Subgroups — #26106908

Order 12032.

578 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×Dic₉₄	C₂	1	1	yes	maximal, characteristic
2	#20243134	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688853	C₂	1	2	yes	maximal
4	#20241227	C₂	1	2	yes	maximal
5	C₄₇×#1767	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₉₄	C₄	1	2	yes	—
7	C₂×C₄×Dic₉₄	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
9	C₄×Dic₄₇⋊C₄	—	2	1	no	—
10	C₄×Dic₄₇⋊C₄	—	2	2	no	—
11	C₄×C₄⋊Dic₄₇	C₄	1	2	yes	—
12	C₄×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
13	C₄²×Dic₄₇	—	2	1	no	—
14	#15688878	—	2	2	no	—
15	#15688853	C₄	1	8	yes	complemented
16	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
17	C₂×#97×C₄₇	—	2	1	no	—
18	C₄³×C₄₇	C₂²	1	1	yes	characteristic
19	C₄₇×#336	—	2	1	no	—
20	C₂²×Dic₉₄	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄₇⋊C₈	—	2	2	no	—
22	C₄×Dic₉₄	—	2	4	no	—
23	C₄×Dic₉₄	—	2	2	no	—
24	C₄×Dic₉₄	—	2	4	no	—
25	C₄×Dic₉₄	C₂×C₄	1	4	yes	—
26	C₂×C₄⋊Dic₄₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
28	C₂×Dic₄₇⋊C₄	—	2	2	no	—
29	C₂×Dic₄₇⋊C₄	—	2	1	no	—
30	C₂×C₄×Dic₄₇	—	2	2	no	—
31	C₂×C₄×Dic₄₇	—	2	1	no	—
32	C₂×C₄×Dic₄₇	—	2	2	no	—
33	C₂×C₄×Dic₄₇	—	2	1	no	—
34	C₄×C₄₇⋊C₈	—	2	2	no	—
35	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
36	C₄₇⋊#95	—	2	2	no	—
37	C₄₇⋊#95	—	2	1	no	—
38	C₂²×C₈×C₄₇	—	2	2	no	—
39	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₄₇	D₄	1	2	yes	—
42	C₂×C₄²×C₄₇	D₄	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	#97×C₄₇	—	4	2	no	—
47	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
48	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
49	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
52	C₂²×Dic₄₇	—	2	1	no	—
53	C₂²×Dic₄₇	—	2	1	no	—
54	C₂²×Dic₄₇	—	2	2	no	—
55	C₂×Dic₉₄	—	2	1	no	—
56	C₂×Dic₉₄	—	2	2	no	—
57	C₂×Dic₉₄	C₄²	1	2	yes	—
58	C₂×Dic₉₄	M₁₆	1	4	yes	—
59	C₂×C₈×C₄₇	—	2	4	no	—
60	C₂×C₈×C₄₇	—	4	2	no	—
61	C₂×C₈×C₄₇	—	4	2	no	—
62	C₄²×C₄₇	—	2	2	no	—
63	C₄²×C₄₇	—	2	2	no	—
64	C₄²×C₄₇	—	2	2	no	—
65	C₄²×C₄₇	—	4	2	no	—
66	C₄²×C₄₇	SD₁₆	1	2	yes	—
67	C₄²×C₄₇	Q₁₆	1	2	yes	—
68	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
69	C₄²×C₄₇	#41	1	2	yes	—
70	C₄⋊Dic₄₇	—	2	1	no	—
71	C₄⋊Dic₄₇	—	2	2	no	—
72	C₄⋊Dic₄₇	C₄²	1	2	yes	—
73	C₄⋊Dic₄₇	M₁₆	1	4	yes	—
74	Dic₄₇⋊C₄	—	2	8	no	—
75	Dic₄₇⋊C₄	—	2	4	no	—
76	C₄×Dic₄₇	—	2	8	no	—
77	C₄×Dic₄₇	—	2	4	no	—
78	C₄×Dic₄₇	—	4	4	no	—
79	C₄×Dic₄₇	—	4	2	no	—
80	C₂×C₄₇⋊C₈	—	2	4	no	—
81	C₂×C₄₇⋊C₈	—	4	4	no	—
82	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₄₇	—	2	2	no	—
84	C₂×C₄×C₄₇	—	2	2	no	—
85	C₂×C₄×C₄₇	—	2	2	no	—
86	C₂×C₄×C₄₇	—	2	2	no	—
87	C₂×C₄×C₄₇	—	2	2	no	—
88	C₂×C₄×C₄₇	—	2	2	no	—
89	C₂×C₄×C₄₇	#103	1	4	yes	—
90	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
91	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
92	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
93	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
94	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
95	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
96	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
97	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
98	C₂×C₄×C₄₇	#135	1	2	yes	—
99	C₂×Dic₄₇	—	2	2	no	—
100	C₂×Dic₄₇	—	2	4	no	—
101	C₂×Dic₄₇	—	2	2	no	—
102	C₂×Dic₄₇	—	2	4	no	—
103	C₂×Dic₄₇	—	4	2	no	—
104	C₂×Dic₄₇	—	4	2	no	—
105	C₂×Dic₄₇	—	4	2	no	—
106	Dic₉₄	—	2	2	no	—
107	Dic₉₄	—	2	4	no	—
108	Dic₉₄	#97	1	4	yes	complemented
109	C₈×C₄₇	—	4	4	no	—
110	C₄₇⋊C₈	—	4	4	no	—
111	#8175	—	47	1	no	maximal, Sylow
112	C₂²×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
113	C₂²×C₄₇	#407	1	2	yes	—
114	C₂²×C₄₇	C₂×#103	1	1	yes	characteristic
115	C₂²×C₄₇	#416	1	1	yes	characteristic
116	C₂²×C₄₇	#421	1	2	yes	—
117	C₄×C₄₇	—	2	4	no	—
118	C₄×C₄₇	—	2	2	no	—
119	C₄×C₄₇	—	4	2	no	—
120	C₄×C₄₇	C₂×#97	1	1	yes	characteristic
121	C₄×C₄₇	C₄×M₁₆	1	1	yes	characteristic
122	C₄×C₄₇	#432	1	2	yes	—
123	C₄×C₄₇	C₄×SD₁₆	1	2	yes	—
124	C₄×C₄₇	C₄×Q₁₆	1	2	yes	—
125	Dic₄₇	—	2	4	no	—
126	Dic₄₇	—	4	4	no	—
127	Dic₄₇	—	4	2	no	—
128	C₄²×Q₈	—	47	1	no	—
129	#1767	D₄₇	1	1	yes	characteristic
130	#1744	—	47	1	no	—
131	C₂×Q₈⋊C₈	—	47	2	no	—
132	#1279	—	47	2	no	—
133	C₂×C₄₇	#1487	1	2	yes	—
134	C₂×C₄₇	#1488	1	2	yes	—
135	C₂×C₄₇	#1749	1	1	yes	characteristic
136	C₂×C₄₇	#1759	1	1	yes	characteristic
137	C₂×C₄₇	C₄×#103	1	1	yes	characteristic
138	C₂×C₄×Q₈	—	47	2	no	—
139	C₂×C₄×Q₈	—	47	1	no	—
140	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
141	C₂×#97	—	2	1	no	—
142	C₂×C₄×C₈	—	47	2	no	—
143	C₄×C₄⋊C₄	—	47	2	no	—
144	C₄×C₄⋊C₄	—	47	1	no	—
145	C₄×C₄⋊C₄	—	94	2	no	—
146	C₄×C₄⋊C₄	—	94	1	no	—
147	C₄³	—	94	1	no	—
148	C₄³	C₂×D₄₇	1	1	yes	characteristic
149	#336	—	2	1	no	—
150	#336	—	94	2	no	—
151	Q₈⋊C₈	—	47	8	no	—
152	C₄₇	#8175	1	1	yes	characteristic, complemented, Sylow
153	C₂²×Q₈	—	47	1	no	—
154	C₂²×C₈	—	2	2	no	—
155	C₂²×C₈	—	94	2	no	—
156	C₄×Q₈	—	47	4	no	—
157	C₄×Q₈	—	94	4	no	—
158	C₄×Q₈	—	94	4	no	—
159	C₄×Q₈	—	94	2	no	—
160	C₂×C₄⋊C₄	—	47	1	no	—
161	C₂×C₄⋊C₄	—	47	2	no	—
162	C₂×C₄⋊C₄	—	94	2	no	—
163	C₂×C₄⋊C₄	—	94	1	no	—
164	C₂×C₄²	—	94	2	no	—
165	C₂×C₄²	—	94	1	no	—
166	C₂×C₄²	—	94	2	no	—
167	C₂×C₄²	—	94	1	no	—
168	C₂×C₄²	C₄×D₄₇	1	2	yes	—
169	C₂×C₄²	D₁₈₈	1	2	yes	—
170	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
171	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
172	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
173	C₄⋊C₈	—	2	2	no	—
174	C₄⋊C₈	C₄×D₄₇	1	4	yes	—
175	#97	—	4	2	no	—
176	C₄×C₈	—	47	4	no	—
177	C₄×C₈	—	94	2	no	—
178	#95	—	94	2	no	—
179	#95	—	94	1	no	—
180	C₂×Q₈	—	47	2	no	—
181	C₂×Q₈	—	47	4	no	—
182	C₂×Q₈	—	94	1	no	—
183	C₂×Q₈	—	94	2	no	—
184	C₂²×C₄	—	94	1	no	—
185	C₂²×C₄	—	94	1	no	—
186	C₂²×C₄	—	94	2	no	—
187	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
188	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
189	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
190	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
191	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
192	C₂×C₈	—	2	4	no	—
193	C₂×C₈	—	4	2	no	—
194	C₂×C₈	—	4	2	no	—
195	C₂×C₈	—	94	4	no	—
196	C₂×C₈	—	188	4	no	—
197	C₄⋊C₄	—	47	2	no	—
198	C₄⋊C₄	—	47	4	no	—
199	C₄⋊C₄	—	94	8	no	—
200	C₄⋊C₄	—	94	1	no	—
201	C₄⋊C₄	—	94	4	no	—
202	C₄⋊C₄	—	94	2	no	—
203	C₄²	—	2	2	no	—
204	C₄²	—	2	2	no	—
205	C₄²	—	2	2	no	—
206	C₄²	—	4	2	no	—
207	C₄²	—	94	8	no	—
208	C₄²	—	94	4	no	—
209	C₄²	—	188	4	no	—
210	C₄²	—	188	2	no	—
211	C₄²	#10625335	1	2	yes	—
212	C₄²	#10625337	1	2	yes	—
213	C₄²	C₂×C₄×D₄₇	1	2	yes	—
214	C₄²	#10625350	1	2	yes	—
215	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
216	Q₈	—	47	4	no	—
217	Q₈	—	94	2	no	—
218	Q₈	—	94	4	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	94	2	no	—
226	C₂×C₄	—	94	4	no	—
227	C₂×C₄	—	94	2	no	—
228	C₂×C₄	—	94	4	no	—
229	C₂×C₄	—	188	2	no	—
230	C₂×C₄	—	188	2	no	—
231	C₂×C₄	—	188	2	no	—
232	C₂×C₄	#13093947	1	4	yes	—
233	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
234	C₂×C₄	#13093981	1	1	yes	characteristic
235	C₂×C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
236	C₂×C₄	C₄×D₁₈₈	1	2	yes	—
237	C₂×C₄	M₁₆×D₄₇	1	2	yes	—
238	C₂×C₄	M₁₆×D₄₇	1	2	yes	—
239	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
240	C₂×C₄	C₂×#10625337	1	1	yes	characteristic
241	C₂×C₄	#13094044	1	2	yes	—
242	C₈	—	4	4	no	—
243	C₈	—	188	4	no	—
244	C₂²	C₄×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
245	C₂²	#15689086	1	2	yes	—
246	C₂²	C₂×#13093947	1	1	yes	characteristic
247	C₂²	#15689117	1	1	yes	characteristic
248	C₂²	#15689139	1	2	yes	—
249	C₄	—	2	2	no	—
250	C₄	—	2	4	no	—
251	C₄	—	4	2	no	—
252	C₄	—	94	4	no	—
253	C₄	—	188	2	no	—
254	C₄	—	188	4	no	—
255	C₄	#97×D₄₇	1	1	yes	characteristic
256	C₄	#15689075	1	1	yes	characteristic
257	C₄	#15689178	1	2	yes	—
258	C₄	C₄×#10625335	1	2	yes	—
259	C₄	C₄×#10625337	1	2	yes	—
260	C₂	#20241816	1	2	yes	—
261	C₂	#20241820	1	2	yes	—
262	C₂	#20243151	1	1	yes	characteristic
263	C₂	#20243201	1	1	yes	characteristic
264	C₂	C₄×#13093947	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	C₄₇×#1767	C₂
socle, prime radical	C₂³×C₄₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1767	D₄₇
hypocenter	C₄₇	#8175
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1767

Complements

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

N	H ≅ G/N	type
15. #15688853	C₄	semidirect
108. Dic₉₄	#97	semidirect
152. C₄₇	#8175	semidirect

Maximal subgroups

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

subgroup	index
C₄²×Dic₉₄	2
#20243134	2
C₂×#15688853	2
#20241227	2
C₄₇×#1767	2
#8175	47