Subgroups — #26109939

Order 12032.

422 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20243540	C₂	1	2	yes	maximal
2	#20241223	C₂	1	4	yes	maximal
3	C₄²×C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
5	C₂×C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
6	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
7	C₂×C₄×C₈×C₄₇	C₄	1	2	yes	—
8	C₂×C₄×C₈×C₄₇	C₄	1	1	yes	characteristic
9	C₂×C₄×C₈×C₄₇	C₄	1	1	yes	characteristic
10	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
11	C₄³×C₄₇	C₂²	1	1	yes	characteristic
12	C₂²×C₄₇⋊C₈	—	2	4	no	—
13	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
14	C₄⋊(C₄₇⋊C₈)	—	4	4	no	—
15	C₄⋊(C₄₇⋊C₈)	—	4	4	no	—
16	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
17	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
18	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
21	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
22	C₂×C₄²×C₄₇	Q₈	1	1	yes	characteristic
23	C₂×C₄²×C₄₇	Q₈	1	1	yes	characteristic
24	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
25	C₄×C₈×C₄₇	—	2	4	no	—
26	C₄×C₈×C₄₇	—	2	2	no	—
27	C₄×C₈×C₄₇	—	2	4	no	—
28	C₄×C₈×C₄₇	C₈	1	4	yes	complemented
29	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
30	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
31	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
32	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
33	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
36	C₂×C₈×C₄₇	—	2	2	no	—
37	C₂×C₈×C₄₇	—	2	2	no	—
38	C₂×C₈×C₄₇	—	2	2	no	—
39	C₂×C₈×C₄₇	—	2	2	no	—
40	C₂×C₈×C₄₇	C₂×C₈	1	4	yes	—
41	C₂×C₈×C₄₇	M₁₆	1	4	yes	—
42	C₄²×C₄₇	—	2	2	no	—
43	C₄²×C₄₇	—	2	2	no	—
44	C₄²×C₄₇	—	2	2	no	—
45	C₄²×C₄₇	C₂×C₈	1	2	yes	—
46	C₄²×C₄₇	M₁₆	1	2	yes	—
47	C₄²×C₄₇	D₈	1	2	yes	—
48	C₄²×C₄₇	SD₁₆	1	4	yes	—
49	C₄²×C₄₇	Q₁₆	1	2	yes	—
50	C₄²×C₄₇	#41	1	4	yes	—
51	C₂×C₄₇⋊C₈	—	4	4	no	—
52	C₂×C₄₇⋊C₈	—	4	4	no	—
53	C₂×C₄₇⋊C₈	—	4	4	no	—
54	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₄₇	—	2	1	no	—
56	C₂×C₄×C₄₇	—	2	1	no	—
57	C₂×C₄×C₄₇	—	2	1	no	—
58	C₂×C₄×C₄₇	—	2	1	no	—
59	C₂×C₄×C₄₇	—	2	1	no	—
60	C₂×C₄×C₄₇	—	2	1	no	—
61	C₂×C₄×C₄₇	C₄⋊C₈	1	2	yes	—
62	C₂×C₄×C₄₇	C₄⋊C₈	1	2	yes	—
63	C₂×C₄×C₄₇	#106	1	2	yes	—
64	C₂×C₄×C₄₇	#106	1	2	yes	—
65	C₂×C₄×C₄₇	#107	1	2	yes	—
66	C₂×C₄×C₄₇	#107	1	2	yes	—
67	C₂×C₄×C₄₇	#108	1	2	yes	—
68	C₂×C₄×C₄₇	#108	1	2	yes	—
69	C₂×C₄×C₄₇	#117	1	2	yes	—
70	C₂×C₄×C₄₇	#124	1	2	yes	—
71	C₂×C₄×C₄₇	#125	1	2	yes	—
72	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
73	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
74	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
75	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
76	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
77	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
78	C₈×C₄₇	—	2	4	no	—
79	C₈×C₄₇	—	2	4	no	—
80	C₄₇⋊C₈	—	8	4	no	—
81	#8784	—	47	1	no	maximal, Sylow
82	C₂²×C₄₇	#382	1	1	yes	characteristic
83	C₂²×C₄₇	C₂×C₄⋊C₈	1	1	yes	characteristic
84	C₂²×C₄₇	#423	1	1	yes	characteristic
85	C₂²×C₄₇	C₂×#106	1	1	yes	characteristic
86	C₂²×C₄₇	C₂×#107	1	1	yes	characteristic
87	C₂²×C₄₇	C₂×#108	1	1	yes	characteristic
88	C₂²×C₄₇	C₂×#108	1	1	yes	characteristic
89	C₄×C₄₇	—	2	2	no	—
90	C₄×C₄₇	—	2	2	no	—
91	C₄×C₄₇	—	2	2	no	—
92	C₄×C₄₇	#334	1	4	yes	—
93	C₄×C₄₇	#335	1	4	yes	—
94	C₄×C₄₇	#431	1	2	yes	—
95	C₄×C₄₇	#432	1	2	yes	—
96	C₄×C₄₇	#486	1	2	yes	—
97	C₄×C₄₇	#487	1	2	yes	—
98	#1842	—	47	2	no	—
99	C₄²×C₈	D₄₇	1	1	yes	characteristic
100	#1277	—	47	4	no	—
101	C₂×C₄₇	C₂×#334	1	1	yes	characteristic
102	C₂×C₄₇	C₂×#335	1	1	yes	characteristic
103	C₂×C₄₇	#1566	1	1	yes	characteristic
104	C₂×C₄₇	#1567	1	1	yes	characteristic
105	C₂×C₄₇	#1831	1	1	yes	characteristic
106	C₂×C₄₇	#1834	1	1	yes	characteristic
107	C₂×C₄₇	#1837	1	1	yes	characteristic
108	C₂×C₄⋊C₈	—	47	4	no	—
109	C₂×C₄⋊C₈	—	94	2	no	—
110	C₂×C₄⋊C₈	—	94	2	no	—
111	C₂×C₄×C₈	Dic₄₇	1	2	yes	—
112	C₂×C₄×C₈	Dic₄₇	1	1	yes	characteristic
113	C₂×C₄×C₈	Dic₄₇	1	1	yes	characteristic
114	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
115	C₄³	C₂×D₄₇	1	1	yes	characteristic
116	C₄₇	#8784	1	1	yes	characteristic, complemented, Sylow
117	C₂²×C₈	—	94	4	no	—
118	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
119	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
120	C₂×C₄²	Dic₉₄	1	1	yes	characteristic
121	C₂×C₄²	Dic₉₄	1	1	yes	characteristic
122	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
123	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
124	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
125	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
126	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
127	C₄⋊C₈	—	94	8	no	—
128	C₄⋊C₈	—	188	4	no	—
129	C₄⋊C₈	—	188	4	no	—
130	C₄×C₈	—	2	4	no	—
131	C₄×C₈	—	2	2	no	—
132	C₄×C₈	—	2	4	no	—
133	C₄×C₈	C₄₇⋊C₈	1	4	yes	complemented
134	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
135	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
136	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
137	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
138	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
139	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
140	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	—	188	4	no	—
146	C₂×C₈	—	188	4	no	—
147	C₂×C₈	—	188	4	no	—
148	C₂×C₈	C₂×C₄₇⋊C₈	1	4	yes	—
149	C₂×C₈	#10625329	1	4	yes	—
150	C₄²	—	2	2	no	—
151	C₄²	—	2	2	no	—
152	C₄²	—	2	2	no	—
153	C₄²	#10625325	1	4	yes	—
154	C₄²	D₃₇₆	1	2	yes	—
155	C₄²	Dic₁₈₈	1	2	yes	—
156	C₄²	C₂×C₄₇⋊C₈	1	2	yes	—
157	C₄²	#10625329	1	2	yes	—
158	C₄²	#10625350	1	4	yes	—
159	C₂³	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	C₄⋊(C₄₇⋊C₈)	1	2	yes	—
167	C₂×C₄	C₄⋊(C₄₇⋊C₈)	1	2	yes	—
168	C₂×C₄	#13093956	1	2	yes	—
169	C₂×C₄	#13093956	1	2	yes	—
170	C₂×C₄	#13093958	1	2	yes	—
171	C₂×C₄	#13093958	1	2	yes	—
172	C₂×C₄	#13093960	1	2	yes	—
173	C₂×C₄	#13093960	1	2	yes	—
174	C₂×C₄	#13093983	1	2	yes	—
175	C₂×C₄	#13094012	1	2	yes	—
176	C₂×C₄	#13094016	1	2	yes	—
177	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
178	C₂×C₄	C₂×#10625329	1	2	yes	—
179	C₂×C₄	C₂×#10625329	1	1	yes	characteristic
180	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
181	C₂×C₄	C₂×#10625325	1	2	yes	—
182	C₂×C₄	C₂×Dic₁₈₈	1	1	yes	characteristic
183	C₈	—	2	4	no	—
184	C₈	—	2	4	no	—
185	C₈	—	376	4	no	—
186	C₂²	#15688989	1	1	yes	characteristic
187	C₂²	C₂×C₄⋊(C₄₇⋊C₈)	1	1	yes	characteristic
188	C₂²	#15689147	1	1	yes	characteristic
189	C₂²	C₂×#13093956	1	1	yes	characteristic
190	C₂²	C₂×#13093958	1	1	yes	characteristic
191	C₂²	C₂×#13093960	1	1	yes	characteristic
192	C₂²	C₂×#13093960	1	1	yes	characteristic
193	C₄	—	2	2	no	—
194	C₄	—	2	2	no	—
195	C₄	—	2	2	no	—
196	C₄	#15688874	1	4	yes	—
197	C₄	#15688876	1	4	yes	—
198	C₄	#15689176	1	2	yes	—
199	C₄	#15689180	1	2	yes	—
200	C₄	#15689470	1	2	yes	—
201	C₄	#15689474	1	2	yes	—
202	C₂	C₂×#15688874	1	1	yes	characteristic
203	C₂	C₂×#15688876	1	1	yes	characteristic
204	C₂	#20242144	1	1	yes	characteristic
205	C₂	#20242147	1	1	yes	characteristic
206	C₂	#20243505	1	1	yes	characteristic
207	C₂	#20243516	1	1	yes	characteristic
208	C₂	#20243526	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄²×C₈×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₄²×C₈	D₄₇
hypocenter	C₄₇	#8784
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
28. C₄×C₈×C₄₇	C₈	semidirect
116. C₄₇	#8784	semidirect
133. C₄×C₈	C₄₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#20243540	2
#20241223	2
C₄²×C₈×C₄₇	2
#8784	47