Subgroups — #26105880

Order 12032.

710 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689180	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688969	C₂	1	1	yes	maximal, characteristic
3	#20241230	C₂	1	4	yes	maximal
4	C₂×C₄₇×#432	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄×Dic₄₇	C₄	1	2	yes	—
6	#15689180	—	2	2	no	—
7	#15689180	C₄	1	4	yes	complemented
8	C₂×C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
9	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
10	C₂×#13093935	—	2	1	no	—
11	#15688969	—	2	1	no	—
12	#15688969	—	2	1	no	—
13	#15688969	—	2	1	no	—
14	#15688969	—	2	1	no	—
15	C₂×C₄₇⋊#95	C₄	1	2	yes	—
16	C₂×C₄₇⋊#95	C₂²	1	2	yes	—
17	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
18	C₄₇×#432	—	2	2	no	—
19	C₄₇×#432	C₄	1	4	yes	complemented
20	C₂×C₄⋊C₈×C₄₇	—	2	1	no	—
21	C₂×C₂²⋊C₈×C₄₇	C₂²	1	2	yes	—
22	C₂×#97×C₄₇	—	2	1	no	—
23	C₂³×Dic₄₇	C₂×C₄	1	4	yes	—
24	C₂²×C₄₇⋊C₈	—	2	2	no	—
25	C₂×C₄×Dic₄₇	—	2	2	no	—
26	C₂×C₄×Dic₄₇	—	2	2	no	—
27	C₂×C₄×Dic₄₇	—	2	8	no	—
28	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
29	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
30	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
31	#13093935	—	2	4	no	—
32	C₄₇⋊#95	—	2	2	no	—
33	C₄₇⋊#95	—	2	2	no	—
34	C₄₇⋊#95	—	2	2	no	—
35	C₄₇⋊#95	—	2	2	no	—
36	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
37	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
38	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
39	C₂²×C₈×C₄₇	—	2	2	no	—
40	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₄₇	D₄	1	2	yes	—
43	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
48	C₄⋊C₈×C₄₇	—	2	4	no	—
49	C₂²⋊C₈×C₄₇	—	2	2	no	—
50	C₂²⋊C₈×C₄₇	C₂×C₄	1	4	yes	—
51	#97×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	—	4	2	no	—
56	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
57	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
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	C₄⋊C₄	1	2	yes	—
63	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
64	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
70	C₂²×Dic₄₇	—	2	4	no	—
71	C₂²×Dic₄₇	—	2	4	no	—
72	C₂²×Dic₄₇	—	4	4	no	—
73	C₂²×Dic₄₇	—	4	4	no	—
74	C₂²×Dic₄₇	M₁₆	1	8	yes	—
75	C₂×C₈×C₄₇	—	2	4	no	—
76	C₂×C₈×C₄₇	—	4	4	no	—
77	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
78	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
79	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
80	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
81	C₄²×C₄₇	C₄⋊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	C₄×Dic₄₇	—	2	8	no	—
86	C₄×Dic₄₇	—	2	8	no	—
87	C₂×C₄₇⋊C₈	—	2	4	no	—
88	C₂×C₄₇⋊C₈	—	4	4	no	—
89	C₂³×C₄₇	—	2	2	no	—
90	C₂³×C₄₇	—	2	2	no	—
91	C₂³×C₄₇	C₂×C₄²	1	1	yes	characteristic
92	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂³×C₄₇	C₂×M₁₆	1	4	yes	—
95	C₂×C₄×C₄₇	—	2	4	no	—
96	C₂×C₄×C₄₇	—	2	2	no	—
97	C₂×C₄×C₄₇	—	2	2	no	—
98	C₂×C₄×C₄₇	—	2	2	no	—
99	C₂×C₄×C₄₇	—	2	2	no	—
100	C₂×C₄×C₄₇	—	2	2	no	—
101	C₂×C₄×C₄₇	—	2	2	no	—
102	C₂×C₄×C₄₇	—	2	4	no	—
103	C₂×C₄×C₄₇	—	4	2	no	—
104	C₂×C₄×C₄₇	—	4	2	no	—
105	C₂×C₄×C₄₇	—	4	1	no	—
106	C₂×C₄×C₄₇	—	4	1	no	—
107	C₂×C₄×C₄₇	—	4	1	no	—
108	C₂×C₄×C₄₇	—	4	1	no	—
109	C₂×C₄×C₄₇	#95	1	2	yes	—
110	C₂×C₄×C₄₇	#95	1	2	yes	—
111	C₂×C₄×C₄₇	#95	1	2	yes	—
112	C₂×C₄×C₄₇	#95	1	2	yes	—
113	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂×Dic₄₇	—	2	8	no	—
118	C₂×Dic₄₇	—	4	8	no	—
119	C₂×Dic₄₇	—	4	4	no	—
120	C₂×Dic₄₇	—	4	4	no	—
121	C₂×Dic₄₇	—	4	8	no	—
122	C₈×C₄₇	—	4	4	no	—
123	C₄₇⋊C₈	—	4	4	no	—
124	#7910	—	47	1	no	maximal, Sylow
125	C₂²×C₄₇	—	2	4	no	—
126	C₂²×C₄₇	—	2	4	no	—
127	C₂²×C₄₇	—	2	1	no	—
128	C₂²×C₄₇	—	2	1	no	—
129	C₂²×C₄₇	—	2	1	no	—
130	C₂²×C₄₇	—	2	1	no	—
131	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
132	C₂²×C₄₇	C₄×M₁₆	1	2	yes	—
133	C₂²×C₄₇	#407	1	2	yes	—
134	C₂²×C₄₇	#410	1	2	yes	—
135	C₂²×C₄₇	#413	1	2	yes	—
136	C₂²×C₄₇	#423	1	2	yes	—
137	C₄×C₄₇	—	2	2	no	—
138	C₄×C₄₇	—	2	2	no	—
139	C₄×C₄₇	—	2	1	no	—
140	C₄×C₄₇	—	2	1	no	—
141	C₄×C₄₇	—	2	1	no	—
142	C₄×C₄₇	—	2	1	no	—
143	C₄×C₄₇	—	2	2	no	—
144	C₄×C₄₇	—	2	2	no	—
145	Dic₄₇	—	4	8	no	—
146	C₂×#432	—	47	1	no	—
147	C₂×#432	D₄₇	1	1	yes	characteristic
148	C₂×#376	—	47	1	no	—
149	#1280	—	47	4	no	—
150	C₂×C₄₇	—	2	2	no	—
151	C₂×C₄₇	—	2	2	no	—
152	C₂×C₄₇	#1464	1	4	yes	—
153	C₂×C₄₇	#1728	1	1	yes	characteristic
154	C₂×C₄₇	#1740	1	1	yes	characteristic
155	C₂×C₄₇	#1742	1	1	yes	characteristic
156	C₂²×C₄²	—	47	2	no	—
157	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
158	#432	—	2	2	no	—
159	#432	—	47	4	no	—
160	#432	—	94	2	no	—
161	#432	Dic₄₇	1	4	yes	complemented
162	C₂×C₄⋊C₈	—	2	1	no	—
163	C₂×C₄⋊C₈	—	94	1	no	—
164	C₂×C₂²⋊C₈	—	47	2	no	—
165	C₂×C₂²⋊C₈	C₂×D₄₇	1	2	yes	—
166	C₂×#97	—	2	1	no	—
167	C₂×#97	—	94	1	no	—
168	#376	—	94	1	no	—
169	#376	—	94	1	no	—
170	#376	—	94	1	no	—
171	#376	—	94	1	no	—
172	C₂×#95	—	47	2	no	—
173	C₂×#95	—	47	2	no	—
174	C₄₇	#7910	1	1	yes	characteristic, complemented, Sylow
175	C₂³×C₄	—	47	4	no	—
176	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
177	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
178	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
179	C₂²×C₈	—	2	2	no	—
180	C₂²×C₈	—	94	2	no	—
181	C₂×C₄²	—	94	8	no	—
182	C₂×C₄²	—	94	2	no	—
183	C₂×C₄²	—	94	2	no	—
184	C₂×C₄²	Dic₉₄	1	2	yes	—
185	C₂×C₄²	C₄×D₄₇	1	2	yes	—
186	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
187	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
188	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
189	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
190	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
191	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
192	C₄⋊C₈	—	2	4	no	—
193	C₄⋊C₈	—	94	4	no	—
194	C₂²⋊C₈	—	2	2	no	—
195	C₂²⋊C₈	—	47	4	no	—
196	C₂²⋊C₈	—	94	2	no	—
197	C₂²⋊C₈	C₂×Dic₄₇	1	4	yes	—
198	#97	—	2	4	no	—
199	#97	—	94	4	no	—
200	#95	—	94	2	no	—
201	#95	—	94	2	no	—
202	#95	—	94	2	no	—
203	#95	—	94	2	no	—
204	C₂⁴	C₂×C₄×D₄₇	1	1	yes	characteristic
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	4	2	no	—
207	C₂²×C₄	—	4	2	no	—
208	C₂²×C₄	—	47	8	no	—
209	C₂²×C₄	—	94	4	no	—
210	C₂²×C₄	—	94	4	no	—
211	C₂²×C₄	—	188	4	no	—
212	C₂²×C₄	—	188	4	no	—
213	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
214	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
215	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
216	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
217	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
218	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
219	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
220	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
221	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
222	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
223	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
224	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
225	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
226	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
227	C₂×C₈	—	2	4	no	—
228	C₂×C₈	—	4	4	no	—
229	C₂×C₈	—	94	4	no	—
230	C₂×C₈	—	188	4	no	—
231	C₄²	—	94	8	no	—
232	C₄²	—	94	8	no	—
233	C₄²	Dic₄₇⋊C₄	1	2	yes	—
234	C₄²	Dic₄₇⋊C₄	1	2	yes	—
235	C₄²	C₄⋊Dic₄₇	1	2	yes	—
236	C₄²	C₄⋊Dic₄₇	1	2	yes	—
237	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
238	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
239	C₄²	C₂²⋊Dic₄₇	1	2	yes	—
240	C₄²	C₂²⋊Dic₄₇	1	2	yes	—
241	C₂³	—	2	2	no	—
242	C₂³	—	2	2	no	—
243	C₂³	C₂×C₄×Dic₄₇	1	1	yes	characteristic
244	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
245	C₂³	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
246	C₂³	M₁₆×D₄₇	1	4	yes	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	1	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	—	94	8	no	—
262	C₂×C₄	—	188	8	no	—
263	C₂×C₄	—	188	4	no	—
264	C₂×C₄	—	188	4	no	—
265	C₂×C₄	—	188	8	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	C₄₇⋊#95	1	2	yes	—
270	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
271	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
272	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
273	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
274	C₈	—	4	4	no	—
275	C₈	—	188	4	no	—
276	C₂²	—	2	4	no	—
277	C₂²	—	2	4	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	2	1	no	—
280	C₂²	—	2	1	no	—
281	C₂²	—	2	1	no	—
282	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
283	C₂²	M₁₆×Dic₄₇	1	2	yes	—
284	C₂²	#15689086	1	2	yes	—
285	C₂²	#15689096	1	2	yes	—
286	C₂²	#15689105	1	2	yes	—
287	C₂²	#15689146	1	2	yes	—
288	C₄	—	2	2	no	—
289	C₄	—	2	2	no	—
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	2	1	no	—
293	C₄	—	2	1	no	—
294	C₄	—	2	2	no	—
295	C₄	—	2	2	no	—
296	C₄	—	188	8	no	—
297	C₂	—	2	2	no	—
298	C₂	—	2	2	no	—
299	C₂	#20241720	1	4	yes	—
300	C₂	#20243081	1	1	yes	characteristic
301	C₂	#20243120	1	1	yes	characteristic
302	C₂	#20243126	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₄₇)⋊C₄
commutator	C₂³×C₄₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₂×C₄₇×#432	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂×#432	D₄₇
hypocenter	C₄₇	#7910
LCS	G ▷ C₂³×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×#432

Complements

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

N	H ≅ G/N	type
7. #15689180	C₄	semidirect
19. C₄₇×#432	C₄	semidirect
161. #432	Dic₄₇	semidirect
174. C₄₇	#7910	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689180	2
C₂×#15688969	2
#20241230	2
C₂×C₄₇×#432	2
#7910	47