Subgroups — #26216804

Order 12032.

862 conjugacy classes of subgroups.

The subgroup lattice has 286 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	#20243540	C₂	1	2	yes	maximal, complemented
4	#20243505	C₂	1	2	yes	maximal, complemented
5	#20243372	C₂	1	4	yes	maximal
6	#20243245	C₂	1	4	yes	maximal, complemented
7	C₄×#117×C₄₇	C₂	1	1	yes	maximal, characteristic
8	#15689180	—	2	4	no	—
9	#15689176	—	2	4	no	—
10	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	8	yes	—
11	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
12	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
13	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
14	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
15	C₂×#13093935	C₂²	1	2	yes	—
16	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
17	#15688878	C₂²	1	8	yes	—
18	C₂×#117×C₄₇	C₂²	1	2	yes	—
19	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
20	C₄×C₄⋊C₄×C₄₇	C₄	1	2	yes	—
21	C₄×C₄⋊C₄×C₄₇	C₄	1	2	yes	—
22	C₄×C₂²⋊C₄×C₄₇	C₄	1	2	yes	—
23	C₄×C₂²⋊C₄×C₄₇	C₄	1	2	yes	—
24	C₄₇×#376	C₂²	1	2	yes	—
25	C₄³×C₄₇	C₂²	1	1	yes	characteristic
26	C₄³×C₄₇	C₂²	1	1	yes	characteristic
27	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
28	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
29	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
30	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
31	C₄⋊(C₄₇⋊C₈)	—	2	16	no	—
32	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
33	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
34	#13093935	—	2	4	no	—
35	C₄×C₄₇⋊C₈	—	2	4	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	#117×C₄₇	—	2	8	no	—
40	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
41	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
42	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₄₇	D₄	1	4	yes	—
49	C₂×C₄²×C₄₇	Q₈	1	4	yes	—
50	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
51	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
54	#95×C₄₇	C₂×C₄	1	4	yes	—
55	#95×C₄₇	C₂×C₄	1	4	yes	—
56	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	—	2	4	no	—
58	C₂²×C₄×C₄₇	—	2	4	no	—
59	C₂²×C₄×C₄₇	—	4	2	no	—
60	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
61	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
62	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
63	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
68	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
71	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
72	C₄⋊C₄×C₄₇	—	2	8	no	—
73	C₂²⋊C₄×C₄₇	—	2	8	no	—
74	C₄²×C₄₇	—	2	4	no	—
75	C₄²×C₄₇	—	2	4	no	—
76	C₄²×C₄₇	—	2	4	no	—
77	C₄²×C₄₇	—	4	2	no	—
78	C₄²×C₄₇	—	4	2	no	—
79	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
80	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
81	C₄²×C₄₇	C₂×D₄	1	2	yes	—
82	C₄²×C₄₇	C₂×D₄	1	2	yes	—
83	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
84	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
85	C₄²×C₄₇	#41	1	8	yes	—
86	C₂×C₄₇⋊C₈	—	2	4	no	—
87	C₂×C₄₇⋊C₈	—	2	4	no	—
88	C₂×C₄₇⋊C₈	—	2	8	no	—
89	C₂×C₄₇⋊C₈	—	2	8	no	—
90	C₂³×C₄₇	—	2	2	no	—
91	C₂³×C₄₇	—	2	2	no	—
92	C₂³×C₄₇	—	2	2	no	—
93	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
94	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
95	C₂×C₄×C₄₇	—	2	2	no	—
96	C₂×C₄×C₄₇	—	2	4	no	—
97	C₂×C₄×C₄₇	—	2	2	no	—
98	C₂×C₄×C₄₇	—	2	2	no	—
99	C₂×C₄×C₄₇	—	2	4	no	—
100	C₂×C₄×C₄₇	—	2	2	no	—
101	C₂×C₄×C₄₇	—	2	4	no	—
102	C₂×C₄×C₄₇	—	2	4	no	—
103	C₂×C₄×C₄₇	—	2	8	no	—
104	C₂×C₄×C₄₇	—	2	4	no	—
105	C₂×C₄×C₄₇	—	2	4	no	—
106	C₂×C₄×C₄₇	—	2	4	no	—
107	C₂×C₄×C₄₇	—	4	4	no	—
108	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
109	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
110	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
111	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
112	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
113	C₂×C₄×C₄₇	#117	1	8	yes	—
114	C₂×C₄×C₄₇	#131	1	4	yes	—
115	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
116	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
117	C₂×C₄×C₄₇	C₂×#41	1	4	yes	—
118	C₄₇⋊C₈	—	4	4	no	—
119	C₄₇⋊C₈	—	4	4	no	—
120	#20815	—	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	C₂²×C₄⋊C₄	1	1	yes	characteristic
126	C₂²×C₄₇	C₂×#117	1	2	yes	—
127	C₂²×C₄₇	C₂×#131	1	2	yes	—
128	C₂²×C₄₇	#568	1	2	yes	—
129	C₄×C₄₇	—	2	2	no	—
130	C₄×C₄₇	—	2	2	no	—
131	C₄×C₄₇	—	2	2	no	—
132	C₄×C₄₇	—	2	8	no	—
133	C₄×C₄₇	—	2	2	no	—
134	C₄×C₄₇	—	4	4	no	—
135	C₄×C₄₇	#424	1	4	yes	—
136	C₄×C₄₇	#532	1	2	yes	—
137	C₄×C₄₇	#533	1	2	yes	—
138	C₂×#432	—	47	1	no	—
139	C₂×#431	—	47	1	no	—
140	C₄×#117	D₄₇	1	1	yes	characteristic
141	#1842	—	47	2	no	—
142	#1831	—	47	2	no	—
143	#1798	—	47	4	no	—
144	#1768	—	47	4	no	—
145	C₂×C₄₇	—	4	2	no	—
146	C₂×C₄₇	#2285	1	1	yes	characteristic
147	C₂×C₄₇	C₂×#424	1	1	yes	characteristic
148	C₂×C₄₇	#2905	1	1	yes	characteristic
149	C₂×C₄₇	#2925	1	4	yes	—
150	C₂×#117	C₂×D₄₇	1	2	yes	—
151	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
152	#432	—	94	4	no	—
153	#431	—	94	4	no	—
154	C₂×C₄⋊C₈	—	47	8	no	—
155	C₂×C₄⋊C₈	—	47	2	no	—
156	C₂×C₄⋊C₈	—	47	2	no	—
157	C₂×C₂²⋊C₈	—	47	2	no	—
158	C₂×C₂²⋊C₈	—	47	2	no	—
159	C₂×#97	—	47	2	no	—
160	C₂×C₄×C₈	—	47	2	no	—
161	C₄×C₄⋊C₄	Dic₄₇	1	2	yes	—
162	C₄×C₄⋊C₄	Dic₄₇	1	2	yes	—
163	C₄×C₂²⋊C₄	Dic₄₇	1	2	yes	—
164	C₄×C₂²⋊C₄	Dic₄₇	1	2	yes	—
165	#376	C₂×D₄₇	1	2	yes	—
166	C₄³	C₂×D₄₇	1	1	yes	characteristic
167	C₄³	C₂×D₄₇	1	1	yes	characteristic
168	#336	—	47	8	no	—
169	C₄₇	#20815	1	1	yes	characteristic, complemented, Sylow
170	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
171	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
172	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
173	C₂²×C₈	—	47	4	no	—
174	C₂²×C₈	—	47	4	no	—
175	#117	—	2	8	no	—
176	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
177	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	4	yes	—
178	C₂×C₄²	Dic₉₄	1	4	yes	—
179	C₂×C₄²	D₁₈₈	1	4	yes	—
180	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
181	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
182	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
183	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
184	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
185	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
186	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
187	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
188	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
189	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
190	C₄⋊C₈	—	94	4	no	—
191	C₄⋊C₈	—	94	4	no	—
192	C₄⋊C₈	—	94	16	no	—
193	C₂²⋊C₈	—	94	4	no	—
194	C₂²⋊C₈	—	94	4	no	—
195	#97	—	94	4	no	—
196	C₄×C₈	—	94	4	no	—
197	#95	C₂×Dic₄₇	1	4	yes	—
198	#95	C₂×Dic₄₇	1	4	yes	—
199	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
200	C₂²×C₄	—	2	4	no	—
201	C₂²×C₄	—	2	4	no	—
202	C₂²×C₄	—	4	2	no	—
203	C₂²×C₄	C₄⋊Dic₄₇	1	4	yes	—
204	C₂²×C₄	C₄⋊Dic₄₇	1	4	yes	—
205	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
206	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
207	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
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	C₂²×Dic₄₇	1	1	yes	characteristic
212	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
213	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
214	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
215	C₂×C₈	—	94	4	no	—
216	C₂×C₈	—	94	4	no	—
217	C₂×C₈	—	94	8	no	—
218	C₂×C₈	—	94	8	no	—
219	C₄⋊C₄	—	2	8	no	—
220	C₂²⋊C₄	—	2	8	no	—
221	C₄²	—	2	4	no	—
222	C₄²	—	2	4	no	—
223	C₄²	—	2	4	no	—
224	C₄²	—	4	2	no	—
225	C₄²	—	4	2	no	—
226	C₄²	C₄⋊Dic₄₇	1	4	yes	—
227	C₄²	C₄⋊Dic₄₇	1	4	yes	—
228	C₄²	C₂×Dic₉₄	1	2	yes	—
229	C₄²	C₂×Dic₉₄	1	2	yes	—
230	C₄²	C₂×D₁₈₈	1	2	yes	—
231	C₄²	C₂×D₁₈₈	1	2	yes	—
232	C₄²	#10625350	1	8	yes	—
233	C₂³	—	2	2	no	—
234	C₂³	—	2	2	no	—
235	C₂³	—	2	2	no	—
236	C₂³	C₂×C₄⋊Dic₄₇	1	2	yes	—
237	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	8	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	4	4	no	—
251	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
252	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
253	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
254	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
255	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
256	C₂×C₄	#13093983	1	8	yes	—
257	C₂×C₄	#13094031	1	4	yes	—
258	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
259	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
260	C₂×C₄	C₂×#10625350	1	4	yes	—
261	C₈	—	188	4	no	—
262	C₈	—	188	4	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	4	4	no	—
267	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
268	C₂²	C₂×#13093983	1	2	yes	—
269	C₂²	C₂×#13094031	1	2	yes	—
270	C₂²	#15689890	1	2	yes	—
271	C₄	—	2	2	no	—
272	C₄	—	2	2	no	—
273	C₄	—	2	2	no	—
274	C₄	—	2	2	no	—
275	C₄	—	2	8	no	—
276	C₄	—	4	4	no	—
277	C₄	#15689150	1	4	yes	—
278	C₄	#15689673	1	2	yes	—
279	C₄	#15689680	1	2	yes	—
280	C₂	—	4	2	no	—
281	C₂	#20245795	1	1	yes	characteristic
282	C₂	C₂×#15689150	1	1	yes	characteristic
283	C₂	#20251952	1	1	yes	characteristic
284	C₂	#20252054	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂³×C₄
Fitting	C₄×#117×C₄₇	C₂
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₄×#117	D₄₇
hypocenter	C₄₇	#20815
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117

Complements

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

N	H ≅ G/N	type
3. #20243540	C₂	semidirect
4. #20243505	C₂	semidirect
6. #20243245	C₂	semidirect
169. C₄₇	#20815	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
#20243540	2
#20243505	2
#20243372	2
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C₄×#117×C₄₇	2
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