Subgroups — #26105855

Order 12032.

686 conjugacy classes of subgroups.

The subgroup lattice has 288 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×#15689176	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	#15689180	—	2	2	no	—
6	#15689180	C₄	1	4	yes	complemented
7	#15689176	—	2	2	no	—
8	#15689176	C₄	1	4	yes	complemented
9	C₂×C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
10	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
11	C₂×C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
12	C₂×C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
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	C₂²×C₄₇⋊C₈	—	2	2	no	—
24	C₂²×C₄₇⋊C₈	—	2	2	no	—
25	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
26	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
27	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
28	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
29	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
30	#13093935	—	2	4	no	—
31	C₄×C₄₇⋊C₈	—	2	2	no	—
32	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
33	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
34	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
35	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
36	#117×C₄₇	—	2	8	no	—
37	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
38	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
39	C₂×C₄²×C₄₇	—	2	2	no	—
40	C₂×C₄²×C₄₇	—	2	2	no	—
41	C₂×C₄²×C₄₇	—	2	2	no	—
42	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₄₇	D₄	1	2	yes	—
48	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
49	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
50	#95×C₄₇	—	2	2	no	—
51	#95×C₄₇	—	2	2	no	—
52	C₂⁴×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	—	2	4	no	—
54	C₂²×C₄×C₄₇	—	2	4	no	—
55	C₂²×C₄×C₄₇	—	2	4	no	—
56	C₂²×C₄×C₄₇	—	4	2	no	—
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	2	yes	—
62	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
64	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
65	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
66	C₂²×C₄×C₄₇	C₂²×C₄	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	C₄⋊C₄×C₄₇	—	2	4	no	—
70	C₄⋊C₄×C₄₇	M₁₆	1	8	yes	—
71	C₂²⋊C₄×C₄₇	—	2	8	no	—
72	C₄²×C₄₇	—	2	8	no	—
73	C₄²×C₄₇	—	4	4	no	—
74	C₄²×C₄₇	—	4	4	no	—
75	C₄²×C₄₇	C₄²	1	2	yes	—
76	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
77	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
78	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
79	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
80	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
81	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
85	C₂×C₄₇⋊C₈	—	2	4	no	—
86	C₂×C₄₇⋊C₈	—	2	4	no	—
87	C₂×C₄₇⋊C₈	—	4	4	no	—
88	C₂×C₄₇⋊C₈	—	4	4	no	—
89	C₂³×C₄₇	—	2	2	no	—
90	C₂³×C₄₇	—	2	2	no	—
91	C₂³×C₄₇	—	2	2	no	—
92	C₂³×C₄₇	#95	1	2	yes	—
93	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₄₇	—	2	2	no	—
95	C₂×C₄×C₄₇	—	2	2	no	—
96	C₂×C₄×C₄₇	—	2	2	no	—
97	C₂×C₄×C₄₇	—	2	2	no	—
98	C₂×C₄×C₄₇	—	2	4	no	—
99	C₂×C₄×C₄₇	—	2	4	no	—
100	C₂×C₄×C₄₇	—	2	4	no	—
101	C₂×C₄×C₄₇	—	2	4	no	—
102	C₂×C₄×C₄₇	—	2	8	no	—
103	C₂×C₄×C₄₇	—	4	2	no	—
104	C₂×C₄×C₄₇	—	4	2	no	—
105	C₂×C₄×C₄₇	—	4	4	no	—
106	C₂×C₄×C₄₇	—	4	4	no	—
107	C₂×C₄×C₄₇	#95	1	2	yes	—
108	C₂×C₄×C₄₇	#95	1	2	yes	—
109	C₂×C₄×C₄₇	#95	1	2	yes	—
110	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
111	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₄₇	C₂×M₁₆	1	4	yes	—
117	C₂×C₄×C₄₇	#135	1	4	yes	—
118	C₄₇⋊C₈	—	4	4	no	—
119	C₄₇⋊C₈	—	4	4	no	—
120	#7902	—	47	1	no	maximal, Sylow
121	C₂²×C₄₇	—	2	2	no	—
122	C₂²×C₄₇	—	2	2	no	—
123	C₂²×C₄₇	—	2	2	no	—
124	C₂²×C₄₇	—	4	4	no	—
125	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
126	C₂²×C₄₇	#407	1	2	yes	—
127	C₂²×C₄₇	#416	1	2	yes	—
128	C₂²×C₄₇	#421	1	2	yes	—
129	C₄×C₄₇	—	2	2	no	—
130	C₄×C₄₇	—	2	1	no	—
131	C₄×C₄₇	—	2	2	no	—
132	C₄×C₄₇	—	2	1	no	—
133	C₄×C₄₇	—	2	2	no	—
134	C₄×C₄₇	—	4	8	no	—
135	C₄×C₄₇	C₄×M₁₆	1	2	yes	—
136	C₄×C₄₇	#416	1	2	yes	—
137	C₄×C₄₇	#423	1	2	yes	—
138	C₄×C₄₇	#427	1	2	yes	—
139	C₂×#432	—	47	1	no	—
140	C₂×#431	—	47	1	no	—
141	C₄×#117	D₄₇	1	1	yes	characteristic
142	#1279	—	47	4	no	—
143	C₂×C₄₇	—	4	2	no	—
144	C₂×C₄₇	#1488	1	4	yes	—
145	C₂×C₄₇	#1728	1	1	yes	characteristic
146	C₂×C₄₇	#1733	1	1	yes	characteristic
147	C₂×C₄₇	#1735	1	1	yes	characteristic
148	C₂×#117	Dic₄₇	1	2	yes	—
149	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
150	#432	—	47	4	no	—
151	#432	—	94	2	no	—
152	#431	—	47	4	no	—
153	#431	—	94	2	no	—
154	C₂×C₄⋊C₈	—	47	2	no	—
155	C₂×C₄⋊C₈	—	94	1	no	—
156	C₂×C₂²⋊C₈	—	94	1	no	—
157	C₂×C₂²⋊C₈	—	94	1	no	—
158	C₂×#97	—	94	1	no	—
159	C₂×C₄×C₈	—	47	2	no	—
160	C₄×C₄⋊C₄	Dic₄₇	1	2	yes	—
161	C₄×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
162	C₄×C₂²⋊C₄	—	2	1	no	—
163	C₄×C₂²⋊C₄	—	2	1	no	—
164	#376	—	2	1	no	—
165	C₄³	—	2	1	no	—
166	C₄₇	#7902	1	1	yes	characteristic, complemented, Sylow
167	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
168	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
169	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
170	C₂²×C₈	—	94	2	no	—
171	C₂²×C₈	—	94	2	no	—
172	#117	—	2	8	no	—
173	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
174	C₂×C₂²⋊C₄	—	2	2	no	—
175	C₂×C₄²	—	2	2	no	—
176	C₂×C₄²	—	2	2	no	—
177	C₂×C₄²	—	2	2	no	—
178	C₂×C₄²	Dic₉₄	1	2	yes	—
179	C₂×C₄²	C₄×D₄₇	1	2	yes	—
180	C₂×C₄²	C₄×D₄₇	1	2	yes	—
181	C₂×C₄²	D₁₈₈	1	2	yes	—
182	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
183	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
184	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
185	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
186	C₄⋊C₈	—	47	4	no	—
187	C₄⋊C₈	—	94	4	no	—
188	C₄⋊C₈	—	94	2	no	—
189	C₂²⋊C₈	—	94	4	no	—
190	C₂²⋊C₈	—	94	4	no	—
191	#97	—	94	4	no	—
192	C₄×C₈	—	47	4	no	—
193	C₄×C₈	—	94	2	no	—
194	#95	—	2	2	no	—
195	#95	—	2	2	no	—
196	C₂⁴	C₂²⋊Dic₄₇	1	1	yes	characteristic
197	C₂²×C₄	—	2	4	no	—
198	C₂²×C₄	—	2	4	no	—
199	C₂²×C₄	—	2	4	no	—
200	C₂²×C₄	—	4	2	no	—
201	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
202	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
203	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
204	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
205	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
206	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
207	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
208	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
209	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
210	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
211	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
212	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
213	C₂×C₈	—	94	4	no	—
214	C₂×C₈	—	94	4	no	—
215	C₂×C₈	—	188	4	no	—
216	C₂×C₈	—	188	4	no	—
217	C₄⋊C₄	—	2	4	no	—
218	C₄⋊C₄	#10625329	1	8	yes	—
219	C₂²⋊C₄	—	2	8	no	—
220	C₄²	—	2	8	no	—
221	C₄²	—	4	4	no	—
222	C₄²	—	4	4	no	—
223	C₄²	C₄×Dic₄₇	1	2	yes	—
224	C₄²	Dic₄₇⋊C₄	1	2	yes	—
225	C₄²	Dic₄₇⋊C₄	1	2	yes	—
226	C₄²	C₄⋊Dic₄₇	1	2	yes	—
227	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
228	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
229	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
230	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
231	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
232	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
233	C₂³	—	2	2	no	—
234	C₂³	—	2	2	no	—
235	C₂³	—	2	2	no	—
236	C₂³	C₄₇⋊#95	1	2	yes	—
237	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	2	8	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	—	4	4	no	—
251	C₂×C₄	C₄₇⋊#95	1	2	yes	—
252	C₂×C₄	C₄₇⋊#95	1	2	yes	—
253	C₂×C₄	C₄₇⋊#95	1	2	yes	—
254	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
255	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
256	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
257	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
258	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
259	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
260	C₂×C₄	C₂×#10625329	1	4	yes	—
261	C₂×C₄	#13094044	1	4	yes	—
262	C₈	—	188	4	no	—
263	C₈	—	188	4	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	2	2	no	—
267	C₂²	—	4	4	no	—
268	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
269	C₂²	#15689087	1	2	yes	—
270	C₂²	#15689119	1	2	yes	—
271	C₂²	#15689140	1	2	yes	—
272	C₄	—	2	2	no	—
273	C₄	—	2	1	no	—
274	C₄	—	2	2	no	—
275	C₄	—	2	1	no	—
276	C₄	—	2	2	no	—
277	C₄	—	4	8	no	—
278	C₄	C₄×#10625329	1	2	yes	—
279	C₄	#15689117	1	2	yes	—
280	C₄	#15689147	1	2	yes	—
281	C₄	#15689159	1	2	yes	—
282	C₂	—	4	2	no	—
283	C₂	#20241822	1	4	yes	—
284	C₂	#20243082	1	1	yes	characteristic
285	C₂	#20243097	1	1	yes	characteristic
286	C₂	#20243104	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	#7902
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
6. #15689180	C₄	semidirect
8. #15689176	C₄	semidirect
166. C₄₇	#7902	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×#15689176	2
#20241229	2
C₄×#117×C₄₇	2
#7902	47