Subgroups — #26376298

Order 12032.

5430 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689598	C₂	1	2	yes	maximal, complemented
2	#20246218	C₂	1	16	yes	maximal, complemented
3	#20245974	C₂	1	2	yes	maximal, complemented
4	#20245966	C₂	1	4	yes	maximal, complemented
5	#20245771	C₂	1	2	yes	maximal, complemented
6	C₂×#15688986	C₂	1	4	yes	maximal, complemented
7	C₂³×#41×C₄₇	C₂	1	1	yes	maximal, characteristic
8	#15689598	—	2	16	no	—
9	C₂×D₄×Dic₄₇	C₂²	1	16	yes	complemented
10	C₂×#13093983	C₂²	1	4	yes	—
11	C₂²×C₂²⋊Dic₄₇	C₂²	1	8	yes	—
12	#15689030	C₂²	1	16	yes	complemented
13	#15689003	C₂²	1	32	yes	complemented
14	#15688986	C₂²	1	4	yes	—
15	#15688986	C₂²	1	8	yes	—
16	#15688986	C₂²	1	16	yes	complemented
17	#15688983	C₂²	1	8	yes	complemented
18	#15688979	C₂²	1	4	yes	—
19	C₄×C₂²⋊Dic₄₇	C₂²	1	8	yes	complemented
20	C₂×C₄₇⋊#95	C₂²	1	16	yes	complemented
21	C₂²×#41×C₄₇	—	2	16	no	—
22	C₂²×#41×C₄₇	C₄	1	16	yes	complemented
23	C₂²×#41×C₄₇	C₂²	1	8	yes	—
24	C₂³×Q₈×C₄₇	C₂²	1	1	yes	characteristic
25	C₂³×D₄×C₄₇	C₂²	1	2	yes	—
26	C₂³×D₄×C₄₇	C₂²	1	1	yes	characteristic
27	C₂⁴×C₄×C₄₇	C₂²	1	2	yes	—
28	C₂⁴×C₄×C₄₇	C₂²	1	1	yes	characteristic
29	C₂³×Dic₄₇	C₂³	1	16	yes	—
30	D₄×Dic₄₇	—	2	64	no	—
31	#13093983	—	2	16	no	—
32	C₂×C₄⋊Dic₄₇	C₂³	1	8	yes	—
33	C₂×C₂²⋊Dic₄₇	—	2	32	no	—
34	C₂×C₂²⋊Dic₄₇	C₂³	1	8	yes	—
35	C₂×C₂²⋊Dic₄₇	C₂³	1	16	yes	—
36	C₂×C₂²⋊Dic₄₇	C₂³	1	16	yes	—
37	C₂×C₄×Dic₄₇	C₂³	1	8	yes	—
38	C₄₇⋊#95	C₂³	1	16	yes	—
39	C₄₇⋊#95	C₂³	1	32	yes	—
40	C₂⁵×C₄₇	C₂³	1	1	yes	characteristic
41	C₂⁵×C₄₇	C₂³	1	2	yes	—
42	C₂×#41×C₄₇	—	2	128	no	—
43	C₂×#41×C₄₇	—	2	64	no	—
44	C₂×#41×C₄₇	C₂×C₄	1	64	yes	complemented
45	C₂²×Q₈×C₄₇	—	2	8	no	—
46	C₂²×Q₈×C₄₇	C₂×C₄	1	8	yes	complemented
47	C₂²×Q₈×C₄₇	C₂³	1	4	yes	—
48	C₂²×D₄×C₄₇	—	2	16	no	—
49	C₂²×D₄×C₄₇	C₂×C₄	1	8	yes	—
50	C₂²×D₄×C₄₇	C₂×C₄	1	16	yes	complemented
51	C₂²×D₄×C₄₇	D₄	1	16	yes	—
52	C₂²×D₄×C₄₇	C₂³	1	8	yes	—
53	C₂²×D₄×C₄₇	C₂³	1	4	yes	—
54	C₂³×C₄×C₄₇	—	2	16	no	—
55	C₂³×C₄×C₄₇	—	2	8	no	—
56	C₂³×C₄×C₄₇	C₂×C₄	1	16	yes	complemented
57	C₂³×C₄×C₄₇	C₂×C₄	1	8	yes	complemented
58	C₂³×C₄×C₄₇	C₂³	1	8	yes	—
59	C₂³×C₄×C₄₇	C₂³	1	4	yes	—
60	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
61	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
62	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
63	C₂⁴×C₄₇	—	2	4	no	—
64	C₂⁴×C₄₇	—	4	8	no	—
65	C₂⁴×C₄₇	—	4	4	no	—
66	C₂⁴×C₄₇	C₂²×C₄	1	4	yes	—
67	C₂⁴×C₄₇	C₂²×C₄	1	8	yes	—
68	C₂⁴×C₄₇	C₂×D₄	1	16	yes	—
69	C₂⁴×C₄₇	C₂⁴	1	2	yes	—
70	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
71	C₂⁴×C₄₇	C₂⁴	1	4	yes	—
72	#41×C₄₇	—	2	256	no	—
73	C₂×Q₈×C₄₇	—	2	32	no	—
74	C₂×Q₈×C₄₇	—	2	16	no	—
75	C₂×Q₈×C₄₇	C₂²×C₄	1	16	yes	—
76	C₂×D₄×C₄₇	—	2	64	no	—
77	C₂×D₄×C₄₇	—	2	32	no	—
78	C₂×D₄×C₄₇	C₂²⋊C₄	1	64	yes	complemented
79	C₂×D₄×C₄₇	C₂²×C₄	1	32	yes	—
80	C₂×D₄×C₄₇	C₂²×C₄	1	16	yes	—
81	C₂×D₄×C₄₇	C₂×D₄	1	32	yes	—
82	C₂²×C₄×C₄₇	—	2	64	no	—
83	C₂²×C₄×C₄₇	—	2	4	no	—
84	C₂²×C₄×C₄₇	—	2	8	no	—
85	C₂²×C₄×C₄₇	—	2	32	no	—
86	C₂²×C₄×C₄₇	—	2	32	no	—
87	C₂²×C₄×C₄₇	—	2	16	no	—
88	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
89	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
90	C₂²×C₄×C₄₇	C₂²×C₄	1	32	yes	—
91	C₂²×C₄×C₄₇	C₂²×C₄	1	16	yes	—
92	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
93	C₂²×C₄×C₄₇	C₂×D₄	1	8	yes	—
94	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
95	C₂²×C₄×C₄₇	C₂⁴	1	4	yes	—
96	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
97	C₂²×Dic₄₇	—	2	32	no	—
98	C₂²×Dic₄₇	—	4	32	no	—
99	C₂²×Dic₄₇	C₂⁴	1	16	yes	—
100	C₂²⋊Dic₄₇	—	2	16	no	—
101	C₂²⋊Dic₄₇	—	2	32	no	—
102	C₂²⋊Dic₄₇	—	4	16	no	—
103	C₄⋊Dic₄₇	—	2	16	no	—
104	C₄×Dic₄₇	—	2	16	no	—
105	C₂³×C₄₇	—	2	16	no	—
106	C₂³×C₄₇	—	2	4	no	—
107	C₂³×C₄₇	—	2	8	no	—
108	C₂³×C₄₇	—	2	4	no	—
109	C₂³×C₄₇	—	2	8	no	—
110	C₂³×C₄₇	—	4	16	no	—
111	C₂³×C₄₇	—	4	8	no	—
112	C₂³×C₄₇	—	4	32	no	—
113	C₂³×C₄₇	—	4	16	no	—
114	C₂³×C₄₇	C₂×C₂²⋊C₄	1	32	yes	—
115	C₂³×C₄₇	C₂³×C₄	1	4	yes	—
116	C₂³×C₄₇	C₂³×C₄	1	2	yes	—
117	C₂³×C₄₇	C₂³×C₄	1	8	yes	—
118	C₂³×C₄₇	C₂²×D₄	1	4	yes	—
119	C₂³×C₄₇	C₂²×D₄	1	16	yes	—
120	C₂³×C₄₇	C₂⁵	1	1	yes	characteristic
121	Q₈×C₄₇	—	2	32	no	—
122	D₄×C₄₇	—	2	64	no	—
123	D₄×C₄₇	C₂×C₂²⋊C₄	1	64	yes	—
124	C₂×C₄×C₄₇	—	2	8	no	—
125	C₂×C₄×C₄₇	—	2	16	no	—
126	C₂×C₄×C₄₇	—	2	64	no	—
127	C₂×C₄×C₄₇	—	2	4	no	—
128	C₂×C₄×C₄₇	—	2	8	no	—
129	C₂×C₄×C₄₇	—	2	32	no	—
130	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	16	yes	—
131	C₂×C₄×C₄₇	C₂³×C₄	1	4	yes	—
132	C₂×C₄×C₄₇	C₂³×C₄	1	8	yes	—
133	C₂×C₄×C₄₇	C₂³×C₄	1	4	yes	—
134	C₂×C₄×C₄₇	C₂²×D₄	1	8	yes	—
135	C₂×Dic₄₇	—	2	16	no	—
136	C₂×Dic₄₇	—	2	32	no	—
137	C₂×Dic₄₇	—	8	32	no	—
138	#34172	—	47	1	no	maximal, Sylow
139	C₂²×C₄₇	—	2	4	no	—
140	C₂²×C₄₇	—	2	16	no	—
141	C₂²×C₄₇	—	2	32	no	—
142	C₂²×C₄₇	—	2	8	no	—
143	C₂²×C₄₇	—	2	4	no	—
144	C₂²×C₄₇	—	2	8	no	—
145	C₂²×C₄₇	—	4	32	no	—
146	C₂²×C₄₇	—	4	16	no	—
147	C₂²×C₄₇	—	4	8	no	—
148	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	4	yes	—
149	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	16	yes	—
150	C₂²×C₄₇	C₂⁴×C₄	1	1	yes	characteristic
151	C₂²×C₄₇	C₂³×D₄	1	2	yes	—
152	C₂²×C₄₇	#585	1	4	yes	—
153	C₄×C₄₇	—	2	4	no	—
154	C₄×C₄₇	—	2	8	no	—
155	C₄×C₄₇	C₂²×C₂²⋊C₄	1	8	yes	—
156	Dic₄₇	—	4	16	no	—
157	C₂³×#41	D₄₇	1	1	yes	characteristic
158	C₂×#518	—	47	2	no	—
159	#2339	—	47	16	no	—
160	#2310	—	47	2	no	—
161	#2308	—	47	4	no	—
162	#2280	—	47	2	no	—
163	C₂×#381	—	47	4	no	—
164	C₂×C₄₇	—	2	4	no	—
165	C₂×C₄₇	—	2	16	no	—
166	C₂×C₄₇	—	4	4	no	—
167	C₂×C₄₇	C₂³×C₂²⋊C₄	1	1	yes	characteristic
168	C₂×C₄₇	#3428	1	2	yes	—
169	C₂×C₄₇	#3485	1	4	yes	—
170	C₂²×#41	—	2	16	no	—
171	C₂²×#41	Dic₄₇	1	16	yes	complemented
172	C₂²×#41	C₂×D₄₇	1	8	yes	—
173	C₂³×Q₈	C₂×D₄₇	1	1	yes	characteristic
174	C₂³×D₄	C₂×D₄₇	1	2	yes	—
175	C₂³×D₄	C₂×D₄₇	1	1	yes	characteristic
176	C₂⁴×C₄	C₂×D₄₇	1	2	yes	—
177	C₂⁴×C₄	C₂×D₄₇	1	1	yes	characteristic
178	#518	—	94	16	no	—
179	C₂×C₄×D₄	—	47	16	no	—
180	C₂×#117	—	47	4	no	—
181	C₂²×C₂²⋊C₄	—	47	8	no	—
182	#391	—	47	16	no	—
183	#386	—	47	32	no	—
184	#381	—	47	8	no	—
185	#381	—	47	4	no	—
186	#381	—	47	16	no	—
187	#380	—	47	8	no	—
188	#379	—	47	4	no	—
189	C₄×C₂²⋊C₄	—	47	8	no	—
190	C₂×#95	—	47	16	no	—
191	C₄₇	#34172	1	1	yes	characteristic, complemented, Sylow
192	C₂⁵	C₂²×D₄₇	1	1	yes	characteristic
193	C₂⁵	C₂²×D₄₇	1	2	yes	—
194	C₂×#41	—	2	128	no	—
195	C₂×#41	—	2	64	no	—
196	C₂×#41	C₂×Dic₄₇	1	64	yes	complemented
197	C₂²×Q₈	—	2	8	no	—
198	C₂²×Q₈	C₂×Dic₄₇	1	8	yes	complemented
199	C₂²×Q₈	C₂²×D₄₇	1	4	yes	—
200	C₂²×D₄	—	2	16	no	—
201	C₂²×D₄	C₂×Dic₄₇	1	8	yes	—
202	C₂²×D₄	C₂×Dic₄₇	1	16	yes	complemented
203	C₂²×D₄	C₂²⋊D₄₇	1	16	yes	—
204	C₂²×D₄	C₂²×D₄₇	1	8	yes	—
205	C₂²×D₄	C₂²×D₄₇	1	4	yes	—
206	C₂³×C₄	—	2	16	no	—
207	C₂³×C₄	—	2	8	no	—
208	C₂³×C₄	—	47	16	no	—
209	C₂³×C₄	C₂×Dic₄₇	1	16	yes	complemented
210	C₂³×C₄	C₂×Dic₄₇	1	8	yes	complemented
211	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
212	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
213	C₂³×C₄	C₂²×D₄₇	1	8	yes	—
214	C₂³×C₄	C₂²×D₄₇	1	4	yes	—
215	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
216	C₄×D₄	—	94	64	no	—
217	#117	—	94	16	no	—
218	C₂×C₄⋊C₄	—	47	8	no	—
219	C₂×C₂²⋊C₄	—	47	8	no	—
220	C₂×C₂²⋊C₄	—	47	16	no	—
221	C₂×C₂²⋊C₄	—	47	16	no	—
222	C₂×C₂²⋊C₄	—	94	32	no	—
223	C₂×C₄²	—	47	8	no	—
224	#95	—	47	16	no	—
225	#95	—	47	32	no	—
226	C₂⁴	—	2	4	no	—
227	C₂⁴	—	4	8	no	—
228	C₂⁴	—	4	4	no	—
229	C₂⁴	C₂²×Dic₄₇	1	4	yes	—
230	C₂⁴	C₂²×Dic₄₇	1	8	yes	—
231	C₂⁴	C₂×C₂²⋊D₄₇	1	16	yes	—
232	C₂⁴	C₂³×D₄₇	1	2	yes	—
233	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
234	C₂⁴	C₂³×D₄₇	1	4	yes	—
235	#41	—	2	256	no	—
236	C₂×Q₈	—	2	32	no	—
237	C₂×Q₈	—	2	16	no	—
238	C₂×Q₈	C₂²×Dic₄₇	1	16	yes	—
239	C₂×D₄	—	2	64	no	—
240	C₂×D₄	—	2	32	no	—
241	C₂×D₄	C₂²⋊Dic₄₇	1	64	yes	complemented
242	C₂×D₄	C₂²×Dic₄₇	1	32	yes	—
243	C₂×D₄	C₂²×Dic₄₇	1	16	yes	—
244	C₂×D₄	C₂×C₂²⋊D₄₇	1	32	yes	—
245	C₂²×C₄	—	2	64	no	—
246	C₂²×C₄	—	2	4	no	—
247	C₂²×C₄	—	2	8	no	—
248	C₂²×C₄	—	2	32	no	—
249	C₂²×C₄	—	2	32	no	—
250	C₂²×C₄	—	2	16	no	—
251	C₂²×C₄	—	47	16	no	—
252	C₂²×C₄	—	94	32	no	—
253	C₂²×C₄	—	188	32	no	—
254	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
255	C₂²×C₄	C₂²×Dic₄₇	1	8	yes	—
256	C₂²×C₄	C₂²×Dic₄₇	1	32	yes	—
257	C₂²×C₄	C₂²×Dic₄₇	1	16	yes	—
258	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
259	C₂²×C₄	C₂×C₂²⋊D₄₇	1	8	yes	—
260	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
261	C₂²×C₄	C₂³×D₄₇	1	4	yes	—
262	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
263	C₄⋊C₄	—	94	16	no	—
264	C₂²⋊C₄	—	94	16	no	—
265	C₂²⋊C₄	—	94	32	no	—
266	C₂²⋊C₄	—	188	16	no	—
267	C₄²	—	94	16	no	—
268	C₂³	—	2	16	no	—
269	C₂³	—	2	4	no	—
270	C₂³	—	2	8	no	—
271	C₂³	—	2	4	no	—
272	C₂³	—	2	8	no	—
273	C₂³	—	4	16	no	—
274	C₂³	—	4	8	no	—
275	C₂³	—	4	32	no	—
276	C₂³	—	4	16	no	—
277	C₂³	C₂×C₂²⋊Dic₄₇	1	32	yes	—
278	C₂³	C₂³×Dic₄₇	1	4	yes	—
279	C₂³	C₂³×Dic₄₇	1	2	yes	—
280	C₂³	C₂³×Dic₄₇	1	8	yes	—
281	C₂³	C₂²×C₂²⋊D₄₇	1	4	yes	—
282	C₂³	C₂²×C₂²⋊D₄₇	1	16	yes	—
283	C₂³	C₂⁴×D₄₇	1	1	yes	characteristic
284	Q₈	—	2	32	no	—
285	D₄	—	2	64	no	—
286	D₄	C₂×C₂²⋊Dic₄₇	1	64	yes	—
287	C₂×C₄	—	2	8	no	—
288	C₂×C₄	—	2	16	no	—
289	C₂×C₄	—	2	64	no	—
290	C₂×C₄	—	2	4	no	—
291	C₂×C₄	—	2	8	no	—
292	C₂×C₄	—	2	32	no	—
293	C₂×C₄	—	94	16	no	—
294	C₂×C₄	—	94	32	no	—
295	C₂×C₄	—	376	32	no	—
296	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	16	yes	—
297	C₂×C₄	C₂³×Dic₄₇	1	4	yes	—
298	C₂×C₄	C₂³×Dic₄₇	1	8	yes	—
299	C₂×C₄	C₂³×Dic₄₇	1	4	yes	—
300	C₂×C₄	C₂²×C₂²⋊D₄₇	1	8	yes	—
301	C₂²	—	2	32	no	—
302	C₂²	—	2	8	no	—
303	C₂²	—	2	4	no	—
304	C₂²	—	2	16	no	—
305	C₂²	—	2	4	no	—
306	C₂²	—	2	8	no	—
307	C₂²	—	4	32	no	—
308	C₂²	—	4	16	no	—
309	C₂²	—	4	8	no	—
310	C₂²	C₂²×C₂²⋊Dic₄₇	1	4	yes	—
311	C₂²	C₂²×C₂²⋊Dic₄₇	1	16	yes	—
312	C₂²	C₂⁴×Dic₄₇	1	1	yes	characteristic
313	C₂²	C₂³×C₂²⋊D₄₇	1	2	yes	—
314	C₂²	#15689959	1	4	yes	—
315	C₄	—	2	4	no	—
316	C₄	—	2	8	no	—
317	C₄	—	188	16	no	—
318	C₄	C₂²×C₂²⋊Dic₄₇	1	8	yes	—
319	C₂	—	2	4	no	—
320	C₂	—	2	16	no	—
321	C₂	—	4	4	no	—
322	C₂	C₂³×C₂²⋊Dic₄₇	1	1	yes	characteristic
323	C₂	#20256591	1	2	yes	—
324	C₂	#20257032	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂⁴×D₄₇
commutator	C₂²×C₄₇	C₂⁴×C₄
Fitting, prime radical	C₂³×#41×C₄₇	C₂
socle, cosocle	C₂³×C₄₇	C₂⁵
hypercenter	C₂³×#41	D₄₇
hypocenter	C₄₇	#34172
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂³×#41

Complements

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

N	H ≅ G/N	type
1. C₂×#15689598	C₂	semidirect
2. #20246218	C₂	semidirect
3. #20245974	C₂	semidirect
4. #20245966	C₂	semidirect
5. #20245771	C₂	semidirect
6. C₂×#15688986	C₂	semidirect
9. C₂×D₄×Dic₄₇	C₂²	semidirect
12. #15689030	C₂²	semidirect
13. #15689003	C₂²	semidirect
16. #15688986	C₂²	semidirect
17. #15688983	C₂²	semidirect
19. C₄×C₂²⋊Dic₄₇	C₂²	semidirect
20. C₂×C₄₇⋊#95	C₂²	semidirect
22. C₂²×#41×C₄₇	C₄	semidirect
44. C₂×#41×C₄₇	C₂×C₄	semidirect
46. C₂²×Q₈×C₄₇	C₂×C₄	semidirect
50. C₂²×D₄×C₄₇	C₂×C₄	semidirect
56. C₂³×C₄×C₄₇	C₂×C₄	semidirect
57. C₂³×C₄×C₄₇	C₂×C₄	semidirect
78. C₂×D₄×C₄₇	C₂²⋊C₄	semidirect
171. C₂²×#41	Dic₄₇	semidirect
191. C₄₇	#34172	semidirect
196. C₂×#41	C₂×Dic₄₇	semidirect
198. C₂²×Q₈	C₂×Dic₄₇	semidirect
202. C₂²×D₄	C₂×Dic₄₇	semidirect
209. C₂³×C₄	C₂×Dic₄₇	semidirect
210. C₂³×C₄	C₂×Dic₄₇	semidirect
241. C₂×D₄	C₂²⋊Dic₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689598	2
#20246218	2
#20245974	2
#20245966	2
#20245771	2
C₂×#15688986	2
C₂³×#41×C₄₇	2
#34172	47