Subgroups — #26132325

Order 12032.

446 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×C₄×Dic₁₈₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688928	C₂	1	2	yes	maximal
3	#20243548	C₂	1	1	yes	maximal, characteristic
4	#20241480	C₂	1	2	yes	maximal
5	C₂×C₄₇×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₈₈	C₂²	1	1	yes	characteristic
7	C₂×C₄×Dic₉₄	—	2	1	no	—
8	C₄×Dic₁₈₈	—	2	2	no	—
9	C₄×Dic₁₈₈	C₄	1	4	yes	complemented
10	C₂×#13093958	—	2	1	no	—
11	C₂×#13093958	C₂²	1	2	yes	—
12	C₂×#13093958	C₂²	1	1	yes	characteristic
13	C₂×#13093949	—	2	1	no	—
14	#15688993	—	2	1	no	—
15	#15688928	—	2	4	no	—
16	C₂²×C₁₆×C₄₇	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
18	C₄₇×C₄⋊C₁₆	—	2	2	no	—
19	C₂²×Dic₉₄	—	2	1	no	—
20	C₂×Dic₁₈₈	—	2	1	no	—
21	C₂×Dic₁₈₈	—	2	1	no	—
22	C₂×Dic₁₈₈	—	2	1	no	—
23	C₂×Dic₁₈₈	C₂×C₄	1	2	yes	—
24	C₂×Dic₁₈₈	D₄	1	2	yes	—
25	C₂×Dic₁₈₈	Q₈	1	2	yes	—
26	C₄×Dic₉₄	—	2	4	no	—
27	C₄×Dic₉₄	—	4	2	no	—
28	C₂×C₄⋊Dic₄₇	—	2	1	no	—
29	C₂×C₄⋊Dic₄₇	—	2	2	no	—
30	C₂×C₄⋊Dic₄₇	—	4	1	no	—
31	C₂×C₄⋊Dic₄₇	—	4	1	no	—
32	C₂×Dic₄₇⋊C₄	—	4	1	no	—
33	C₂×C₄×Dic₄₇	—	4	1	no	—
34	#13093958	—	2	1	no	—
35	#13093958	—	2	2	no	—
36	#13093958	—	2	2	no	—
37	#13093958	—	4	2	no	—
38	#13093958	C₂×C₄	1	2	yes	—
39	#13093949	—	2	4	no	—
40	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
42	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
43	C₂×C₁₆×C₄₇	—	2	2	no	—
44	C₂×C₁₆×C₄₇	—	2	2	no	—
45	C₂×C₁₆×C₄₇	—	2	2	no	—
46	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
47	C₄×C₈×C₄₇	D₄	1	2	yes	—
48	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
49	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
50	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
51	C₂²×Dic₄₇	—	4	1	no	—
52	C₂²×Dic₄₇	—	4	1	no	—
53	C₂²×Dic₄₇	—	4	2	no	—
54	C₂×Dic₉₄	—	2	2	no	—
55	C₂×Dic₉₄	—	2	2	no	—
56	C₂×Dic₉₄	—	2	2	no	—
57	C₂×Dic₉₄	—	4	1	no	—
58	C₂×Dic₉₄	—	4	1	no	—
59	C₂×Dic₉₄	—	4	1	no	—
60	C₂×C₈×C₄₇	—	2	1	no	—
61	C₂×C₈×C₄₇	—	2	1	no	—
62	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
63	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
67	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
68	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
69	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
70	C₄²×C₄₇	SD₁₆	1	2	yes	—
71	C₄⋊Dic₄₇	—	2	2	no	—
72	C₄⋊Dic₄₇	—	4	1	no	—
73	C₄⋊Dic₄₇	—	4	2	no	—
74	C₄⋊Dic₄₇	—	4	2	no	—
75	C₄⋊Dic₄₇	—	8	2	no	—
76	C₄⋊Dic₄₇	—	8	2	no	—
77	Dic₄₇⋊C₄	—	4	4	no	—
78	C₄×Dic₄₇	—	4	4	no	—
79	Dic₁₈₈	—	2	2	no	—
80	Dic₁₈₈	—	2	2	no	—
81	Dic₁₈₈	—	2	2	no	—
82	Dic₁₈₈	C₄⋊C₄	1	4	yes	complemented
83	C₁₆×C₄₇	—	4	2	no	—
84	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
85	C₂×C₄×C₄₇	—	2	1	no	—
86	C₂×C₄×C₄₇	—	2	1	no	—
87	C₂×C₄×C₄₇	#102	1	2	yes	—
88	C₂×C₄×C₄₇	#102	1	2	yes	—
89	C₂×C₄×C₄₇	SD₃₂	1	2	yes	—
90	C₂×C₄×C₄₇	Q₃₂	1	2	yes	—
91	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
93	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
94	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
95	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
96	C₂×C₄×C₄₇	#137	1	1	yes	characteristic
97	C₂×Dic₄₇	—	4	2	no	—
98	C₂×Dic₄₇	—	4	2	no	—
99	C₂×Dic₄₇	—	4	2	no	—
100	C₂×Dic₄₇	—	4	2	no	—
101	C₂×Dic₄₇	—	8	1	no	—
102	C₂×Dic₄₇	—	8	1	no	—
103	C₂×Dic₄₇	—	8	2	no	—
104	C₂×Dic₄₇	—	8	2	no	—
105	C₂×Dic₄₇	—	8	2	no	—
106	Dic₉₄	—	2	4	no	—
107	Dic₉₄	—	4	2	no	—
108	Dic₉₄	—	4	2	no	—
109	Dic₉₄	—	4	2	no	—
110	C₈×C₄₇	—	2	2	no	—
111	C₈×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₈×C₄₇	#117	1	1	yes	characteristic
113	C₈×C₄₇	#121	1	1	yes	characteristic
114	C₈×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
115	#12812	—	47	1	no	maximal, Sylow
116	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
117	C₂²×C₄₇	#459	1	1	yes	characteristic
118	C₂²×C₄₇	#474	1	1	yes	characteristic
119	C₂²×C₄₇	C₂×SD₃₂	1	1	yes	characteristic
120	C₂²×C₄₇	C₂×Q₃₂	1	1	yes	characteristic
121	C₂²×C₄₇	#509	1	1	yes	characteristic
122	C₂²×C₄₇	#510	1	1	yes	characteristic
123	C₄×C₄₇	—	2	2	no	—
124	C₄×C₄₇	#358	1	4	yes	complemented
125	C₄×C₄₇	#380	1	1	yes	characteristic
126	C₄×C₄₇	#417	1	1	yes	characteristic
127	C₄×C₄₇	#461	1	1	yes	characteristic
128	C₄×C₄₇	#476	1	1	yes	characteristic
129	Dic₄₇	—	4	4	no	—
130	Dic₄₇	—	8	2	no	—
131	Dic₄₇	—	16	2	no	—
132	C₂×C₄×Q₁₆	—	47	1	no	—
133	C₂×C₄⋊C₁₆	D₄₇	1	1	yes	characteristic
134	C₂×#358	—	47	2	no	—
135	#1845	—	47	1	no	—
136	#1380	—	47	2	no	—
137	C₂×C₄₇	#1802	1	1	yes	characteristic
138	C₂×C₄₇	C₂×#358	1	1	yes	characteristic
139	C₂×C₄₇	#2139	1	1	yes	characteristic
140	C₂×C₄₇	#2207	1	1	yes	characteristic
141	C₂×C₄₇	#2209	1	1	yes	characteristic
142	C₂×C₄₇	#2225	1	1	yes	characteristic
143	C₂×C₄₇	#2227	1	1	yes	characteristic
144	C₂²×Q₁₆	—	47	1	no	—
145	C₂×C₄×Q₈	—	94	1	no	—
146	C₂²×C₁₆	C₂×D₄₇	1	2	yes	—
147	C₄×Q₁₆	—	47	4	no	—
148	C₄×Q₁₆	—	94	2	no	—
149	C₂×#107	—	47	1	no	—
150	C₂×#107	—	47	2	no	—
151	C₂×#107	—	94	1	no	—
152	C₂×#103	—	94	1	no	—
153	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
154	#384	—	94	1	no	—
155	C₄⋊C₁₆	—	2	2	no	—
156	#358	—	94	4	no	—
157	C₄₇	#12812	1	1	yes	characteristic, complemented, Sylow
158	C₂²×Q₈	—	94	1	no	—
159	C₂×Q₁₆	—	47	2	no	—
160	C₂×Q₁₆	—	47	2	no	—
161	C₂×Q₁₆	—	47	2	no	—
162	C₂×Q₁₆	—	94	1	no	—
163	C₂×Q₁₆	—	94	1	no	—
164	C₂×Q₁₆	—	94	1	no	—
165	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
166	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
167	C₄×Q₈	—	94	4	no	—
168	C₄×Q₈	—	188	2	no	—
169	C₂×C₄⋊C₄	—	94	1	no	—
170	C₂×C₄⋊C₄	—	94	2	no	—
171	C₂×C₄⋊C₄	—	188	1	no	—
172	C₂×C₄⋊C₄	—	188	1	no	—
173	C₂×C₄⋊C₄	—	188	1	no	—
174	C₂×C₄²	—	188	1	no	—
175	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
176	C₂×C₁₆	—	2	2	no	—
177	C₂×C₁₆	—	2	2	no	—
178	C₂×C₁₆	—	2	2	no	—
179	#107	—	47	2	no	—
180	#107	—	94	1	no	—
181	#107	—	94	2	no	—
182	#107	—	94	2	no	—
183	#107	—	188	2	no	—
184	#103	—	94	4	no	—
185	C₄×C₈	C₄×D₄₇	1	2	yes	—
186	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
187	C₂×Q₈	—	94	2	no	—
188	C₂×Q₈	—	94	2	no	—
189	C₂×Q₈	—	94	2	no	—
190	C₂×Q₈	—	188	1	no	—
191	C₂×Q₈	—	188	1	no	—
192	C₂×Q₈	—	188	1	no	—
193	C₂²×C₄	—	188	2	no	—
194	C₂²×C₄	—	188	1	no	—
195	C₂²×C₄	—	188	1	no	—
196	C₂²×C₄	D₃₇₆	1	1	yes	characteristic
197	C₂²×C₄	D₃₇₆	1	1	yes	characteristic
198	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
199	Q₁₆	—	47	4	no	—
200	Q₁₆	—	94	2	no	—
201	Q₁₆	—	94	2	no	—
202	Q₁₆	—	94	2	no	—
203	C₂×C₈	—	2	1	no	—
204	C₂×C₈	—	2	1	no	—
205	C₂×C₈	(C₂×D₄₇)⋊C₄	1	2	yes	—
206	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
207	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
208	C₂×C₈	#10625352	1	1	yes	characteristic
209	C₂×C₈	Q₈×D₄₇	1	1	yes	characteristic
210	C₂×C₈	#10625354	1	1	yes	characteristic
211	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
212	C₄⋊C₄	—	94	2	no	—
213	C₄⋊C₄	—	376	2	no	—
214	C₄⋊C₄	—	376	2	no	—
215	C₄⋊C₄	—	188	2	no	—
216	C₄⋊C₄	—	188	2	no	—
217	C₄⋊C₄	—	188	4	no	—
218	C₄⋊C₄	—	188	1	no	—
219	C₄²	—	188	4	no	—
220	C₄²	#10625325	1	2	yes	—
221	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
222	C₁₆	—	4	2	no	—
223	C₂³	C₂×D₃₇₆	1	1	yes	characteristic
224	Q₈	—	94	4	no	—
225	Q₈	—	188	2	no	—
226	Q₈	—	188	2	no	—
227	Q₈	—	188	2	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	376	1	no	—
231	C₂×C₄	—	376	1	no	—
232	C₂×C₄	—	376	2	no	—
233	C₂×C₄	—	376	2	no	—
234	C₂×C₄	—	376	2	no	—
235	C₂×C₄	—	188	2	no	—
236	C₂×C₄	—	188	2	no	—
237	C₂×C₄	—	188	2	no	—
238	C₂×C₄	—	188	2	no	—
239	C₂×C₄	#13093946	1	2	yes	—
240	C₂×C₄	#13093946	1	2	yes	—
241	C₂×C₄	#13093969	1	2	yes	—
242	C₂×C₄	Dic₃₇₆	1	2	yes	—
243	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
244	C₂×C₄	#13093999	1	1	yes	characteristic
245	C₂×C₄	#13094004	1	1	yes	characteristic
246	C₂×C₄	C₂×#10625325	1	1	yes	characteristic
247	C₂×C₄	#13094052	1	1	yes	characteristic
248	C₂×C₄	#13094057	1	1	yes	characteristic
249	C₈	—	2	2	no	—
250	C₈	C₄⋊C₄×D₄₇	1	1	yes	characteristic
251	C₈	#13093982	1	1	yes	characteristic
252	C₈	#13093995	1	1	yes	characteristic
253	C₈	#13094000	1	1	yes	characteristic
254	C₂²	C₂×#13093946	1	1	yes	characteristic
255	C₂²	#15689319	1	1	yes	characteristic
256	C₂²	#15689404	1	1	yes	characteristic
257	C₂²	C₂×#13093969	1	1	yes	characteristic
258	C₂²	C₂×Dic₃₇₆	1	1	yes	characteristic
259	C₂²	#15689561	1	1	yes	characteristic
260	C₂²	#15689566	1	1	yes	characteristic
261	C₄	—	2	2	no	—
262	C₄	—	376	2	no	—
263	C₄	—	188	4	no	—
264	C₄	—	752	2	no	—
265	C₄	#15688928	1	4	yes	complemented
266	C₄	#15688980	1	1	yes	characteristic
267	C₄	#15689125	1	1	yes	characteristic
268	C₄	#15689329	1	1	yes	characteristic
269	C₄	#15689414	1	1	yes	characteristic
270	C₂	#20243389	1	1	yes	characteristic
271	C₂	C₂×#15688928	1	1	yes	characteristic
272	C₂	#20245101	1	1	yes	characteristic
273	C₂	#20245429	1	1	yes	characteristic
274	C₂	#20245439	1	1	yes	characteristic
275	C₂	#20245531	1	1	yes	characteristic
276	C₂	#20245541	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₃₇₆
commutator	C₂×C₈×C₄₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₂×C₄₇×C₄⋊C₁₆	C₂
socle, prime radical	C₂³×C₄₇	C₂×D₈
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₄₇
hypocenter	C₄₇	#12812
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
9. C₄×Dic₁₈₈	C₄	semidirect
82. Dic₁₈₈	C₄⋊C₄	semidirect
124. C₄×C₄₇	#358	semidirect
157. C₄₇	#12812	semidirect
265. C₄	#15688928	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₁₈₈	2
C₂×#15688928	2
#20243548	2
#20241480	2
C₂×C₄₇×C₄⋊C₁₆	2
#12812	47