Subgroups — #26126085

Order 12032.

362 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20245144	C₂	1	1	yes	maximal, characteristic
2	C₄×#13093964	C₂	1	1	yes	maximal, characteristic
3	Q₈⋊(C₄₇⋊C₁₆)	C₂	1	4	yes	maximal, complemented
4	C₂×C₈×Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#13093964	—	2	1	no	—
6	C₂×#13093964	—	2	1	no	—
7	C₄⋊(C₄₇⋊C₁₆)	—	2	1	no	—
8	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	2	yes	—
9	#15688899	—	2	1	no	—
10	C₄×C₄₇⋊C₁₆	C₂²	1	2	yes	—
11	C₂×C₄×Q₈×C₄₇	C₄	1	1	yes	characteristic
12	C₈×Q₈×C₄₇	—	2	1	no	—
13	C₈×Q₈×C₄₇	—	2	1	no	—
14	C₈×Q₈×C₄₇	C₄	1	2	yes	—
15	C₈×Q₈×C₄₇	C₂²	1	2	yes	—
16	C₂×C₄⋊C₈×C₄₇	—	2	1	no	—
17	C₂×C₄⋊C₈×C₄₇	C₄	1	1	yes	characteristic
18	C₂×C₄×C₈×C₄₇	—	2	1	no	—
19	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
20	#13093964	—	2	4	no	—
21	#13093964	—	4	2	no	—
22	C₂×C₄₇⋊C₁₆	—	2	2	no	—
23	C₂×C₄₇⋊C₁₆	—	2	2	no	—
24	C₂²×Q₈×C₄₇	C₈	1	1	yes	characteristic
25	C₂²×C₈×C₄₇	—	2	1	no	—
26	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
28	C₄×Q₈×C₄₇	—	2	1	no	—
29	C₄×Q₈×C₄₇	—	2	1	no	—
30	C₄×Q₈×C₄₇	C₈	1	2	yes	—
31	C₄×Q₈×C₄₇	C₂×C₄	1	2	yes	—
32	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
33	C₂×C₄⋊C₄×C₄₇	C₈	1	1	yes	characteristic
34	C₂×C₄²×C₄₇	—	2	1	no	—
35	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
36	C₄⋊C₈×C₄₇	—	2	2	no	—
37	C₄⋊C₈×C₄₇	—	2	2	no	—
38	C₄⋊C₈×C₄₇	—	2	1	no	—
39	C₄⋊C₈×C₄₇	C₂×C₄	1	2	yes	—
40	C₄×C₈×C₄₇	—	2	2	no	—
41	C₄×C₈×C₄₇	—	2	2	no	—
42	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
44	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
45	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
46	C₂×Q₈×C₄₇	—	2	4	no	—
47	C₂×Q₈×C₄₇	—	2	1	no	—
48	C₂×Q₈×C₄₇	C₂×C₈	1	2	yes	—
49	C₂²×C₄×C₄₇	—	2	1	no	—
50	C₂²×C₄×C₄₇	—	2	1	no	—
51	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	M₁₆	1	1	yes	characteristic
54	C₂×C₈×C₄₇	—	2	2	no	—
55	C₂×C₈×C₄₇	—	2	2	no	—
56	C₂×C₈×C₄₇	—	4	2	no	—
57	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₈×C₄₇	SD₁₆	1	2	yes	—
60	C₂×C₈×C₄₇	Q₁₆	1	2	yes	—
61	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
63	C₄⋊C₄×C₄₇	—	2	2	no	—
64	C₄⋊C₄×C₄₇	—	2	1	no	—
65	C₄⋊C₄×C₄₇	—	2	2	no	—
66	C₄⋊C₄×C₄₇	C₂×C₈	1	2	yes	—
67	C₄²×C₄₇	—	2	2	no	—
68	C₄²×C₄₇	—	2	2	no	—
69	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
70	C₄²×C₄₇	C₂×C₈	1	1	yes	characteristic
71	C₄²×C₄₇	M₁₆	1	1	yes	characteristic
72	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
73	C₄₇⋊C₁₆	—	2	4	no	—
74	C₄₇⋊C₁₆	—	4	2	no	—
75	C₂³×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
76	Q₈×C₄₇	—	2	4	no	—
77	Q₈×C₄₇	—	2	2	no	—
78	Q₈×C₄₇	M₃₂	1	4	yes	complemented
79	C₂×C₄×C₄₇	—	2	4	no	—
80	C₂×C₄×C₄₇	—	2	2	no	—
81	C₂×C₄×C₄₇	—	2	2	no	—
82	C₂×C₄×C₄₇	—	2	2	no	—
83	C₂×C₄×C₄₇	—	4	1	no	—
84	C₂×C₄×C₄₇	—	4	2	no	—
85	C₂×C₄×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
86	C₂×C₄×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
87	C₂×C₄×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
88	C₂×C₄×C₄₇	#103	1	2	yes	—
89	C₂×C₄×C₄₇	#104	1	2	yes	—
90	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
92	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
93	C₈×C₄₇	—	2	1	no	—
94	C₈×C₄₇	—	2	1	no	—
95	C₈×C₄₇	—	4	2	no	—
96	C₈×C₄₇	#103	1	2	yes	—
97	C₈×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
98	C₈×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
99	#11607	—	47	1	no	maximal, Sylow
100	C₂²×C₄₇	—	2	1	no	—
101	C₂²×C₄₇	—	2	1	no	—
102	C₂²×C₄₇	Q₈⋊C₈	1	2	yes	—
103	C₂²×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
104	C₄×C₄₇	—	2	4	no	—
105	C₄×C₄₇	—	2	1	no	—
106	C₄×C₄₇	—	2	1	no	—
107	C₄×C₄₇	—	2	1	no	—
108	C₄×C₄₇	—	4	2	no	—
109	C₄×C₄₇	Q₈⋊C₈	1	2	yes	—
110	C₄×C₄₇	C₂×#103	1	1	yes	characteristic
111	C₄×C₄₇	C₂×#104	1	1	yes	characteristic
112	C₄×C₄₇	C₂×M₃₂	1	2	yes	—
113	C₂×C₈×Q₈	D₄₇	1	1	yes	characteristic
114	#2150	—	47	1	no	—
115	C₄×M₃₂	—	47	1	no	—
116	Q₈⋊C₁₆	—	47	4	no	—
117	C₂×C₄₇	—	2	2	no	—
118	C₂×C₄₇	C₂×Q₈⋊C₈	1	1	yes	characteristic
119	C₂×C₄₇	#2112	1	1	yes	characteristic
120	C₂×C₄₇	#2117	1	1	yes	characteristic
121	C₂×C₄×Q₈	Dic₄₇	1	1	yes	characteristic
122	C₂×M₃₂	—	94	1	no	—
123	C₂×M₃₂	—	94	1	no	—
124	C₈×Q₈	—	2	1	no	—
125	C₈×Q₈	—	2	1	no	—
126	C₈×Q₈	Dic₄₇	1	2	yes	—
127	C₈×Q₈	C₂×D₄₇	1	2	yes	—
128	C₂×C₄⋊C₈	—	2	1	no	—
129	C₂×C₄⋊C₈	Dic₄₇	1	1	yes	characteristic
130	C₂×C₄×C₈	—	2	1	no	—
131	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
132	C₄⋊C₁₆	—	47	2	no	—
133	C₄⋊C₁₆	—	94	1	no	—
134	#346	—	94	1	no	—
135	C₄×C₁₆	—	47	2	no	—
136	C₄₇	#11607	1	1	yes	characteristic, complemented, Sylow
137	C₂²×Q₈	C₄₇⋊C₈	1	1	yes	characteristic
138	C₂²×C₈	—	2	1	no	—
139	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
140	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
141	C₄×Q₈	—	2	1	no	—
142	C₄×Q₈	—	2	1	no	—
143	C₄×Q₈	C₄₇⋊C₈	1	2	yes	—
144	C₄×Q₈	C₂×Dic₄₇	1	2	yes	—
145	C₂×C₄⋊C₄	—	2	1	no	—
146	C₂×C₄⋊C₄	C₄₇⋊C₈	1	1	yes	characteristic
147	C₂×C₄²	—	2	1	no	—
148	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
149	M₃₂	—	94	4	no	—
150	M₃₂	—	188	2	no	—
151	C₂×C₁₆	—	94	2	no	—
152	C₂×C₁₆	—	94	2	no	—
153	C₄⋊C₈	—	2	2	no	—
154	C₄⋊C₈	—	2	2	no	—
155	C₄⋊C₈	—	2	1	no	—
156	C₄⋊C₈	C₂×Dic₄₇	1	2	yes	—
157	C₄×C₈	—	2	2	no	—
158	C₄×C₈	—	2	2	no	—
159	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
160	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
161	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
162	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
163	C₂×Q₈	—	2	4	no	—
164	C₂×Q₈	—	2	1	no	—
165	C₂×Q₈	C₂×C₄₇⋊C₈	1	2	yes	—
166	C₂²×C₄	—	2	1	no	—
167	C₂²×C₄	—	2	1	no	—
168	C₂²×C₄	C₂×C₄₇⋊C₈	1	1	yes	characteristic
169	C₂²×C₄	#10625329	1	1	yes	characteristic
170	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
171	C₂×C₈	—	2	2	no	—
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	—	4	2	no	—
174	C₂×C₈	#10625336	1	2	yes	—
175	C₂×C₈	#10625337	1	2	yes	—
176	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
177	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
178	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
179	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
180	C₄⋊C₄	—	2	2	no	—
181	C₄⋊C₄	—	2	1	no	—
182	C₄⋊C₄	—	2	2	no	—
183	C₄⋊C₄	C₂×C₄₇⋊C₈	1	2	yes	—
184	C₄²	—	2	2	no	—
185	C₄²	—	2	2	no	—
186	C₄²	C₂×C₄₇⋊C₈	1	1	yes	characteristic
187	C₄²	#10625329	1	1	yes	characteristic
188	C₄²	C₂²⋊Dic₄₇	1	1	yes	characteristic
189	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
190	C₁₆	—	94	4	no	—
191	C₁₆	—	188	2	no	—
192	C₂³	C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
193	Q₈	—	2	4	no	—
194	Q₈	—	2	2	no	—
195	Q₈	#13093964	1	4	yes	complemented
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	4	1	no	—
201	C₂×C₄	—	4	2	no	—
202	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
203	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
204	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
205	C₂×C₄	#13093948	1	2	yes	—
206	C₂×C₄	#13093951	1	2	yes	—
207	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
208	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
209	C₂×C₄	C₂×#10625329	1	1	yes	characteristic
210	C₈	—	2	1	no	—
211	C₈	—	2	1	no	—
212	C₈	—	4	2	no	—
213	C₈	#13093948	1	2	yes	—
214	C₈	C₂×#10625336	1	1	yes	characteristic
215	C₈	C₂×#10625337	1	1	yes	characteristic
216	C₂²	—	2	1	no	—
217	C₂²	—	2	1	no	—
218	C₂²	Q₈⋊(C₄₇⋊C₈)	1	2	yes	—
219	C₂²	C₂×C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
220	C₄	—	2	4	no	—
221	C₄	—	2	1	no	—
222	C₄	—	2	1	no	—
223	C₄	—	2	1	no	—
224	C₄	—	4	2	no	—
225	C₄	Q₈⋊(C₄₇⋊C₈)	1	2	yes	—
226	C₄	C₂×#13093948	1	1	yes	characteristic
227	C₄	C₂×#13093951	1	1	yes	characteristic
228	C₄	C₂×#13093964	1	2	yes	—
229	C₂	—	2	2	no	—
230	C₂	C₂×Q₈⋊(C₄₇⋊C₈)	1	1	yes	characteristic
231	C₂	#20244999	1	1	yes	characteristic
232	C₂	#20245016	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	C₂×C₈×Q₈×C₄₇	C₂
socle	C₂²×C₄₇	C₂×C₂²⋊C₈
prime radical	C₂³×C₄₇	C₂²⋊C₈
cosocle	C₄×C₈×C₄₇	C₂³
hypercenter	C₂×C₈×Q₈	D₄₇
hypocenter	C₄₇	#11607
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
3. Q₈⋊(C₄₇⋊C₁₆)	C₂	semidirect
78. Q₈×C₄₇	M₃₂	semidirect
136. C₄₇	#11607	semidirect
195. Q₈	#13093964	semidirect

Maximal subgroups

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

subgroup	index
#20245144	2
C₄×#13093964	2
Q₈⋊(C₄₇⋊C₁₆)	2
C₂×C₈×Q₈×C₄₇	2
#11607	47