Subgroups — #26110022

Order 12032.

422 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20243504	C₂	1	2	yes	maximal
2	C₄×C₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688876	C₂	1	1	yes	maximal, characteristic
4	C₂×#15688873	C₂	1	2	yes	maximal
5	C₂×C₄₇×#335	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₈×Dic₄₇	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₄₇	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₄₇	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₄₇	C₄	1	1	yes	characteristic
12	C₄²×Dic₄₇	C₂²	1	1	yes	characteristic
13	#15688877	—	2	4	no	—
14	#15688876	—	2	2	no	—
15	#15688873	—	2	4	no	—
16	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
18	C₄₇×#335	—	2	2	no	—
19	C₂²×C₄₇⋊C₈	—	2	2	no	—
20	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
25	C₂×C₄×Dic₄₇	Q₈	1	2	yes	—
26	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
27	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
28	C₄×C₄₇⋊C₈	D₄	1	2	yes	—
29	C₄×C₄₇⋊C₈	Q₈	1	2	yes	—
30	C₈×Dic₄₇	—	2	2	no	—
31	C₈×Dic₄₇	—	2	2	no	—
32	C₈×Dic₄₇	—	2	2	no	—
33	C₈×Dic₄₇	C₈	1	4	yes	complemented
34	C₂²×C₈×C₄₇	—	2	2	no	—
35	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
38	C₄⋊C₈×C₄₇	—	2	2	no	—
39	C₄⋊C₈×C₄₇	—	2	2	no	—
40	C₄×C₈×C₄₇	D₄	1	2	yes	—
41	C₄×C₈×C₄₇	Q₈	1	2	yes	—
42	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
45	C₂²×Dic₄₇	C₄⋊C₄	1	2	yes	—
46	C₂²×Dic₄₇	C₄⋊C₄	1	2	yes	—
47	C₂×C₈×C₄₇	—	2	1	no	—
48	C₂×C₈×C₄₇	—	2	1	no	—
49	C₂×C₈×C₄₇	—	4	2	no	—
50	C₂×C₈×C₄₇	—	4	4	no	—
51	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
52	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
53	C₂×C₈×C₄₇	C₂×C₈	1	2	yes	—
54	C₂×C₈×C₄₇	M₁₆	1	2	yes	—
55	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
56	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
57	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
58	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
59	C₄×Dic₄₇	—	2	1	no	—
60	C₄×Dic₄₇	—	2	1	no	—
61	C₄×Dic₄₇	—	2	2	no	—
62	C₄×Dic₄₇	—	2	2	no	—
63	C₄×Dic₄₇	—	2	4	no	—
64	C₄×Dic₄₇	C₂×C₈	1	2	yes	—
65	C₄×Dic₄₇	M₁₆	1	2	yes	—
66	C₂×C₄₇⋊C₈	—	2	1	no	—
67	C₂×C₄₇⋊C₈	—	2	1	no	—
68	C₂×C₄₇⋊C₈	—	4	2	no	—
69	C₂×C₄₇⋊C₈	—	4	4	no	—
70	C₂×C₄₇⋊C₈	C₄⋊C₄	1	2	yes	—
71	C₂×C₄₇⋊C₈	C₄⋊C₄	1	2	yes	—
72	C₂×C₄₇⋊C₈	C₂×C₈	1	2	yes	—
73	C₂×C₄₇⋊C₈	M₁₆	1	2	yes	—
74	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₄₇	—	2	2	no	—
76	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
79	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
80	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
81	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
82	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
83	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
84	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
85	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
86	C₂×C₄×C₄₇	#135	1	2	yes	—
87	C₂×C₄×C₄₇	#135	1	2	yes	—
88	C₂×Dic₄₇	—	2	2	no	—
89	C₂×Dic₄₇	—	2	2	no	—
90	C₂×Dic₄₇	—	2	2	no	—
91	C₂×Dic₄₇	—	2	2	no	—
92	C₂×Dic₄₇	C₄⋊C₈	1	4	yes	—
93	C₂×Dic₄₇	#108	1	4	yes	—
94	C₈×C₄₇	—	2	2	no	—
95	C₈×C₄₇	—	8	2	no	—
96	C₈×C₄₇	C₄⋊C₈	1	4	yes	—
97	C₄₇⋊C₈	—	2	2	no	—
98	C₄₇⋊C₈	—	8	2	no	—
99	C₄₇⋊C₈	C₄⋊C₈	1	4	yes	—
100	#8807	—	47	1	no	maximal, Sylow
101	C₂²×C₄₇	#384	1	1	yes	characteristic
102	C₂²×C₄₇	C₂×C₄⋊C₈	1	1	yes	characteristic
103	C₂²×C₄₇	#423	1	1	yes	characteristic
104	C₂²×C₄₇	#427	1	2	yes	—
105	C₂²×C₄₇	C₂×#108	1	2	yes	—
106	C₄×C₄₇	—	2	1	no	—
107	C₄×C₄₇	—	2	1	no	—
108	C₄×C₄₇	C₂×C₄⋊C₈	1	1	yes	characteristic
109	C₄×C₄₇	C₂×C₄⋊C₈	1	1	yes	characteristic
110	C₄×C₄₇	#423	1	1	yes	characteristic
111	C₄×C₄₇	#423	1	1	yes	characteristic
112	C₄×C₄₇	#495	1	1	yes	characteristic
113	C₄×C₄₇	#495	1	1	yes	characteristic
114	C₄×C₄₇	#499	1	2	yes	—
115	Dic₄₇	—	2	4	no	—
116	Dic₄₇	—	2	4	no	—
117	#1831	—	47	2	no	—
118	C₄²×C₈	—	47	1	no	—
119	C₂×#335	—	47	1	no	—
120	C₂×#335	D₄₇	1	1	yes	characteristic
121	C₂×#334	—	47	2	no	—
122	C₂×C₄₇	#1564	1	2	yes	—
123	C₂×C₄₇	#1568	1	2	yes	—
124	C₂×C₄₇	#1842	1	1	yes	characteristic
125	C₂×C₄₇	#1846	1	1	yes	characteristic
126	C₂×C₄₇	#1848	1	1	yes	characteristic
127	C₂×C₄⋊C₈	—	47	2	no	—
128	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
129	C₂×C₄×C₈	—	47	1	no	—
130	C₂×C₄×C₈	—	47	1	no	—
131	C₂×C₄×C₈	—	47	1	no	—
132	C₂×C₄×C₈	—	47	1	no	—
133	C₂×C₄×C₈	—	47	1	no	—
134	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
135	C₄³	—	47	1	no	—
136	#336	—	94	4	no	—
137	#335	—	2	2	no	—
138	#335	—	94	2	no	—
139	#334	—	94	4	no	—
140	C₄₇	#8807	1	1	yes	characteristic, complemented, Sylow
141	C₂²×C₈	—	2	2	no	—
142	C₂²×C₈	—	47	1	no	—
143	C₂²×C₈	—	47	1	no	—
144	C₂²×C₈	—	94	2	no	—
145	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
146	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
147	C₂×C₄²	—	47	1	no	—
148	C₂×C₄²	—	47	1	no	—
149	C₂×C₄²	—	47	2	no	—
150	C₂×C₄²	—	47	2	no	—
151	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
152	C₄⋊C₈	—	2	2	no	—
153	C₄⋊C₈	—	2	2	no	—
154	C₄⋊C₈	—	94	2	no	—
155	C₄⋊C₈	—	94	2	no	—
156	C₄×C₈	—	47	4	no	—
157	C₄×C₈	—	47	2	no	—
158	C₄×C₈	—	47	2	no	—
159	C₄×C₈	—	94	2	no	—
160	C₄×C₈	—	94	2	no	—
161	C₄×C₈	—	94	2	no	—
162	C₄×C₈	Dic₉₄	1	2	yes	—
163	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
164	C₂²×C₄	—	47	2	no	—
165	C₂²×C₄	—	47	2	no	—
166	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
167	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
168	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
169	C₂×C₈	—	2	1	no	—
170	C₂×C₈	—	2	1	no	—
171	C₂×C₈	—	4	2	no	—
172	C₂×C₈	—	4	4	no	—
173	C₂×C₈	—	47	2	no	—
174	C₂×C₈	—	47	2	no	—
175	C₂×C₈	—	47	2	no	—
176	C₂×C₈	—	47	2	no	—
177	C₂×C₈	—	94	1	no	—
178	C₂×C₈	—	94	1	no	—
179	C₂×C₈	—	188	2	no	—
180	C₂×C₈	—	188	4	no	—
181	C₂×C₈	C₈×D₄₇	1	2	yes	—
182	C₂×C₈	#10625324	1	2	yes	—
183	C₂×C₈	Dic₄₇⋊C₄	1	2	yes	—
184	C₂×C₈	Dic₄₇⋊C₄	1	2	yes	—
185	C₄²	—	47	2	no	—
186	C₄²	—	47	2	no	—
187	C₄²	—	94	4	no	—
188	C₄²	—	94	1	no	—
189	C₄²	—	94	1	no	—
190	C₄²	—	94	2	no	—
191	C₄²	—	94	2	no	—
192	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
193	C₄²	D₄×D₄₇	1	1	yes	characteristic
194	C₄²	Q₈×D₄₇	1	1	yes	characteristic
195	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
196	C₂³	C₄⋊C₄×D₄₇	1	1	yes	characteristic
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	47	4	no	—
199	C₂×C₄	—	47	4	no	—
200	C₂×C₄	—	94	2	no	—
201	C₂×C₄	—	94	2	no	—
202	C₂×C₄	—	94	2	no	—
203	C₂×C₄	—	94	2	no	—
204	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
205	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
206	C₂×C₄	#13094020	1	1	yes	characteristic
207	C₂×C₄	#13094022	1	1	yes	characteristic
208	C₂×C₄	#13094023	1	1	yes	characteristic
209	C₂×C₄	#13094023	1	1	yes	characteristic
210	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
211	C₂×C₄	M₁₆×D₄₇	1	1	yes	characteristic
212	C₂×C₄	M₁₆×D₄₇	1	1	yes	characteristic
213	C₂×C₄	C₂×#10625324	1	1	yes	characteristic
214	C₂×C₄	#13094043	1	2	yes	—
215	C₂×C₄	#13094047	1	2	yes	—
216	C₈	—	2	2	no	—
217	C₈	—	8	2	no	—
218	C₈	—	47	4	no	—
219	C₈	—	94	2	no	—
220	C₈	—	376	2	no	—
221	C₈	Dic₄₇⋊C₈	1	4	yes	—
222	C₂²	#15688992	1	1	yes	characteristic
223	C₂²	C₄⋊C₈×D₄₇	1	1	yes	characteristic
224	C₂²	#15689145	1	1	yes	characteristic
225	C₂²	#15689160	1	2	yes	—
226	C₂²	#108×D₄₇	1	2	yes	—
227	C₄	—	2	1	no	—
228	C₄	—	2	1	no	—
229	C₄	—	94	4	no	—
230	C₄	—	94	4	no	—
231	C₄	C₄⋊C₈×D₄₇	1	1	yes	characteristic
232	C₄	C₂×Dic₄₇⋊C₈	1	1	yes	characteristic
233	C₄	#15689145	1	1	yes	characteristic
234	C₄	#15689146	1	1	yes	characteristic
235	C₄	#15689506	1	1	yes	characteristic
236	C₄	#15689508	1	1	yes	characteristic
237	C₄	#15689524	1	2	yes	—
238	C₂	#20242135	1	2	yes	—
239	C₂	#20242150	1	2	yes	—
240	C₂	#20243539	1	1	yes	characteristic
241	C₂	#20243551	1	1	yes	characteristic
242	C₂	#20243558	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₂×C₄₇×#335	C₂
socle, prime radical	C₂³×C₄₇	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂×#335	D₄₇
hypocenter	C₄₇	#8807
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
33. C₈×Dic₄₇	C₈	semidirect
140. C₄₇	#8807	semidirect

Maximal subgroups

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

subgroup	index
#20243504	2
C₄×C₈×Dic₄₇	2
C₂×#15688876	2
C₂×#15688873	2
C₂×C₄₇×#335	2
#8807	47