄₇	C₂×#498	1	1	yes	characteristic
174	C₂×C₄₇	C₂×#501	1	1	yes	characteristic
175	C₂³×C₈	C₂×D₄₇	1	2	yes	—
176	C₂²×C₄⋊C₄	—	47	1	no	—
177	C₂²×C₄⋊C₄	—	47	2	no	—
178	C₂²×C₄⋊C₄	—	47	1	no	—
179	C₂²×C₄⋊C₄	—	94	1	no	—
180	C₂²×C₄⋊C₄	—	94	1	no	—
181	C₂²×C₄⋊C₄	—	94	1	no	—
182	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
183	C₂×#106	—	47	8	no	—
184	C₂×#106	—	94	8	no	—
185	#389	—	47	4	no	—
186	#389	—	94	2	no	—
187	#384	—	47	4	no	—
188	#384	—	94	2	no	—
189	C₂×#95	—	94	1	no	—
190	#340	—	47	8	no	—
191	#336	C₂×D₄₇	1	4	yes	—
192	C₄₇	C₂×#2084	1	1	yes	characteristic, complemented, Sylow
193	C₂³×C₄	—	94	1	no	—
194	C₂³×C₄	—	94	2	no	—
195	C₂³×C₄	—	94	1	no	—
196	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
197	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
198	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
199	C₂²×C₈	—	2	4	no	—
200	C₂²×C₈	—	2	4	no	—
201	C₂²×C₈	Dic₉₄	1	8	yes	—
202	C₂²×C₈	C₂²×D₄₇	1	4	yes	—
203	C₂×C₄⋊C₄	—	47	4	no	—
204	C₂×C₄⋊C₄	—	47	2	no	—
205	C₂×C₄⋊C₄	—	47	2	no	—
206	C₂×C₄⋊C₄	—	47	4	no	—
207	C₂×C₄⋊C₄	—	94	1	no	—
208	C₂×C₄⋊C₄	—	94	2	no	—
209	C₂×C₄⋊C₄	—	94	4	no	—
210	C₂×C₄⋊C₄	—	94	4	no	—
211	C₂×C₄⋊C₄	—	94	4	no	—
212	C₂×C₄⋊C₄	—	94	2	no	—
213	C₂×C₄⋊C₄	—	94	1	no	—
214	C₂×C₄⋊C₄	—	94	4	no	—
215	C₂×C₄⋊C₄	—	94	4	no	—
216	C₂×C₄⋊C₄	—	94	4	no	—
217	C₂×C₄⋊C₄	—	188	4	no	—
218	C₂×C₄⋊C₄	—	188	4	no	—
219	C₂×C₄⋊C₄	—	188	4	no	—
220	C₂×C₄²	—	2	4	no	—
221	C₂×C₄²	D₁₈₈	1	2	yes	—
222	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
223	#106	—	94	16	no	—
224	#95	—	94	4	no	—
225	C₂⁴	C₂×D₁₈₈	1	1	yes	characteristic
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	94	4	no	—
231	C₂²×C₄	—	94	2	no	—
232	C₂²×C₄	—	94	2	no	—
233	C₂²×C₄	—	94	4	no	—
234	C₂²×C₄	—	188	4	no	—
235	C₂²×C₄	—	188	2	no	—
236	C₂²×C₄	—	188	4	no	—
237	C₂²×C₄	—	188	2	no	—
238	C₂²×C₄	—	188	4	no	—
239	C₂²×C₄	—	188	2	no	—
240	C₂²×C₄	—	188	2	no	—
241	C₂²×C₄	—	188	2	no	—
242	C₂²×C₄	#10625325	1	4	yes	—
243	C₂²×C₄	#10625325	1	4	yes	—
244	C₂²×C₄	C₂×Dic₉₄	1	4	yes	—
245	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
246	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
247	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
248	C₂²×C₄	D₄×D₄₇	1	2	yes	—
249	C₂²×C₄	#10625352	1	4	yes	—
250	C₂²×C₄	#10625354	1	2	yes	—
251	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
252	C₂×C₈	—	2	16	no	—
253	C₂×C₈	—	2	4	no	—
254	C₂×C₈	—	2	4	no	—
255	C₂×C₈	C₂×Dic₉₄	1	8	yes	—
256	C₄⋊C₄	—	47	4	no	—
257	C₄⋊C₄	—	94	4	no	—
258	C₄⋊C₄	—	94	2	no	—
259	C₄⋊C₄	—	94	4	no	—
260	C₄⋊C₄	—	94	4	no	—
261	C₄⋊C₄	—	94	4	no	—
262	C₄⋊C₄	—	188	8	no	—
263	C₄⋊C₄	—	188	4	no	—
264	C₄⋊C₄	—	188	2	no	—
265	C₄⋊C₄	—	188	8	no	—
266	C₄⋊C₄	—	188	8	no	—
267	C₄²	—	2	4	no	—
268	C₄²	—	2	4	no	—
269	C₂³	#13093999	1	2	yes	—
270	C₂³	#13094008	1	2	yes	—
271	C₂³	#13094024	1	2	yes	—
272	C₂³	C₂×#10625325	1	2	yes	—
273	C₂³	C₂×#10625325	1	2	yes	—
274	C₂³	#13094052	1	2	yes	—
275	C₂³	#13094057	1	2	yes	—
276	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	8	no	—
282	C₂×C₄	—	2	8	no	—
283	C₂×C₄	—	94	4	no	—
284	C₂×C₄	—	376	8	no	—
285	C₂×C₄	—	376	4	no	—
286	C₂×C₄	—	188	4	no	—
287	C₂×C₄	—	188	2	no	—
288	C₂×C₄	—	188	2	no	—
289	C₂×C₄	—	188	4	no	—
290	C₂×C₄	—	188	4	no	—
291	C₂×C₄	—	188	2	no	—
292	C₂×C₄	—	188	8	no	—
293	C₂×C₄	—	188	2	no	—
294	C₂×C₄	—	188	2	no	—
295	C₂×C₄	#13093999	1	2	yes	—
296	C₂×C₄	#13094001	1	8	yes	—
297	C₂×C₄	#13094008	1	2	yes	—
298	C₂×C₄	#13094015	1	8	yes	—
299	C₂×C₄	#13094024	1	2	yes	—
300	C₂×C₄	C₂×#10625325	1	4	yes	—
301	C₂×C₄	C₂×#10625325	1	4	yes	—
302	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
303	C₂×C₄	C₂²×Dic₉₄	1	2	yes	—
304	C₂×C₄	C₂×#10625352	1	2	yes	—
305	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
306	C₈	—	2	8	no	—
307	C₂²	#15689055	1	4	yes	—
308	C₂²	#15689324	1	4	yes	—
309	C₂²	#15689329	1	4	yes	—
310	C₂²	#15689442	1	4	yes	—
311	C₂²	#15689454	1	4	yes	—
312	C₂²	#15689521	1	4	yes	—
313	C₂²	#15689532	1	4	yes	—
314	C₂²	C₂×#13093999	1	1	yes	characteristic
315	C₂²	C₂×#13094008	1	1	yes	characteristic
316	C₂²	C₂×#13094024	1	1	yes	characteristic
317	C₂²	C₂²×#10625325	1	1	yes	characteristic
318	C₂²	C₂²×#10625325	1	1	yes	characteristic
319	C₂²	C₂×#13094052	1	1	yes	characteristic
320	C₂²	C₂×#13094057	1	1	yes	characteristic
321	C₄	—	2	4	no	—
322	C₄	—	2	4	no	—
323	C₄	—	376	2	no	—
324	C₄	—	376	4	no	—
325	C₄	—	188	4	no	—
326	C₄	#15689055	1	4	yes	—
327	C₄	C₂×#13094001	1	2	yes	—
328	C₄	C₂×#13094015	1	2	yes	—
329	C₂	#20244866	1	8	yes	direct product factor
330	C₂	C₂×#15689055	1	1	yes	characteristic
331	C₂	C₂×#15689329	1	1	yes	characteristic
332	C₂	C₂×#15689324	1	1	yes	characteristic
333	C₂	C₂×#15689454	1	1	yes	characteristic
334	C₂	C₂×#15689442	1	1	yes	characteristic
335	C₂	C₂×#15689521	1	1	yes	characteristic
336	C₂	C₂×#15689532	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂×D₁₈₈
commutator, cosocle	C₂²×C₄×C₄₇	C₂⁴
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×#336	C₂
socle, prime radical	C₂⁴×C₄₇	C₂×D₄
hypercenter	C₂×#336	D₄₇
hypocenter	C₄₇	C₂×#2084
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂×#336

Complements

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

N	H ≅ G/N	type
4. #20244866	C₂	direct (N × H)
167. C₂×C₄₇	#2084	semidirect
192. C₄₇	C₂×#2084	semidirect
329. C₂	#20244866	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×#13093956	2
C₂×#15689021	2
C₂×#15688993	2
#20244866	2
C₂×#15688886	2
C₂×C₄₇×#336	2
C₂×#2084	47

Subgroups — C2×#20244866 #26245253

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20244866 #26245253