Subgroups — #26105703

Order 12032.

614 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252021	C₂	1	2	yes	maximal
2	#20241219	C₂	1	4	yes	maximal, complemented
3	C₂×C₄×M₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
4	#15689180	—	2	2	no	—
5	#15689180	—	2	2	no	—
6	#15689176	C₄	1	4	yes	—
7	#15689176	C₂²	1	4	yes	—
8	#15689147	—	2	2	no	—
9	#15689087	—	2	2	no	—
10	C₄×#10625329	C₄	1	4	yes	—
11	C₂²×M₁₆×C₄₇	C₄	1	2	yes	—
12	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
13	C₄×M₁₆×C₄₇	—	2	2	no	—
14	C₄×M₁₆×C₄₇	—	2	2	no	—
15	C₂×#97×C₄₇	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₄₇	C₄	1	2	yes	—
17	C₂×#10625329	—	2	4	no	—
18	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
19	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
20	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
21	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
22	#13093935	—	2	4	no	—
23	C₄×C₄₇⋊C₈	C₂×C₄	1	8	yes	—
24	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
26	C₂³×C₄×C₄₇	Q₈	1	1	yes	characteristic
27	C₂×M₁₆×C₄₇	—	2	4	no	—
28	C₂×M₁₆×C₄₇	—	2	8	no	—
29	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
30	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₄₇	D₄	1	2	yes	—
35	C₂×C₄²×C₄₇	D₄	1	2	yes	—
36	C₂×C₄²×C₄₇	Q₈	1	1	yes	characteristic
37	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
38	#97×C₄₇	—	2	4	no	—
39	C₄×C₈×C₄₇	—	2	2	no	—
40	C₄×C₈×C₄₇	—	2	2	no	—
41	C₂⁴×C₄₇	C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₄×C₄₇	—	2	4	no	—
43	C₂²×C₄×C₄₇	—	4	2	no	—
44	C₂²×C₄×C₄₇	—	4	2	no	—
45	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
46	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
47	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
48	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
49	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
54	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
57	M₁₆×C₄₇	—	2	16	no	—
58	C₂×C₈×C₄₇	—	2	4	no	—
59	C₂×C₈×C₄₇	—	2	4	no	—
60	C₂×C₈×C₄₇	M₁₆	1	8	yes	complemented
61	C₄²×C₄₇	C₄²	1	1	yes	characteristic
62	C₄²×C₄₇	C₄²	1	1	yes	characteristic
63	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
64	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
65	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
66	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
67	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
68	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
69	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
70	C₄²×C₄₇	C₂×D₄	1	2	yes	—
71	#10625329	—	4	8	no	—
72	C₂×C₄₇⋊C₈	—	2	8	no	—
73	C₂³×C₄₇	—	2	2	no	—
74	C₂³×C₄₇	—	4	1	no	—
75	C₂³×C₄₇	—	4	1	no	—
76	C₂³×C₄₇	#95	1	2	yes	—
77	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₄₇	—	2	2	no	—
79	C₂×C₄×C₄₇	—	2	4	no	—
80	C₂×C₄×C₄₇	—	2	2	no	—
81	C₂×C₄×C₄₇	—	2	4	no	—
82	C₂×C₄×C₄₇	—	2	2	no	—
83	C₂×C₄×C₄₇	—	2	4	no	—
84	C₂×C₄×C₄₇	—	2	4	no	—
85	C₂×C₄×C₄₇	—	4	2	no	—
86	C₂×C₄×C₄₇	—	4	2	no	—
87	C₂×C₄×C₄₇	—	4	1	no	—
88	C₂×C₄×C₄₇	—	4	1	no	—
89	C₂×C₄×C₄₇	#95	1	1	yes	characteristic
90	C₂×C₄×C₄₇	#95	1	1	yes	characteristic
91	C₂×C₄×C₄₇	#95	1	2	yes	—
92	C₂×C₄×C₄₇	#95	1	1	yes	characteristic
93	C₂×C₄×C₄₇	#95	1	1	yes	characteristic
94	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
95	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
96	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
98	C₂×C₄×C₄₇	C₂×M₁₆	1	4	yes	—
99	C₈×C₄₇	—	2	8	no	—
100	C₄₇⋊C₈	—	4	8	no	—
101	#7862	—	47	1	no	maximal, Sylow
102	C₂²×C₄₇	—	2	1	no	—
103	C₂²×C₄₇	—	2	1	no	—
104	C₂²×C₄₇	—	2	2	no	—
105	C₂²×C₄₇	—	2	2	no	—
106	C₂²×C₄₇	—	2	2	no	—
107	C₂²×C₄₇	—	4	4	no	—
108	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
109	C₂²×C₄₇	#423	1	2	yes	—
110	C₄×C₄₇	—	2	2	no	—
111	C₄×C₄₇	—	2	2	no	—
112	C₄×C₄₇	—	2	1	no	—
113	C₄×C₄₇	—	2	1	no	—
114	C₄×C₄₇	—	2	2	no	—
115	C₄×C₄₇	H(3,ℤ/4)	1	2	yes	—
116	C₄×C₄₇	#338	1	2	yes	—
117	C₄×C₄₇	C₄×M₁₆	1	2	yes	—
118	C₄×C₄₇	#407	1	2	yes	—
119	#2920	—	47	2	no	—
120	C₂×C₄×M₁₆	D₄₇	1	1	yes	characteristic
121	#1275	—	47	4	no	—
122	C₂×C₄₇	—	2	2	no	—
123	C₂×C₄₇	—	4	2	no	—
124	C₂×C₄₇	#1728	1	1	yes	characteristic
125	C₂×C₄₇	C₂×H(3,ℤ/4)	1	1	yes	characteristic
126	C₂×C₄₇	C₂×#338	1	1	yes	characteristic
127	C₂²×M₁₆	Dic₄₇	1	2	yes	—
128	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
129	#432	—	94	2	no	—
130	#432	—	94	2	no	—
131	#431	—	47	4	no	—
132	#431	—	47	4	no	—
133	#423	—	94	2	no	—
134	#407	—	94	2	no	—
135	C₄×M₁₆	—	2	2	no	—
136	C₄×M₁₆	—	2	2	no	—
137	C₄×M₁₆	—	47	4	no	—
138	C₂×#97	C₂×D₄₇	1	2	yes	—
139	C₂×C₄×C₈	Dic₄₇	1	2	yes	—
140	C₄₇	#7862	1	1	yes	characteristic, complemented, Sylow
141	C₂³×C₄	Dic₉₄	1	1	yes	characteristic
142	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
143	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
144	C₂×M₁₆	—	2	4	no	—
145	C₂×M₁₆	—	2	8	no	—
146	C₂×M₁₆	—	94	4	no	—
147	C₂²×C₈	C₂×Dic₄₇	1	4	yes	—
148	C₂×C₄²	Dic₉₄	1	1	yes	characteristic
149	C₂×C₄²	C₄×D₄₇	1	2	yes	—
150	C₂×C₄²	C₄×D₄₇	1	2	yes	—
151	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
152	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
153	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
154	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
155	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
156	C₄⋊C₈	—	94	4	no	—
157	C₄⋊C₈	—	94	4	no	—
158	C₂²⋊C₈	—	94	4	no	—
159	C₂²⋊C₈	—	94	4	no	—
160	#97	—	2	4	no	—
161	#97	—	94	4	no	—
162	C₄×C₈	—	2	2	no	—
163	C₄×C₈	—	2	2	no	—
164	C₄×C₈	—	47	8	no	—
165	C₂⁴	C₄⋊Dic₄₇	1	1	yes	characteristic
166	C₂²×C₄	—	2	4	no	—
167	C₂²×C₄	—	4	2	no	—
168	C₂²×C₄	—	4	2	no	—
169	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
170	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
171	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
172	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
173	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
174	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
175	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
176	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
177	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
178	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
179	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
180	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
181	M₁₆	—	2	16	no	—
182	M₁₆	—	188	8	no	—
183	C₂×C₈	—	2	4	no	—
184	C₂×C₈	—	2	4	no	—
185	C₂×C₈	—	94	8	no	—
186	C₂×C₈	#10625329	1	8	yes	complemented
187	C₄²	C₄×Dic₄₇	1	1	yes	characteristic
188	C₄²	C₄×Dic₄₇	1	1	yes	characteristic
189	C₄²	Dic₄₇⋊C₄	1	2	yes	—
190	C₄²	Dic₄₇⋊C₄	1	2	yes	—
191	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
192	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
193	C₄²	C₂²⋊Dic₄₇	1	1	yes	characteristic
194	C₄²	C₂²⋊Dic₄₇	1	1	yes	characteristic
195	C₄²	C₂×C₄×D₄₇	1	2	yes	—
196	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
197	C₂³	—	2	2	no	—
198	C₂³	—	4	1	no	—
199	C₂³	—	4	1	no	—
200	C₂³	C₄₇⋊#95	1	2	yes	—
201	C₂³	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	2	4	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	4	2	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	—	4	1	no	—
212	C₂×C₄	—	4	1	no	—
213	C₂×C₄	C₄₇⋊#95	1	1	yes	characteristic
214	C₂×C₄	C₄₇⋊#95	1	1	yes	characteristic
215	C₂×C₄	C₄₇⋊#95	1	2	yes	—
216	C₂×C₄	C₄₇⋊#95	1	1	yes	characteristic
217	C₂×C₄	C₄₇⋊#95	1	1	yes	characteristic
218	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
219	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	2	yes	—
220	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
221	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
222	C₂×C₄	C₂×#10625329	1	4	yes	—
223	C₈	—	2	8	no	—
224	C₈	—	188	8	no	—
225	C₂²	—	2	1	no	—
226	C₂²	—	2	1	no	—
227	C₂²	—	2	2	no	—
228	C₂²	—	2	2	no	—
229	C₂²	—	2	2	no	—
230	C₂²	—	4	4	no	—
231	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
232	C₂²	#15689147	1	2	yes	—
233	C₄	—	2	2	no	—
234	C₄	—	2	2	no	—
235	C₄	—	2	1	no	—
236	C₄	—	2	1	no	—
237	C₄	—	2	2	no	—
238	C₄	#15688880	1	2	yes	—
239	C₄	C₄₇⋊#338	1	2	yes	—
240	C₄	C₄×#10625329	1	2	yes	—
241	C₄	#15689087	1	2	yes	—
242	C₂	—	2	2	no	—
243	C₂	—	4	2	no	—
244	C₂	#20243082	1	1	yes	characteristic
245	C₂	C₂×#15688880	1	1	yes	characteristic
246	C₂	C₂×C₄₇⋊#338	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₄₇
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₂×C₄×M₁₆×C₄₇	C₂
socle	C₂²×C₄₇	C₂×#95
prime radical	C₂⁴×C₄₇	C₄⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂×C₄×M₁₆	D₄₇
hypocenter	C₄₇	#7862
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. #20241219	C₂	semidirect
60. C₂×C₈×C₄₇	M₁₆	semidirect
140. C₄₇	#7862	semidirect
186. C₂×C₈	#10625329	semidirect

Maximal subgroups

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

subgroup	index
#20252021	2
#20241219	2
C₂×C₄×M₁₆×C₄₇	2
#7862	47