Subgroups — #26216446

Order 12032.

922 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₄×#13093983	C₂	1	1	yes	maximal, characteristic
3	#20243507	C₂	1	4	yes	maximal, complemented
4	#20243504	C₂	1	4	yes	maximal, complemented
5	#20243086	C₂	1	4	yes	maximal
6	C₂×C₄₇×#423	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₄₇⋊C₈	C₂²	1	2	yes	—
8	C₂×#13093983	C₂²	1	2	yes	—
9	C₂×C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
10	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
11	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
12	C₄×C₄⋊Dic₄₇	C₄	1	2	yes	—
13	C₄×C₄⋊Dic₄₇	C₄	1	2	yes	—
14	C₄×C₂²⋊Dic₄₇	C₄	1	2	yes	—
15	C₄×C₂²⋊Dic₄₇	C₄	1	2	yes	—
16	#15688969	C₂²	1	2	yes	—
17	C₄²×Dic₄₇	C₂²	1	2	yes	—
18	#15688877	C₂²	1	16	yes	—
19	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
20	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
21	C₄₇×#423	—	2	4	no	—
22	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
23	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
24	C₂²×C₄₇⋊C₈	—	2	4	no	—
25	C₂²×C₄₇⋊C₈	—	2	8	no	—
26	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
27	#13093983	—	2	4	no	—
28	#13093983	—	2	4	no	—
29	C₂×C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
30	C₂×C₂²⋊Dic₄₇	C₂×C₄	1	4	yes	—
31	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
32	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
33	C₂×C₄×Dic₄₇	C₂³	1	4	yes	—
34	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
35	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
36	C₄₇⋊#95	C₂×C₄	1	4	yes	—
37	C₄₇⋊#95	C₂×C₄	1	4	yes	—
38	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
39	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
40	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
41	C₂×M₁₆×C₄₇	—	2	4	no	—
42	C₂×M₁₆×C₄₇	—	2	8	no	—
43	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
44	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₄₇	D₄	1	4	yes	—
47	C₂×C₄²×C₄₇	Q₈	1	4	yes	—
48	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
50	C₄⋊C₈×C₄₇	—	2	4	no	—
51	C₄⋊C₈×C₄₇	—	2	4	no	—
52	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	—	2	4	no	—
54	C₂²×C₄×C₄₇	—	4	2	no	—
55	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
56	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
57	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
64	C₂²×C₄×C₄₇	#41	1	4	yes	—
65	C₂²×C₄×C₄₇	#41	1	4	yes	—
66	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
67	C₂²×Dic₄₇	C₂²×C₄	1	8	yes	—
68	M₁₆×C₄₇	—	4	8	no	—
69	C₂×C₈×C₄₇	—	2	4	no	—
70	C₂×C₈×C₄₇	—	2	8	no	—
71	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
72	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
73	C₄²×C₄₇	C₂×D₄	1	2	yes	—
74	C₄²×C₄₇	C₂×D₄	1	2	yes	—
75	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
76	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
77	C₂²⋊Dic₄₇	—	2	8	no	—
78	C₄⋊Dic₄₇	—	2	8	no	—
79	C₄×Dic₄₇	—	2	8	no	—
80	C₄×Dic₄₇	—	2	4	no	—
81	C₄×Dic₄₇	—	2	4	no	—
82	C₄×Dic₄₇	—	4	4	no	—
83	C₂×C₄₇⋊C₈	—	2	4	no	—
84	C₂×C₄₇⋊C₈	—	2	8	no	—
85	C₂×C₄₇⋊C₈	—	4	8	no	—
86	C₂³×C₄₇	—	2	2	no	—
87	C₂³×C₄₇	—	2	2	no	—
88	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
89	C₂³×C₄₇	#117	1	4	yes	—
90	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
91	C₂×C₄×C₄₇	—	2	2	no	—
92	C₂×C₄×C₄₇	—	2	2	no	—
93	C₂×C₄×C₄₇	—	2	2	no	—
94	C₂×C₄×C₄₇	—	2	8	no	—
95	C₂×C₄×C₄₇	—	4	4	no	—
96	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
97	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
98	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
99	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
100	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
101	C₂×C₄×C₄₇	#117	1	4	yes	—
102	C₂×C₄×C₄₇	#124	1	4	yes	—
103	C₂×C₄×C₄₇	#124	1	4	yes	—
104	C₂×C₄×C₄₇	#125	1	8	yes	—
105	C₂×C₄×C₄₇	#131	1	4	yes	—
106	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
107	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
108	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
109	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
110	C₂×Dic₄₇	—	2	8	no	—
111	C₂×Dic₄₇	—	2	8	no	—
112	C₂×Dic₄₇	—	2	8	no	—
113	C₈×C₄₇	—	4	4	no	—
114	C₄₇⋊C₈	—	4	4	no	—
115	#20755	—	47	1	no	maximal, Sylow
116	C₂²×C₄₇	—	2	2	no	—
117	C₂²×C₄₇	—	2	2	no	—
118	C₂²×C₄₇	—	2	8	no	—
119	C₂²×C₄₇	#382	1	4	yes	—
120	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
121	C₂²×C₄₇	C₂×#117	1	2	yes	—
122	C₂²×C₄₇	C₂×#131	1	2	yes	—
123	C₂²×C₄₇	#568	1	2	yes	—
124	C₄×C₄₇	—	2	2	no	—
125	C₄×C₄₇	—	2	2	no	—
126	C₄×C₄₇	—	2	2	no	—
127	C₄×C₄₇	#382	1	4	yes	—
128	C₄×C₄₇	#424	1	4	yes	—
129	C₄×C₄₇	C₂×#124	1	1	yes	characteristic
130	C₄×C₄₇	C₂×#124	1	1	yes	characteristic
131	C₄×C₄₇	C₂×#125	1	2	yes	—
132	Dic₄₇	—	4	8	no	—
133	C₂×#423	D₄₇	1	1	yes	characteristic
134	C₂²×C₄⋊C₈	—	47	1	no	—
135	C₄×#117	—	47	1	no	—
136	#1832	—	47	4	no	—
137	#1831	—	47	4	no	—
138	#1729	—	47	4	no	—
139	C₂×C₄₇	—	2	4	no	—
140	C₂×C₄₇	C₂×#382	1	1	yes	characteristic
141	C₂×C₄₇	C₂×#424	1	1	yes	characteristic
142	C₂×C₄₇	#2905	1	1	yes	characteristic
143	C₂×C₄₇	#2925	1	4	yes	—
144	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
145	C₂³×C₈	—	47	2	no	—
146	C₂×#117	—	47	2	no	—
147	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
148	#423	—	2	4	no	—
149	C₂×C₄⋊C₈	—	47	2	no	—
150	C₂×C₄⋊C₈	—	47	2	no	—
151	C₂×C₄⋊C₈	—	94	4	no	—
152	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
153	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
154	C₄×C₄⋊C₄	—	47	2	no	—
155	C₄×C₄⋊C₄	—	47	2	no	—
156	C₄×C₂²⋊C₄	—	47	2	no	—
157	C₄×C₂²⋊C₄	—	47	2	no	—
158	#376	—	47	2	no	—
159	C₄³	—	47	2	no	—
160	#336	—	47	16	no	—
161	C₄₇	#20755	1	1	yes	characteristic, complemented, Sylow
162	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
163	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
164	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
165	C₂×M₁₆	—	2	8	no	—
166	C₂×M₁₆	—	2	4	no	—
167	C₂²×C₈	—	47	4	no	—
168	C₂²×C₈	—	94	8	no	—
169	C₂²×C₈	—	94	4	no	—
170	C₂²×C₈	C₂²×D₄₇	1	4	yes	—
171	#117	—	94	4	no	—
172	#117	—	94	4	no	—
173	C₂×C₄⋊C₄	—	47	4	no	—
174	C₂×C₂²⋊C₄	—	47	4	no	—
175	C₂×C₄²	—	47	4	no	—
176	C₂×C₄²	—	47	4	no	—
177	C₂×C₄²	—	47	4	no	—
178	C₂×C₄²	Dic₉₄	1	4	yes	—
179	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
180	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
181	C₂×C₄²	C₂²⋊D₄₇	1	4	yes	—
182	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
183	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
184	C₄⋊C₈	—	2	4	no	—
185	C₄⋊C₈	—	2	4	no	—
186	C₄⋊C₈	—	94	4	no	—
187	C₄⋊C₈	—	94	4	no	—
188	#95	—	47	4	no	—
189	#95	—	47	4	no	—
190	C₂⁴	C₂×C₄×D₄₇	1	1	yes	characteristic
191	C₂²×C₄	—	2	4	no	—
192	C₂²×C₄	—	4	2	no	—
193	C₂²×C₄	—	47	8	no	—
194	C₂²×C₄	Dic₄₇⋊C₄	1	4	yes	—
195	C₂²×C₄	Dic₄₇⋊C₄	1	4	yes	—
196	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
197	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
198	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
199	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
200	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
201	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
202	C₂²×C₄	#10625352	1	4	yes	—
203	C₂²×C₄	#10625354	1	4	yes	—
204	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
205	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
206	M₁₆	—	4	8	no	—
207	C₂×C₈	—	2	4	no	—
208	C₂×C₈	—	2	8	no	—
209	C₂×C₈	—	94	4	no	—
210	C₂×C₈	—	94	8	no	—
211	C₂×C₈	—	188	8	no	—
212	C₄⋊C₄	—	94	8	no	—
213	C₂²⋊C₄	—	94	8	no	—
214	C₄²	—	94	8	no	—
215	C₄²	—	94	4	no	—
216	C₄²	—	94	4	no	—
217	C₄²	—	188	4	no	—
218	C₄²	Dic₄₇⋊C₄	1	4	yes	—
219	C₄²	Dic₄₇⋊C₄	1	4	yes	—
220	C₄²	C₂×Dic₉₄	1	2	yes	—
221	C₄²	C₂×Dic₉₄	1	2	yes	—
222	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
223	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
224	C₂³	—	2	2	no	—
225	C₂³	—	2	2	no	—
226	C₂³	C₂×Dic₄₇⋊C₄	1	2	yes	—
227	C₂³	#13093982	1	4	yes	—
228	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	8	no	—
233	C₂×C₄	—	4	4	no	—
234	C₂×C₄	—	94	8	no	—
235	C₂×C₄	—	94	8	no	—
236	C₂×C₄	—	94	8	no	—
237	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
238	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
239	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
240	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
241	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
242	C₂×C₄	#13093982	1	4	yes	—
243	C₂×C₄	#13094011	1	4	yes	—
244	C₂×C₄	#13094013	1	4	yes	—
245	C₂×C₄	#13094015	1	8	yes	—
246	C₂×C₄	#13094032	1	4	yes	—
247	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
248	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
249	C₂×C₄	C₂×#10625352	1	2	yes	—
250	C₂×C₄	C₂×#10625354	1	2	yes	—
251	C₈	—	4	4	no	—
252	C₈	—	188	4	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	2	2	no	—
255	C₂²	—	2	8	no	—
256	C₂²	#15688988	1	4	yes	—
257	C₂²	C₂²×Dic₄₇⋊C₄	1	1	yes	characteristic
258	C₂²	C₂×#13093982	1	2	yes	—
259	C₂²	C₂×#13094032	1	2	yes	—
260	C₂²	#15689889	1	2	yes	—
261	C₄	—	2	2	no	—
262	C₄	—	2	2	no	—
263	C₄	—	2	2	no	—
264	C₄	—	188	8	no	—
265	C₄	#15688988	1	4	yes	—
266	C₄	#15689151	1	4	yes	—
267	C₄	C₂×#13094011	1	1	yes	characteristic
268	C₄	C₂×#13094013	1	1	yes	characteristic
269	C₄	C₂×#13094015	1	2	yes	—
270	C₂	—	2	4	no	—
271	C₂	C₂×#15688988	1	1	yes	characteristic
272	C₂	C₂×#15689151	1	1	yes	characteristic
273	C₂	#20251951	1	1	yes	characteristic
274	C₂	#20252056	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂³×C₄
Fitting	C₂×C₄₇×#423	C₂
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#423	D₄₇
hypocenter	C₄₇	#20755
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
3. #20243507	C₂	semidirect
4. #20243504	C₂	semidirect
161. C₄₇	#20755	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊(C₄₇⋊C₈)	2
C₄×#13093983	2
#20243507	2
#20243504	2
#20243086	2
C₂×C₄₇×#423	2
#20755	47