Subgroups — #26121431

Order 12032.

1046 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093935	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688878	C₂	1	4	yes	maximal
4	C₂×C₄₇×#376	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₄₇⋊C₈	C₂²	1	2	yes	—
6	C₂³×C₄₇⋊C₈	C₂²	1	2	yes	—
7	C₂×#13093935	—	2	2	no	—
8	C₂×#13093935	—	2	2	no	—
9	C₂×#13093935	C₄	1	4	yes	complemented
10	C₂×C₄×C₄₇⋊C₈	—	2	2	no	—
11	C₂×C₄×C₄₇⋊C₈	—	2	2	no	—
12	C₂×C₄×C₄₇⋊C₈	C₄	1	4	yes	complemented
13	#15688878	C₄	1	16	yes	complemented
14	C₂²×C₄²×C₄₇	C₂²	1	2	yes	—
15	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
16	C₄₇×#376	—	2	2	no	—
17	C₄₇×#376	—	2	2	no	—
18	C₂×#95×C₄₇	C₄	1	4	yes	—
19	C₂²×C₄₇⋊C₈	—	2	2	no	—
20	C₂²×C₄₇⋊C₈	—	2	2	no	—
21	C₂²×C₄₇⋊C₈	—	2	2	no	—
22	C₂²×C₄₇⋊C₈	—	2	2	no	—
23	C₂²×C₄₇⋊C₈	—	2	4	no	—
24	C₂²×C₄₇⋊C₈	—	2	4	no	—
25	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
26	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
27	C₂²×C₄₇⋊C₈	C₂×C₄	1	8	yes	—
28	C₂²×C₄₇⋊C₈	C₂×C₄	1	8	yes	—
29	#13093935	—	2	8	no	—
30	C₄×C₄₇⋊C₈	—	2	8	no	—
31	C₂³×C₄×C₄₇	C₂×C₄	1	4	yes	—
32	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
33	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₄₇	—	2	2	no	—
35	C₂×C₄²×C₄₇	—	2	2	no	—
36	C₂×C₄²×C₄₇	—	2	4	no	—
37	C₂×C₄²×C₄₇	C₂×C₄	1	8	yes	—
38	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₄₇	D₄	1	2	yes	—
41	C₂×C₄²×C₄₇	D₄	1	2	yes	—
42	C₂×C₄²×C₄₇	D₄	1	2	yes	—
43	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
44	#95×C₄₇	—	2	8	no	—
45	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₄₇	—	2	8	no	—
47	C₂²×C₄×C₄₇	—	2	4	no	—
48	C₂²×C₄×C₄₇	—	2	4	no	—
49	C₂²×C₄×C₄₇	—	2	16	no	—
50	C₂²×C₄×C₄₇	C₄²	1	8	yes	—
51	C₂²×C₄×C₄₇	C₄²	1	4	yes	—
52	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
53	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
54	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
57	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	#41	1	4	yes	—
62	C₄²×C₄₇	—	2	8	no	—
63	C₄²×C₄₇	—	2	8	no	—
64	C₄²×C₄₇	C₂²⋊C₄	1	4	yes	—
65	C₄²×C₄₇	C₂²⋊C₄	1	4	yes	—
66	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
67	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
68	C₂×C₄₇⋊C₈	—	2	4	no	—
69	C₂×C₄₇⋊C₈	—	2	4	no	—
70	C₂×C₄₇⋊C₈	—	2	4	no	—
71	C₂×C₄₇⋊C₈	—	2	4	no	—
72	C₂×C₄₇⋊C₈	—	2	8	no	—
73	C₂×C₄₇⋊C₈	—	2	8	no	—
74	C₂×C₄₇⋊C₈	—	2	8	no	—
75	C₂×C₄₇⋊C₈	—	2	8	no	—
76	C₂×C₄₇⋊C₈	C₄²	1	8	yes	—
77	C₂×C₄₇⋊C₈	C₄²	1	8	yes	—
78	C₂³×C₄₇	C₂×C₄²	1	1	yes	characteristic
79	C₂³×C₄₇	C₂×C₄²	1	2	yes	—
80	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂³×C₄₇	#117	1	2	yes	—
83	C₂³×C₄₇	#131	1	8	yes	—
84	C₂×C₄×C₄₇	—	2	8	no	—
85	C₂×C₄×C₄₇	—	2	8	no	—
86	C₂×C₄×C₄₇	—	2	8	no	—
87	C₂×C₄×C₄₇	—	2	8	no	—
88	C₂×C₄×C₄₇	—	2	16	no	—
89	C₂×C₄×C₄₇	—	4	8	no	—
90	C₂×C₄×C₄₇	—	4	8	no	—
91	C₂×C₄×C₄₇	#95	1	8	yes	—
92	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
93	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
94	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₄₇	#117	1	2	yes	—
99	C₂×C₄×C₄₇	#117	1	2	yes	—
100	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
101	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
102	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
103	C₂×C₄×C₄₇	C₄×Q₈	1	4	yes	—
104	C₄₇⋊C₈	—	2	8	no	—
105	C₄₇⋊C₈	—	2	8	no	—
106	#10694	—	47	1	no	maximal, Sylow
107	C₂²×C₄₇	C₄³	1	1	yes	characteristic
108	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
109	C₂²×C₄₇	#376	1	1	yes	characteristic
110	C₂²×C₄₇	C₄×C₂²⋊C₄	1	2	yes	—
111	C₂²×C₄₇	C₄×C₄⋊C₄	1	2	yes	—
112	C₂²×C₄₇	#405	1	4	yes	—
113	C₂²×C₄₇	#405	1	8	yes	—
114	C₂²×C₄₇	#408	1	4	yes	—
115	C₂²×C₄₇	#424	1	4	yes	—
116	C₂²×C₄₇	#433	1	8	yes	—
117	C₄×C₄₇	—	2	8	no	—
118	C₄×C₄₇	—	4	8	no	—
119	C₄×C₄₇	C₄×C₂²⋊C₄	1	2	yes	—
120	C₄×C₄₇	C₄×C₂²⋊C₄	1	2	yes	—
121	C₄×C₄₇	C₄×C₄⋊C₄	1	2	yes	—
122	C₄×C₄₇	C₄×C₄⋊C₄	1	2	yes	—
123	C₂²×#97	—	47	1	no	—
124	C₂²×C₄×C₈	—	47	1	no	—
125	C₂×#376	D₄₇	1	1	yes	characteristic
126	C₂×#336	—	47	4	no	—
127	C₂×C₄₇	C₄×#95	1	1	yes	characteristic
128	C₂×C₄₇	#1726	1	2	yes	—
129	C₂×C₄₇	#1729	1	2	yes	—
130	C₂×C₄₇	#1746	1	2	yes	—
131	C₂×C₄₇	#1750	1	4	yes	—
132	C₂×C₄₇	#1768	1	4	yes	—
133	C₂³×C₈	—	47	2	no	—
134	C₂³×C₈	—	47	2	no	—
135	C₂²×C₄²	C₂×D₄₇	1	2	yes	—
136	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
137	C₂×#97	—	47	4	no	—
138	C₂×#97	—	94	2	no	—
139	C₂×#97	—	94	2	no	—
140	C₂×C₄×C₈	—	47	4	no	—
141	C₂×C₄×C₈	—	94	2	no	—
142	C₂×C₄×C₈	—	94	2	no	—
143	#376	—	2	2	no	—
144	#376	—	2	2	no	—
145	C₂×#95	Dic₄₇	1	4	yes	—
146	#336	—	47	16	no	—
147	C₄₇	#10694	1	1	yes	characteristic, complemented, Sylow
148	C₂³×C₄	C₂×Dic₄₇	1	4	yes	—
149	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
150	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
151	C₂²×C₈	—	47	8	no	—
152	C₂²×C₈	—	47	8	no	—
153	C₂²×C₈	—	47	4	no	—
154	C₂²×C₈	—	47	4	no	—
155	C₂²×C₈	—	94	2	no	—
156	C₂²×C₈	—	94	2	no	—
157	C₂²×C₈	—	94	4	no	—
158	C₂²×C₈	—	94	4	no	—
159	C₂²×C₈	—	94	2	no	—
160	C₂²×C₈	—	94	2	no	—
161	C₂×C₄²	—	2	4	no	—
162	C₂×C₄²	—	2	2	no	—
163	C₂×C₄²	—	2	2	no	—
164	C₂×C₄²	Dic₉₄	1	2	yes	—
165	C₂×C₄²	C₄×D₄₇	1	8	yes	—
166	C₂×C₄²	C₄×D₄₇	1	2	yes	—
167	C₂×C₄²	C₄×D₄₇	1	2	yes	—
168	C₂×C₄²	D₁₈₈	1	2	yes	—
169	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
170	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
171	#97	—	94	8	no	—
172	C₄×C₈	—	94	8	no	—
173	#95	—	2	8	no	—
174	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
175	C₂²×C₄	—	2	8	no	—
176	C₂²×C₄	—	2	4	no	—
177	C₂²×C₄	—	2	4	no	—
178	C₂²×C₄	—	2	16	no	—
179	C₂²×C₄	C₄×Dic₄₇	1	8	yes	—
180	C₂²×C₄	C₄×Dic₄₇	1	4	yes	—
181	C₂²×C₄	C₄⋊Dic₄₇	1	4	yes	—
182	C₂²×C₄	C₂²⋊Dic₄₇	1	4	yes	—
183	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
184	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
185	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
186	C₂²×C₄	C₂×C₄×D₄₇	1	4	yes	—
187	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
188	C₂²×C₄	#10625350	1	4	yes	—
189	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
190	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
191	C₂×C₈	—	47	8	no	—
192	C₂×C₈	—	47	8	no	—
193	C₂×C₈	—	94	4	no	—
194	C₂×C₈	—	94	4	no	—
195	C₂×C₈	—	94	4	no	—
196	C₂×C₈	—	94	4	no	—
197	C₂×C₈	—	94	8	no	—
198	C₂×C₈	—	94	8	no	—
199	C₂×C₈	—	94	8	no	—
200	C₂×C₈	—	94	8	no	—
201	C₄²	—	2	8	no	—
202	C₄²	—	2	8	no	—
203	C₄²	Dic₄₇⋊C₄	1	4	yes	—
204	C₄²	Dic₄₇⋊C₄	1	4	yes	—
205	C₄²	(C₂×D₄₇)⋊C₄	1	4	yes	—
206	C₄²	(C₂×D₄₇)⋊C₄	1	4	yes	—
207	C₂³	C₂×C₄×Dic₄₇	1	1	yes	characteristic
208	C₂³	C₂×C₄×Dic₄₇	1	2	yes	—
209	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
210	C₂³	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
211	C₂³	#13093983	1	2	yes	—
212	C₂³	#13094031	1	8	yes	—
213	C₂×C₄	—	2	8	no	—
214	C₂×C₄	—	2	8	no	—
215	C₂×C₄	—	2	8	no	—
216	C₂×C₄	—	2	8	no	—
217	C₂×C₄	—	2	16	no	—
218	C₂×C₄	—	4	8	no	—
219	C₂×C₄	—	4	8	no	—
220	C₂×C₄	C₄₇⋊#95	1	8	yes	—
221	C₂×C₄	C₄²×D₄₇	1	2	yes	—
222	C₂×C₄	C₄²×D₄₇	1	2	yes	—
223	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
224	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
225	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
226	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
227	C₂×C₄	#13093981	1	2	yes	—
228	C₂×C₄	#13093981	1	2	yes	—
229	C₂×C₄	C₄×C₂²⋊D₄₇	1	4	yes	—
230	C₂×C₄	C₄×C₂²⋊D₄₇	1	4	yes	—
231	C₂×C₄	C₄×D₁₈₈	1	4	yes	—
232	C₂×C₄	C₄×Dic₉₄	1	4	yes	—
233	C₈	—	94	8	no	—
234	C₈	—	94	8	no	—
235	C₂²	C₄²×Dic₄₇	1	1	yes	characteristic
236	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
237	C₂²	#15688969	1	1	yes	characteristic
238	C₂²	C₄×C₂²⋊Dic₄₇	1	2	yes	—
239	C₂²	C₄×C₄⋊Dic₄₇	1	2	yes	—
240	C₂²	#15689079	1	8	yes	—
241	C₂²	#15689079	1	4	yes	—
242	C₂²	#15689090	1	4	yes	—
243	C₂²	#15689150	1	4	yes	—
244	C₂²	#15689184	1	8	yes	—
245	C₄	—	2	8	no	—
246	C₄	—	4	8	no	—
247	C₄	C₄×(C₂×D₄₇)⋊C₄	1	2	yes	—
248	C₄	C₄×(C₂×D₄₇)⋊C₄	1	2	yes	—
249	C₄	C₄×Dic₄₇⋊C₄	1	2	yes	—
250	C₄	C₄×Dic₄₇⋊C₄	1	2	yes	—
251	C₂	C₄×C₄₇⋊#95	1	1	yes	characteristic
252	C₂	#20243075	1	2	yes	—
253	C₂	#20243085	1	2	yes	—
254	C₂	#20243140	1	2	yes	—
255	C₂	#20243154	1	4	yes	—
256	C₂	#20243245	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂³×C₄	C₂²×D₄₇
commutator	C₂²×C₄₇	C₄³
Fitting	C₂×C₄₇×#376	C₂
socle, prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂×#376	D₄₇
hypocenter	C₄₇	#10694
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³×C₄ ◁ C₂×#376

Complements

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

N	H ≅ G/N	type
9. C₂×#13093935	C₄	semidirect
12. C₂×C₄×C₄₇⋊C₈	C₄	semidirect
13. #15688878	C₄	semidirect
147. C₄₇	#10694	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13093935	2
C₂²×C₄×C₄₇⋊C₈	2
C₂×#15688878	2
C₂×C₄₇×#376	2
#10694	47