Subgroups — #26376308

Order 12032.

5318 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689601	C₂	1	2	yes	maximal, complemented
2	#20246229	C₂	1	16	yes	maximal, complemented
3	#20245970	C₂	1	6	yes	maximal, complemented
4	#20245771	C₂	1	6	yes	maximal, complemented
5	C₂³×#41×C₄₇	C₂	1	1	yes	maximal, characteristic
6	#15689601	—	2	16	no	—
7	C₂×Q₈×Dic₄₇	C₂²	1	16	yes	complemented
8	C₂×#13093983	C₂²	1	12	yes	—
9	#15689026	C₂²	1	48	yes	complemented
10	#15688986	C₂²	1	4	yes	—
11	#15688983	C₂²	1	24	yes	complemented
12	#15688979	C₂²	1	12	yes	—
13	C₄×C₂²⋊Dic₄₇	C₂²	1	24	yes	complemented
14	C₂²×#41×C₄₇	—	2	16	no	—
15	C₂²×#41×C₄₇	C₄	1	16	yes	complemented
16	C₂²×#41×C₄₇	C₂²	1	8	yes	—
17	C₂³×Q₈×C₄₇	C₂²	1	1	yes	characteristic
18	C₂³×D₄×C₄₇	C₂²	1	3	yes	—
19	C₂⁴×C₄×C₄₇	C₂²	1	3	yes	—
20	Q₈×Dic₄₇	—	2	64	no	—
21	#13093983	—	2	48	no	—
22	C₂×C₄⋊Dic₄₇	C₂³	1	24	yes	—
23	C₂×C₂²⋊Dic₄₇	C₂³	1	8	yes	—
24	C₂×C₂²⋊Dic₄₇	C₂³	1	48	yes	—
25	C₂×C₄×Dic₄₇	C₂³	1	24	yes	—
26	C₄₇⋊#95	C₂³	1	16	yes	—
27	C₂⁵×C₄₇	C₂³	1	3	yes	—
28	C₂×#41×C₄₇	—	2	128	no	—
29	C₂×#41×C₄₇	—	2	64	no	—
30	C₂×#41×C₄₇	C₂×C₄	1	64	yes	—
31	C₂²×Q₈×C₄₇	C₂×C₄	1	8	yes	—
32	C₂²×Q₈×C₄₇	D₄	1	16	yes	—
33	C₂²×Q₈×C₄₇	C₂³	1	4	yes	—
34	C₂²×D₄×C₄₇	—	2	24	no	—
35	C₂²×D₄×C₄₇	C₂×C₄	1	24	yes	—
36	C₂²×D₄×C₄₇	C₂³	1	12	yes	—
37	C₂³×C₄×C₄₇	—	2	24	no	—
38	C₂³×C₄×C₄₇	C₂×C₄	1	24	yes	—
39	C₂³×C₄×C₄₇	C₂³	1	12	yes	—
40	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
41	C₂³×C₄×C₄₇	C₂³	1	3	yes	—
42	C₂⁴×C₄₇	—	2	12	no	—
43	C₂⁴×C₄₇	—	4	12	no	—
44	C₂⁴×C₄₇	C₂²×C₄	1	12	yes	—
45	C₂⁴×C₄₇	C₂⁴	1	6	yes	—
46	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
47	#41×C₄₇	—	2	256	no	—
48	C₂×Q₈×C₄₇	C₂²⋊C₄	1	64	yes	complemented
49	C₂×Q₈×C₄₇	C₂²×C₄	1	16	yes	—
50	C₂×Q₈×C₄₇	C₂×D₄	1	32	yes	—
51	C₂×D₄×C₄₇	—	2	96	no	—
52	C₂×D₄×C₄₇	—	2	48	no	—
53	C₂×D₄×C₄₇	C₂²×C₄	1	48	yes	—
54	C₂²×C₄×C₄₇	—	2	4	no	—
55	C₂²×C₄×C₄₇	—	2	96	no	—
56	C₂²×C₄×C₄₇	—	2	48	no	—
57	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
58	C₂²×C₄×C₄₇	C₂²×C₄	1	48	yes	—
59	C₂²×C₄×C₄₇	C₂²×C₄	1	12	yes	—
60	C₂²×C₄×C₄₇	C₂×D₄	1	24	yes	—
61	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
62	C₂²×C₄×C₄₇	C₂⁴	1	6	yes	—
63	C₂²×Dic₄₇	C₂⁴	1	16	yes	—
64	C₂²⋊Dic₄₇	—	2	16	no	—
65	C₂²⋊Dic₄₇	—	4	48	no	—
66	C₄⋊Dic₄₇	—	2	48	no	—
67	C₄×Dic₄₇	—	2	48	no	—
68	C₂³×C₄₇	—	2	4	no	—
69	C₂³×C₄₇	—	2	24	no	—
70	C₂³×C₄₇	—	2	12	no	—
71	C₂³×C₄₇	—	2	24	no	—
72	C₂³×C₄₇	—	4	24	no	—
73	C₂³×C₄₇	—	4	48	no	—
74	C₂³×C₄₇	C₂³×C₄	1	12	yes	—
75	C₂³×C₄₇	C₂³×C₄	1	2	yes	—
76	C₂³×C₄₇	C₂²×D₄	1	4	yes	—
77	C₂³×C₄₇	C₂⁵	1	1	yes	characteristic
78	Q₈×C₄₇	C₂×C₂²⋊C₄	1	64	yes	—
79	D₄×C₄₇	—	2	96	no	—
80	C₂×C₄×C₄₇	—	2	8	no	—
81	C₂×C₄×C₄₇	—	2	4	no	—
82	C₂×C₄×C₄₇	—	2	96	no	—
83	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	48	yes	—
84	C₂×C₄×C₄₇	C₂³×C₄	1	4	yes	—
85	C₂×C₄×C₄₇	C₂³×C₄	1	12	yes	—
86	C₂×C₄×C₄₇	C₂²×D₄	1	24	yes	—
87	C₂×Dic₄₇	—	2	16	no	—
88	C₂×Dic₄₇	—	2	32	no	—
89	#34174	—	47	1	no	maximal, Sylow
90	C₂²×C₄₇	—	2	4	no	—
91	C₂²×C₄₇	—	2	8	no	—
92	C₂²×C₄₇	—	2	12	no	—
93	C₂²×C₄₇	—	2	24	no	—
94	C₂²×C₄₇	—	4	48	no	—
95	C₂²×C₄₇	—	4	24	no	—
96	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	4	yes	—
97	C₂²×C₄₇	C₂⁴×C₄	1	1	yes	characteristic
98	C₂²×C₄₇	C₂³×D₄	1	2	yes	—
99	C₂²×C₄₇	#585	1	4	yes	—
100	C₄×C₄₇	—	2	4	no	—
101	C₄×C₄₇	C₂²×C₂²⋊C₄	1	24	yes	—
102	Dic₄₇	—	4	16	no	—
103	C₂³×#41	D₄₇	1	1	yes	characteristic
104	C₂×#519	—	47	2	no	—
105	#2341	—	47	16	no	—
106	#2309	—	47	6	no	—
107	#2280	—	47	6	no	—
108	C₂×C₄₇	—	2	4	no	—
109	C₂×C₄₇	—	4	12	no	—
110	C₂×C₄₇	C₂³×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄₇	#3428	1	2	yes	—
112	C₂×C₄₇	#3487	1	4	yes	—
113	C₂²×#41	—	2	16	no	—
114	C₂²×#41	Dic₄₇	1	16	yes	complemented
115	C₂²×#41	C₂×D₄₇	1	8	yes	—
116	C₂³×Q₈	C₂×D₄₇	1	1	yes	characteristic
117	C₂³×D₄	C₂×D₄₇	1	3	yes	—
118	C₂⁴×C₄	C₂×D₄₇	1	3	yes	—
119	#519	—	94	16	no	—
120	C₂×C₄×Q₈	—	47	16	no	—
121	C₂×#117	—	47	12	no	—
122	#390	—	47	48	no	—
123	#381	—	47	4	no	—
124	#380	—	47	24	no	—
125	#379	—	47	12	no	—
126	C₄×C₂²⋊C₄	—	47	24	no	—
127	C₄₇	#34174	1	1	yes	characteristic, complemented, Sylow
128	C₂⁵	C₂²×D₄₇	1	3	yes	—
129	C₂×#41	—	2	128	no	—
130	C₂×#41	—	2	64	no	—
131	C₂×#41	C₂×Dic₄₇	1	64	yes	—
132	C₂²×Q₈	C₂×Dic₄₇	1	8	yes	—
133	C₂²×Q₈	C₂²⋊D₄₇	1	16	yes	—
134	C₂²×Q₈	C₂²×D₄₇	1	4	yes	—
135	C₂²×D₄	—	2	24	no	—
136	C₂²×D₄	C₂×Dic₄₇	1	24	yes	—
137	C₂²×D₄	C₂²×D₄₇	1	12	yes	—
138	C₂³×C₄	—	2	24	no	—
139	C₂³×C₄	C₂×Dic₄₇	1	24	yes	—
140	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
141	C₂³×C₄	C₂²×D₄₇	1	12	yes	—
142	C₂³×C₄	C₂²×D₄₇	1	3	yes	—
143	C₄×Q₈	—	94	64	no	—
144	#117	—	94	48	no	—
145	C₂×C₄⋊C₄	—	47	24	no	—
146	C₂×C₂²⋊C₄	—	47	8	no	—
147	C₂×C₂²⋊C₄	—	47	48	no	—
148	C₂×C₄²	—	47	24	no	—
149	#95	—	47	16	no	—
150	C₂⁴	—	2	12	no	—
151	C₂⁴	—	4	12	no	—
152	C₂⁴	C₂²×Dic₄₇	1	12	yes	—
153	C₂⁴	C₂³×D₄₇	1	6	yes	—
154	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
155	#41	—	2	256	no	—
156	C₂×Q₈	C₂²⋊Dic₄₇	1	64	yes	complemented
157	C₂×Q₈	C₂²×Dic₄₇	1	16	yes	—
158	C₂×Q₈	C₂×C₂²⋊D₄₇	1	32	yes	—
159	C₂×D₄	—	2	96	no	—
160	C₂×D₄	—	2	48	no	—
161	C₂×D₄	C₂²×Dic₄₇	1	48	yes	—
162	C₂²×C₄	—	2	4	no	—
163	C₂²×C₄	—	2	96	no	—
164	C₂²×C₄	—	2	48	no	—
165	C₂²×C₄	—	47	16	no	—
166	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
167	C₂²×C₄	C₂²×Dic₄₇	1	48	yes	—
168	C₂²×C₄	C₂²×Dic₄₇	1	12	yes	—
169	C₂²×C₄	C₂×C₂²⋊D₄₇	1	24	yes	—
170	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
171	C₂²×C₄	C₂³×D₄₇	1	6	yes	—
172	C₄⋊C₄	—	94	48	no	—
173	C₂²⋊C₄	—	94	16	no	—
174	C₂²⋊C₄	—	188	48	no	—
175	C₄²	—	94	48	no	—
176	C₂³	—	2	4	no	—
177	C₂³	—	2	24	no	—
178	C₂³	—	2	12	no	—
179	C₂³	—	2	24	no	—
180	C₂³	—	4	24	no	—
181	C₂³	—	4	48	no	—
182	C₂³	C₂³×Dic₄₇	1	12	yes	—
183	C₂³	C₂³×Dic₄₇	1	2	yes	—
184	C₂³	C₂²×C₂²⋊D₄₇	1	4	yes	—
185	C₂³	C₂⁴×D₄₇	1	1	yes	characteristic
186	Q₈	C₂×C₂²⋊Dic₄₇	1	64	yes	—
187	D₄	—	2	96	no	—
188	C₂×C₄	—	2	8	no	—
189	C₂×C₄	—	2	4	no	—
190	C₂×C₄	—	2	96	no	—
191	C₂×C₄	—	94	16	no	—
192	C₂×C₄	—	94	32	no	—
193	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	48	yes	—
194	C₂×C₄	C₂³×Dic₄₇	1	4	yes	—
195	C₂×C₄	C₂³×Dic₄₇	1	12	yes	—
196	C₂×C₄	C₂²×C₂²⋊D₄₇	1	24	yes	—
197	C₂²	—	2	8	no	—
198	C₂²	—	2	4	no	—
199	C₂²	—	2	12	no	—
200	C₂²	—	2	24	no	—
201	C₂²	—	4	48	no	—
202	C₂²	—	4	24	no	—
203	C₂²	C₂²×C₂²⋊Dic₄₇	1	4	yes	—
204	C₂²	C₂⁴×Dic₄₇	1	1	yes	characteristic
205	C₂²	C₂³×C₂²⋊D₄₇	1	2	yes	—
206	C₂²	#15689959	1	4	yes	—
207	C₄	—	2	4	no	—
208	C₄	—	188	16	no	—
209	C₄	C₂²×C₂²⋊Dic₄₇	1	24	yes	—
210	C₂	—	2	4	no	—
211	C₂	—	4	12	no	—
212	C₂	C₂³×C₂²⋊Dic₄₇	1	1	yes	characteristic
213	C₂	#20256591	1	2	yes	—
214	C₂	#20257046	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂⁴×D₄₇
commutator	C₂²×C₄₇	C₂⁴×C₄
Fitting, prime radical	C₂³×#41×C₄₇	C₂
socle, cosocle	C₂³×C₄₇	C₂⁵
hypercenter	C₂³×#41	D₄₇
hypocenter	C₄₇	#34174
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂³×#41

Complements

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

N	H ≅ G/N	type
1. C₂×#15689601	C₂	semidirect
2. #20246229	C₂	semidirect
3. #20245970	C₂	semidirect
4. #20245771	C₂	semidirect
7. C₂×Q₈×Dic₄₇	C₂²	semidirect
9. #15689026	C₂²	semidirect
11. #15688983	C₂²	semidirect
13. C₄×C₂²⋊Dic₄₇	C₂²	semidirect
15. C₂²×#41×C₄₇	C₄	semidirect
48. C₂×Q₈×C₄₇	C₂²⋊C₄	semidirect
114. C₂²×#41	Dic₄₇	semidirect
127. C₄₇	#34174	semidirect
156. C₂×Q₈	C₂²⋊Dic₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689601	2
#20246229	2
#20245970	2
#20245771	2
C₂³×#41×C₄₇	2
#34174	47