Subgroups — #27079979

Order 12032.

2564 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689890	C₂	1	1	yes	maximal, characteristic
2	C₂×#15689890	C₂	1	1	yes	maximal, characteristic
3	#20251997	C₂	1	16	yes	maximal, complemented
4	C₂×#15689171	C₂	1	12	yes	maximal, complemented
5	C₂×C₄₇×#568	C₂	1	1	yes	maximal, characteristic
6	#15689890	—	2	4	no	—
7	#15689890	C₄	1	8	yes	—
8	#15689890	C₄	1	8	yes	—
9	#15689890	C₂²	1	8	yes	—
10	C₂×#13094031	—	2	4	no	—
11	C₂×#13094031	C₂²	1	8	yes	—
12	C₂²×#10625329	C₂²	1	6	yes	—
13	C₂²×#10625329	C₂²	1	6	yes	—
14	#15689171	C₂²	1	16	yes	—
15	#15689171	C₂²	1	48	yes	complemented
16	C₂×#13093959	C₂²	1	48	yes	complemented
17	C₂²×#41×C₄₇	C₂²	1	1	yes	characteristic
18	C₄₇×#568	—	2	4	no	—
19	C₄₇×#568	C₂²	1	8	yes	—
20	C₂×#131×C₄₇	—	2	4	no	—
21	C₂²×M₁₆×C₄₇	C₂²	1	6	yes	—
22	#13094031	—	2	32	no	—
23	#13094031	C₂×C₄	1	64	yes	—
24	C₂×#10625329	—	2	6	no	—
25	C₂×#10625329	—	2	12	no	—
26	C₂×#10625329	C₂×C₄	1	24	yes	—
27	C₂×#10625329	C₂×C₄	1	24	yes	—
28	C₂×#10625329	C₂³	1	4	yes	—
29	C₂×#10625329	C₂³	1	24	yes	—
30	C₂×#10625329	C₂³	1	4	yes	—
31	C₂×#10625329	C₂³	1	12	yes	—
32	C₂²×C₄₇⋊C₈	—	2	6	no	—
33	C₂²×C₄₇⋊C₈	C₂³	1	12	yes	—
34	#13093959	C₂³	1	64	yes	—
35	C₂×#41×C₄₇	C₂×C₄	1	4	yes	—
36	C₂×#41×C₄₇	C₂×C₄	1	4	yes	—
37	C₂×#41×C₄₇	D₄	1	4	yes	—
38	C₂×#41×C₄₇	D₄	1	4	yes	—
39	C₂×#41×C₄₇	Q₈	1	4	yes	—
40	C₂×#41×C₄₇	C₂³	1	4	yes	—
41	C₂²×Q₈×C₄₇	Q₈	1	1	yes	characteristic
42	C₂²×D₄×C₄₇	Q₈	1	3	yes	—
43	C₂³×C₄×C₄₇	C₂³	1	3	yes	—
44	#131×C₄₇	—	2	32	no	—
45	C₂×M₁₆×C₄₇	—	2	6	no	—
46	C₂×M₁₆×C₄₇	—	2	12	no	—
47	C₂×M₁₆×C₄₇	C₂³	1	4	yes	—
48	C₂×M₁₆×C₄₇	C₂³	1	24	yes	—
49	C₂²×C₈×C₄₇	—	2	6	no	—
50	C₂⁴×C₄₇	C₂×Q₈	1	3	yes	—
51	#41×C₄₇	C₄⋊C₄	1	16	yes	—
52	#41×C₄₇	C₄⋊C₄	1	16	yes	—
53	#41×C₄₇	C₂²×C₄	1	16	yes	—
54	#41×C₄₇	C₂×D₄	1	16	yes	—
55	C₂×Q₈×C₄₇	C₄⋊C₄	1	4	yes	—
56	C₂×Q₈×C₄₇	C₄⋊C₄	1	4	yes	—
57	C₂×Q₈×C₄₇	C₂×Q₈	1	4	yes	—
58	C₂×D₄×C₄₇	C₄⋊C₄	1	12	yes	—
59	C₂×D₄×C₄₇	C₄⋊C₄	1	12	yes	—
60	C₂×D₄×C₄₇	C₂×Q₈	1	12	yes	—
61	C₂²×C₄×C₄₇	C₂²×C₄	1	6	yes	—
62	C₂²×C₄×C₄₇	C₂²×C₄	1	6	yes	—
63	C₂²×C₄×C₄₇	C₂×D₄	1	6	yes	—
64	C₂²×C₄×C₄₇	C₂×D₄	1	6	yes	—
65	C₂²×C₄×C₄₇	C₂×Q₈	1	3	yes	—
66	C₂²×C₄×C₄₇	C₂×Q₈	1	6	yes	—
67	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	C₂⁴	1	6	yes	—
69	M₁₆×C₄₇	—	2	4	no	—
70	M₁₆×C₄₇	—	2	24	no	—
71	M₁₆×C₄₇	C₂⁴	1	8	yes	—
72	C₂×C₈×C₄₇	—	2	4	no	—
73	C₂×C₈×C₄₇	—	2	24	no	—
74	#10625329	—	2	4	no	—
75	#10625329	—	2	24	no	—
76	#10625329	C₂²×C₄	1	8	yes	—
77	#10625329	C₂²×C₄	1	8	yes	—
78	#10625329	C₂²×C₄	1	48	yes	—
79	#10625329	C₂⁴	1	8	yes	—
80	C₂×C₄₇⋊C₈	—	2	4	no	—
81	C₂×C₄₇⋊C₈	—	2	24	no	—
82	C₂×C₄₇⋊C₈	C₂²×C₄	1	48	yes	—
83	C₂×C₄₇⋊C₈	C₂⁴	1	8	yes	—
84	C₂³×C₄₇	—	8	3	no	—
85	C₂³×C₄₇	C₂×C₄⋊C₄	1	6	yes	—
86	C₂³×C₄₇	C₂×C₄⋊C₄	1	6	yes	—
87	C₂³×C₄₇	C₂²×Q₈	1	1	yes	characteristic
88	C₂³×C₄₇	C₂²×Q₈	1	6	yes	—
89	Q₈×C₄₇	C₂×C₄⋊C₄	1	16	yes	—
90	D₄×C₄₇	C₂×C₄⋊C₄	1	48	yes	—
91	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	6	yes	—
92	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	6	yes	—
93	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	12	yes	—
94	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	12	yes	—
95	C₂×C₄×C₄₇	C₂³×C₄	1	12	yes	—
96	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
97	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
98	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
99	C₂×C₄×C₄₇	C₂²×D₄	1	12	yes	—
100	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
101	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
102	C₂×C₄×C₄₇	C₂²×Q₈	1	6	yes	—
103	C₂×C₄×C₄₇	C₂⁵	1	1	yes	characteristic
104	C₈×C₄₇	—	2	8	no	—
105	C₄₇⋊C₈	—	2	8	no	—
106	C₄₇⋊C₈	C₂³×C₄	1	16	yes	—
107	#60113	—	47	1	no	maximal, Sylow
108	C₂²×C₄₇	—	4	1	no	—
109	C₂²×C₄₇	—	4	6	no	—
110	C₂²×C₄₇	—	4	6	no	—
111	C₂²×C₄₇	—	4	6	no	—
112	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
113	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
114	C₂²×C₄₇	C₂²×C₄⋊C₄	1	12	yes	—
115	C₂²×C₄₇	C₂³×Q₈	1	1	yes	characteristic
116	C₄×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
117	C₄×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
118	C₄×C₄₇	C₂²×C₄⋊C₄	1	12	yes	—
119	C₄×C₄₇	C₂⁴×C₄	1	1	yes	characteristic
120	C₄×C₄₇	C₂³×D₄	1	1	yes	characteristic
121	C₂×#568	—	47	1	no	—
122	C₂×#568	—	47	1	no	—
123	C₂×#568	D₄₇	1	1	yes	characteristic
124	#2916	—	47	16	no	—
125	C₂×#430	—	47	12	no	—
126	C₂×C₄₇	—	2	1	no	—
127	C₂×C₄₇	—	2	1	no	—
128	C₂×C₄₇	—	2	1	no	—
129	C₂×C₄₇	—	2	12	no	—
130	C₂×C₄₇	C₂³×C₄⋊C₄	1	1	yes	characteristic
131	C₂²×#41	C₂×D₄₇	1	1	yes	characteristic
132	#568	—	2	4	no	—
133	#568	—	47	8	no	—
134	#568	—	47	8	no	—
135	#568	—	47	8	no	—
136	#568	—	94	4	no	—
137	#568	C₂×D₄₇	1	8	yes	—
138	C₂×#131	—	2	4	no	—
139	C₂×#131	—	47	8	no	—
140	C₂×#131	—	94	4	no	—
141	C₂²×M₁₆	—	47	6	no	—
142	C₂²×M₁₆	—	47	6	no	—
143	C₂²×M₁₆	C₂×D₄₇	1	6	yes	—
144	#430	—	47	16	no	—
145	#430	—	47	48	no	—
146	C₂×#108	—	47	48	no	—
147	C₄₇	#60113	1	1	yes	characteristic, complemented, Sylow
148	C₂×#41	Dic₉₄	1	4	yes	—
149	C₂×#41	C₄×D₄₇	1	4	yes	—
150	C₂×#41	C₄×D₄₇	1	4	yes	—
151	C₂×#41	C₂²⋊D₄₇	1	4	yes	—
152	C₂×#41	C₂²⋊D₄₇	1	4	yes	—
153	C₂×#41	C₂²×D₄₇	1	4	yes	—
154	C₂²×Q₈	Dic₉₄	1	1	yes	characteristic
155	C₂²×D₄	Dic₉₄	1	3	yes	—
156	C₂³×C₄	C₂²×D₄₇	1	3	yes	—
157	#131	—	2	32	no	—
158	#131	—	47	64	no	—
159	#131	—	94	32	no	—
160	C₂×M₁₆	—	2	6	no	—
161	C₂×M₁₆	—	2	12	no	—
162	C₂×M₁₆	—	47	24	no	—
163	C₂×M₁₆	—	47	24	no	—
164	C₂×M₁₆	—	47	4	no	—
165	C₂×M₁₆	—	47	12	no	—
166	C₂×M₁₆	—	47	4	no	—
167	C₂×M₁₆	—	47	24	no	—
168	C₂×M₁₆	—	94	6	no	—
169	C₂×M₁₆	—	94	12	no	—
170	C₂×M₁₆	C₂²×D₄₇	1	4	yes	—
171	C₂×M₁₆	C₂²×D₄₇	1	24	yes	—
172	C₂²×C₈	—	2	6	no	—
173	C₂²×C₈	—	47	12	no	—
174	C₂²×C₈	—	94	6	no	—
175	#108	—	47	64	no	—
176	C₂⁴	C₂×Dic₉₄	1	3	yes	—
177	#41	Dic₄₇⋊C₄	1	16	yes	—
178	#41	Dic₄₇⋊C₄	1	16	yes	—
179	#41	C₂×C₄×D₄₇	1	16	yes	—
180	#41	C₂×C₂²⋊D₄₇	1	16	yes	—
181	C₂×Q₈	Dic₄₇⋊C₄	1	4	yes	—
182	C₂×Q₈	Dic₄₇⋊C₄	1	4	yes	—
183	C₂×Q₈	C₂×Dic₉₄	1	4	yes	—
184	C₂×D₄	Dic₄₇⋊C₄	1	12	yes	—
185	C₂×D₄	Dic₄₇⋊C₄	1	12	yes	—
186	C₂×D₄	C₂×Dic₉₄	1	12	yes	—
187	C₂²×C₄	C₂×Dic₉₄	1	3	yes	—
188	C₂²×C₄	C₂×Dic₉₄	1	6	yes	—
189	C₂²×C₄	C₂×C₄×D₄₇	1	6	yes	—
190	C₂²×C₄	C₂×C₄×D₄₇	1	6	yes	—
191	C₂²×C₄	C₂×C₂²⋊D₄₇	1	6	yes	—
192	C₂²×C₄	C₂×C₂²⋊D₄₇	1	6	yes	—
193	C₂²×C₄	C₂³×D₄₇	1	6	yes	—
194	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
195	M₁₆	—	2	4	no	—
196	M₁₆	—	2	24	no	—
197	M₁₆	—	47	8	no	—
198	M₁₆	—	47	8	no	—
199	M₁₆	—	47	48	no	—
200	M₁₆	—	47	8	no	—
201	M₁₆	—	94	4	no	—
202	M₁₆	—	94	24	no	—
203	M₁₆	C₂³×D₄₇	1	8	yes	—
204	C₂×C₈	—	2	4	no	—
205	C₂×C₈	—	2	24	no	—
206	C₂×C₈	—	47	48	no	—
207	C₂×C₈	—	47	8	no	—
208	C₂×C₈	—	94	4	no	—
209	C₂×C₈	—	94	24	no	—
210	C₂³	—	8	3	no	—
211	C₂³	C₂×Dic₄₇⋊C₄	1	6	yes	—
212	C₂³	C₂×Dic₄₇⋊C₄	1	6	yes	—
213	C₂³	C₂²×Dic₉₄	1	1	yes	characteristic
214	C₂³	C₂²×Dic₉₄	1	6	yes	—
215	Q₈	C₂×Dic₄₇⋊C₄	1	16	yes	—
216	D₄	C₂×Dic₄₇⋊C₄	1	48	yes	—
217	C₂×C₄	C₂×Dic₄₇⋊C₄	1	6	yes	—
218	C₂×C₄	C₂×Dic₄₇⋊C₄	1	6	yes	—
219	C₂×C₄	C₂×Dic₄₇⋊C₄	1	12	yes	—
220	C₂×C₄	C₂×Dic₄₇⋊C₄	1	12	yes	—
221	C₂×C₄	C₂²×C₄×D₄₇	1	12	yes	—
222	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
223	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
224	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
225	C₂×C₄	C₂²×C₂²⋊D₄₇	1	12	yes	—
226	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
227	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
228	C₂×C₄	C₂²×Dic₉₄	1	6	yes	—
229	C₂×C₄	C₂⁴×D₄₇	1	1	yes	characteristic
230	C₈	—	2	8	no	—
231	C₈	—	47	16	no	—
232	C₈	—	94	8	no	—
233	C₂²	—	4	1	no	—
234	C₂²	—	4	6	no	—
235	C₂²	—	4	6	no	—
236	C₂²	—	4	6	no	—
237	C₂²	C₂²×Dic₄₇⋊C₄	1	1	yes	characteristic
238	C₂²	C₂²×Dic₄₇⋊C₄	1	1	yes	characteristic
239	C₂²	C₂²×Dic₄₇⋊C₄	1	12	yes	—
240	C₂²	C₂³×Dic₉₄	1	1	yes	characteristic
241	C₄	C₂²×Dic₄₇⋊C₄	1	1	yes	characteristic
242	C₄	C₂²×Dic₄₇⋊C₄	1	1	yes	characteristic
243	C₄	C₂²×Dic₄₇⋊C₄	1	12	yes	—
244	C₄	C₂³×C₄×D₄₇	1	1	yes	characteristic
245	C₄	C₂³×C₂²⋊D₄₇	1	1	yes	characteristic
246	C₂	—	2	1	no	—
247	C₂	—	2	1	no	—
248	C₂	—	2	1	no	—
249	C₂	—	2	12	no	—
250	C₂	C₂³×Dic₄₇⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂³×C₂²⋊D₄₇
commutator	C₄×C₄₇	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂×C₄₇×#568	C₂
socle	C₂×C₄₇	C₂³×C₄⋊C₄
prime radical	C₂²×#41×C₄₇	C₂²
cosocle	C₂×C₄×C₄₇	C₂⁵
hypercenter	C₂×#568	D₄₇
hypocenter	C₄₇	#60113
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄ ◁ C₂²×#41 ◁ C₂×#568

Complements

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

N	H ≅ G/N	type
3. #20251997	C₂	semidirect
4. C₂×#15689171	C₂	semidirect
15. #15689171	C₂²	semidirect
16. C₂×#13093959	C₂²	semidirect
147. C₄₇	#60113	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689890	2
#20251997	2
C₂×#15689171	2
C₂×C₄₇×#568	2
#60113	47