Subgroups — #26214251

Order 12032.

1110 conjugacy classes of subgroups.

The subgroup lattice has 278 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×#15689078	C₂	1	2	yes	maximal
3	#20245795	C₂	1	1	yes	maximal, characteristic
4	#20243260	C₂	1	8	yes	maximal, complemented
5	#20243084	C₂	1	2	yes	maximal
6	C₂²×#97×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×#10625329	C₂²	1	2	yes	—
8	C₂×#13093983	C₂²	1	2	yes	—
9	#15689147	C₄	1	8	yes	complemented
10	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
11	#15689078	—	2	8	no	—
12	C₂×#13093934	C₂²	1	4	yes	—
13	C₂×C₈×Dic₄₇	C₂²	1	4	yes	—
14	#15688993	C₄	1	2	yes	—
15	#15688989	C₄	1	2	yes	—
16	#15688983	C₄	1	4	yes	—
17	C₄×C₄⋊Dic₄₇	C₂²	1	4	yes	—
18	#15688877	C₂²	1	8	yes	—
19	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
20	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
21	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
22	C₂×#97×C₄₇	C₄	1	8	yes	complemented
23	C₂×#97×C₄₇	C₂²	1	4	yes	—
24	C₂×#10625329	—	2	4	no	—
25	C₂×#10625329	—	2	4	no	—
26	C₂×#10625329	C₂×C₄	1	8	yes	—
27	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
28	#13093983	—	2	4	no	—
29	#13093983	—	2	4	no	—
30	C₂×C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
31	C₂×C₄⋊Dic₄₇	C₂×C₄	1	8	yes	—
32	C₂×C₂²⋊Dic₄₇	C₂×C₄	1	4	yes	—
33	C₂×C₄×Dic₄₇	C₂³	1	4	yes	—
34	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	16	yes	—
35	#13093934	—	2	8	no	—
36	C₈×Dic₄₇	—	2	8	no	—
37	C₄₇⋊#95	C₂×C₄	1	8	yes	—
38	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
39	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₄₇	—	2	2	no	—
41	C₂²×C₈×C₄₇	—	2	2	no	—
42	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
43	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
44	C₂²×C₈×C₄₇	D₄	1	4	yes	—
45	C₂²×C₈×C₄₇	Q₈	1	4	yes	—
46	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
47	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
48	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
51	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
52	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
53	#97×C₄₇	C₂×C₄	1	16	yes	—
54	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	—	4	4	no	—
56	C₂²×C₄×C₄₇	C₄²	1	8	yes	—
57	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
58	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
62	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
64	C₂²×C₄×C₄₇	#41	1	2	yes	—
65	C₂²×C₄×C₄₇	#41	1	2	yes	—
66	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
67	C₂²×Dic₄₇	C₂²×C₄	1	8	yes	—
68	C₂×C₈×C₄₇	—	2	8	no	—
69	C₂×C₈×C₄₇	—	2	2	no	—
70	C₂×C₈×C₄₇	—	2	2	no	—
71	C₂×C₈×C₄₇	C₄⋊C₄	1	16	yes	complemented
72	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
73	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
74	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
75	C₂×C₈×C₄₇	C₂×Q₈	1	4	yes	—
76	C₄²×C₄₇	C₄²	1	4	yes	—
77	C₄²×C₄₇	C₄²	1	4	yes	—
78	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
79	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
80	C₂²⋊Dic₄₇	—	2	8	no	—
81	C₄⋊Dic₄₇	—	2	8	no	—
82	C₄⋊Dic₄₇	—	4	8	no	—
83	C₄×Dic₄₇	—	2	4	no	—
84	C₄×Dic₄₇	—	2	4	no	—
85	#10625329	—	2	16	no	—
86	C₂×C₄₇⋊C₈	—	2	4	no	—
87	C₂×C₄₇⋊C₈	—	2	4	no	—
88	C₂×C₄₇⋊C₈	C₂²×C₄	1	8	yes	—
89	C₂³×C₄₇	—	2	4	no	—
90	C₂³×C₄₇	C₂×C₄²	1	2	yes	—
91	C₂³×C₄₇	C₄×D₄	1	2	yes	—
92	C₂³×C₄₇	C₄×Q₈	1	2	yes	—
93	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
94	C₂×C₄×C₄₇	—	2	4	no	—
95	C₂×C₄×C₄₇	—	2	4	no	—
96	C₂×C₄×C₄₇	—	4	8	no	—
97	C₂×C₄×C₄₇	C₂×C₄²	1	4	yes	—
98	C₂×C₄×C₄₇	C₂×C₄²	1	4	yes	—
99	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
100	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
101	C₂×C₄×C₄₇	#117	1	4	yes	—
102	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
103	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
104	C₂×C₄×C₄₇	C₄×Q₈	1	4	yes	—
105	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
106	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
107	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
108	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
110	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
111	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
112	C₂×Dic₄₇	—	2	8	no	—
113	C₂×Dic₄₇	—	2	16	no	—
114	C₈×C₄₇	—	2	4	no	—
115	C₈×C₄₇	C₂×C₄⋊C₄	1	8	yes	—
116	C₄₇⋊C₈	—	2	8	no	—
117	#20438	—	47	1	no	maximal, Sylow
118	C₂²×C₄₇	—	2	4	no	—
119	C₂²×C₄₇	—	2	4	no	—
120	C₂²×C₄₇	—	2	4	no	—
121	C₂²×C₄₇	C₄×C₄⋊C₄	1	4	yes	—
122	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
123	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
124	C₂²×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
125	C₂²×C₄₇	#568	1	4	yes	—
126	C₄×C₄₇	—	2	4	no	—
127	C₄×C₄₇	—	2	4	no	—
128	C₄×C₄₇	C₄×C₄⋊C₄	1	4	yes	—
129	C₄×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
130	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
131	C₄×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
132	C₄×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
133	Dic₄₇	—	4	8	no	—
134	C₂×#423	—	47	1	no	—
135	C₂×#405	—	47	2	no	—
136	C₂²×#97	D₄₇	1	1	yes	characteristic
137	#2285	—	47	1	no	—
138	#1771	—	47	8	no	—
139	#1729	—	47	2	no	—
140	C₂×C₄₇	—	2	4	no	—
141	C₂×C₄₇	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
142	C₂×C₄₇	#2873	1	2	yes	—
143	C₂×C₄₇	#2932	1	2	yes	—
144	C₂×C₄₇	#2961	1	2	yes	—
145	C₂²×M₁₆	—	47	2	no	—
146	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
147	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
148	C₂×#117	—	47	2	no	—
149	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
150	#423	—	47	8	no	—
151	C₂×C₄⋊C₈	—	47	4	no	—
152	#405	—	94	8	no	—
153	C₂×#97	—	47	4	no	—
154	C₂×#97	Dic₄₇	1	8	yes	complemented
155	C₂×#97	C₂×D₄₇	1	4	yes	—
156	C₂×C₄×C₈	—	47	4	no	—
157	#384	—	47	2	no	—
158	#382	—	47	2	no	—
159	#380	—	47	4	no	—
160	C₄×C₄⋊C₄	—	47	4	no	—
161	#336	—	47	8	no	—
162	C₄₇	#20438	1	1	yes	characteristic, complemented, Sylow
163	C₂³×C₄	C₄×D₄₇	1	2	yes	—
164	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
165	C₂×M₁₆	—	47	8	no	—
166	C₂×M₁₆	—	94	4	no	—
167	C₂×M₁₆	—	94	4	no	—
168	C₂²×C₈	—	2	2	no	—
169	C₂²×C₈	—	2	2	no	—
170	C₂²×C₈	—	47	4	no	—
171	C₂²×C₈	Dic₉₄	1	4	yes	—
172	C₂²×C₈	D₁₈₈	1	4	yes	—
173	C₂²×C₈	C₂×Dic₄₇	1	4	yes	—
174	C₂²×C₈	C₂×Dic₄₇	1	4	yes	—
175	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
176	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
177	#117	—	94	4	no	—
178	#117	—	94	4	no	—
179	C₂×C₄⋊C₄	—	47	4	no	—
180	C₂×C₄⋊C₄	—	47	8	no	—
181	C₂×C₂²⋊C₄	—	47	4	no	—
182	C₂×C₄²	—	47	4	no	—
183	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
184	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
185	C₂×C₄²	C₄×D₄₇	1	4	yes	—
186	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
187	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
188	C₄⋊C₈	—	47	16	no	—
189	#97	—	94	8	no	—
190	#97	C₂×Dic₄₇	1	16	yes	—
191	C₄×C₈	—	94	8	no	—
192	#95	—	47	8	no	—
193	C₂⁴	C₂×C₄×D₄₇	1	1	yes	characteristic
194	C₂²×C₄	—	4	4	no	—
195	C₂²×C₄	—	47	8	no	—
196	C₂²×C₄	C₄×Dic₄₇	1	8	yes	—
197	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
198	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
199	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
200	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
201	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
202	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
203	C₂²×C₄	#10625350	1	2	yes	—
204	C₂²×C₄	#10625350	1	2	yes	—
205	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
206	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
207	M₁₆	—	94	16	no	—
208	C₂×C₈	—	2	8	no	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	47	8	no	—
212	C₂×C₈	—	94	4	no	—
213	C₂×C₈	—	94	4	no	—
214	C₂×C₈	C₄⋊Dic₄₇	1	16	yes	complemented
215	C₂×C₈	C₂×Dic₉₄	1	4	yes	—
216	C₂×C₈	C₂×D₁₈₈	1	4	yes	—
217	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
218	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
219	C₄⋊C₄	—	94	8	no	—
220	C₄⋊C₄	—	188	8	no	—
221	C₂²⋊C₄	—	94	8	no	—
222	C₄²	—	94	4	no	—
223	C₄²	—	94	4	no	—
224	C₄²	C₄×Dic₄₇	1	4	yes	—
225	C₄²	C₄×Dic₄₇	1	4	yes	—
226	C₄²	C₂×C₄×D₄₇	1	4	yes	—
227	C₄²	C₂²×Dic₄₇	1	4	yes	—
228	C₂³	—	2	4	no	—
229	C₂³	C₂×C₄×Dic₄₇	1	2	yes	—
230	C₂³	C₄×D₁₈₈	1	2	yes	—
231	C₂³	C₄×Dic₉₄	1	2	yes	—
232	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	4	8	no	—
236	C₂×C₄	—	94	8	no	—
237	C₂×C₄	—	94	16	no	—
238	C₂×C₄	C₂×C₄×Dic₄₇	1	4	yes	—
239	C₂×C₄	C₂×C₄×Dic₄₇	1	4	yes	—
240	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
241	C₂×C₄	C₂×C₄⋊Dic₄₇	1	4	yes	—
242	C₂×C₄	#13093983	1	4	yes	—
243	C₂×C₄	C₄×D₁₈₈	1	2	yes	—
244	C₂×C₄	C₄×D₁₈₈	1	4	yes	—
245	C₂×C₄	C₄×Dic₉₄	1	2	yes	—
246	C₂×C₄	C₄×Dic₉₄	1	4	yes	—
247	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
248	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
249	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
250	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
251	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
252	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
253	C₈	—	2	4	no	—
254	C₈	—	94	8	no	—
255	C₈	C₂×C₄⋊Dic₄₇	1	8	yes	—
256	C₂²	—	2	4	no	—
257	C₂²	—	2	4	no	—
258	C₂²	—	2	4	no	—
259	C₂²	C₄×C₄⋊Dic₄₇	1	4	yes	—
260	C₂²	C₂²×C₄×Dic₄₇	1	1	yes	characteristic
261	C₂²	C₂×C₄×D₁₈₈	1	1	yes	characteristic
262	C₂²	C₂×C₄×Dic₉₄	1	1	yes	characteristic
263	C₂²	#15689889	1	4	yes	—
264	C₄	—	2	4	no	—
265	C₄	—	2	4	no	—
266	C₄	—	188	8	no	—
267	C₄	C₄×C₄⋊Dic₄₇	1	4	yes	—
268	C₄	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
269	C₄	C₂×#13093983	1	1	yes	characteristic
270	C₄	C₂×C₄×D₁₈₈	1	1	yes	characteristic
271	C₄	C₂×C₄×Dic₉₄	1	1	yes	characteristic
272	C₂	—	2	4	no	—
273	C₂	C₂×C₄×C₄⋊Dic₄₇	1	1	yes	characteristic
274	C₂	#20251806	1	2	yes	—
275	C₂	#20252107	1	2	yes	—
276	C₂	#20252328	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂²×C₄²
Fitting	C₂²×#97×C₄₇	C₂
socle	C₂³×C₄₇	C₂³×C₄
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂²×#97	D₄₇
hypocenter	C₄₇	#20438
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#97

Complements

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

N	H ≅ G/N	type
4. #20243260	C₂	semidirect
9. #15689147	C₄	semidirect
22. C₂×#97×C₄₇	C₄	semidirect
71. C₂×C₈×C₄₇	C₄⋊C₄	semidirect
154. C₂×#97	Dic₄₇	semidirect
162. C₄₇	#20438	semidirect
214. C₂×C₈	C₄⋊Dic₄₇	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×#15689078	2
#20245795	2
#20243260	2
#20243084	2
C₂²×#97×C₄₇	2
#20438	47