Subgroups — #20246468

Order 6016.

548 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13094003	C₂	1	1	yes	maximal, characteristic
2	#15689027	C₂	1	2	yes	maximal, complemented
3	#15689008	C₂	1	2	yes	maximal, complemented
4	#15689005	C₂	1	2	yes	maximal, complemented
5	#15688996	C₂	1	2	yes	maximal
6	#15688974	C₂	1	2	yes	maximal, complemented
7	C₄×C₂²⋊Dic₄₇	C₂	1	2	yes	maximal
8	C₄×C₂²⋊Dic₄₇	C₂	1	1	yes	maximal, characteristic
9	C₂×#95×C₄₇	C₂	1	1	yes	maximal, characteristic
10	C₂²×Dic₉₄	C₂²	1	1	yes	characteristic
11	#13094003	C₄	1	8	yes	complemented
12	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
13	C₂×Dic₄₇⋊C₄	C₂²	1	2	yes	—
14	C₂×Dic₄₇⋊C₄	C₂²	1	2	yes	—
15	C₂×Dic₄₇⋊C₄	C₂²	1	2	yes	—
16	C₂×C₂²⋊Dic₄₇	C₂²	1	1	yes	characteristic
17	C₂×C₂²⋊Dic₄₇	C₂²	1	2	yes	—
18	C₂×C₂²⋊Dic₄₇	C₂²	1	1	yes	characteristic
19	C₂×C₄×Dic₄₇	C₂²	1	2	yes	—
20	C₂×C₄×Dic₄₇	C₂²	1	2	yes	—
21	C₂×C₄×Dic₄₇	C₂²	1	2	yes	—
22	C₄₇⋊#95	C₂²	1	2	yes	—
23	C₄₇⋊#95	C₂²	1	2	yes	—
24	C₄₇⋊#95	C₂²	1	2	yes	—
25	C₄₇⋊#95	C₂²	1	2	yes	—
26	C₄₇⋊#95	C₂²	1	2	yes	—
27	C₂³×C₄×C₄₇	C₂²	1	2	yes	—
28	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
29	#95×C₄₇	C₂²	1	1	yes	characteristic
30	#95×C₄₇	C₂²	1	1	yes	characteristic
31	#95×C₄₇	C₂²	1	2	yes	—
32	C₂⁴×C₄₇	C₂³	1	1	yes	characteristic
33	C₂²×C₄×C₄₇	—	2	4	no	—
34	C₂²×C₄×C₄₇	—	2	4	no	—
35	C₂²×C₄×C₄₇	—	4	2	no	—
36	C₂²×C₄×C₄₇	—	4	2	no	—
37	C₂²×C₄×C₄₇	C₂×C₄	1	4	yes	—
38	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
39	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
42	C₂²×Dic₄₇	C₂³	1	2	yes	—
43	C₂²×Dic₄₇	C₂³	1	2	yes	—
44	C₂²×Dic₄₇	C₂³	1	2	yes	—
45	C₂²×Dic₄₇	C₂³	1	2	yes	—
46	C₂×Dic₉₄	—	4	2	no	—
47	C₂×Dic₉₄	C₂×C₄	1	4	yes	complemented
48	C₂²⋊Dic₄₇	—	2	2	no	—
49	C₂²⋊Dic₄₇	C₂×C₄	1	8	yes	—
50	C₂²⋊Dic₄₇	Q₈	1	4	yes	complemented
51	C₄⋊Dic₄₇	C₂×C₄	1	4	yes	complemented
52	Dic₄₇⋊C₄	—	2	4	no	—
53	Dic₄₇⋊C₄	—	4	2	no	—
54	Dic₄₇⋊C₄	C₂×C₄	1	8	yes	complemented
55	C₄×Dic₄₇	—	2	4	no	—
56	C₄×Dic₄₇	—	4	2	no	—
57	C₄×Dic₄₇	—	4	2	no	—
58	C₂³×C₄₇	—	2	2	no	—
59	C₂³×C₄₇	—	2	2	no	—
60	C₂³×C₄₇	C₂²×C₄	1	2	yes	—
61	C₂³×C₄₇	C₂×Q₈	1	2	yes	—
62	C₂³×C₄₇	#41	1	2	yes	—
63	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
64	C₂×C₄×C₄₇	—	2	2	no	—
65	C₂×C₄×C₄₇	—	2	2	no	—
66	C₂×C₄×C₄₇	—	2	2	no	—
67	C₂×C₄×C₄₇	—	2	2	no	—
68	C₂×C₄×C₄₇	—	2	2	no	—
69	C₂×C₄×C₄₇	—	2	2	no	—
70	C₂×C₄×C₄₇	—	2	2	no	—
71	C₂×C₄×C₄₇	—	4	4	no	—
72	C₂×C₄×C₄₇	—	4	8	no	—
73	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
74	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
75	C₂×C₄×C₄₇	#41	1	4	yes	—
76	C₂×Dic₄₇	—	2	2	no	—
77	C₂×Dic₄₇	—	2	2	no	—
78	C₂×Dic₄₇	—	2	2	no	—
79	C₂×Dic₄₇	—	2	2	no	—
80	C₂×Dic₄₇	—	2	2	no	—
81	C₂×Dic₄₇	—	2	2	no	—
82	C₂×Dic₄₇	—	2	2	no	—
83	C₂×Dic₄₇	—	2	2	no	—
84	C₂×Dic₄₇	—	2	2	no	—
85	C₂×Dic₄₇	C₂²×C₄	1	4	yes	—
86	C₂×Dic₄₇	C₂²×C₄	1	4	yes	—
87	C₂×Dic₄₇	C₂×Q₈	1	4	yes	—
88	Dic₉₄	—	4	4	no	—
89	C₂²×C₄₇	—	2	2	no	—
90	C₂²×C₄₇	—	2	2	no	—
91	C₂²×C₄₇	—	2	4	no	—
92	C₂²×C₄₇	—	2	4	no	—
93	C₂²×C₄₇	C₄×Q₈	1	4	yes	—
94	C₂²×C₄₇	C₂³×C₄	1	1	yes	characteristic
95	C₂²×C₄₇	C₂²×Q₈	1	1	yes	characteristic
96	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
97	C₂²×C₄₇	C₂×#41	1	2	yes	—
98	C₂²×C₄₇	H(5,𝔽₂)	1	2	yes	—
99	C₄×C₄₇	—	2	2	no	—
100	C₄×C₄₇	—	2	2	no	—
101	C₄×C₄₇	—	2	4	no	—
102	C₄×C₄₇	—	4	2	no	—
103	Dic₄₇	—	2	4	no	—
104	Dic₄₇	—	2	4	no	—
105	Dic₄₇	—	2	4	no	—
106	Dic₄₇	—	4	2	no	—
107	#2368	—	47	1	no	maximal, Sylow
108	C₂×C₄₇	—	2	4	no	—
109	C₂×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
110	C₂×C₄₇	C₄×#41	1	1	yes	characteristic
111	C₂×C₄₇	#518	1	1	yes	characteristic
112	C₂×C₄₇	#551	1	2	yes	—
113	C₂×C₄₇	#554	1	2	yes	—
114	C₂×C₂²⋊Q₈	—	47	1	no	—
115	#391	—	47	2	no	—
116	#387	—	47	2	no	—
117	#387	—	47	2	no	—
118	#385	—	47	2	no	—
119	C₄×C₄⋊C₄	—	47	2	no	—
120	C₄×C₂²⋊C₄	—	47	2	no	—
121	C₄×C₂²⋊C₄	—	47	1	no	—
122	C₂×#95	D₄₇	1	1	yes	characteristic
123	C₄₇	#2368	1	1	yes	characteristic, complemented, Sylow
124	C₂²×Q₈	—	47	1	no	—
125	C₂³×C₄	C₂×D₄₇	1	2	yes	—
126	C₂³×C₄	C₂×D₄₇	1	1	yes	characteristic
127	C₂²⋊Q₈	—	47	8	no	—
128	C₂×C₄⋊C₄	—	47	2	no	—
129	C₂×C₄⋊C₄	—	47	1	no	—
130	C₂×C₄⋊C₄	—	47	2	no	—
131	C₂×C₄⋊C₄	—	47	2	no	—
132	C₂×C₂²⋊C₄	—	47	2	no	—
133	C₂×C₂²⋊C₄	—	47	1	no	—
134	C₂×C₂²⋊C₄	—	47	1	no	—
135	C₂×C₄²	—	47	2	no	—
136	C₂×C₄²	—	47	2	no	—
137	C₂×C₄²	—	47	2	no	—
138	#95	—	47	2	no	—
139	#95	—	47	2	no	—
140	#95	—	47	2	no	—
141	#95	—	47	2	no	—
142	#95	—	47	2	no	—
143	#95	C₂×D₄₇	1	1	yes	characteristic
144	#95	C₂×D₄₇	1	1	yes	characteristic
145	#95	C₂×D₄₇	1	2	yes	—
146	C₂⁴	C₂²×D₄₇	1	1	yes	characteristic
147	C₂×Q₈	—	47	4	no	—
148	C₂×Q₈	—	188	2	no	—
149	C₂²×C₄	—	2	4	no	—
150	C₂²×C₄	—	2	4	no	—
151	C₂²×C₄	—	4	2	no	—
152	C₂²×C₄	—	4	2	no	—
153	C₂²×C₄	—	47	2	no	—
154	C₂²×C₄	—	47	2	no	—
155	C₂²×C₄	—	47	2	no	—
156	C₂²×C₄	—	47	2	no	—
157	C₂²×C₄	C₄×D₄₇	1	4	yes	—
158	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
159	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
160	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
161	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
162	C₄⋊C₄	—	47	8	no	—
163	C₄⋊C₄	—	47	4	no	—
164	C₄⋊C₄	—	94	4	no	—
165	C₄⋊C₄	—	188	2	no	—
166	C₂²⋊C₄	—	47	8	no	—
167	C₂²⋊C₄	—	47	4	no	—
168	C₂²⋊C₄	—	94	2	no	—
169	C₄²	—	94	4	no	—
170	C₄²	—	188	2	no	—
171	C₄²	—	188	2	no	—
172	C₂³	—	2	2	no	—
173	C₂³	—	2	2	no	—
174	C₂³	C₂×C₄×D₄₇	1	2	yes	—
175	C₂³	#10625352	1	2	yes	—
176	C₂³	Q₈×D₄₇	1	2	yes	—
177	C₂³	C₂³×D₄₇	1	1	yes	characteristic
178	Q₈	—	188	4	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	4	4	no	—
187	C₂×C₄	—	4	8	no	—
188	C₂×C₄	—	47	4	no	—
189	C₂×C₄	—	47	4	no	—
190	C₂×C₄	—	47	4	no	—
191	C₂×C₄	—	94	2	no	—
192	C₂×C₄	—	94	2	no	—
193	C₂×C₄	—	94	2	no	—
194	C₂×C₄	—	94	2	no	—
195	C₂×C₄	—	94	2	no	—
196	C₂×C₄	—	94	2	no	—
197	C₂×C₄	—	94	2	no	—
198	C₂×C₄	—	94	2	no	—
199	C₂×C₄	—	94	2	no	—
200	C₂×C₄	C₂×C₄×D₄₇	1	2	yes	—
201	C₂×C₄	C₂×C₄×D₄₇	1	2	yes	—
202	C₂×C₄	#10625350	1	4	yes	—
203	C₂²	—	2	4	no	—
204	C₂²	—	2	4	no	—
205	C₂²	—	2	2	no	—
206	C₂²	—	2	2	no	—
207	C₂²	#13093989	1	4	yes	—
208	C₂²	C₂²×C₄×D₄₇	1	1	yes	characteristic
209	C₂²	C₂×Q₈×D₄₇	1	1	yes	characteristic
210	C₂²	C₂×#10625352	1	1	yes	characteristic
211	C₂²	C₂×#10625350	1	2	yes	—
212	C₂²	#13094069	1	2	yes	—
213	C₄	—	2	4	no	—
214	C₄	—	2	2	no	—
215	C₄	—	2	2	no	—
216	C₄	—	4	2	no	—
217	C₄	—	94	4	no	—
218	C₄	—	94	4	no	—
219	C₄	—	94	4	no	—
220	C₄	—	188	2	no	—
221	C₂	—	2	4	no	—
222	C₂	C₂×#13093989	1	1	yes	characteristic
223	C₂	C₄×#10625350	1	1	yes	characteristic
224	C₂	#15689596	1	1	yes	characteristic
225	C₂	#15689799	1	2	yes	—
226	C₂	#15689828	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂³×C₄
Fitting	C₂×#95×C₄₇	C₂
socle	C₂³×C₄₇	C₂⁴
hypercenter	C₂×#95	D₄₇
hypocenter	C₄₇	#2368
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×#95

Complements

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

N	H ≅ G/N	type
2. #15689027	C₂	semidirect
3. #15689008	C₂	semidirect
4. #15689005	C₂	semidirect
6. #15688974	C₂	semidirect
11. #13094003	C₄	semidirect
47. C₂×Dic₉₄	C₂×C₄	semidirect
50. C₂²⋊Dic₄₇	Q₈	semidirect
51. C₄⋊Dic₄₇	C₂×C₄	semidirect
54. Dic₄₇⋊C₄	C₂×C₄	semidirect
123. C₄₇	#2368	semidirect