Subgroups — #26121428

Order 12032.

1222 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093935	C₂	1	2	yes	maximal
2	C₂×#15688878	C₂	1	4	yes	maximal
3	C₂×C₄³×C₄₇	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₄₇⋊C₈	C₂²	1	4	yes	—
5	C₂×#13093935	—	2	4	no	—
6	C₂×#13093935	—	2	4	no	—
7	C₂×#13093935	C₄	1	8	yes	complemented
8	#15688878	C₄	1	16	yes	complemented
9	C₂²×C₄²×C₄₇	C₄	1	4	yes	—
10	C₂²×C₄²×C₄₇	C₂²	1	2	yes	—
11	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
12	C₄³×C₄₇	—	2	4	no	—
13	C₂²×C₄₇⋊C₈	—	2	4	no	—
14	C₂²×C₄₇⋊C₈	—	2	4	no	—
15	C₂²×C₄₇⋊C₈	—	2	8	no	—
16	C₂²×C₄₇⋊C₈	C₂×C₄	1	8	yes	—
17	C₂²×C₄₇⋊C₈	C₂×C₄	1	16	yes	—
18	#13093935	—	2	16	no	—
19	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
20	C₂³×C₄×C₄₇	C₂×C₄	1	4	yes	—
21	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
22	C₂×C₄²×C₄₇	—	2	8	no	—
23	C₂×C₄²×C₄₇	—	2	16	no	—
24	C₂×C₄²×C₄₇	—	2	4	no	—
25	C₂×C₄²×C₄₇	—	2	4	no	—
26	C₂×C₄²×C₄₇	C₂×C₄	1	8	yes	—
27	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
28	C₂×C₄²×C₄₇	D₄	1	2	yes	—
29	C₂×C₄²×C₄₇	D₄	1	4	yes	—
30	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
31	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₄₇	—	2	4	no	—
33	C₂²×C₄×C₄₇	—	2	4	no	—
34	C₂²×C₄×C₄₇	—	2	8	no	—
35	C₂²×C₄×C₄₇	—	2	8	no	—
36	C₂²×C₄×C₄₇	C₄²	1	4	yes	—
37	C₂²×C₄×C₄₇	C₄²	1	8	yes	—
38	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
39	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
40	C₂²×C₄×C₄₇	M₁₆	1	16	yes	—
41	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
42	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
43	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
45	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
46	C₂²×C₄×C₄₇	#41	1	4	yes	—
47	C₄²×C₄₇	—	2	32	no	—
48	C₄²×C₄₇	—	2	16	no	—
49	C₄²×C₄₇	C₂²⋊C₄	1	8	yes	—
50	C₄²×C₄₇	C₄⋊C₄	1	8	yes	—
51	C₂×C₄₇⋊C₈	—	2	8	no	—
52	C₂×C₄₇⋊C₈	—	2	8	no	—
53	C₂×C₄₇⋊C₈	—	2	16	no	—
54	C₂×C₄₇⋊C₈	—	2	16	no	—
55	C₂×C₄₇⋊C₈	C₄²	1	16	yes	complemented
56	C₂³×C₄₇	C₂×C₄²	1	1	yes	characteristic
57	C₂³×C₄₇	C₂×C₄²	1	2	yes	—
58	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂³×C₄₇	#117	1	2	yes	—
61	C₂³×C₄₇	C₂×M₁₆	1	8	yes	—
62	C₂×C₄×C₄₇	—	2	8	no	—
63	C₂×C₄×C₄₇	—	2	8	no	—
64	C₂×C₄×C₄₇	—	2	8	no	—
65	C₂×C₄×C₄₇	—	2	8	no	—
66	C₂×C₄×C₄₇	—	2	16	no	—
67	C₂×C₄×C₄₇	—	2	8	no	—
68	C₂×C₄×C₄₇	—	2	16	no	—
69	C₂×C₄×C₄₇	#95	1	8	yes	—
70	C₂×C₄×C₄₇	#97	1	16	yes	complemented
71	C₂×C₄×C₄₇	C₂×C₄²	1	4	yes	—
72	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
73	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
74	C₂×C₄×C₄₇	#117	1	4	yes	—
75	C₂×C₄×C₄₇	C₄×D₄	1	8	yes	—
76	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
77	C₂×C₄×C₄₇	C₄×Q₈	1	4	yes	—
78	C₄₇⋊C₈	—	2	16	no	—
79	#10693	—	47	1	no	maximal, Sylow
80	C₂²×C₄₇	C₄³	1	1	yes	characteristic
81	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
82	C₂²×C₄₇	#376	1	1	yes	characteristic
83	C₂²×C₄₇	C₄×C₂²⋊C₄	1	2	yes	—
84	C₂²×C₄₇	C₄×C₄⋊C₄	1	2	yes	—
85	C₂²×C₄₇	C₂×#97	1	4	yes	—
86	C₂²×C₄₇	C₄×M₁₆	1	4	yes	—
87	C₂²×C₄₇	C₄×M₁₆	1	4	yes	—
88	C₂²×C₄₇	#407	1	4	yes	—
89	C₂²×C₄₇	#423	1	4	yes	—
90	C₂²×C₄₇	#432	1	8	yes	—
91	C₄×C₄₇	—	2	8	no	—
92	C₄×C₄₇	—	2	16	no	—
93	C₄×C₄₇	C₄×C₂²⋊C₄	1	4	yes	—
94	C₄×C₄₇	C₄×C₄⋊C₄	1	4	yes	—
95	C₂²×#97	—	47	2	no	—
96	C₂×C₄³	D₄₇	1	1	yes	characteristic
97	C₂×#336	—	47	4	no	—
98	C₂×C₄₇	C₄×#95	1	1	yes	characteristic
99	C₂×C₄₇	C₄×#97	1	2	yes	—
100	C₂×C₄₇	#1728	1	2	yes	—
101	C₂×C₄₇	#1745	1	2	yes	—
102	C₂×C₄₇	#1749	1	4	yes	—
103	C₂×C₄₇	#1767	1	4	yes	—
104	C₂³×C₈	—	47	4	no	—
105	C₂²×C₄²	Dic₄₇	1	4	yes	—
106	C₂²×C₄²	C₂×D₄₇	1	2	yes	—
107	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
108	C₂×#97	—	47	8	no	—
109	C₂×#97	—	94	4	no	—
110	C₂×#97	—	94	4	no	—
111	C₄³	—	2	4	no	—
112	#336	—	47	16	no	—
113	C₄₇	#10693	1	1	yes	characteristic, complemented, Sylow
114	C₂³×C₄	C₂×Dic₄₇	1	4	yes	—
115	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
116	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
117	C₂²×C₈	—	47	16	no	—
118	C₂²×C₈	—	47	8	no	—
119	C₂²×C₈	—	94	4	no	—
120	C₂²×C₈	—	94	8	no	—
121	C₂²×C₈	—	94	4	no	—
122	C₂×C₄²	—	2	16	no	—
123	C₂×C₄²	—	2	8	no	—
124	C₂×C₄²	—	2	4	no	—
125	C₂×C₄²	—	2	4	no	—
126	C₂×C₄²	Dic₉₄	1	2	yes	—
127	C₂×C₄²	C₄×D₄₇	1	8	yes	—
128	C₂×C₄²	C₄×D₄₇	1	4	yes	—
129	C₂×C₄²	D₁₈₈	1	2	yes	—
130	C₂×C₄²	C₂²⋊D₄₇	1	4	yes	—
131	#97	—	94	16	no	—
132	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
133	C₂²×C₄	—	2	4	no	—
134	C₂²×C₄	—	2	4	no	—
135	C₂²×C₄	—	2	8	no	—
136	C₂²×C₄	—	2	8	no	—
137	C₂²×C₄	#10625329	1	16	yes	—
138	C₂²×C₄	C₄×Dic₄₇	1	4	yes	—
139	C₂²×C₄	C₄×Dic₄₇	1	8	yes	—
140	C₂²×C₄	C₄⋊Dic₄₇	1	4	yes	—
141	C₂²×C₄	C₂²⋊Dic₄₇	1	4	yes	—
142	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
143	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
144	C₂²×C₄	C₂×C₄×D₄₇	1	4	yes	—
145	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
146	C₂²×C₄	#10625350	1	4	yes	—
147	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
148	C₂×C₈	—	47	16	no	—
149	C₂×C₈	—	94	8	no	—
150	C₂×C₈	—	94	8	no	—
151	C₂×C₈	—	94	16	no	—
152	C₂×C₈	—	94	16	no	—
153	C₄²	—	2	32	no	—
154	C₄²	—	2	16	no	—
155	C₄²	Dic₄₇⋊C₄	1	8	yes	—
156	C₄²	(C₂×D₄₇)⋊C₄	1	8	yes	—
157	C₂³	C₂×C₄×Dic₄₇	1	1	yes	characteristic
158	C₂³	C₂×C₄×Dic₄₇	1	2	yes	—
159	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
160	C₂³	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
161	C₂³	#13093983	1	2	yes	—
162	C₂³	C₂×#10625329	1	8	yes	—
163	C₂×C₄	—	2	8	no	—
164	C₂×C₄	—	2	8	no	—
165	C₂×C₄	—	2	8	no	—
166	C₂×C₄	—	2	16	no	—
167	C₂×C₄	—	2	8	no	—
168	C₂×C₄	—	2	8	no	—
169	C₂×C₄	—	2	16	no	—
170	C₂×C₄	C₄₇⋊#95	1	8	yes	—
171	C₂×C₄	#13093935	1	16	yes	complemented
172	C₂×C₄	C₄²×D₄₇	1	4	yes	—
173	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	2	yes	—
174	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
175	C₂×C₄	#13093981	1	4	yes	—
176	C₂×C₄	C₄×C₂²⋊D₄₇	1	8	yes	—
177	C₂×C₄	C₄×D₁₈₈	1	4	yes	—
178	C₂×C₄	C₄×Dic₉₄	1	4	yes	—
179	C₈	—	94	16	no	—
180	C₂²	C₄²×Dic₄₇	1	1	yes	characteristic
181	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
182	C₂²	#15688969	1	1	yes	characteristic
183	C₂²	C₄×C₂²⋊Dic₄₇	1	2	yes	—
184	C₂²	C₄×C₄⋊Dic₄₇	1	2	yes	—
185	C₂²	C₂×#13093935	1	4	yes	—
186	C₂²	C₄×#10625329	1	4	yes	—
187	C₂²	C₄×#10625329	1	4	yes	—
188	C₂²	#15689087	1	4	yes	—
189	C₂²	#15689147	1	4	yes	—
190	C₂²	#15689180	1	8	yes	—
191	C₄	—	2	8	no	—
192	C₄	—	2	16	no	—
193	C₄	C₄×(C₂×D₄₇)⋊C₄	1	4	yes	—
194	C₄	C₄×Dic₄₇⋊C₄	1	4	yes	—
195	C₂	C₄×C₄₇⋊#95	1	1	yes	characteristic
196	C₂	C₄×#13093935	1	2	yes	—
197	C₂	#20243082	1	2	yes	—
198	C₂	#20243137	1	2	yes	—
199	C₂	#20243150	1	4	yes	—
200	C₂	#20243241	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂³×C₄	C₂²×D₄₇
commutator	C₂²×C₄₇	C₄³
Fitting	C₂×C₄³×C₄₇	C₂
socle, prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂×C₄³	D₄₇
hypocenter	C₄₇	#10693
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³×C₄ ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
7. C₂×#13093935	C₄	semidirect
8. #15688878	C₄	semidirect
55. C₂×C₄₇⋊C₈	C₄²	semidirect
70. C₂×C₄×C₄₇	#97	semidirect
113. C₄₇	#10693	semidirect
171. C₂×C₄	#13093935	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13093935	2
C₂×#15688878	2
C₂×C₄³×C₄₇	2
#10693	47