Subgroups — C₄₇×#29794 #26069492

Order 12032.

1254 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₇×#459	C₂	1	2	yes	maximal, complemented
2	C₄₇×#2867	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₇×#2293	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₇×#2041	C₂	1	4	yes	maximal, complemented
5	C₄₇×#2011	C₂	1	4	yes	maximal, complemented
6	C₄₇×#1982	C₂	1	2	yes	maximal, complemented
7	C₄₇×#1831	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₈×C₄₇	C₂²	1	4	yes	—
9	C₂×#127×C₄₇	—	2	1	no	—
10	C₂×#127×C₄₇	—	2	2	no	—
11	C₂×#127×C₄₇	—	2	1	no	—
12	C₂×#127×C₄₇	C₂²	1	2	yes	—
13	C₂×#127×C₄₇	C₂²	1	4	yes	—
14	C₂×C₄×D₄×C₄₇	C₂²	1	1	yes	characteristic
15	C₂×#117×C₄₇	—	2	1	no	—
16	C₄₇×#459	—	2	8	no	—
17	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
18	C₂×#102×C₄₇	C₂²	1	4	yes	—
19	C₂×#102×C₄₇	C₂²	1	8	yes	—
20	C₄₇×#390	C₂²	1	2	yes	—
21	C₄₇×#389	C₂²	1	2	yes	—
22	C₄₇×#388	—	2	2	no	—
23	C₄×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
24	C₄×C₂²⋊C₄×C₄₇	—	2	1	no	—
25	C₄³×C₄₇	C₂²	1	1	yes	characteristic
26	C₄₇×#336	C₂²	1	4	yes	—
27	C₂²×D₄×C₄₇	—	2	2	no	—
28	C₂²×D₄×C₄₇	—	2	4	no	—
29	C₂²×D₄×C₄₇	—	2	4	no	—
30	C₂²×D₄×C₄₇	—	2	2	no	—
31	C₂²×D₄×C₄₇	C₂³	1	4	yes	—
32	C₂²×D₄×C₄₇	C₂³	1	1	yes	characteristic
33	C₂³×C₄×C₄₇	—	2	1	no	—
34	C₂×D₈×C₄₇	—	2	8	no	—
35	C₂×D₈×C₄₇	—	4	8	no	—
36	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
37	#127×C₄₇	—	2	4	no	—
38	#127×C₄₇	—	2	4	no	—
39	#127×C₄₇	—	4	8	no	—
40	#127×C₄₇	—	4	8	no	—
41	#127×C₄₇	—	4	4	no	—
42	#127×C₄₇	—	4	4	no	—
43	C₄×D₄×C₄₇	—	2	2	no	—
44	C₄×D₄×C₄₇	D₄	1	4	yes	complemented
45	#117×C₄₇	—	4	4	no	—
46	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
47	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
48	C₂×C₄⋊C₄×C₄₇	C₂³	1	1	yes	characteristic
49	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
50	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
51	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
52	C₂×C₄²×C₄₇	—	2	1	no	—
53	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
54	C₂×C₄²×C₄₇	D₄	1	2	yes	—
55	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
56	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
57	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
58	C₄⋊C₈×C₄₇	—	2	4	no	—
59	#102×C₄₇	—	2	8	no	—
60	#102×C₄₇	—	4	8	no	—
61	#95×C₄₇	—	2	2	no	—
62	#95×C₄₇	—	2	1	no	—
63	C₂⁴×C₄₇	—	2	1	no	—
64	C₂⁴×C₄₇	—	2	4	no	—
65	C₂×D₄×C₄₇	—	2	4	no	—
66	C₂×D₄×C₄₇	—	2	1	no	—
67	C₂×D₄×C₄₇	—	2	4	no	—
68	C₂×D₄×C₄₇	—	2	4	no	—
69	C₂×D₄×C₄₇	—	2	8	no	—
70	C₂×D₄×C₄₇	—	2	8	no	—
71	C₂×D₄×C₄₇	—	4	8	no	—
72	C₂×D₄×C₄₇	—	4	8	no	—
73	C₂×D₄×C₄₇	—	4	8	no	—
74	C₂×D₄×C₄₇	—	4	8	no	—
75	C₂×D₄×C₄₇	—	4	8	no	—
76	C₂×D₄×C₄₇	—	4	4	no	—
77	C₂×D₄×C₄₇	—	4	4	no	—
78	C₂×D₄×C₄₇	—	4	8	no	—
79	C₂×D₄×C₄₇	—	4	4	no	—
80	C₂×D₄×C₄₇	—	4	8	no	—
81	C₂×D₄×C₄₇	—	4	4	no	—
82	C₂×D₄×C₄₇	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₄₇	—	2	1	no	—
84	C₂²×C₄×C₄₇	—	2	4	no	—
85	C₂²×C₄×C₄₇	—	2	2	no	—
86	C₂²×C₄×C₄₇	—	4	2	no	—
87	C₂²×C₄×C₄₇	—	4	2	no	—
88	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
90	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
91	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
93	D₈×C₄₇	—	4	16	no	—
94	C₂×C₈×C₄₇	—	2	4	no	—
95	C₂×C₈×C₄₇	—	2	8	no	—
96	C₄⋊C₄×C₄₇	—	2	1	no	—
97	C₄⋊C₄×C₄₇	—	2	4	no	—
98	C₄⋊C₄×C₄₇	—	4	4	no	—
99	C₄⋊C₄×C₄₇	C₂×D₄	1	2	yes	—
100	C₂²⋊C₄×C₄₇	—	2	4	no	—
101	C₂²⋊C₄×C₄₇	—	4	4	no	—
102	C₂²⋊C₄×C₄₇	—	4	4	no	—
103	C₄²×C₄₇	—	2	4	no	—
104	C₄²×C₄₇	—	2	2	no	—
105	C₄²×C₄₇	—	2	4	no	—
106	C₄²×C₄₇	—	4	2	no	—
107	C₄²×C₄₇	D₈	1	4	yes	complemented
108	C₄²×C₄₇	C₂×D₄	1	2	yes	—
109	C₄²×C₄₇	#41	1	2	yes	—
110	C₂³×C₄₇	—	2	2	no	—
111	C₂³×C₄₇	—	4	2	no	—
112	C₂³×C₄₇	—	4	1	no	—
113	C₂³×C₄₇	—	4	2	no	—
114	C₂³×C₄₇	—	4	1	no	—
115	C₂³×C₄₇	—	4	4	no	—
116	C₂³×C₄₇	—	4	8	no	—
117	C₂³×C₄₇	—	4	4	no	—
118	C₂³×C₄₇	—	4	8	no	—
119	C₂³×C₄₇	—	4	4	no	—
120	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
121	D₄×C₄₇	—	2	4	no	—
122	D₄×C₄₇	—	4	4	no	—
123	D₄×C₄₇	—	4	4	no	—
124	D₄×C₄₇	—	4	2	no	—
125	D₄×C₄₇	—	4	8	no	—
126	D₄×C₄₇	—	8	16	no	—
127	D₄×C₄₇	—	8	16	no	—
128	D₄×C₄₇	—	8	8	no	—
129	D₄×C₄₇	—	8	8	no	—
130	C₂×C₄×C₄₇	—	2	2	no	—
131	C₂×C₄×C₄₇	—	2	2	no	—
132	C₂×C₄×C₄₇	—	2	2	no	—
133	C₂×C₄×C₄₇	—	2	2	no	—
134	C₂×C₄×C₄₇	—	2	2	no	—
135	C₂×C₄×C₄₇	—	2	2	no	—
136	C₂×C₄×C₄₇	—	4	1	no	—
137	C₂×C₄×C₄₇	—	4	1	no	—
138	C₂×C₄×C₄₇	—	4	8	no	—
139	C₂×C₄×C₄₇	—	4	2	no	—
140	C₂×C₄×C₄₇	—	4	2	no	—
141	C₂×C₄×C₄₇	—	4	2	no	—
142	C₂×C₄×C₄₇	#121	1	2	yes	—
143	C₂×C₄×C₄₇	#121	1	2	yes	—
144	C₂×C₄×C₄₇	C₂×D₈	1	4	yes	—
145	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
146	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	4	yes	—
147	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
148	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
149	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
150	C₂×C₄×C₄₇	H(5,𝔽₂)	1	4	yes	—
151	C₈×C₄₇	—	4	4	no	—
152	#29794	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
153	C₂²×C₄₇	—	4	2	no	—
154	C₂²×C₄₇	—	4	2	no	—
155	C₂²×C₄₇	—	4	2	no	—
156	C₂²×C₄₇	—	8	2	no	—
157	C₂²×C₄₇	—	8	2	no	—
158	C₂²×C₄₇	—	8	4	no	—
159	C₂²×C₄₇	—	8	8	no	—
160	C₂²×C₄₇	—	8	8	no	—
161	C₂²×C₄₇	—	8	4	no	—
162	C₂²×C₄₇	—	8	4	no	—
163	C₂²×C₄₇	C₂×#121	1	1	yes	characteristic
164	C₂²×C₄₇	#535	1	2	yes	—
165	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
166	C₂²×C₄₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
167	C₂²×C₄₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
168	C₄×C₄₇	—	2	4	no	—
169	C₄×C₄₇	—	2	4	no	—
170	C₄×C₄₇	—	2	2	no	—
171	C₄×C₄₇	—	4	2	no	—
172	C₄×C₄₇	—	8	2	no	—
173	C₄×C₄₇	#459	1	4	yes	—
174	C₄×C₄₇	#535	1	2	yes	—
175	C₄×C₄₇	#540	1	2	yes	—
176	C₂×#459	C₂×C₄₇	1	2	yes	complemented
177	#2867	C₂×C₄₇	1	1	yes	characteristic, complemented
178	#2293	C₂×C₄₇	1	1	yes	characteristic, complemented
179	#2041	C₂×C₄₇	1	4	yes	complemented
180	#2011	C₂×C₄₇	1	4	yes	complemented
181	#1982	C₂×C₄₇	1	2	yes	complemented
182	#1831	C₂×C₄₇	1	1	yes	characteristic, complemented
183	C₂×C₄₇	—	8	2	no	—
184	C₂×C₄₇	—	16	4	no	—
185	C₂×C₄₇	#2619	1	1	yes	characteristic
186	C₂×C₄₇	C₂×#459	1	1	yes	characteristic
187	C₂×C₄₇	#3038	1	1	yes	characteristic
188	C₂×C₄₇	#3205	1	2	yes	—
189	C₂×C₄₇	#3217	1	2	yes	—
190	C₂²×D₈	C₂²×C₄₇	1	4	yes	—
191	C₂×#127	—	2	1	no	—
192	C₂×#127	—	2	2	no	—
193	C₂×#127	—	2	1	no	—
194	C₂×#127	C₂²×C₄₇	1	2	yes	—
195	C₂×#127	C₂²×C₄₇	1	4	yes	—
196	C₂×C₄×D₄	C₂²×C₄₇	1	1	yes	characteristic
197	C₂×#117	—	2	1	no	—
198	#459	—	2	8	no	—
199	C₂×C₄⋊C₈	C₂²×C₄₇	1	2	yes	—
200	C₂×#102	C₂²×C₄₇	1	4	yes	—
201	C₂×#102	C₂²×C₄₇	1	8	yes	—
202	#390	C₂²×C₄₇	1	2	yes	—
203	#389	C₂²×C₄₇	1	2	yes	—
204	#388	—	2	2	no	—
205	C₄×C₄⋊C₄	C₂²×C₄₇	1	1	yes	characteristic
206	C₄×C₂²⋊C₄	—	2	1	no	—
207	C₄³	C₂²×C₄₇	1	1	yes	characteristic
208	#336	C₂²×C₄₇	1	4	yes	—
209	C₄₇	#29794	1	1	yes	characteristic, direct product factor, Sylow
210	C₂²×D₄	—	2	2	no	—
211	C₂²×D₄	—	2	4	no	—
212	C₂²×D₄	—	2	4	no	—
213	C₂²×D₄	—	2	2	no	—
214	C₂²×D₄	C₂³×C₄₇	1	4	yes	—
215	C₂²×D₄	C₂³×C₄₇	1	1	yes	characteristic
216	C₂³×C₄	—	2	1	no	—
217	C₂×D₈	—	2	8	no	—
218	C₂×D₈	—	4	8	no	—
219	C₂²×C₈	C₂³×C₄₇	1	4	yes	—
220	#127	—	2	4	no	—
221	#127	—	2	4	no	—
222	#127	—	4	8	no	—
223	#127	—	4	8	no	—
224	#127	—	4	4	no	—
225	#127	—	4	4	no	—
226	C₄×D₄	—	2	2	no	—
227	C₄×D₄	D₄×C₄₇	1	4	yes	complemented
228	#117	—	4	4	no	—
229	C₂×C₄⋊C₄	—	2	2	no	—
230	C₂×C₄⋊C₄	C₂³×C₄₇	1	2	yes	—
231	C₂×C₄⋊C₄	C₂³×C₄₇	1	1	yes	characteristic
232	C₂×C₂²⋊C₄	—	2	1	no	—
233	C₂×C₂²⋊C₄	—	2	2	no	—
234	C₂×C₂²⋊C₄	—	2	2	no	—
235	C₂×C₄²	—	2	1	no	—
236	C₂×C₄²	D₄×C₄₇	1	1	yes	characteristic
237	C₂×C₄²	D₄×C₄₇	1	2	yes	—
238	C₂×C₄²	D₄×C₄₇	1	1	yes	characteristic
239	C₂×C₄²	C₂³×C₄₇	1	2	yes	—
240	C₂×C₄²	C₂³×C₄₇	1	1	yes	characteristic
241	C₄⋊C₈	—	2	4	no	—
242	#102	—	2	8	no	—
243	#102	—	4	8	no	—
244	#95	—	2	2	no	—
245	#95	—	2	1	no	—
246	C₂⁴	—	2	1	no	—
247	C₂⁴	—	2	4	no	—
248	C₂×D₄	—	2	4	no	—
249	C₂×D₄	—	2	1	no	—
250	C₂×D₄	—	2	4	no	—
251	C₂×D₄	—	2	4	no	—
252	C₂×D₄	—	2	8	no	—
253	C₂×D₄	—	2	8	no	—
254	C₂×D₄	—	4	8	no	—
255	C₂×D₄	—	4	8	no	—
256	C₂×D₄	—	4	8	no	—
257	C₂×D₄	—	4	8	no	—
258	C₂×D₄	—	4	8	no	—
259	C₂×D₄	—	4	4	no	—
260	C₂×D₄	—	4	4	no	—
261	C₂×D₄	—	4	8	no	—
262	C₂×D₄	—	4	4	no	—
263	C₂×D₄	—	4	8	no	—
264	C₂×D₄	—	4	4	no	—
265	C₂×D₄	C₂×D₄×C₄₇	1	2	yes	—
266	C₂²×C₄	—	2	1	no	—
267	C₂²×C₄	—	2	4	no	—
268	C₂²×C₄	—	2	2	no	—
269	C₂²×C₄	—	4	2	no	—
270	C₂²×C₄	—	4	2	no	—
271	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
272	C₂²×C₄	C₂×D₄×C₄₇	1	2	yes	—
273	C₂²×C₄	C₂×D₄×C₄₇	1	2	yes	—
274	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
275	C₂²×C₄	C₂⁴×C₄₇	1	1	yes	characteristic
276	D₈	—	4	16	no	—
277	C₂×C₈	—	2	4	no	—
278	C₂×C₈	—	2	8	no	—
279	C₄⋊C₄	—	2	1	no	—
280	C₄⋊C₄	—	2	4	no	—
281	C₄⋊C₄	—	4	4	no	—
282	C₄⋊C₄	C₂×D₄×C₄₇	1	2	yes	—
283	C₂²⋊C₄	—	2	4	no	—
284	C₂²⋊C₄	—	4	4	no	—
285	C₂²⋊C₄	—	4	4	no	—
286	C₄²	—	2	4	no	—
287	C₄²	—	2	2	no	—
288	C₄²	—	2	4	no	—
289	C₄²	—	4	2	no	—
290	C₄²	D₈×C₄₇	1	4	yes	complemented
291	C₄²	C₂×D₄×C₄₇	1	2	yes	—
292	C₄²	#41×C₄₇	1	2	yes	—
293	C₂³	—	2	2	no	—
294	C₂³	—	4	2	no	—
295	C₂³	—	4	1	no	—
296	C₂³	—	4	2	no	—
297	C₂³	—	4	1	no	—
298	C₂³	—	4	4	no	—
299	C₂³	—	4	8	no	—
300	C₂³	—	4	4	no	—
301	C₂³	—	4	8	no	—
302	C₂³	—	4	4	no	—
303	C₂³	C₂²×D₄×C₄₇	1	1	yes	characteristic
304	D₄	—	2	4	no	—
305	D₄	—	4	4	no	—
306	D₄	—	4	4	no	—
307	D₄	—	4	2	no	—
308	D₄	—	4	8	no	—
309	D₄	—	8	8	no	—
310	D₄	—	8	16	no	—
311	D₄	—	8	16	no	—
312	D₄	—	8	8	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	4	1	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	4	8	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	#121×C₄₇	1	2	yes	—
326	C₂×C₄	#121×C₄₇	1	2	yes	—
327	C₂×C₄	C₂×D₈×C₄₇	1	4	yes	—
328	C₂×C₄	C₂×D₈×C₄₇	1	2	yes	—
329	C₂×C₄	AGL(1,ℤ/8)×C₄₇	1	4	yes	—
330	C₂×C₄	AGL(1,ℤ/8)×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	4	yes	—
334	C₈	—	4	4	no	—
335	C₂²	—	4	2	no	—
336	C₂²	—	4	2	no	—
337	C₂²	—	4	2	no	—
338	C₂²	—	8	2	no	—
339	C₂²	—	8	2	no	—
340	C₂²	—	8	4	no	—
341	C₂²	—	8	4	no	—
342	C₂²	—	8	8	no	—
343	C₂²	—	8	8	no	—
344	C₂²	—	8	4	no	—
345	C₂²	C₂×#121×C₄₇	1	1	yes	characteristic
346	C₂²	C₄₇×#535	1	2	yes	—
347	C₂²	C₂²×D₈×C₄₇	1	1	yes	characteristic
348	C₂²	C₂×AGL(1,ℤ/8)×C₄₇	1	2	yes	—
349	C₂²	C₂×AGL(1,ℤ/8)×C₄₇	1	1	yes	characteristic
350	C₄	—	2	2	no	—
351	C₄	—	2	4	no	—
352	C₄	—	2	4	no	—
353	C₄	—	4	2	no	—
354	C₄	—	8	2	no	—
355	C₄	C₄₇×#459	1	4	yes	—
356	C₄	C₄₇×#535	1	2	yes	—
357	C₄	C₄₇×#540	1	2	yes	—
358	C₂	—	8	2	no	—
359	C₂	—	16	4	no	—
360	C₂	C₄₇×#2619	1	1	yes	characteristic
361	C₂	C₂×C₄₇×#459	1	1	yes	characteristic
362	C₂	C₄₇×#3038	1	1	yes	characteristic
363	C₂	C₄₇×#3205	1	2	yes	—
364	C₂	C₄₇×#3217	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₄₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₇
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₇ ◁ C₄³×C₄₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₄₇×#459	C₂	semidirect
2. C₄₇×#2867	C₂	semidirect
3. C₄₇×#2293	C₂	semidirect
4. C₄₇×#2041	C₂	semidirect
5. C₄₇×#2011	C₂	semidirect
6. C₄₇×#1982	C₂	semidirect
7. C₄₇×#1831	C₂	semidirect
44. C₄×D₄×C₄₇	D₄	semidirect
107. C₄²×C₄₇	D₈	semidirect
152. #29794	C₄₇	direct (N × H)
176. C₂×#459	C₂×C₄₇	semidirect
177. #2867	C₂×C₄₇	semidirect
178. #2293	C₂×C₄₇	semidirect
179. #2041	C₂×C₄₇	semidirect
180. #2011	C₂×C₄₇	semidirect
181. #1982	C₂×C₄₇	semidirect
182. #1831	C₂×C₄₇	semidirect
209. C₄₇	#29794	direct (N × H)
227. C₄×D₄	D₄×C₄₇	semidirect
290. C₄²	D₈×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×#459	2
C₄₇×#2867	2
C₄₇×#2293	2
C₄₇×#2041	2
C₄₇×#2011	2
C₄₇×#1982	2
C₄₇×#1831	2
#29794	47

Subgroups — C47×#29794 #26069492

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#29794 #26069492