Subgroups — #26216885

Order 12032.

810 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	#20252021	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#20243559	C₂	1	2	yes	maximal, complemented
4	#20243525	C₂	1	2	yes	maximal, complemented
5	#20243424	C₂	1	4	yes	maximal
6	C₄×#13093959	C₂	1	4	yes	maximal, complemented
7	C₄₇×#2920	C₂	1	1	yes	maximal, characteristic
8	C₂³×C₄₇⋊C₈	C₂²	1	2	yes	—
9	#15689180	—	2	2	no	—
10	#15689176	—	2	2	no	—
11	C₂×#13093959	C₂²	1	8	yes	—
12	#15689147	C₂²	1	2	yes	—
13	#15689087	C₂²	1	2	yes	—
14	C₄×#10625329	C₂²	1	2	yes	—
15	C₂×C₄×C₄₇⋊C₈	C₄	1	4	yes	—
16	C₂×C₄×C₄₇⋊C₈	C₄	1	4	yes	—
17	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
19	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
20	#15688887	C₂²	1	8	yes	—
21	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
22	C₄₇×#432	—	2	2	no	—
23	C₄₇×#431	—	2	2	no	—
24	C₄₇×#423	C₂²	1	2	yes	—
25	C₄₇×#407	C₂²	1	2	yes	—
26	C₄×M₁₆×C₄₇	C₂²	1	2	yes	—
27	C₂×#10625329	C₂³	1	4	yes	—
28	C₂²×C₄₇⋊C₈	—	2	8	no	—
29	C₂²×C₄₇⋊C₈	C₂×C₄	1	8	yes	—
30	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
31	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
32	#13093959	—	2	16	no	—
33	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
34	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
35	#13093935	—	2	2	no	—
36	C₄×C₄₇⋊C₈	—	2	2	no	—
37	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
38	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
39	C₄×C₄₇⋊C₈	C₂×C₄	1	8	yes	—
40	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
41	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
42	C₂³×C₄×C₄₇	Q₈	1	2	yes	—
43	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
44	C₂×M₁₆×C₄₇	C₂³	1	4	yes	—
45	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₄₇	D₄	1	2	yes	—
48	C₂×C₄²×C₄₇	D₄	1	2	yes	—
49	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
50	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
52	C₄⋊C₈×C₄₇	—	2	4	no	—
53	C₂²⋊C₈×C₄₇	—	2	4	no	—
54	#97×C₄₇	—	2	2	no	—
55	C₄×C₈×C₄₇	—	2	2	no	—
56	C₂⁴×C₄₇	C₂×Q₈	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	—	2	8	no	—
58	C₂²×C₄×C₄₇	—	2	4	no	—
59	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
60	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
61	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
62	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
63	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
67	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
68	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
69	M₁₆×C₄₇	—	2	8	no	—
70	C₂×C₈×C₄₇	—	2	4	no	—
71	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
72	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
73	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
74	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
75	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
76	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
77	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
78	C₄²×C₄₇	C₂×D₄	1	2	yes	—
79	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
81	#10625329	—	2	8	no	—
82	C₂×C₄₇⋊C₈	—	2	4	no	—
83	C₂×C₄₇⋊C₈	—	2	16	no	—
84	C₂×C₄₇⋊C₈	—	2	4	no	—
85	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
86	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
87	C₂×C₄₇⋊C₈	#41	1	8	yes	—
88	C₂³×C₄₇	—	2	2	no	—
89	C₂³×C₄₇	—	2	4	no	—
90	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
91	C₂³×C₄₇	C₂²×Q₈	1	1	yes	characteristic
92	C₂×C₄×C₄₇	—	2	2	no	—
93	C₂×C₄×C₄₇	—	2	4	no	—
94	C₂×C₄×C₄₇	—	2	4	no	—
95	C₂×C₄×C₄₇	—	2	4	no	—
96	C₂×C₄×C₄₇	—	2	4	no	—
97	C₂×C₄×C₄₇	—	2	4	no	—
98	C₂×C₄×C₄₇	—	2	4	no	—
99	C₂×C₄×C₄₇	—	2	4	no	—
100	C₂×C₄×C₄₇	#108	1	8	yes	complemented
101	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
102	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
103	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
104	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
105	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
106	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
107	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
108	C₂×C₄×C₄₇	C₂×#41	1	4	yes	—
109	C₈×C₄₇	—	4	4	no	—
110	C₄₇⋊C₈	—	2	4	no	—
111	C₄₇⋊C₈	—	4	4	no	—
112	C₄₇⋊C₈	#117	1	8	yes	—
113	#20826	—	47	1	no	maximal, Sylow
114	C₂²×C₄₇	—	2	8	no	—
115	C₂²×C₄₇	—	2	2	no	—
116	C₂²×C₄₇	—	2	2	no	—
117	C₂²×C₄₇	—	2	2	no	—
118	C₂²×C₄₇	C₂×#108	1	4	yes	—
119	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
120	C₂²×C₄₇	#533	1	2	yes	—
121	C₄×C₄₇	—	2	2	no	—
122	C₄×C₄₇	—	2	2	no	—
123	C₄×C₄₇	—	2	2	no	—
124	C₄×C₄₇	C₂×#108	1	4	yes	—
125	C₄×C₄₇	C₂×#108	1	4	yes	—
126	C₄×C₄₇	C₂×#117	1	2	yes	—
127	C₄×C₄₇	#532	1	2	yes	—
128	#2920	—	47	1	no	—
129	#2920	D₄₇	1	1	yes	characteristic
130	C₂²×C₄×C₈	—	47	1	no	—
131	#1848	—	47	2	no	—
132	#1837	—	47	2	no	—
133	#1810	—	47	4	no	—
134	C₄×#108	—	47	4	no	—
135	C₂×C₄₇	—	2	2	no	—
136	C₂×C₄₇	—	2	4	no	—
137	C₂×C₄₇	#2285	1	1	yes	characteristic
138	C₂×C₄₇	C₂²×#108	1	2	yes	—
139	C₂³×C₈	—	47	2	no	—
140	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
141	#432	—	2	2	no	—
142	#432	—	94	2	no	—
143	#431	—	2	2	no	—
144	#431	—	94	2	no	—
145	C₂×#108	—	47	8	no	—
146	#423	—	47	2	no	—
147	#423	C₂×D₄₇	1	2	yes	—
148	#407	—	47	2	no	—
149	#407	C₂×D₄₇	1	2	yes	—
150	C₄×M₁₆	—	47	2	no	—
151	C₄×M₁₆	C₂×D₄₇	1	2	yes	—
152	C₂×C₄×C₈	—	47	4	no	—
153	C₂×C₄×C₈	—	47	4	no	—
154	C₂×C₄×C₈	—	47	1	no	—
155	C₂×C₄×C₈	—	47	1	no	—
156	C₂×C₄×C₈	—	47	2	no	—
157	#341	—	47	8	no	—
158	C₄₇	#20826	1	1	yes	characteristic, complemented, Sylow
159	C₂³×C₄	Dic₉₄	1	2	yes	—
160	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
161	C₂×M₁₆	—	47	4	no	—
162	C₂×M₁₆	C₂²×D₄₇	1	4	yes	—
163	C₂²×C₈	—	47	2	no	—
164	C₂²×C₈	—	47	8	no	—
165	C₂²×C₈	—	47	2	no	—
166	C₂²×C₈	—	94	8	no	—
167	C₂×C₄²	Dic₉₄	1	2	yes	—
168	C₂×C₄²	C₄×D₄₇	1	2	yes	—
169	C₂×C₄²	C₄×D₄₇	1	2	yes	—
170	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
171	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
172	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
173	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
174	#108	—	94	16	no	—
175	C₄⋊C₈	—	2	4	no	—
176	C₄⋊C₈	—	94	4	no	—
177	C₂²⋊C₈	—	2	4	no	—
178	C₂²⋊C₈	—	94	4	no	—
179	#97	—	2	2	no	—
180	#97	—	94	2	no	—
181	C₄×C₈	—	2	2	no	—
182	C₄×C₈	—	47	2	no	—
183	C₄×C₈	—	47	2	no	—
184	C₄×C₈	—	47	8	no	—
185	C₄×C₈	—	47	2	no	—
186	C₄×C₈	—	47	2	no	—
187	C₄×C₈	—	94	2	no	—
188	C₂⁴	C₂×Dic₉₄	1	1	yes	characteristic
189	C₂²×C₄	—	2	8	no	—
190	C₂²×C₄	—	2	4	no	—
191	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
192	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
193	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
194	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
195	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
196	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
197	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
198	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
199	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
200	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
201	M₁₆	—	2	8	no	—
202	M₁₆	—	94	8	no	—
203	C₂×C₈	—	2	4	no	—
204	C₂×C₈	—	47	4	no	—
205	C₂×C₈	—	47	8	no	—
206	C₂×C₈	—	47	4	no	—
207	C₂×C₈	—	94	16	no	—
208	C₂×C₈	—	94	4	no	—
209	C₂×C₈	—	94	4	no	—
210	C₄²	Dic₄₇⋊C₄	1	2	yes	—
211	C₄²	Dic₄₇⋊C₄	1	2	yes	—
212	C₄²	Dic₄₇⋊C₄	1	2	yes	—
213	C₄²	Dic₄₇⋊C₄	1	2	yes	—
214	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
215	C₄²	C₂×C₄×D₄₇	1	2	yes	—
216	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
217	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
218	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
219	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
220	C₂³	—	2	2	no	—
221	C₂³	—	2	4	no	—
222	C₂³	C₂×Dic₄₇⋊C₄	1	2	yes	—
223	C₂³	C₂²×Dic₉₄	1	1	yes	characteristic
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	#13093959	1	8	yes	complemented
233	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
234	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
235	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
236	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
237	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
238	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
239	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
240	C₂×C₄	#41×D₄₇	1	4	yes	—
241	C₈	—	4	4	no	—
242	C₈	—	47	8	no	—
243	C₈	—	94	4	no	—
244	C₈	—	188	4	no	—
245	C₂²	—	2	8	no	—
246	C₂²	—	2	2	no	—
247	C₂²	—	2	2	no	—
248	C₂²	—	2	2	no	—
249	C₂²	C₂×#13093959	1	4	yes	—
250	C₂²	C₂²×Dic₄₇⋊C₄	1	1	yes	characteristic
251	C₂²	#15689676	1	2	yes	—
252	C₄	—	2	2	no	—
253	C₄	—	2	2	no	—
254	C₄	—	2	2	no	—
255	C₄	C₂×#13093959	1	4	yes	—
256	C₄	C₂×#13093959	1	4	yes	—
257	C₄	#117×D₄₇	1	2	yes	—
258	C₄	#15689669	1	2	yes	—
259	C₂	—	2	2	no	—
260	C₂	—	2	4	no	—
261	C₂	#20245791	1	1	yes	characteristic
262	C₂	C₂²×#13093959	1	2	yes	—

	subgroup	G/H
center	C₄²	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#2920	C₂
socle	C₂²×C₄₇	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₄₇	C₂×Q₈
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#2920	D₄₇
hypocenter	C₄₇	#20826
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ #2920

Complements

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

N	H ≅ G/N	type
3. #20243559	C₂	semidirect
4. #20243525	C₂	semidirect
6. C₄×#13093959	C₂	semidirect
100. C₂×C₄×C₄₇	#108	semidirect
158. C₄₇	#20826	semidirect
232. C₂×C₄	#13093959	semidirect

Maximal subgroups

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

subgroup	index
#20252021	2
C₂²×C₄×C₄₇⋊C₈	2
#20243559	2
#20243525	2
#20243424	2
C₄×#13093959	2
C₄₇×#2920	2
#20826	47