Subgroups — C₄₇×#29221 #26068919

Order 12032.

638 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₇×#2859	C₂	1	1	yes	maximal, characteristic
2	C₄×#125×C₄₇	C₂	1	1	yes	maximal, characteristic
3	C₄₇×#2100	C₂	1	2	yes	maximal
4	C₄₇×#2099	C₂	1	2	yes	maximal
5	C₄₇×#1991	C₂	1	2	yes	maximal
6	C₄₇×#1989	C₂	1	4	yes	maximal
7	C₄₇×#1835	C₂	1	2	yes	maximal
8	C₄×C₄⋊C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
9	C₂×#125×C₄₇	C₂²	1	2	yes	—
10	C₂×#125×C₄₇	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
13	C₄₇×#389	—	2	2	no	—
14	C₄₇×#389	—	2	2	no	—
15	C₄₇×#389	C₂²	1	2	yes	—
16	C₄₇×#387	—	2	2	no	—
17	C₄₇×#384	C₂²	1	1	yes	characteristic
18	C₄₇×#384	C₂²	1	1	yes	characteristic
19	C₄₇×#382	C₂²	1	1	yes	characteristic
20	C₄₇×#382	C₂²	1	2	yes	—
21	C₄×C₄⋊C₄×C₄₇	—	2	2	no	—
22	C₄×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
23	C₄³×C₄₇	C₂²	1	1	yes	characteristic
24	C₄₇×#340	C₂²	1	4	yes	—
25	C₄₇×#340	C₂²	1	4	yes	—
26	C₄₇×#340	C₂²	1	4	yes	—
27	C₄₇×#340	C₂²	1	4	yes	—
28	C₄₇×#336	C₂²	1	2	yes	—
29	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
30	#125×C₄₇	—	2	2	no	—
31	#125×C₄₇	—	2	8	no	—
32	#125×C₄₇	Q₈	1	4	yes	—
33	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
34	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
35	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
36	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
37	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
38	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
39	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
40	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
41	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
42	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
43	C₂×C₄²×C₄₇	—	2	2	no	—
44	C₂×C₄²×C₄₇	D₄	1	2	yes	—
45	C₂×C₄²×C₄₇	D₄	1	2	yes	—
46	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
49	C₄⋊C₈×C₄₇	—	2	4	no	—
50	C₄×C₈×C₄₇	—	2	4	no	—
51	#95×C₄₇	—	2	2	no	—
52	#95×C₄₇	—	2	2	no	—
53	#95×C₄₇	—	2	2	no	—
54	#95×C₄₇	—	2	4	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	2	no	—
59	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
60	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
64	C₂×C₈×C₄₇	—	2	4	no	—
65	C₂×C₈×C₄₇	—	2	4	no	—
66	C₂×C₈×C₄₇	—	2	4	no	—
67	C₄⋊C₄×C₄₇	—	2	8	no	—
68	C₄⋊C₄×C₄₇	—	2	2	no	—
69	C₄⋊C₄×C₄₇	—	2	4	no	—
70	C₄⋊C₄×C₄₇	—	2	4	no	—
71	C₄⋊C₄×C₄₇	—	2	4	no	—
72	C₄⋊C₄×C₄₇	—	4	4	no	—
73	C₄⋊C₄×C₄₇	—	4	4	no	—
74	C₄⋊C₄×C₄₇	—	4	4	no	—
75	C₄⋊C₄×C₄₇	—	4	4	no	—
76	C₄⋊C₄×C₄₇	—	4	4	no	—
77	C₄⋊C₄×C₄₇	C₂×Q₈	1	4	yes	—
78	C₄²×C₄₇	—	2	4	no	—
79	C₄²×C₄₇	—	2	2	no	—
80	C₄²×C₄₇	—	4	2	no	—
81	C₄²×C₄₇	—	4	4	no	—
82	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
83	C₄²×C₄₇	#41	1	2	yes	—
84	C₄²×C₄₇	#41	1	4	yes	—
85	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
86	C₂×C₄×C₄₇	—	2	2	no	—
87	C₂×C₄×C₄₇	—	2	2	no	—
88	C₂×C₄×C₄₇	—	2	2	no	—
89	C₂×C₄×C₄₇	—	2	2	no	—
90	C₂×C₄×C₄₇	—	2	2	no	—
91	C₂×C₄×C₄₇	—	2	2	no	—
92	C₂×C₄×C₄₇	—	2	4	no	—
93	C₂×C₄×C₄₇	—	4	2	no	—
94	C₂×C₄×C₄₇	—	4	2	no	—
95	C₂×C₄×C₄₇	—	4	2	no	—
96	C₂×C₄×C₄₇	—	4	2	no	—
97	C₂×C₄×C₄₇	—	4	2	no	—
98	C₂×C₄×C₄₇	—	4	4	no	—
99	C₂×C₄×C₄₇	—	4	2	no	—
100	C₂×C₄×C₄₇	—	4	2	no	—
101	C₂×C₄×C₄₇	—	4	4	no	—
102	C₂×C₄×C₄₇	C₂²⋊Q₈	1	4	yes	—
103	C₂×C₄×C₄₇	#124	1	2	yes	—
104	C₂×C₄×C₄₇	#124	1	2	yes	—
105	C₂×C₄×C₄₇	#135	1	4	yes	—
106	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
107	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
108	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
109	C₂×C₄×C₄₇	2₋¹⁺⁴	1	2	yes	—
110	C₈×C₄₇	—	4	4	no	—
111	#29221	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
112	C₂²×C₄₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
113	C₂²×C₄₇	C₂×#124	1	1	yes	characteristic
114	C₂²×C₄₇	#536	1	1	yes	characteristic
115	C₂²×C₄₇	C₂×#135	1	2	yes	—
116	C₂²×C₄₇	#575	1	2	yes	—
117	C₄×C₄₇	—	2	4	no	—
118	C₄×C₄₇	—	2	2	no	—
119	C₄×C₄₇	—	2	2	no	—
120	C₄×C₄₇	—	2	4	no	—
121	C₄×C₄₇	—	4	4	no	—
122	C₄×C₄₇	—	8	2	no	—
123	C₄×C₄₇	—	8	2	no	—
124	C₄×C₄₇	—	8	2	no	—
125	C₄×C₄₇	#556	1	2	yes	—
126	C₄×C₄₇	#557	1	2	yes	—
127	#2859	C₂×C₄₇	1	1	yes	characteristic
128	C₄×#125	C₂×C₄₇	1	1	yes	characteristic
129	#2100	C₂×C₄₇	1	2	yes	—
130	#2099	C₂×C₄₇	1	2	yes	—
131	#1991	C₂×C₄₇	1	2	yes	—
132	#1989	C₂×C₄₇	1	4	yes	—
133	#1835	C₂×C₄₇	1	2	yes	—
134	C₄×C₄⋊C₈	C₂×C₄₇	1	1	yes	characteristic
135	C₂×C₄₇	#2568	1	1	yes	characteristic
136	C₂×C₄₇	#3076	1	2	yes	—
137	C₂×C₄₇	#3131	1	1	yes	characteristic
138	C₂×C₄₇	#3142	1	1	yes	characteristic
139	C₂×C₄₇	#3232	1	2	yes	—
140	C₂×#125	C₂²×C₄₇	1	2	yes	—
141	C₂×#125	C₂²×C₄₇	1	1	yes	characteristic
142	C₂×C₄⋊C₈	C₂²×C₄₇	1	2	yes	—
143	C₂×C₄×C₈	C₂²×C₄₇	1	2	yes	—
144	#389	—	2	2	no	—
145	#389	—	2	2	no	—
146	#389	C₂²×C₄₇	1	2	yes	—
147	#387	—	2	2	no	—
148	#384	C₂²×C₄₇	1	1	yes	characteristic
149	#384	C₂²×C₄₇	1	1	yes	characteristic
150	#382	C₂²×C₄₇	1	1	yes	characteristic
151	#382	C₂²×C₄₇	1	2	yes	—
152	C₄×C₄⋊C₄	—	2	2	no	—
153	C₄×C₄⋊C₄	C₂²×C₄₇	1	2	yes	—
154	C₄³	C₂²×C₄₇	1	1	yes	characteristic
155	#340	C₂²×C₄₇	1	4	yes	—
156	#340	C₂²×C₄₇	1	4	yes	—
157	#340	C₂²×C₄₇	1	4	yes	—
158	#340	C₂²×C₄₇	1	4	yes	—
159	#336	C₂²×C₄₇	1	2	yes	—
160	C₄₇	#29221	1	1	yes	characteristic, direct product factor, Sylow
161	C₂²×C₈	C₂³×C₄₇	1	4	yes	—
162	#125	—	2	2	no	—
163	#125	—	2	8	no	—
164	#125	Q₈×C₄₇	1	4	yes	—
165	C₂×C₄⋊C₄	—	2	2	no	—
166	C₂×C₄⋊C₄	—	2	2	no	—
167	C₂×C₄⋊C₄	—	2	2	no	—
168	C₂×C₄⋊C₄	—	2	2	no	—
169	C₂×C₄⋊C₄	—	2	2	no	—
170	C₂×C₄⋊C₄	—	2	2	no	—
171	C₂×C₄⋊C₄	C₂³×C₄₇	1	2	yes	—
172	C₂×C₄⋊C₄	C₂³×C₄₇	1	2	yes	—
173	C₂×C₄⋊C₄	C₂³×C₄₇	1	2	yes	—
174	C₂×C₄⋊C₄	C₂³×C₄₇	1	2	yes	—
175	C₂×C₄²	—	2	2	no	—
176	C₂×C₄²	D₄×C₄₇	1	2	yes	—
177	C₂×C₄²	D₄×C₄₇	1	2	yes	—
178	C₂×C₄²	C₂³×C₄₇	1	1	yes	characteristic
179	C₂×C₄²	C₂³×C₄₇	1	1	yes	characteristic
180	C₂×C₄²	C₂³×C₄₇	1	1	yes	characteristic
181	C₄⋊C₈	—	2	4	no	—
182	C₄×C₈	—	2	4	no	—
183	#95	—	2	2	no	—
184	#95	—	2	2	no	—
185	#95	—	2	2	no	—
186	#95	—	2	4	no	—
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	C₂×D₄×C₄₇	1	2	yes	—
192	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
193	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
194	C₂²×C₄	C₂×D₄×C₄₇	1	2	yes	—
195	C₂²×C₄	C₂⁴×C₄₇	1	1	yes	characteristic
196	C₂×C₈	—	2	4	no	—
197	C₂×C₈	—	2	4	no	—
198	C₂×C₈	—	2	4	no	—
199	C₄⋊C₄	—	2	8	no	—
200	C₄⋊C₄	—	2	2	no	—
201	C₄⋊C₄	—	2	4	no	—
202	C₄⋊C₄	—	2	4	no	—
203	C₄⋊C₄	—	2	4	no	—
204	C₄⋊C₄	—	4	4	no	—
205	C₄⋊C₄	—	4	4	no	—
206	C₄⋊C₄	—	4	4	no	—
207	C₄⋊C₄	—	4	4	no	—
208	C₄⋊C₄	—	4	4	no	—
209	C₄⋊C₄	C₂×Q₈×C₄₇	1	4	yes	—
210	C₄²	—	2	4	no	—
211	C₄²	—	2	2	no	—
212	C₄²	—	4	2	no	—
213	C₄²	—	4	4	no	—
214	C₄²	C₂×Q₈×C₄₇	1	2	yes	—
215	C₄²	#41×C₄₇	1	2	yes	—
216	C₄²	#41×C₄₇	1	4	yes	—
217	C₂³	C₂²×D₄×C₄₇	1	1	yes	characteristic
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	4	4	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	4	4	no	—
234	C₂×C₄	C₂²⋊Q₈×C₄₇	1	4	yes	—
235	C₂×C₄	#124×C₄₇	1	2	yes	—
236	C₂×C₄	#124×C₄₇	1	2	yes	—
237	C₂×C₄	#135×C₄₇	1	4	yes	—
238	C₂×C₄	C₂²×Q₈×C₄₇	1	1	yes	characteristic
239	C₂×C₄	C₂×#41×C₄₇	1	1	yes	characteristic
240	C₂×C₄	C₂×#41×C₄₇	1	2	yes	—
241	C₂×C₄	2₋¹⁺⁴×C₄₇	1	2	yes	—
242	C₈	—	4	4	no	—
243	C₂²	C₂×C₂²⋊Q₈×C₄₇	1	1	yes	characteristic
244	C₂²	C₂×#124×C₄₇	1	1	yes	characteristic
245	C₂²	C₄₇×#536	1	1	yes	characteristic
246	C₂²	C₂×#135×C₄₇	1	2	yes	—
247	C₂²	C₄₇×#575	1	2	yes	—
248	C₄	—	2	4	no	—
249	C₄	—	2	2	no	—
250	C₄	—	2	2	no	—
251	C₄	—	2	4	no	—
252	C₄	—	4	4	no	—
253	C₄	—	8	2	no	—
254	C₄	—	8	2	no	—
255	C₄	—	8	2	no	—
256	C₄	C₄₇×#556	1	2	yes	—
257	C₄	C₄₇×#557	1	2	yes	—
258	C₂	C₄₇×#2568	1	1	yes	characteristic
259	C₂	C₄₇×#3076	1	2	yes	—
260	C₂	C₄₇×#3131	1	1	yes	characteristic
261	C₂	C₄₇×#3142	1	1	yes	characteristic
262	C₂	C₄₇×#3232	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	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
111. #29221	C₄₇	direct (N × H)
160. C₄₇	#29221	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄₇×#2859	2
C₄×#125×C₄₇	2
C₄₇×#2100	2
C₄₇×#2099	2
C₄₇×#1991	2
C₄₇×#1989	2
C₄₇×#1835	2
C₄×C₄⋊C₈×C₄₇	2
#29221	47

Subgroups — C47×#29221 #26068919

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#29221 #26068919