Subgroups — #26104402

Order 12032.

440 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	#20243531	C₂	1	1	yes	maximal, characteristic
2	#20243531	C₂	1	1	yes	maximal, characteristic
3	#20241211	C₂	1	4	yes	maximal
4	C₄×C₄⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
5	#15688878	—	2	1	no	—
6	#15688878	—	2	1	no	—
7	#15688878	—	2	1	no	—
8	#15688878	—	2	1	no	—
9	#15688878	C₂²	1	2	yes	—
10	#15688878	C₂²	1	2	yes	—
11	C₂×C₄⋊Q₈×C₄₇	C₄	1	1	yes	characteristic
12	C₂×C₄×Q₈×C₄₇	—	2	1	no	—
13	C₄₇×#391	—	2	1	no	—
14	C₄₇×#389	—	2	1	no	—
15	C₄₇×#389	C₂²	1	2	yes	—
16	C₄₇×#384	C₄	1	1	yes	characteristic
17	C₄×C₄⋊C₄×C₄₇	—	2	1	no	—
18	C₄×C₄⋊C₄×C₄₇	C₄	1	2	yes	—
19	C₄³×C₄₇	C₂²	1	1	yes	characteristic
20	C₂²×C₄₇⋊C₈	—	2	2	no	—
21	C₂²×C₄₇⋊C₈	—	2	2	no	—
22	C₂²×Q₈×C₄₇	—	2	1	no	—
23	C₄⋊Q₈×C₄₇	—	2	2	no	—
24	C₄⋊Q₈×C₄₇	C₈	1	4	yes	complemented
25	C₄×Q₈×C₄₇	—	4	4	no	—
26	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
27	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
28	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
29	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
30	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
31	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₄₇	—	2	2	no	—
33	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
40	#95×C₄₇	—	2	2	no	—
41	#95×C₄₇	—	2	2	no	—
42	C₂×Q₈×C₄₇	—	2	4	no	—
43	C₂×Q₈×C₄₇	—	4	2	no	—
44	C₂×Q₈×C₄₇	—	4	2	no	—
45	C₂²×C₄×C₄₇	—	2	2	no	—
46	C₂²×C₄×C₄₇	—	2	2	no	—
47	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
54	C₄⋊C₄×C₄₇	—	2	4	no	—
55	C₄⋊C₄×C₄₇	—	2	2	no	—
56	C₄⋊C₄×C₄₇	—	4	2	no	—
57	C₄⋊C₄×C₄₇	—	4	2	no	—
58	C₄⋊C₄×C₄₇	—	4	2	no	—
59	C₄⋊C₄×C₄₇	—	4	4	no	—
60	C₄⋊C₄×C₄₇	C₂×C₈	1	4	yes	—
61	C₄²×C₄₇	—	2	2	no	—
62	C₄²×C₄₇	—	2	2	no	—
63	C₄²×C₄₇	—	2	2	no	—
64	C₄²×C₄₇	—	4	1	no	—
65	C₄²×C₄₇	—	4	1	no	—
66	C₄²×C₄₇	—	4	1	no	—
67	C₄²×C₄₇	—	4	4	no	—
68	C₄²×C₄₇	C₂×C₈	1	2	yes	—
69	C₄²×C₄₇	M₁₆	1	2	yes	—
70	C₂×C₄₇⋊C₈	—	4	2	no	—
71	C₂×C₄₇⋊C₈	—	4	2	no	—
72	C₂×C₄₇⋊C₈	—	4	2	no	—
73	C₂×C₄₇⋊C₈	—	4	2	no	—
74	C₂×C₄₇⋊C₈	—	4	2	no	—
75	C₂×C₄₇⋊C₈	—	4	2	no	—
76	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	Q₈×C₄₇	—	4	8	no	—
78	C₂×C₄×C₄₇	—	2	1	no	—
79	C₂×C₄×C₄₇	—	2	1	no	—
80	C₂×C₄×C₄₇	—	2	1	no	—
81	C₂×C₄×C₄₇	—	2	1	no	—
82	C₂×C₄×C₄₇	—	2	1	no	—
83	C₂×C₄×C₄₇	—	2	1	no	—
84	C₂×C₄×C₄₇	—	2	1	no	—
85	C₂×C₄×C₄₇	—	2	1	no	—
86	C₂×C₄×C₄₇	—	2	1	no	—
87	C₂×C₄×C₄₇	—	2	1	no	—
88	C₂×C₄×C₄₇	—	2	1	no	—
89	C₂×C₄×C₄₇	—	2	1	no	—
90	C₂×C₄×C₄₇	—	2	4	no	—
91	C₂×C₄×C₄₇	—	2	1	no	—
92	C₂×C₄×C₄₇	—	2	1	no	—
93	C₂×C₄×C₄₇	—	2	1	no	—
94	C₂×C₄×C₄₇	—	4	2	no	—
95	C₂×C₄×C₄₇	—	4	2	no	—
96	C₂×C₄×C₄₇	—	4	2	no	—
97	C₂×C₄×C₄₇	—	4	2	no	—
98	C₂×C₄×C₄₇	—	4	2	no	—
99	C₂×C₄×C₄₇	C₂²⋊C₈	1	2	yes	—
100	C₂×C₄×C₄₇	C₂²⋊C₈	1	2	yes	—
101	C₂×C₄×C₄₇	#101	1	2	yes	—
102	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
103	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
104	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
105	C₂×C₄×C₄₇	#137	1	2	yes	—
106	C₄₇⋊C₈	—	8	2	no	—
107	C₄₇⋊C₈	—	8	2	no	—
108	#7604	—	47	1	no	maximal, Sylow
109	C₂²×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
110	C₂²×C₄₇	#410	1	1	yes	characteristic
111	C₂²×C₄₇	C₂×#101	1	1	yes	characteristic
112	C₂²×C₄₇	#418	1	1	yes	characteristic
113	C₂²×C₄₇	#419	1	1	yes	characteristic
114	C₂²×C₄₇	#421	1	1	yes	characteristic
115	C₂²×C₄₇	#421	1	1	yes	characteristic
116	C₄×C₄₇	—	2	2	no	—
117	C₄×C₄₇	—	2	1	no	—
118	C₄×C₄₇	—	2	1	no	—
119	C₄×C₄₇	—	2	2	no	—
120	C₄×C₄₇	—	2	2	no	—
121	C₄×C₄₇	—	2	2	no	—
122	C₄×C₄₇	—	2	2	no	—
123	C₄×C₄₇	—	4	1	no	—
124	C₄×C₄₇	—	4	4	no	—
125	C₄×C₄₇	—	8	2	no	—
126	C₄×C₄⋊Q₈	D₄₇	1	1	yes	characteristic
127	#1839	—	47	1	no	—
128	#1839	—	47	1	no	—
129	#1271	—	47	4	no	—
130	C₂×C₄₇	#1461	1	1	yes	characteristic
131	C₂×C₄₇	#1478	1	1	yes	characteristic
132	C₂×C₄₇	#1479	1	1	yes	characteristic
133	C₂×C₄₇	#1514	1	1	yes	characteristic
134	C₂×C₄₇	#1515	1	1	yes	characteristic
135	C₂×C₄₇	#1537	1	1	yes	characteristic
136	C₂×C₄₇	#1556	1	1	yes	characteristic
137	C₂×C₄⋊Q₈	Dic₄₇	1	1	yes	characteristic
138	C₂×C₄×Q₈	—	2	1	no	—
139	#391	—	2	1	no	—
140	#389	—	2	1	no	—
141	#389	C₂×D₄₇	1	2	yes	—
142	#384	Dic₄₇	1	1	yes	characteristic
143	C₄×C₄⋊C₄	—	2	1	no	—
144	C₄×C₄⋊C₄	Dic₄₇	1	2	yes	—
145	C₄³	C₂×D₄₇	1	1	yes	characteristic
146	#336	—	47	2	no	—
147	#336	—	47	2	no	—
148	#336	—	94	1	no	—
149	#336	—	94	1	no	—
150	#336	—	94	1	no	—
151	#336	—	94	1	no	—
152	C₄₇	#7604	1	1	yes	characteristic, complemented, Sylow
153	C₂²×Q₈	—	2	1	no	—
154	C₂²×C₈	—	94	2	no	—
155	C₂²×C₈	—	94	2	no	—
156	C₄⋊Q₈	—	2	2	no	—
157	C₄⋊Q₈	C₄₇⋊C₈	1	4	yes	complemented
158	C₄×Q₈	—	4	4	no	—
159	C₂×C₄⋊C₄	—	2	1	no	—
160	C₂×C₄⋊C₄	—	2	1	no	—
161	C₂×C₄⋊C₄	—	2	1	no	—
162	C₂×C₄⋊C₄	—	2	2	no	—
163	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
164	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
165	C₂×C₄²	—	2	2	no	—
166	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
167	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
168	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
169	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
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	#95	—	2	2	no	—
174	#95	—	2	2	no	—
175	C₂×Q₈	—	2	4	no	—
176	C₂×Q₈	—	4	2	no	—
177	C₂×Q₈	—	4	2	no	—
178	C₂²×C₄	—	2	2	no	—
179	C₂²×C₄	—	2	2	no	—
180	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
181	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
182	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
183	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
184	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
185	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
186	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
187	C₂×C₈	—	188	2	no	—
188	C₂×C₈	—	188	2	no	—
189	C₂×C₈	—	188	2	no	—
190	C₂×C₈	—	188	2	no	—
191	C₂×C₈	—	188	2	no	—
192	C₂×C₈	—	188	2	no	—
193	C₄⋊C₄	—	2	4	no	—
194	C₄⋊C₄	—	2	2	no	—
195	C₄⋊C₄	—	4	2	no	—
196	C₄⋊C₄	—	4	2	no	—
197	C₄⋊C₄	—	4	2	no	—
198	C₄⋊C₄	—	4	4	no	—
199	C₄⋊C₄	C₂×C₄₇⋊C₈	1	4	yes	—
200	C₄²	—	2	2	no	—
201	C₄²	—	2	2	no	—
202	C₄²	—	2	2	no	—
203	C₄²	—	4	1	no	—
204	C₄²	—	4	1	no	—
205	C₄²	—	4	1	no	—
206	C₄²	—	4	4	no	—
207	C₄²	C₂×C₄₇⋊C₈	1	2	yes	—
208	C₄²	#10625329	1	2	yes	—
209	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
210	Q₈	—	4	8	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
233	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
234	C₂×C₄	#13093943	1	2	yes	—
235	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
236	C₂×C₄	C₂×#10625329	1	1	yes	characteristic
237	C₂×C₄	#13094050	1	2	yes	—
238	C₂×C₄	#13094055	1	2	yes	—
239	C₈	—	376	2	no	—
240	C₈	—	376	2	no	—
241	C₂²	C₂×C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
242	C₂²	#15689097	1	1	yes	characteristic
243	C₂²	C₂×#13093943	1	1	yes	characteristic
244	C₂²	#15689128	1	1	yes	characteristic
245	C₂²	#15689132	1	1	yes	characteristic
246	C₂²	#15689140	1	1	yes	characteristic
247	C₂²	#15689140	1	1	yes	characteristic
248	C₄	—	2	2	no	—
249	C₄	—	2	2	no	—
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	2	2	no	—
253	C₄	—	2	2	no	—
254	C₄	—	2	2	no	—
255	C₄	—	4	1	no	—
256	C₄	—	4	4	no	—
257	C₄	—	8	2	no	—
258	C₂	#20241712	1	1	yes	characteristic
259	C₂	#20241779	1	1	yes	characteristic
260	C₂	#20241783	1	1	yes	characteristic
261	C₂	#20241935	1	1	yes	characteristic
262	C₂	#20241939	1	1	yes	characteristic
263	C₂	#20242026	1	1	yes	characteristic
264	C₂	#20242105	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₄₇
commutator	C₂×C₄×C₄₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄×C₄⋊Q₈×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₄×C₄⋊Q₈	D₄₇
hypocenter	C₄₇	#7604
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
24. C₄⋊Q₈×C₄₇	C₈	semidirect
152. C₄₇	#7604	semidirect
157. C₄⋊Q₈	C₄₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#20243531	2
#20241211	2
C₄×C₄⋊Q₈×C₄₇	2
#7604	47