Subgroups — C₄₇×#29722 #26069420

Order 12032.

1022 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₇×#448	C₂	1	2	yes	maximal, complemented
2	C₄₇×#2619	C₂	1	1	yes	maximal, characteristic
3	C₄₇×#2310	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₇×#2074	C₂	1	2	yes	maximal, complemented
5	C₄₇×#2055	C₂	1	4	yes	maximal, complemented
6	C₄₇×#2014	C₂	1	2	yes	maximal, complemented
7	C₄₇×#1875	C₂	1	2	yes	maximal, complemented
8	C₄₇×#1798	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂³×Q₈×C₄₇	C₂²	1	1	yes	characteristic
10	C₂²×SD₁₆×C₄₇	C₂²	1	4	yes	—
11	C₂×#127×C₄₇	C₂²	1	1	yes	characteristic
12	C₂×#121×C₄₇	C₂²	1	2	yes	—
13	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
14	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
15	C₄₇×#448	—	2	8	no	—
16	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
17	C₂×#103×C₄₇	C₂²	1	4	yes	—
18	C₂×#102×C₄₇	C₂²	1	4	yes	—
19	C₂×C₂²⋊C₈×C₄₇	C₂²	1	2	yes	—
20	C₄₇×#394	—	2	2	no	—
21	C₄₇×#392	—	2	2	no	—
22	C₄₇×#391	—	2	2	no	—
23	C₄₇×#391	C₂²	1	2	yes	—
24	C₄₇×#391	C₂²	1	2	yes	—
25	C₄₇×#390	C₂²	1	2	yes	—
26	C₄₇×#382	C₂²	1	1	yes	characteristic
27	C₄₇×#381	—	2	2	no	—
28	C₄₇×#340	C₂²	1	4	yes	—
29	C₄₇×#336	C₂²	1	2	yes	—
30	C₂²×Q₈×C₄₇	—	2	1	no	—
31	C₂²×Q₈×C₄₇	—	2	2	no	—
32	C₂²×Q₈×C₄₇	—	4	2	no	—
33	C₂²×Q₈×C₄₇	—	4	2	no	—
34	C₂²×Q₈×C₄₇	D₄	1	4	yes	complemented
35	C₂²×Q₈×C₄₇	C₂³	1	2	yes	—
36	C₂²×D₄×C₄₇	—	2	2	no	—
37	C₂²×D₄×C₄₇	—	2	2	no	—
38	C₂²×D₄×C₄₇	C₂³	1	2	yes	—
39	C₂³×C₄×C₄₇	—	2	1	no	—
40	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
41	C₂×SD₁₆×C₄₇	—	2	8	no	—
42	C₂×SD₁₆×C₄₇	—	4	4	no	—
43	C₂×SD₁₆×C₄₇	—	4	4	no	—
44	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
45	#127×C₄₇	—	4	2	no	—
46	#121×C₄₇	—	2	4	no	—
47	#121×C₄₇	—	2	4	no	—
48	#117×C₄₇	—	4	2	no	—
49	#117×C₄₇	—	4	2	no	—
50	C₂×C₄⋊C₄×C₄₇	D₄	1	2	yes	—
51	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
52	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
53	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
54	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
55	C₂×C₂²⋊C₄×C₄₇	—	2	4	no	—
56	C₂×C₂²⋊C₄×C₄₇	D₄	1	2	yes	—
57	C₂×C₄²×C₄₇	—	2	1	no	—
58	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
59	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
60	C₄⋊C₈×C₄₇	—	4	2	no	—
61	#103×C₄₇	—	2	8	no	—
62	#102×C₄₇	—	4	4	no	—
63	C₂²⋊C₈×C₄₇	—	2	4	no	—
64	#95×C₄₇	—	2	4	no	—
65	#95×C₄₇	—	2	4	no	—
66	#95×C₄₇	—	2	2	no	—
67	C₂⁴×C₄₇	—	2	2	no	—
68	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
69	C₂×Q₈×C₄₇	—	2	4	no	—
70	C₂×Q₈×C₄₇	—	2	2	no	—
71	C₂×Q₈×C₄₇	—	2	4	no	—
72	C₂×Q₈×C₄₇	—	2	4	no	—
73	C₂×Q₈×C₄₇	—	4	8	no	—
74	C₂×Q₈×C₄₇	—	4	2	no	—
75	C₂×Q₈×C₄₇	—	4	2	no	—
76	C₂×Q₈×C₄₇	—	4	8	no	—
77	C₂×Q₈×C₄₇	C₂×D₄	1	4	yes	—
78	C₂×D₄×C₄₇	—	2	2	no	—
79	C₂×D₄×C₄₇	—	2	2	no	—
80	C₂×D₄×C₄₇	—	2	4	no	—
81	C₂×D₄×C₄₇	—	2	4	no	—
82	C₂×D₄×C₄₇	—	4	4	no	—
83	C₂×D₄×C₄₇	—	4	4	no	—
84	C₂×D₄×C₄₇	—	8	4	no	—
85	C₂×D₄×C₄₇	—	8	4	no	—
86	C₂²×C₄×C₄₇	—	2	2	no	—
87	C₂²×C₄×C₄₇	—	2	2	no	—
88	C₂²×C₄×C₄₇	—	2	4	no	—
89	C₂²×C₄×C₄₇	—	2	2	no	—
90	C₂²×C₄×C₄₇	—	2	2	no	—
91	C₂²×C₄×C₄₇	—	2	2	no	—
92	C₂²×C₄×C₄₇	—	4	2	no	—
93	C₂²×C₄×C₄₇	—	4	2	no	—
94	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
95	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
96	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
98	C₂²×C₄×C₄₇	#41	1	2	yes	—
99	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
100	SD₁₆×C₄₇	—	4	16	no	—
101	C₂×C₈×C₄₇	—	2	4	no	—
102	C₂×C₈×C₄₇	—	2	4	no	—
103	C₂×C₈×C₄₇	—	2	4	no	—
104	C₄⋊C₄×C₄₇	—	2	2	no	—
105	C₄⋊C₄×C₄₇	—	2	2	no	—
106	C₄⋊C₄×C₄₇	—	2	4	no	—
107	C₄⋊C₄×C₄₇	—	4	2	no	—
108	C₂²⋊C₄×C₄₇	—	4	2	no	—
109	C₂²⋊C₄×C₄₇	—	4	2	no	—
110	C₂²⋊C₄×C₄₇	—	4	4	no	—
111	C₂²⋊C₄×C₄₇	—	8	4	no	—
112	C₄²×C₄₇	—	2	2	no	—
113	C₄²×C₄₇	—	2	2	no	—
114	C₄²×C₄₇	—	4	2	no	—
115	C₂³×C₄₇	—	2	1	no	—
116	C₂³×C₄₇	—	2	1	no	—
117	C₂³×C₄₇	—	2	4	no	—
118	C₂³×C₄₇	—	4	2	no	—
119	C₂³×C₄₇	—	4	2	no	—
120	C₂³×C₄₇	—	4	2	no	—
121	C₂³×C₄₇	—	4	4	no	—
122	C₂³×C₄₇	—	4	4	no	—
123	C₂³×C₄₇	#121	1	2	yes	—
124	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
125	Q₈×C₄₇	—	2	8	no	—
126	Q₈×C₄₇	—	4	8	no	—
127	Q₈×C₄₇	—	4	4	no	—
128	D₄×C₄₇	—	4	4	no	—
129	D₄×C₄₇	—	4	4	no	—
130	D₄×C₄₇	—	8	8	no	—
131	D₄×C₄₇	—	8	8	no	—
132	C₂×C₄×C₄₇	—	2	1	no	—
133	C₂×C₄×C₄₇	—	2	1	no	—
134	C₂×C₄×C₄₇	—	2	4	no	—
135	C₂×C₄×C₄₇	—	2	4	no	—
136	C₂×C₄×C₄₇	—	2	4	no	—
137	C₂×C₄×C₄₇	—	2	2	no	—
138	C₂×C₄×C₄₇	—	2	2	no	—
139	C₂×C₄×C₄₇	—	2	4	no	—
140	C₂×C₄×C₄₇	—	2	4	no	—
141	C₂×C₄×C₄₇	—	4	8	no	—
142	C₂×C₄×C₄₇	—	4	2	no	—
143	C₂×C₄×C₄₇	—	4	2	no	—
144	C₂×C₄×C₄₇	—	4	2	no	—
145	C₂×C₄×C₄₇	—	4	2	no	—
146	C₂×C₄×C₄₇	—	4	4	no	—
147	C₂×C₄×C₄₇	—	4	4	no	—
148	C₂×C₄×C₄₇	#121	1	2	yes	—
149	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
150	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
151	C₂×C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
152	C₂×C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
153	C₈×C₄₇	—	4	4	no	—
154	#29722	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
155	C₂²×C₄₇	—	2	2	no	—
156	C₂²×C₄₇	—	2	4	no	—
157	C₂²×C₄₇	—	4	2	no	—
158	C₂²×C₄₇	—	4	2	no	—
159	C₂²×C₄₇	—	8	2	no	—
160	C₂²×C₄₇	—	8	4	no	—
161	C₂²×C₄₇	—	8	4	no	—
162	C₂²×C₄₇	—	8	2	no	—
163	C₂²×C₄₇	—	8	2	no	—
164	C₂²×C₄₇	C₂×#121	1	1	yes	characteristic
165	C₂²×C₄₇	#534	1	1	yes	characteristic
166	C₂²×C₄₇	#535	1	1	yes	characteristic
167	C₂²×C₄₇	#577	1	4	yes	—
168	C₄×C₄₇	—	2	2	no	—
169	C₄×C₄₇	—	4	4	no	—
170	C₄×C₄₇	—	4	2	no	—
171	C₄×C₄₇	—	4	2	no	—
172	C₄×C₄₇	—	8	2	no	—
173	C₄×C₄₇	—	8	2	no	—
174	C₄×C₄₇	#534	1	2	yes	—
175	C₄×C₄₇	#538	1	2	yes	—
176	C₂×#448	C₂×C₄₇	1	2	yes	complemented
177	#2619	C₂×C₄₇	1	1	yes	characteristic
178	#2310	C₂×C₄₇	1	1	yes	characteristic, complemented
179	#2074	C₂×C₄₇	1	2	yes	complemented
180	#2055	C₂×C₄₇	1	4	yes	complemented
181	#2014	C₂×C₄₇	1	2	yes	complemented
182	#1875	C₂×C₄₇	1	2	yes	complemented
183	#1798	C₂×C₄₇	1	1	yes	characteristic, complemented
184	C₂×C₄₇	—	4	2	no	—
185	C₂×C₄₇	—	16	2	no	—
186	C₂×C₄₇	#2617	1	1	yes	characteristic
187	C₂×C₄₇	#3043	1	2	yes	—
188	C₂×C₄₇	#3200	1	2	yes	—
189	C₂×C₄₇	#3221	1	2	yes	—
190	C₂³×Q₈	C₂²×C₄₇	1	1	yes	characteristic
191	C₂²×SD₁₆	C₂²×C₄₇	1	4	yes	—
192	C₂×#127	C₂²×C₄₇	1	1	yes	characteristic
193	C₂×#121	C₂²×C₄₇	1	2	yes	—
194	C₂×#117	C₂²×C₄₇	1	1	yes	characteristic
195	C₂×#117	C₂²×C₄₇	1	1	yes	characteristic
196	#448	—	2	8	no	—
197	C₂×C₄⋊C₈	C₂²×C₄₇	1	2	yes	—
198	C₂×#103	C₂²×C₄₇	1	4	yes	—
199	C₂×#102	C₂²×C₄₇	1	4	yes	—
200	C₂×C₂²⋊C₈	C₂²×C₄₇	1	2	yes	—
201	#394	—	2	2	no	—
202	#392	—	2	2	no	—
203	#391	—	2	2	no	—
204	#391	C₂²×C₄₇	1	2	yes	—
205	#391	C₂²×C₄₇	1	2	yes	—
206	#390	C₂²×C₄₇	1	2	yes	—
207	#382	C₂²×C₄₇	1	1	yes	characteristic
208	#381	—	2	2	no	—
209	#340	C₂²×C₄₇	1	4	yes	—
210	#336	C₂²×C₄₇	1	2	yes	—
211	C₄₇	#29722	1	1	yes	characteristic, direct product factor, Sylow
212	C₂²×Q₈	—	2	1	no	—
213	C₂²×Q₈	—	2	2	no	—
214	C₂²×Q₈	—	4	2	no	—
215	C₂²×Q₈	—	4	2	no	—
216	C₂²×Q₈	D₄×C₄₇	1	4	yes	complemented
217	C₂²×Q₈	C₂³×C₄₇	1	2	yes	—
218	C₂²×D₄	—	2	2	no	—
219	C₂²×D₄	—	2	2	no	—
220	C₂²×D₄	C₂³×C₄₇	1	2	yes	—
221	C₂³×C₄	—	2	1	no	—
222	C₂³×C₄	C₂³×C₄₇	1	1	yes	characteristic
223	C₂×SD₁₆	—	2	8	no	—
224	C₂×SD₁₆	—	4	4	no	—
225	C₂×SD₁₆	—	4	4	no	—
226	C₂²×C₈	C₂³×C₄₇	1	4	yes	—
227	#127	—	4	2	no	—
228	#121	—	2	4	no	—
229	#121	—	2	4	no	—
230	#117	—	4	2	no	—
231	#117	—	4	2	no	—
232	C₂×C₄⋊C₄	D₄×C₄₇	1	2	yes	—
233	C₂×C₄⋊C₄	C₂³×C₄₇	1	2	yes	—
234	C₂×C₄⋊C₄	C₂³×C₄₇	1	2	yes	—
235	C₂×C₂²⋊C₄	—	2	1	no	—
236	C₂×C₂²⋊C₄	—	2	2	no	—
237	C₂×C₂²⋊C₄	—	2	4	no	—
238	C₂×C₂²⋊C₄	D₄×C₄₇	1	2	yes	—
239	C₂×C₄²	—	2	1	no	—
240	C₂×C₄²	C₂³×C₄₇	1	1	yes	characteristic
241	C₂×C₄²	C₂³×C₄₇	1	1	yes	characteristic
242	C₄⋊C₈	—	4	2	no	—
243	#103	—	2	8	no	—
244	#102	—	4	4	no	—
245	C₂²⋊C₈	—	2	4	no	—
246	#95	—	2	4	no	—
247	#95	—	2	4	no	—
248	#95	—	2	2	no	—
249	C₂⁴	—	2	2	no	—
250	C₂⁴	C₂×D₄×C₄₇	1	1	yes	characteristic
251	C₂×Q₈	—	2	4	no	—
252	C₂×Q₈	—	2	2	no	—
253	C₂×Q₈	—	2	4	no	—
254	C₂×Q₈	—	2	4	no	—
255	C₂×Q₈	—	4	8	no	—
256	C₂×Q₈	—	4	2	no	—
257	C₂×Q₈	—	4	2	no	—
258	C₂×Q₈	—	4	8	no	—
259	C₂×Q₈	C₂×D₄×C₄₇	1	4	yes	—
260	C₂×D₄	—	2	2	no	—
261	C₂×D₄	—	2	2	no	—
262	C₂×D₄	—	2	4	no	—
263	C₂×D₄	—	2	4	no	—
264	C₂×D₄	—	4	4	no	—
265	C₂×D₄	—	4	4	no	—
266	C₂×D₄	—	8	4	no	—
267	C₂×D₄	—	8	4	no	—
268	C₂²×C₄	—	2	2	no	—
269	C₂²×C₄	—	2	2	no	—
270	C₂²×C₄	—	2	4	no	—
271	C₂²×C₄	—	2	2	no	—
272	C₂²×C₄	—	2	2	no	—
273	C₂²×C₄	—	2	2	no	—
274	C₂²×C₄	—	4	2	no	—
275	C₂²×C₄	—	4	2	no	—
276	C₂²×C₄	C₂×D₄×C₄₇	1	2	yes	—
277	C₂²×C₄	C₂×D₄×C₄₇	1	2	yes	—
278	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
279	C₂²×C₄	C₂×D₄×C₄₇	1	2	yes	—
280	C₂²×C₄	#41×C₄₇	1	2	yes	—
281	C₂²×C₄	C₂⁴×C₄₇	1	1	yes	characteristic
282	SD₁₆	—	4	16	no	—
283	C₂×C₈	—	2	4	no	—
284	C₂×C₈	—	2	4	no	—
285	C₂×C₈	—	2	4	no	—
286	C₄⋊C₄	—	2	2	no	—
287	C₄⋊C₄	—	2	2	no	—
288	C₄⋊C₄	—	2	4	no	—
289	C₄⋊C₄	—	4	2	no	—
290	C₂²⋊C₄	—	4	2	no	—
291	C₂²⋊C₄	—	4	2	no	—
292	C₂²⋊C₄	—	4	4	no	—
293	C₂²⋊C₄	—	8	4	no	—
294	C₄²	—	2	2	no	—
295	C₄²	—	2	2	no	—
296	C₄²	—	4	2	no	—
297	C₂³	—	2	1	no	—
298	C₂³	—	2	1	no	—
299	C₂³	—	2	4	no	—
300	C₂³	—	4	2	no	—
301	C₂³	—	4	2	no	—
302	C₂³	—	4	2	no	—
303	C₂³	—	4	4	no	—
304	C₂³	—	4	4	no	—
305	C₂³	#121×C₄₇	1	2	yes	—
306	C₂³	C₂²×D₄×C₄₇	1	1	yes	characteristic
307	Q₈	—	2	8	no	—
308	Q₈	—	4	8	no	—
309	Q₈	—	4	4	no	—
310	D₄	—	4	4	no	—
311	D₄	—	4	4	no	—
312	D₄	—	8	8	no	—
313	D₄	—	8	8	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	4	no	—
317	C₂×C₄	—	2	4	no	—
318	C₂×C₄	—	2	4	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	4	no	—
322	C₂×C₄	—	2	4	no	—
323	C₂×C₄	—	4	8	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	4	4	no	—
329	C₂×C₄	—	4	4	no	—
330	C₂×C₄	#121×C₄₇	1	2	yes	—
331	C₂×C₄	C₂²×D₄×C₄₇	1	1	yes	characteristic
332	C₂×C₄	C₂×#41×C₄₇	1	1	yes	characteristic
333	C₂×C₄	H(5,𝔽₂)×C₄₇	1	2	yes	—
334	C₂×C₄	H(5,𝔽₂)×C₄₇	1	2	yes	—
335	C₈	—	4	4	no	—
336	C₂²	—	2	4	no	—
337	C₂²	—	2	2	no	—
338	C₂²	—	4	2	no	—
339	C₂²	—	4	2	no	—
340	C₂²	—	8	2	no	—
341	C₂²	—	8	2	no	—
342	C₂²	—	8	2	no	—
343	C₂²	—	8	4	no	—
344	C₂²	—	8	4	no	—
345	C₂²	C₂×#121×C₄₇	1	1	yes	characteristic
346	C₂²	C₄₇×#534	1	1	yes	characteristic
347	C₂²	C₄₇×#535	1	1	yes	characteristic
348	C₂²	C₄₇×#577	1	4	yes	—
349	C₄	—	2	2	no	—
350	C₄	—	4	4	no	—
351	C₄	—	4	2	no	—
352	C₄	—	4	2	no	—
353	C₄	—	8	2	no	—
354	C₄	—	8	2	no	—
355	C₄	C₄₇×#534	1	2	yes	—
356	C₄	C₄₇×#538	1	2	yes	—
357	C₂	—	4	2	no	—
358	C₂	—	16	2	no	—
359	C₂	C₄₇×#2617	1	1	yes	characteristic
360	C₂	C₄₇×#3043	1	2	yes	—
361	C₂	C₄₇×#3200	1	2	yes	—
362	C₂	C₄₇×#3221	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₄₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₇
prime radical	C₄₇×#2619	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₇ ◁ C₂×#117×C₄₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₄₇×#448	C₂	semidirect
3. C₄₇×#2310	C₂	semidirect
4. C₄₇×#2074	C₂	semidirect
5. C₄₇×#2055	C₂	semidirect
6. C₄₇×#2014	C₂	semidirect
7. C₄₇×#1875	C₂	semidirect
8. C₄₇×#1798	C₂	semidirect
34. C₂²×Q₈×C₄₇	D₄	semidirect
154. #29722	C₄₇	direct (N × H)
176. C₂×#448	C₂×C₄₇	semidirect
178. #2310	C₂×C₄₇	semidirect
179. #2074	C₂×C₄₇	semidirect
180. #2055	C₂×C₄₇	semidirect
181. #2014	C₂×C₄₇	semidirect
182. #1875	C₂×C₄₇	semidirect
183. #1798	C₂×C₄₇	semidirect
211. C₄₇	#29722	direct (N × H)
216. C₂²×Q₈	D₄×C₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄₇×#448	2
C₄₇×#2619	2
C₄₇×#2310	2
C₄₇×#2074	2
C₄₇×#2055	2
C₄₇×#2014	2
C₄₇×#1875	2
C₄₇×#1798	2
#29722	47

Subgroups — C47×#29722 #26069420

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#29722 #26069420