Subgroups — #26105765

Order 12032.

618 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689176	C₂	1	1	yes	maximal, characteristic
2	C₂×#15689147	C₂	1	1	yes	maximal, characteristic
3	#20241222	C₂	1	4	yes	maximal, complemented
4	C₂×C₄₇×#431	C₂	1	1	yes	maximal, characteristic
5	C₂²×#10625329	—	2	1	no	—
6	#15689176	—	2	2	no	—
7	#15689176	—	2	2	no	—
8	#15689147	—	2	2	no	—
9	#15689147	C₄	1	4	yes	—
10	C₂×C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
11	C₂×C₄⋊(C₄₇⋊C₈)	C₄	1	2	yes	—
12	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
13	C₂×C₂²⋊(C₄₇⋊C₈)	C₄	1	2	yes	—
14	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
15	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
16	C₄₇×#431	—	2	2	no	—
17	C₄₇×#431	C₄	1	4	yes	—
18	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
19	C₂×C₂²⋊C₈×C₄₇	—	2	1	no	—
20	C₂×C₄×C₈×C₄₇	—	2	1	no	—
21	C₂×#10625329	—	2	4	no	—
22	C₂×#10625329	—	4	4	no	—
23	C₂²×C₄₇⋊C₈	—	2	2	no	—
24	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
25	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
26	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
27	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
28	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
29	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
30	C₄×C₄₇⋊C₈	—	2	2	no	—
31	C₄×C₄₇⋊C₈	—	2	2	no	—
32	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
33	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
34	C₂³×C₄×C₄₇	Q₈	1	1	yes	characteristic
35	C₂²×C₈×C₄₇	—	2	2	no	—
36	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₄₇	D₄	1	2	yes	—
41	C₂×C₄²×C₄₇	D₄	1	2	yes	—
42	C₂×C₄²×C₄₇	Q₈	1	1	yes	characteristic
43	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
44	C₄⋊C₈×C₄₇	—	2	2	no	—
45	C₄⋊C₈×C₄₇	C₂×C₄	1	4	yes	—
46	C₂²⋊C₈×C₄₇	—	2	4	no	—
47	C₄×C₈×C₄₇	—	2	4	no	—
48	C₂⁴×C₄₇	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₄₇	—	2	4	no	—
50	C₂²×C₄×C₄₇	—	2	4	no	—
51	C₂²×C₄×C₄₇	—	2	2	no	—
52	C₂²×C₄×C₄₇	—	2	2	no	—
53	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
54	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
55	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
60	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
61	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
65	C₂×C₈×C₄₇	—	2	4	no	—
66	C₂×C₈×C₄₇	—	4	2	no	—
67	C₂×C₈×C₄₇	—	4	2	no	—
68	C₄²×C₄₇	C₄²	1	2	yes	—
69	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
70	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
71	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
72	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
73	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
74	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
75	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
76	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
78	#10625329	—	4	8	no	—
79	C₂×C₄₇⋊C₈	—	2	4	no	—
80	C₂×C₄₇⋊C₈	—	2	4	no	—
81	C₂×C₄₇⋊C₈	—	2	8	no	—
82	C₂×C₄₇⋊C₈	—	4	4	no	—
83	C₂³×C₄₇	—	2	2	no	—
84	C₂³×C₄₇	—	2	2	no	—
85	C₂³×C₄₇	—	2	2	no	—
86	C₂³×C₄₇	#95	1	2	yes	—
87	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₄₇	—	2	2	no	—
89	C₂×C₄×C₄₇	—	2	2	no	—
90	C₂×C₄×C₄₇	—	2	2	no	—
91	C₂×C₄×C₄₇	—	2	4	no	—
92	C₂×C₄×C₄₇	—	2	4	no	—
93	C₂×C₄×C₄₇	—	2	4	no	—
94	C₂×C₄×C₄₇	—	2	4	no	—
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	#95	1	2	yes	—
99	C₂×C₄×C₄₇	#95	1	2	yes	—
100	C₂×C₄×C₄₇	#95	1	2	yes	—
101	C₂×C₄×C₄₇	#108	1	4	yes	—
102	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
103	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₄₇	#131	1	4	yes	—
109	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
110	C₂×C₄×C₄₇	#137	1	2	yes	—
111	C₈×C₄₇	—	4	4	no	—
112	C₄₇⋊C₈	—	4	4	no	—
113	C₄₇⋊C₈	—	4	4	no	—
114	#7880	—	47	1	no	maximal, Sylow
115	C₂²×C₄₇	—	2	2	no	—
116	C₂²×C₄₇	—	2	2	no	—
117	C₂²×C₄₇	—	2	2	no	—
118	C₂²×C₄₇	—	4	4	no	—
119	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
120	C₂²×C₄₇	#424	1	2	yes	—
121	C₂²×C₄₇	#428	1	2	yes	—
122	C₂²×C₄₇	C₂×#108	1	2	yes	—
123	C₄×C₄₇	—	2	2	no	—
124	C₄×C₄₇	—	2	2	no	—
125	C₄×C₄₇	—	2	1	no	—
126	C₄×C₄₇	—	2	1	no	—
127	C₄×C₄₇	#341	1	4	yes	—
128	C₄×C₄₇	#405	1	2	yes	—
129	C₄×C₄₇	#408	1	2	yes	—
130	C₄×C₄₇	#417	1	2	yes	—
131	C₄×C₄₇	#418	1	1	yes	characteristic
132	C₄×C₄₇	#419	1	1	yes	characteristic
133	C₂×#431	—	47	1	no	—
134	C₂×#431	D₄₇	1	1	yes	characteristic
135	C₂×#423	—	47	1	no	—
136	#1277	—	47	4	no	—
137	C₂×C₄₇	—	4	2	no	—
138	C₂×C₄₇	#1574	1	4	yes	—
139	C₂×C₄₇	#1729	1	1	yes	characteristic
140	C₂×C₄₇	#1736	1	1	yes	characteristic
141	C₂×C₄₇	C₂×#341	1	1	yes	characteristic
142	C₂²×M₁₆	—	94	1	no	—
143	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
144	#431	—	2	2	no	—
145	#431	—	94	2	no	—
146	#431	—	94	2	no	—
147	#431	Dic₄₇	1	4	yes	—
148	#423	—	47	4	no	—
149	#423	—	94	2	no	—
150	C₂×C₄⋊C₈	—	47	2	no	—
151	C₂×C₄⋊C₈	—	47	2	no	—
152	C₂×C₄⋊C₈	—	94	1	no	—
153	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
154	C₂×C₂²⋊C₈	—	2	1	no	—
155	C₂×C₂²⋊C₈	—	47	2	no	—
156	C₂×C₄×C₈	—	2	1	no	—
157	C₂×C₄×C₈	—	47	2	no	—
158	C₄₇	#7880	1	1	yes	characteristic, complemented, Sylow
159	C₂³×C₄	Dic₉₄	1	1	yes	characteristic
160	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
161	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
162	C₂×M₁₆	—	94	4	no	—
163	C₂×M₁₆	—	188	4	no	—
164	C₂²×C₈	—	2	2	no	—
165	C₂²×C₈	—	47	4	no	—
166	C₂²×C₈	—	94	2	no	—
167	C₂×C₄²	Dic₉₄	1	1	yes	characteristic
168	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
169	C₂×C₄²	C₄×D₄₇	1	2	yes	—
170	C₂×C₄²	D₁₈₈	1	2	yes	—
171	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
172	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
173	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
174	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
175	C₄⋊C₈	—	2	2	no	—
176	C₄⋊C₈	—	47	4	no	—
177	C₄⋊C₈	—	94	4	no	—
178	C₄⋊C₈	—	94	4	no	—
179	C₄⋊C₈	—	94	2	no	—
180	C₄⋊C₈	C₂×Dic₄₇	1	4	yes	—
181	C₂²⋊C₈	—	2	4	no	—
182	C₂²⋊C₈	—	94	4	no	—
183	C₄×C₈	—	2	4	no	—
184	C₄×C₈	—	94	2	no	—
185	C₄×C₈	—	94	2	no	—
186	C₂⁴	Dic₄₇⋊C₄	1	1	yes	characteristic
187	C₂²×C₄	—	2	4	no	—
188	C₂²×C₄	—	2	4	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
192	C₂²×C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
193	C₂²×C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
194	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
195	C₂²×C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
196	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
197	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
198	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
199	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
200	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
201	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
202	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
203	M₁₆	—	188	8	no	—
204	C₂×C₈	—	2	4	no	—
205	C₂×C₈	—	4	2	no	—
206	C₂×C₈	—	4	2	no	—
207	C₂×C₈	—	94	4	no	—
208	C₂×C₈	—	94	4	no	—
209	C₂×C₈	—	94	8	no	—
210	C₂×C₈	—	188	4	no	—
211	C₄²	C₄×Dic₄₇	1	2	yes	—
212	C₄²	Dic₄₇⋊C₄	1	2	yes	—
213	C₄²	C₄⋊Dic₄₇	1	2	yes	—
214	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
215	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
216	C₄²	C₂²⋊Dic₄₇	1	2	yes	—
217	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
218	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
219	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
220	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
221	C₂³	—	2	2	no	—
222	C₂³	—	2	2	no	—
223	C₂³	—	2	2	no	—
224	C₂³	C₄₇⋊#95	1	2	yes	—
225	C₂³	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	C₄₇⋊#95	1	2	yes	—
237	C₂×C₄	C₄₇⋊#95	1	2	yes	—
238	C₂×C₄	C₄₇⋊#95	1	2	yes	—
239	C₂×C₄	#13093959	1	4	yes	—
240	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
241	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
242	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
243	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
244	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
245	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
246	C₂×C₄	#13094030	1	4	yes	—
247	C₂×C₄	#13094049	1	2	yes	—
248	C₂×C₄	#13094054	1	2	yes	—
249	C₈	—	4	4	no	—
250	C₈	—	188	4	no	—
251	C₈	—	188	4	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	2	2	no	—
255	C₂²	—	4	4	no	—
256	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
257	C₂²	#15689149	1	2	yes	—
258	C₂²	#15689166	1	2	yes	—
259	C₂²	C₂×#13093959	1	2	yes	—
260	C₄	—	2	2	no	—
261	C₄	—	2	2	no	—
262	C₄	—	2	1	no	—
263	C₄	—	2	1	no	—
264	C₄	#15688887	1	4	yes	—
265	C₄	#15689078	1	2	yes	—
266	C₄	#15689089	1	2	yes	—
267	C₄	#15689124	1	2	yes	—
268	C₄	#15689127	1	1	yes	characteristic
269	C₄	#15689131	1	1	yes	characteristic
270	C₂	—	4	2	no	—
271	C₂	#20242176	1	4	yes	—
272	C₂	#20243084	1	1	yes	characteristic
273	C₂	#20243109	1	1	yes	characteristic
274	C₂	C₂×#15688887	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	C₂×C₄₇×#431	C₂
socle	C₂³×C₄₇	C₂×C₄⋊C₄
prime radical	C₂⁴×C₄₇	C₄⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂×#431	D₄₇
hypocenter	C₄₇	#7880
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
3. #20241222	C₂	semidirect
158. C₄₇	#7880	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689176	2
C₂×#15689147	2
#20241222	2
C₂×C₄₇×#431	2
#7880	47