Subgroups — #27079957

Order 12032.

2666 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20256591	C₂	1	2	yes	maximal
2	#20251988	C₂	1	8	yes	maximal, complemented
3	#20251983	C₂	1	8	yes	maximal, complemented
4	#20251978	C₂	1	6	yes	maximal, complemented
5	C₂×#15689161	C₂	1	6	yes	maximal, complemented
6	C₂²×#131×C₄₇	C₂	1	1	yes	maximal, characteristic
7	#15689606	C₂²	1	16	yes	—
8	#15689601	—	2	2	no	—
9	#15689598	—	2	6	no	—
10	#41×Dic₄₇	—	2	8	no	—
11	C₂×#13093983	C₂²	1	12	yes	—
12	#15689165	C₂²	1	48	yes	complemented
13	#15689161	C₂²	1	4	yes	—
14	#15689161	C₂²	1	12	yes	—
15	#15689161	C₂²	1	12	yes	complemented
16	#15689161	C₂²	1	12	yes	complemented
17	C₂×#13093958	C₂²	1	12	yes	complemented
18	C₂×#13093956	C₂²	1	12	yes	complemented
19	C₂²×#41×C₄₇	C₂²	1	1	yes	characteristic
20	C₂×#131×C₄₇	C₄	1	16	yes	complemented
21	C₂×#131×C₄₇	C₂²	1	4	yes	—
22	C₂×#131×C₄₇	C₂²	1	4	yes	—
23	C₂²×M₁₆×C₄₇	C₂²	1	3	yes	—
24	C₂³×C₈×C₄₇	C₂²	1	3	yes	—
25	Q₈×Dic₄₇	—	2	16	no	—
26	D₄×Dic₄₇	—	2	48	no	—
27	#13093983	—	2	12	no	—
28	#13093983	—	2	12	no	—
29	#13093983	C₂³	1	8	yes	—
30	#13093983	C₂³	1	24	yes	—
31	C₂×C₄⋊Dic₄₇	C₂³	1	24	yes	—
32	C₂×C₂²⋊Dic₄₇	—	2	12	no	—
33	C₂×C₄×Dic₄₇	—	2	12	no	—
34	#13093958	C₂³	1	8	yes	—
35	#13093958	C₂³	1	24	yes	—
36	#13093956	C₂³	1	8	yes	—
37	#13093956	C₂³	1	24	yes	—
38	C₂×#41×C₄₇	C₂×C₄	1	8	yes	—
39	C₂×#41×C₄₇	D₄	1	4	yes	—
40	C₂×#41×C₄₇	Q₈	1	8	yes	—
41	C₂×#41×C₄₇	C₂³	1	4	yes	—
42	C₂²×Q₈×C₄₇	D₄	1	1	yes	characteristic
43	C₂²×D₄×C₄₇	D₄	1	3	yes	—
44	C₂³×C₄×C₄₇	C₂³	1	3	yes	—
45	#131×C₄₇	C₂×C₄	1	32	yes	complemented
46	#131×C₄₇	C₂×C₄	1	32	yes	complemented
47	C₂×M₁₆×C₄₇	C₂×C₄	1	24	yes	complemented
48	C₂×M₁₆×C₄₇	C₂³	1	12	yes	—
49	C₂²×C₈×C₄₇	C₂×C₄	1	24	yes	complemented
50	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₄₇	C₂³	1	3	yes	—
52	C₂²×C₈×C₄₇	C₂³	1	6	yes	—
53	C₂²×C₈×C₄₇	C₂³	1	6	yes	—
54	C₂⁴×C₄₇	C₂×D₄	1	3	yes	—
55	#41×C₄₇	C₄⋊C₄	1	32	yes	—
56	#41×C₄₇	C₂²×C₄	1	16	yes	—
57	#41×C₄₇	C₂×Q₈	1	16	yes	—
58	C₂×Q₈×C₄₇	C₄⋊C₄	1	8	yes	—
59	C₂×Q₈×C₄₇	C₂×D₄	1	4	yes	—
60	C₂×D₄×C₄₇	C₄⋊C₄	1	24	yes	—
61	C₂×D₄×C₄₇	C₂×D₄	1	12	yes	—
62	C₂²×C₄×C₄₇	C₂²×C₄	1	12	yes	—
63	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
64	C₂²×C₄×C₄₇	C₂×D₄	1	6	yes	—
65	C₂²×C₄×C₄₇	C₂×Q₈	1	12	yes	—
66	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂⁴	1	6	yes	—
68	C₂²×Dic₄₇	—	2	24	no	—
69	M₁₆×C₄₇	C₂²×C₄	1	48	yes	—
70	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
71	C₂×C₈×C₄₇	C₂²×C₄	1	12	yes	—
72	C₂×C₈×C₄₇	C₂²×C₄	1	24	yes	—
73	C₂×C₈×C₄₇	C₂²×C₄	1	24	yes	—
74	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
75	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
76	C₂×C₈×C₄₇	C₂⁴	1	6	yes	—
77	C₂²⋊Dic₄₇	—	2	8	no	—
78	C₂²⋊Dic₄₇	—	2	24	no	—
79	C₄⋊Dic₄₇	—	2	24	no	—
80	C₄⋊Dic₄₇	C₂⁴	1	16	yes	—
81	C₄×Dic₄₇	—	2	8	no	—
82	C₄×Dic₄₇	—	2	24	no	—
83	C₂³×C₄₇	—	4	6	no	—
84	C₂³×C₄₇	C₂×C₄⋊C₄	1	12	yes	—
85	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
86	C₂³×C₄₇	C₂²×D₄	1	6	yes	—
87	Q₈×C₄₇	C₂×C₄⋊C₄	1	16	yes	—
88	D₄×C₄₇	C₂×C₄⋊C₄	1	48	yes	—
89	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	12	yes	—
90	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	24	yes	—
91	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
92	C₂×C₄×C₄₇	C₂³×C₄	1	12	yes	—
93	C₂×C₄×C₄₇	C₂²×D₄	1	6	yes	—
94	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
95	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
96	C₂×C₄×C₄₇	C₂²×Q₈	1	12	yes	—
97	C₂×C₄×C₄₇	C₂⁵	1	1	yes	characteristic
98	C₂×Dic₄₇	—	2	16	no	—
99	C₂×Dic₄₇	—	8	24	no	—
100	C₈×C₄₇	C₂³×C₄	1	2	yes	—
101	C₈×C₄₇	C₂³×C₄	1	2	yes	—
102	C₈×C₄₇	C₂³×C₄	1	12	yes	—
103	#60108	—	47	1	no	maximal, Sylow
104	C₂²×C₄₇	—	2	12	no	—
105	C₂²×C₄₇	—	2	2	no	—
106	C₂²×C₄₇	—	4	12	no	—
107	C₂²×C₄₇	C₂²×C₄⋊C₄	1	2	yes	—
108	C₂²×C₄₇	C₂²×C₄⋊C₄	1	12	yes	—
109	C₂²×C₄₇	C₂³×D₄	1	1	yes	characteristic
110	C₄×C₄₇	C₂²×C₄⋊C₄	1	2	yes	—
111	C₄×C₄₇	C₂²×C₄⋊C₄	1	12	yes	—
112	C₄×C₄₇	C₂⁴×C₄	1	1	yes	characteristic
113	C₄×C₄₇	C₂³×Q₈	1	1	yes	characteristic
114	Dic₄₇	—	4	16	no	—
115	C₂²×#131	D₄₇	1	1	yes	characteristic
116	#3428	—	47	2	no	—
117	#2913	—	47	8	no	—
118	#2912	—	47	8	no	—
119	#2911	—	47	6	no	—
120	C₂×#427	—	47	6	no	—
121	C₂×C₄₇	—	2	2	no	—
122	C₂×C₄₇	—	2	12	no	—
123	C₂×C₄₇	C₂³×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₄₇	#3584	1	2	yes	—
125	C₂²×#41	C₂×D₄₇	1	1	yes	characteristic
126	C₂×#131	Dic₄₇	1	16	yes	complemented
127	C₂×#131	C₂×D₄₇	1	4	yes	—
128	C₂×#131	C₂×D₄₇	1	4	yes	—
129	C₂²×M₁₆	C₂×D₄₇	1	3	yes	—
130	C₂³×C₈	C₂×D₄₇	1	3	yes	—
131	#520	—	47	16	no	—
132	#519	—	94	2	no	—
133	#518	—	94	6	no	—
134	C₄×#41	—	94	8	no	—
135	C₂×#117	—	47	12	no	—
136	#428	—	47	48	no	—
137	#427	—	47	4	no	—
138	#427	—	47	12	no	—
139	#427	—	47	12	no	—
140	#427	—	47	12	no	—
141	C₂×#107	—	47	12	no	—
142	C₂×#106	—	47	12	no	—
143	C₄₇	#60108	1	1	yes	characteristic, complemented, Sylow
144	C₂×#41	Dic₉₄	1	8	yes	—
145	C₂×#41	D₁₈₈	1	4	yes	—
146	C₂×#41	C₂×Dic₄₇	1	8	yes	—
147	C₂×#41	C₂²×D₄₇	1	4	yes	—
148	C₂²×Q₈	D₁₈₈	1	1	yes	characteristic
149	C₂²×D₄	D₁₈₈	1	3	yes	—
150	C₂³×C₄	C₂²×D₄₇	1	3	yes	—
151	#131	C₂×Dic₄₇	1	32	yes	complemented
152	#131	C₂×Dic₄₇	1	32	yes	complemented
153	C₂×M₁₆	C₂×Dic₄₇	1	24	yes	complemented
154	C₂×M₁₆	C₂²×D₄₇	1	12	yes	—
155	C₂²×C₈	C₂×Dic₄₇	1	24	yes	complemented
156	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
157	C₂²×C₈	C₂²×D₄₇	1	3	yes	—
158	C₂²×C₈	C₂²×D₄₇	1	6	yes	—
159	C₂²×C₈	C₂²×D₄₇	1	6	yes	—
160	C₄×Q₈	—	94	16	no	—
161	C₄×D₄	—	94	48	no	—
162	#117	—	47	8	no	—
163	#117	—	47	24	no	—
164	#117	—	94	12	no	—
165	#117	—	94	12	no	—
166	C₂×C₄⋊C₄	—	47	24	no	—
167	C₂×C₂²⋊C₄	—	94	12	no	—
168	C₂×C₄²	—	94	12	no	—
169	#107	—	47	8	no	—
170	#107	—	47	24	no	—
171	#106	—	47	8	no	—
172	#106	—	47	24	no	—
173	C₂⁴	C₂×D₁₈₈	1	3	yes	—
174	#41	C₄⋊Dic₄₇	1	32	yes	—
175	#41	C₂×Dic₉₄	1	16	yes	—
176	#41	C₂²×Dic₄₇	1	16	yes	—
177	C₂×Q₈	C₄⋊Dic₄₇	1	8	yes	—
178	C₂×Q₈	C₂×D₁₈₈	1	4	yes	—
179	C₂×D₄	C₄⋊Dic₄₇	1	24	yes	—
180	C₂×D₄	C₂×D₁₈₈	1	12	yes	—
181	C₂²×C₄	—	94	24	no	—
182	C₂²×C₄	C₂×Dic₉₄	1	12	yes	—
183	C₂²×C₄	C₂×D₁₈₈	1	3	yes	—
184	C₂²×C₄	C₂×D₁₈₈	1	6	yes	—
185	C₂²×C₄	C₂²×Dic₄₇	1	12	yes	—
186	C₂²×C₄	C₂³×D₄₇	1	6	yes	—
187	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
188	M₁₆	C₂²×Dic₄₇	1	48	yes	—
189	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
190	C₂×C₈	C₂²×Dic₄₇	1	12	yes	—
191	C₂×C₈	C₂²×Dic₄₇	1	24	yes	—
192	C₂×C₈	C₂²×Dic₄₇	1	24	yes	—
193	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
194	C₂×C₈	C₂³×D₄₇	1	6	yes	—
195	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
196	C₄⋊C₄	—	47	16	no	—
197	C₄⋊C₄	—	94	24	no	—
198	C₂²⋊C₄	—	94	8	no	—
199	C₂²⋊C₄	—	94	24	no	—
200	C₄²	—	94	24	no	—
201	C₄²	—	94	8	no	—
202	C₂³	—	4	6	no	—
203	C₂³	C₂×C₄⋊Dic₄₇	1	12	yes	—
204	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
205	C₂³	C₂²×D₁₈₈	1	6	yes	—
206	Q₈	C₂×C₄⋊Dic₄₇	1	16	yes	—
207	D₄	C₂×C₄⋊Dic₄₇	1	48	yes	—
208	C₂×C₄	—	94	16	no	—
209	C₂×C₄	—	376	24	no	—
210	C₂×C₄	C₂×C₄⋊Dic₄₇	1	12	yes	—
211	C₂×C₄	C₂×C₄⋊Dic₄₇	1	24	yes	—
212	C₂×C₄	C₂³×Dic₄₇	1	2	yes	—
213	C₂×C₄	C₂³×Dic₄₇	1	12	yes	—
214	C₂×C₄	C₂²×D₁₈₈	1	6	yes	—
215	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
216	C₂×C₄	C₂²×Dic₉₄	1	2	yes	—
217	C₂×C₄	C₂²×Dic₉₄	1	12	yes	—
218	C₂×C₄	C₂⁴×D₄₇	1	1	yes	characteristic
219	C₈	C₂³×Dic₄₇	1	2	yes	—
220	C₈	C₂³×Dic₄₇	1	2	yes	—
221	C₈	C₂³×Dic₄₇	1	12	yes	—
222	C₂²	—	2	12	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	4	12	no	—
225	C₂²	C₂²×C₄⋊Dic₄₇	1	2	yes	—
226	C₂²	C₂²×C₄⋊Dic₄₇	1	12	yes	—
227	C₂²	C₂³×D₁₈₈	1	1	yes	characteristic
228	C₄	—	188	16	no	—
229	C₄	C₂²×C₄⋊Dic₄₇	1	2	yes	—
230	C₄	C₂²×C₄⋊Dic₄₇	1	12	yes	—
231	C₄	C₂⁴×Dic₄₇	1	1	yes	characteristic
232	C₄	C₂³×Dic₉₄	1	1	yes	characteristic
233	C₂	—	2	2	no	—
234	C₂	—	2	12	no	—
235	C₂	C₂³×C₄⋊Dic₄₇	1	1	yes	characteristic
236	C₂	#20258162	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₁₈₈
commutator	C₄×C₄₇	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂²×#131×C₄₇	C₂
socle	C₂²×C₄₇	C₂³×D₄
prime radical	C₂²×#41×C₄₇	C₂²
cosocle	C₂×C₄×C₄₇	C₂⁵
hypercenter	C₂²×#131	D₄₇
hypocenter	C₄₇	#60108
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#41 ◁ C₂²×#131

Complements

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

N	H ≅ G/N	type
2. #20251988	C₂	semidirect
3. #20251983	C₂	semidirect
4. #20251978	C₂	semidirect
5. C₂×#15689161	C₂	semidirect
12. #15689165	C₂²	semidirect
15. #15689161	C₂²	semidirect
16. #15689161	C₂²	semidirect
17. C₂×#13093958	C₂²	semidirect
18. C₂×#13093956	C₂²	semidirect
20. C₂×#131×C₄₇	C₄	semidirect
45. #131×C₄₇	C₂×C₄	semidirect
46. #131×C₄₇	C₂×C₄	semidirect
47. C₂×M₁₆×C₄₇	C₂×C₄	semidirect
49. C₂²×C₈×C₄₇	C₂×C₄	semidirect
126. C₂×#131	Dic₄₇	semidirect
143. C₄₇	#60108	semidirect
151. #131	C₂×Dic₄₇	semidirect
152. #131	C₂×Dic₄₇	semidirect
153. C₂×M₁₆	C₂×Dic₄₇	semidirect
155. C₂²×C₈	C₂×Dic₄₇	semidirect

Maximal subgroups

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

subgroup	index
#20256591	2
#20251988	2
#20251983	2
#20251978	2
C₂×#15689161	2
C₂²×#131×C₄₇	2
#60108	47