Subgroups — #26105859

Order 12032.

794 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689147	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#10625329	C₂	1	1	yes	maximal, characteristic
3	#20241229	C₂	1	4	yes	maximal, complemented
4	C₄×#117×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#10625329	—	2	1	no	—
6	C₂²×#10625329	—	2	1	no	—
7	#15689147	—	2	2	no	—
8	#15689147	C₄	1	4	yes	complemented
9	C₂×C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
10	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
11	C₄×#10625329	—	2	2	no	—
12	C₄×#10625329	C₄	1	4	yes	complemented
13	C₂×#13093935	—	2	1	no	—
14	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
15	C₂×#117×C₄₇	C₄	1	2	yes	—
16	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
17	C₄×C₄⋊C₄×C₄₇	C₄	1	2	yes	—
18	C₄×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
19	C₄×C₂²⋊C₄×C₄₇	—	2	1	no	—
20	C₄×C₂²⋊C₄×C₄₇	—	2	1	no	—
21	C₄₇×#376	—	2	1	no	—
22	C₄³×C₄₇	—	2	1	no	—
23	C₂×#10625329	—	2	8	no	—
24	C₂×#10625329	—	2	4	no	—
25	C₂×#10625329	—	2	4	no	—
26	C₂×#10625329	—	4	4	no	—
27	C₂²×C₄₇⋊C₈	—	2	2	no	—
28	C₂²×C₄₇⋊C₈	—	2	2	no	—
29	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
30	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
31	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
32	#13093935	—	2	4	no	—
33	C₄×C₄₇⋊C₈	—	2	2	no	—
34	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
35	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
36	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
37	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
38	#117×C₄₇	—	2	8	no	—
39	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
40	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
41	C₂×C₄²×C₄₇	—	2	2	no	—
42	C₂×C₄²×C₄₇	—	2	2	no	—
43	C₂×C₄²×C₄₇	—	2	2	no	—
44	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₄₇	D₄	1	2	yes	—
50	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
51	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
52	#95×C₄₇	—	2	2	no	—
53	#95×C₄₇	—	2	2	no	—
54	C₂⁴×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	—	2	4	no	—
56	C₂²×C₄×C₄₇	—	2	4	no	—
57	C₂²×C₄×C₄₇	—	2	2	no	—
58	C₂²×C₄×C₄₇	—	2	2	no	—
59	C₂²×C₄×C₄₇	—	2	4	no	—
60	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
61	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
65	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
67	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
68	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
69	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
72	C₄⋊C₄×C₄₇	—	2	4	no	—
73	C₄⋊C₄×C₄₇	M₁₆	1	8	yes	complemented
74	C₂²⋊C₄×C₄₇	—	2	8	no	—
75	C₄²×C₄₇	—	2	8	no	—
76	C₄²×C₄₇	—	4	4	no	—
77	C₄²×C₄₇	—	4	2	no	—
78	C₄²×C₄₇	—	4	2	no	—
79	C₄²×C₄₇	C₄²	1	2	yes	—
80	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
81	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
82	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
83	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
84	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
85	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
89	#10625329	—	2	16	no	—
90	#10625329	—	4	8	no	—
91	C₂×C₄₇⋊C₈	—	2	8	no	—
92	C₂×C₄₇⋊C₈	—	2	4	no	—
93	C₂×C₄₇⋊C₈	—	2	4	no	—
94	C₂×C₄₇⋊C₈	—	4	4	no	—
95	C₂³×C₄₇	—	2	2	no	—
96	C₂³×C₄₇	—	2	2	no	—
97	C₂³×C₄₇	—	2	2	no	—
98	C₂³×C₄₇	#95	1	2	yes	—
99	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₄₇	—	2	2	no	—
101	C₂×C₄×C₄₇	—	2	2	no	—
102	C₂×C₄×C₄₇	—	2	2	no	—
103	C₂×C₄×C₄₇	—	2	4	no	—
104	C₂×C₄×C₄₇	—	2	4	no	—
105	C₂×C₄×C₄₇	—	2	4	no	—
106	C₂×C₄×C₄₇	—	2	2	no	—
107	C₂×C₄×C₄₇	—	2	2	no	—
108	C₂×C₄×C₄₇	—	2	4	no	—
109	C₂×C₄×C₄₇	—	2	4	no	—
110	C₂×C₄×C₄₇	—	2	8	no	—
111	C₂×C₄×C₄₇	—	4	4	no	—
112	C₂×C₄×C₄₇	—	4	4	no	—
113	C₂×C₄×C₄₇	#95	1	2	yes	—
114	C₂×C₄×C₄₇	#95	1	2	yes	—
115	C₂×C₄×C₄₇	#95	1	2	yes	—
116	C₂×C₄×C₄₇	#104	1	4	yes	—
117	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
118	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₄₇	C₂×M₁₆	1	4	yes	—
124	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
125	C₂×C₄×C₄₇	#137	1	2	yes	—
126	C₄₇⋊C₈	—	2	8	no	—
127	C₄₇⋊C₈	—	4	4	no	—
128	#7903	—	47	1	no	maximal, Sylow
129	C₂²×C₄₇	—	2	2	no	—
130	C₂²×C₄₇	—	2	2	no	—
131	C₂²×C₄₇	—	2	2	no	—
132	C₂²×C₄₇	—	4	4	no	—
133	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
134	C₂²×C₄₇	#407	1	2	yes	—
135	C₂²×C₄₇	#417	1	2	yes	—
136	C₂²×C₄₇	C₂×#104	1	2	yes	—
137	C₄×C₄₇	—	2	2	no	—
138	C₄×C₄₇	—	2	1	no	—
139	C₄×C₄₇	—	2	1	no	—
140	C₄×C₄₇	—	2	2	no	—
141	C₄×C₄₇	—	4	8	no	—
142	C₄×C₄₇	#339	1	4	yes	—
143	C₄×C₄₇	C₄×M₁₆	1	2	yes	—
144	C₄×C₄₇	#418	1	1	yes	characteristic
145	C₄×C₄₇	#419	1	1	yes	characteristic
146	C₄×C₄₇	#423	1	2	yes	—
147	C₄×C₄₇	#428	1	2	yes	—
148	C₂×#423	—	47	1	no	—
149	C₂×C₄×M₁₆	—	47	1	no	—
150	C₄×#117	D₄₇	1	1	yes	characteristic
151	#1279	—	47	4	no	—
152	C₂×C₄₇	—	4	2	no	—
153	C₂×C₄₇	#1489	1	4	yes	—
154	C₂×C₄₇	#1728	1	1	yes	characteristic
155	C₂×C₄₇	C₂×#339	1	1	yes	characteristic
156	C₂×C₄₇	#1736	1	1	yes	characteristic
157	C₂²×M₁₆	—	94	1	no	—
158	C₂²×M₁₆	—	94	1	no	—
159	C₂×#117	Dic₄₇	1	2	yes	—
160	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
161	#423	—	47	4	no	—
162	#423	—	94	2	no	—
163	C₂×C₄⋊C₈	—	47	2	no	—
164	C₂×C₄⋊C₈	—	94	1	no	—
165	C₄×M₁₆	—	47	4	no	—
166	C₄×M₁₆	—	94	2	no	—
167	C₂×#97	—	94	1	no	—
168	C₂×C₄×C₈	—	47	2	no	—
169	C₄×C₄⋊C₄	Dic₄₇	1	2	yes	—
170	C₄×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
171	C₄×C₂²⋊C₄	—	2	1	no	—
172	C₄×C₂²⋊C₄	—	2	1	no	—
173	#376	—	2	1	no	—
174	C₄³	—	2	1	no	—
175	C₄₇	#7903	1	1	yes	characteristic, complemented, Sylow
176	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
177	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
178	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
179	C₂×M₁₆	—	94	8	no	—
180	C₂×M₁₆	—	94	4	no	—
181	C₂×M₁₆	—	94	4	no	—
182	C₂×M₁₆	—	188	4	no	—
183	C₂²×C₈	—	94	2	no	—
184	C₂²×C₈	—	94	2	no	—
185	#117	—	2	8	no	—
186	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
187	C₂×C₂²⋊C₄	—	2	2	no	—
188	C₂×C₄²	—	2	2	no	—
189	C₂×C₄²	—	2	2	no	—
190	C₂×C₄²	—	2	2	no	—
191	C₂×C₄²	Dic₉₄	1	2	yes	—
192	C₂×C₄²	C₄×D₄₇	1	2	yes	—
193	C₂×C₄²	C₄×D₄₇	1	2	yes	—
194	C₂×C₄²	D₁₈₈	1	2	yes	—
195	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
196	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
197	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
198	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
199	C₄⋊C₈	—	47	4	no	—
200	C₄⋊C₈	—	94	4	no	—
201	C₄⋊C₈	—	94	2	no	—
202	#97	—	94	4	no	—
203	C₄×C₈	—	47	4	no	—
204	C₄×C₈	—	94	2	no	—
205	#95	—	2	2	no	—
206	#95	—	2	2	no	—
207	C₂⁴	C₂²⋊Dic₄₇	1	1	yes	characteristic
208	C₂²×C₄	—	2	4	no	—
209	C₂²×C₄	—	2	4	no	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	2	4	no	—
213	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
214	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
215	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
216	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
217	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
218	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
219	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
220	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
221	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
222	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
223	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
224	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
225	M₁₆	—	94	16	no	—
226	M₁₆	—	188	8	no	—
227	C₂×C₈	—	94	8	no	—
228	C₂×C₈	—	94	4	no	—
229	C₂×C₈	—	94	4	no	—
230	C₂×C₈	—	188	4	no	—
231	C₄⋊C₄	—	2	4	no	—
232	C₄⋊C₄	#10625329	1	8	yes	complemented
233	C₂²⋊C₄	—	2	8	no	—
234	C₄²	—	2	8	no	—
235	C₄²	—	4	4	no	—
236	C₄²	—	4	2	no	—
237	C₄²	—	4	2	no	—
238	C₄²	C₄×Dic₄₇	1	2	yes	—
239	C₄²	Dic₄₇⋊C₄	1	2	yes	—
240	C₄²	Dic₄₇⋊C₄	1	2	yes	—
241	C₄²	C₄⋊Dic₄₇	1	2	yes	—
242	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
243	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
244	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
245	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
246	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
247	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
248	C₂³	—	2	2	no	—
249	C₂³	—	2	2	no	—
250	C₂³	—	2	2	no	—
251	C₂³	C₄₇⋊#95	1	2	yes	—
252	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	8	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	C₄₇⋊#95	1	2	yes	—
267	C₂×C₄	C₄₇⋊#95	1	2	yes	—
268	C₂×C₄	C₄₇⋊#95	1	2	yes	—
269	C₂×C₄	#13093951	1	4	yes	—
270	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
271	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
272	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
273	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
274	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
275	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
276	C₂×C₄	C₂×#10625329	1	4	yes	—
277	C₂×C₄	#13094049	1	2	yes	—
278	C₂×C₄	#13094054	1	2	yes	—
279	C₈	—	94	8	no	—
280	C₈	—	188	4	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	2	2	no	—
284	C₂²	—	4	4	no	—
285	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
286	C₂²	#15689087	1	2	yes	—
287	C₂²	#15689124	1	2	yes	—
288	C₂²	C₂×#13093951	1	2	yes	—
289	C₄	—	2	2	no	—
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	2	2	no	—
293	C₄	—	4	8	no	—
294	C₄	#15688882	1	4	yes	—
295	C₄	C₄×#10625329	1	2	yes	—
296	C₄	#15689127	1	1	yes	characteristic
297	C₄	#15689131	1	1	yes	characteristic
298	C₄	#15689147	1	2	yes	—
299	C₄	#15689166	1	2	yes	—
300	C₂	—	4	2	no	—
301	C₂	#20241827	1	4	yes	—
302	C₂	#20243082	1	1	yes	characteristic
303	C₂	C₂×#15688882	1	1	yes	characteristic
304	C₂	#20243109	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄×#117×C₄₇	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₄×#117	D₄₇
hypocenter	C₄₇	#7903
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₄×#117

Complements

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

N	H ≅ G/N	type
3. #20241229	C₂	semidirect
8. #15689147	C₄	semidirect
12. C₄×#10625329	C₄	semidirect
73. C₄⋊C₄×C₄₇	M₁₆	semidirect
175. C₄₇	#7903	semidirect
232. C₄⋊C₄	#10625329	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689147	2
C₂×C₄×#10625329	2
#20241229	2
C₄×#117×C₄₇	2
#7903	47