Subgroups — #26116312

Order 12032.

518 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13094024	C₂	1	1	yes	maximal, characteristic
2	C₄×#13093935	C₂	1	1	yes	maximal, characteristic
3	#20241227	C₂	1	4	yes	maximal
4	C₄₇×#1831	C₂	1	1	yes	maximal, characteristic
5	C₂×#13094024	C₄	1	1	yes	characteristic
6	C₂×C₄×Dic₉₄	—	2	1	no	—
7	C₂×#13093935	—	2	1	no	—
8	C₂×#13093935	—	2	1	no	—
9	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
10	#15689029	—	2	1	no	—
11	#15689021	—	2	1	no	—
12	#15689021	C₄	1	2	yes	—
13	#15688993	C₄	1	1	yes	characteristic
14	C₄×C₄⋊Dic₄₇	—	2	1	no	—
15	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
16	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
17	C₄³×C₄₇	C₂²	1	1	yes	characteristic
18	C₄₇×#336	—	2	1	no	—
19	C₄₇×#336	—	2	1	no	—
20	C₂²×Dic₉₄	—	2	1	no	—
21	C₂²×C₄₇⋊C₈	—	2	2	no	—
22	#13094024	—	2	4	no	—
23	C₄×Dic₉₄	—	4	4	no	—
24	C₂×C₄⋊Dic₄₇	—	2	1	no	—
25	C₂×C₄⋊Dic₄₇	—	2	1	no	—
26	C₂×C₄⋊Dic₄₇	—	2	1	no	—
27	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
28	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
29	C₂×Dic₄₇⋊C₄	—	2	2	no	—
30	C₂×C₄×Dic₄₇	—	2	2	no	—
31	#13093935	—	2	4	no	—
32	#13093935	—	2	4	no	—
33	C₄×C₄₇⋊C₈	—	2	2	no	—
34	C₄×C₄₇⋊C₈	Q₈	1	4	yes	complemented
35	C₄₇⋊#95	—	2	2	no	—
36	C₄₇⋊#95	—	2	2	no	—
37	C₂²×C₈×C₄₇	—	2	2	no	—
38	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
45	C₄⋊C₈×C₄₇	—	2	4	no	—
46	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
53	C₂²×Dic₄₇	—	2	2	no	—
54	C₂²×Dic₄₇	—	2	2	no	—
55	C₂×Dic₉₄	—	2	4	no	—
56	C₂×Dic₉₄	—	4	2	no	—
57	C₂×Dic₉₄	—	4	2	no	—
58	C₂×C₈×C₄₇	—	4	2	no	—
59	C₂×C₈×C₄₇	—	4	2	no	—
60	C₂×C₈×C₄₇	—	4	2	no	—
61	C₄²×C₄₇	—	2	2	no	—
62	C₄²×C₄₇	—	2	2	no	—
63	C₄²×C₄₇	—	2	2	no	—
64	C₄²×C₄₇	—	2	2	no	—
65	C₄²×C₄₇	—	2	2	no	—
66	C₄²×C₄₇	—	2	2	no	—
67	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
68	C₄²×C₄₇	#41	1	2	yes	—
69	C₄⋊Dic₄₇	—	2	4	no	—
70	C₄⋊Dic₄₇	—	2	2	no	—
71	C₄⋊Dic₄₇	—	2	4	no	—
72	C₄⋊Dic₄₇	—	4	2	no	—
73	C₄⋊Dic₄₇	—	4	2	no	—
74	C₄⋊Dic₄₇	M₁₆	1	4	yes	—
75	Dic₄₇⋊C₄	—	4	4	no	—
76	C₄×Dic₄₇	—	4	4	no	—
77	C₂×C₄₇⋊C₈	—	2	4	no	—
78	C₂×C₄₇⋊C₈	—	2	4	no	—
79	C₂×C₄₇⋊C₈	—	2	4	no	—
80	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₄₇	—	2	1	no	—
82	C₂×C₄×C₄₇	—	2	1	no	—
83	C₂×C₄×C₄₇	—	2	1	no	—
84	C₂×C₄×C₄₇	—	2	1	no	—
85	C₂×C₄×C₄₇	—	2	1	no	—
86	C₂×C₄×C₄₇	—	2	1	no	—
87	C₂×C₄×C₄₇	—	2	1	no	—
88	C₂×C₄×C₄₇	—	2	1	no	—
89	C₂×C₄×C₄₇	—	2	1	no	—
90	C₂×C₄×C₄₇	—	2	1	no	—
91	C₂×C₄×C₄₇	—	2	1	no	—
92	C₂×C₄×C₄₇	—	2	1	no	—
93	C₂×C₄×C₄₇	#104	1	2	yes	—
94	C₂×C₄×C₄₇	#104	1	2	yes	—
95	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
96	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
97	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
98	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
99	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
100	C₂×C₄×C₄₇	#131	1	2	yes	—
101	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
102	C₂×C₄×C₄₇	#137	1	2	yes	—
103	C₂×Dic₄₇	—	2	4	no	—
104	C₂×Dic₄₇	—	4	2	no	—
105	C₂×Dic₄₇	—	4	2	no	—
106	C₂×Dic₄₇	—	4	2	no	—
107	C₂×Dic₄₇	—	4	2	no	—
108	C₂×Dic₄₇	—	4	2	no	—
109	Dic₉₄	—	4	8	no	—
110	C₈×C₄₇	—	8	2	no	—
111	C₄₇⋊C₈	—	2	8	no	—
112	#9877	—	47	1	no	maximal, Sylow
113	C₂²×C₄₇	#391	1	1	yes	characteristic
114	C₂²×C₄₇	#407	1	1	yes	characteristic
115	C₂²×C₄₇	#408	1	1	yes	characteristic
116	C₂²×C₄₇	#418	1	1	yes	characteristic
117	C₂²×C₄₇	#419	1	1	yes	characteristic
118	C₂²×C₄₇	C₂×#104	1	1	yes	characteristic
119	C₂²×C₄₇	C₂×#104	1	1	yes	characteristic
120	C₄×C₄₇	—	2	2	no	—
121	C₄×C₄₇	—	2	2	no	—
122	C₄×C₄₇	—	2	2	no	—
123	C₄×C₄₇	—	4	1	no	—
124	C₄×C₄₇	—	4	1	no	—
125	C₄×C₄₇	#446	1	4	yes	—
126	C₄×C₄₇	#489	1	2	yes	—
127	C₄×C₄₇	#501	1	2	yes	—
128	Dic₄₇	—	4	4	no	—
129	Dic₄₇	—	8	2	no	—
130	C₄×C₄⋊Q₈	—	47	1	no	—
131	#1831	D₄₇	1	1	yes	characteristic
132	C₄×#97	—	47	1	no	—
133	#1279	—	47	4	no	—
134	C₂×C₄₇	#1484	1	1	yes	characteristic
135	C₂×C₄₇	#1485	1	1	yes	characteristic
136	C₂×C₄₇	#1490	1	1	yes	characteristic
137	C₂×C₄₇	#1491	1	1	yes	characteristic
138	C₂×C₄₇	#1985	1	1	yes	characteristic
139	C₂×C₄₇	#1993	1	1	yes	characteristic
140	C₂×C₄₇	#1994	1	1	yes	characteristic
141	C₂×C₄⋊Q₈	—	47	1	no	—
142	C₂×C₄×Q₈	—	94	1	no	—
143	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
144	C₂×#97	—	94	1	no	—
145	C₂×#97	—	94	1	no	—
146	C₂×C₄×C₈	—	47	2	no	—
147	#391	—	94	1	no	—
148	#389	—	47	2	no	—
149	#389	—	94	1	no	—
150	#384	—	47	1	no	—
151	C₄×C₄⋊C₄	—	47	2	no	—
152	C₄×C₄⋊C₄	—	94	1	no	—
153	C₄³	C₂×D₄₇	1	1	yes	characteristic
154	#336	—	2	1	no	—
155	#336	—	2	1	no	—
156	C₄₇	#9877	1	1	yes	characteristic, complemented, Sylow
157	C₂²×Q₈	—	94	1	no	—
158	C₂²×C₈	—	2	2	no	—
159	C₂²×C₈	—	94	2	no	—
160	C₄⋊Q₈	—	94	4	no	—
161	C₄×Q₈	—	188	4	no	—
162	C₂×C₄⋊C₄	—	47	2	no	—
163	C₂×C₄⋊C₄	—	47	2	no	—
164	C₂×C₄⋊C₄	—	94	1	no	—
165	C₂×C₄⋊C₄	—	94	2	no	—
166	C₂×C₄⋊C₄	—	94	1	no	—
167	C₂×C₄⋊C₄	—	94	1	no	—
168	C₂×C₄²	—	94	2	no	—
169	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
170	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
171	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
172	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
173	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
174	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
175	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
176	C₄⋊C₈	—	2	4	no	—
177	#97	—	94	4	no	—
178	#97	—	94	4	no	—
179	C₄×C₈	—	47	4	no	—
180	C₄×C₈	—	94	2	no	—
181	#95	—	94	2	no	—
182	#95	—	94	2	no	—
183	C₂×Q₈	—	94	4	no	—
184	C₂×Q₈	—	188	2	no	—
185	C₂×Q₈	—	188	2	no	—
186	C₂²×C₄	—	94	2	no	—
187	C₂²×C₄	—	94	2	no	—
188	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
189	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
190	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
191	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
192	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
193	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
194	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
195	C₂×C₈	—	4	2	no	—
196	C₂×C₈	—	4	2	no	—
197	C₂×C₈	—	4	2	no	—
198	C₂×C₈	—	94	4	no	—
199	C₂×C₈	—	94	4	no	—
200	C₂×C₈	—	94	4	no	—
201	C₄⋊C₄	—	47	4	no	—
202	C₄⋊C₄	—	94	4	no	—
203	C₄⋊C₄	—	94	2	no	—
204	C₄⋊C₄	—	94	4	no	—
205	C₄⋊C₄	—	188	4	no	—
206	C₄⋊C₄	—	188	2	no	—
207	C₄⋊C₄	—	188	2	no	—
208	C₄²	—	2	2	no	—
209	C₄²	—	2	2	no	—
210	C₄²	—	2	2	no	—
211	C₄²	—	2	2	no	—
212	C₄²	—	2	2	no	—
213	C₄²	—	2	2	no	—
214	C₄²	—	188	4	no	—
215	C₄²	#10625352	1	2	yes	—
216	C₄²	Q₈×D₄₇	1	2	yes	—
217	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
218	Q₈	—	188	8	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	94	4	no	—
232	C₂×C₄	—	188	2	no	—
233	C₂×C₄	—	188	2	no	—
234	C₂×C₄	—	188	2	no	—
235	C₂×C₄	—	188	2	no	—
236	C₂×C₄	—	188	2	no	—
237	C₂×C₄	#13093950	1	2	yes	—
238	C₂×C₄	#13093950	1	2	yes	—
239	C₂×C₄	#13093989	1	2	yes	—
240	C₂×C₄	#13094004	1	2	yes	—
241	C₂×C₄	#13094011	1	1	yes	characteristic
242	C₂×C₄	#13094022	1	1	yes	characteristic
243	C₂×C₄	M₁₆×D₄₇	1	2	yes	—
244	C₂×C₄	#13094030	1	2	yes	—
245	C₂×C₄	#13094049	1	2	yes	—
246	C₂×C₄	#13094054	1	2	yes	—
247	C₈	—	8	2	no	—
248	C₈	—	94	8	no	—
249	C₂²	#15689028	1	1	yes	characteristic
250	C₂²	#15689086	1	1	yes	characteristic
251	C₂²	#15689089	1	1	yes	characteristic
252	C₂²	#15689127	1	1	yes	characteristic
253	C₂²	#15689131	1	1	yes	characteristic
254	C₂²	C₂×#13093950	1	1	yes	characteristic
255	C₂²	C₂×#13093950	1	1	yes	characteristic
256	C₄	—	2	2	no	—
257	C₄	—	2	2	no	—
258	C₄	—	2	2	no	—
259	C₄	—	4	1	no	—
260	C₄	—	4	1	no	—
261	C₄	—	376	2	no	—
262	C₄	—	188	4	no	—
263	C₄	#15689252	1	4	yes	—
264	C₄	#15689482	1	2	yes	—
265	C₄	#15689533	1	2	yes	—
266	C₂	#20241804	1	1	yes	characteristic
267	C₂	#20241808	1	1	yes	characteristic
268	C₂	#20241830	1	1	yes	characteristic
269	C₂	#20241834	1	1	yes	characteristic
270	C₂	#20244327	1	1	yes	characteristic
271	C₂	#20244360	1	1	yes	characteristic
272	C₂	#20244364	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₄₇
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1831	C₂
socle, prime radical	C₂³×C₄₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1831	D₄₇
hypocenter	C₄₇	#9877
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1831

Complements

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

N	H ≅ G/N	type
34. C₄×C₄₇⋊C₈	Q₈	semidirect
156. C₄₇	#9877	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13094024	2
C₄×#13093935	2
#20241227	2
C₄₇×#1831	2
#9877	47