Subgroups — #26368876

Order 12032.

894 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689533	C₂	1	4	yes	maximal
2	#20251786	C₂	1	1	yes	maximal, characteristic
3	#20243557	C₂	1	4	yes	maximal
4	#20243547	C₂	1	2	yes	maximal
5	#20243544	C₂	1	2	yes	maximal
6	C₄×#13093935	C₂	1	1	yes	maximal, characteristic
7	C₄₇×#2859	C₂	1	1	yes	maximal, characteristic
8	C₂×#13094024	—	2	2	no	—
9	C₂×#13094024	C₂²	1	2	yes	—
10	#15689533	—	2	16	no	—
11	C₂×#13093957	C₂²	1	8	yes	—
12	C₂×#13093955	C₂²	1	8	yes	—
13	C₂×#13093935	C₂²	1	2	yes	—
14	C₂×#13093935	C₂²	1	2	yes	—
15	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
16	#15688993	—	2	2	no	—
17	#15688993	C₂²	1	2	yes	—
18	#15688993	C₂²	1	2	yes	—
19	C₂×#125×C₄₇	C₂²	1	2	yes	—
20	C₄₇×#389	—	2	2	no	—
21	C₄₇×#389	—	2	2	no	—
22	C₄₇×#384	C₂²	1	1	yes	characteristic
23	C₄₇×#384	C₂²	1	1	yes	characteristic
24	C₄₇×#382	C₂²	1	2	yes	—
25	C₄³×C₄₇	C₂²	1	1	yes	characteristic
26	C₂²×Dic₉₄	—	2	2	no	—
27	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
28	#13094024	—	2	8	no	—
29	#13094024	—	4	8	no	—
30	C₂×C₄⋊Dic₄₇	—	2	2	no	—
31	C₂×C₄⋊Dic₄₇	—	2	4	no	—
32	C₂×C₄⋊Dic₄₇	—	2	4	no	—
33	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
34	#13093957	—	2	16	no	—
35	#13093955	—	2	16	no	—
36	#13093935	D₄	1	8	yes	—
37	#13093935	Q₈	1	8	yes	complemented
38	C₄×C₄₇⋊C₈	Q₈	1	8	yes	complemented
39	#125×C₄₇	—	2	8	no	—
40	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
41	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
42	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
43	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
44	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
45	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
46	C₂×C₄²×C₄₇	D₄	1	2	yes	—
47	C₂×C₄²×C₄₇	D₄	1	2	yes	—
48	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
51	#95×C₄₇	—	2	4	no	—
52	C₂²×C₄×C₄₇	—	2	2	no	—
53	C₂²×C₄×C₄₇	—	2	2	no	—
54	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
58	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
59	C₂²×Dic₄₇	—	2	4	no	—
60	C₂×Dic₉₄	—	4	8	no	—
61	C₂×Dic₉₄	—	4	4	no	—
62	C₄⋊C₄×C₄₇	—	2	4	no	—
63	C₄⋊C₄×C₄₇	—	2	4	no	—
64	C₄⋊C₄×C₄₇	—	4	4	no	—
65	C₄⋊C₄×C₄₇	—	4	4	no	—
66	C₄⋊C₄×C₄₇	—	4	4	no	—
67	C₄⋊C₄×C₄₇	—	4	4	no	—
68	C₄²×C₄₇	—	2	4	no	—
69	C₄²×C₄₇	—	2	4	no	—
70	C₄²×C₄₇	C₂×D₄	1	4	yes	—
71	C₄²×C₄₇	C₂×Q₈	1	4	yes	—
72	C₄²×C₄₇	C₂×Q₈	1	4	yes	—
73	C₄⋊Dic₄₇	—	2	8	no	—
74	C₄⋊Dic₄₇	—	4	8	no	—
75	C₄⋊Dic₄₇	—	4	4	no	—
76	C₄⋊Dic₄₇	—	4	8	no	—
77	C₂×C₄₇⋊C₈	C₂×D₄	1	8	yes	—
78	C₂×C₄₇⋊C₈	C₂×Q₈	1	8	yes	—
79	C₂×C₄₇⋊C₈	C₂×Q₈	1	8	yes	—
80	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
81	C₂×C₄×C₄₇	—	2	2	no	—
82	C₂×C₄×C₄₇	—	2	2	no	—
83	C₂×C₄×C₄₇	—	2	2	no	—
84	C₂×C₄×C₄₇	—	2	2	no	—
85	C₂×C₄×C₄₇	—	4	2	no	—
86	C₂×C₄×C₄₇	—	4	2	no	—
87	C₂×C₄×C₄₇	—	4	4	no	—
88	C₂×C₄×C₄₇	—	4	4	no	—
89	C₂×C₄×C₄₇	#127	1	2	yes	—
90	C₂×C₄×C₄₇	#127	1	2	yes	—
91	C₂×C₄×C₄₇	C₄⋊Q₈	1	4	yes	—
92	C₂×C₄×C₄₇	C₄⋊Q₈	1	4	yes	—
93	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	4	yes	—
94	C₂×C₄×C₄₇	#137	1	4	yes	—
95	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
96	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
97	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
98	C₂×Dic₄₇	—	4	4	no	—
99	C₂×Dic₄₇	—	4	8	no	—
100	Dic₉₄	—	8	8	no	—
101	C₄₇⋊C₈	C₄⋊Q₈	1	16	yes	complemented
102	#33316	—	47	1	no	maximal, Sylow
103	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
104	C₂²×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
105	C₂²×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
106	C₂²×C₄₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
107	C₂²×C₄₇	C₂×#137	1	2	yes	—
108	C₄×C₄₇	—	2	2	no	—
109	C₄×C₄₇	—	2	2	no	—
110	C₄×C₄₇	—	2	4	no	—
111	C₄×C₄₇	—	8	2	no	—
112	C₄×C₄₇	—	8	2	no	—
113	C₄×C₄₇	#501	1	8	yes	complemented
114	C₄×C₄₇	C₂×C₄⋊Q₈	1	4	yes	—
115	Dic₄₇	—	8	4	no	—
116	C₂×#501	—	47	4	no	—
117	#2868	—	47	1	no	—
118	#2859	D₄₇	1	1	yes	characteristic
119	#1847	—	47	4	no	—
120	#1845	—	47	2	no	—
121	#1844	—	47	2	no	—
122	C₄×#97	—	47	1	no	—
123	C₂×C₄₇	#2868	1	1	yes	characteristic
124	C₂×C₄₇	#3151	1	1	yes	characteristic
125	C₂×C₄₇	#3152	1	1	yes	characteristic
126	C₂×C₄₇	C₂×#501	1	2	yes	—
127	C₂×C₄₇	#3165	1	2	yes	—
128	C₂×C₄⋊Q₈	—	47	2	no	—
129	C₂×C₄⋊Q₈	—	94	2	no	—
130	C₂×#125	C₂×D₄₇	1	2	yes	—
131	#501	—	94	16	no	—
132	C₂×#107	—	47	8	no	—
133	C₂×#106	—	47	8	no	—
134	C₂×#97	—	47	2	no	—
135	C₂×#97	—	47	2	no	—
136	C₂×C₄×C₈	—	47	2	no	—
137	#389	—	2	2	no	—
138	#389	—	2	2	no	—
139	#384	—	47	2	no	—
140	#384	—	47	2	no	—
141	#384	—	94	2	no	—
142	#384	C₂×D₄₇	1	1	yes	characteristic
143	#384	C₂×D₄₇	1	1	yes	characteristic
144	#382	C₂×D₄₇	1	2	yes	—
145	C₄³	C₂×D₄₇	1	1	yes	characteristic
146	C₄₇	#33316	1	1	yes	characteristic, complemented, Sylow
147	C₂²×Q₈	—	94	2	no	—
148	C₂²×C₈	—	47	4	no	—
149	C₄⋊Q₈	—	94	8	no	—
150	C₄⋊Q₈	—	188	8	no	—
151	#125	—	2	8	no	—
152	C₂×C₄⋊C₄	—	2	2	no	—
153	C₂×C₄⋊C₄	—	2	2	no	—
154	C₂×C₄⋊C₄	—	2	2	no	—
155	C₂×C₄⋊C₄	—	2	2	no	—
156	C₂×C₄⋊C₄	—	47	4	no	—
157	C₂×C₄⋊C₄	—	94	2	no	—
158	C₂×C₄⋊C₄	—	94	4	no	—
159	C₂×C₄⋊C₄	—	94	4	no	—
160	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
161	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
162	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
163	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
164	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
165	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
166	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
167	#107	—	94	16	no	—
168	#106	—	94	16	no	—
169	#97	—	47	8	no	—
170	#97	—	47	8	no	—
171	C₄×C₈	—	47	8	no	—
172	#95	—	2	4	no	—
173	C₂×Q₈	—	188	8	no	—
174	C₂×Q₈	—	188	4	no	—
175	C₂²×C₄	—	2	2	no	—
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	—	94	4	no	—
178	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
179	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
180	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
181	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
182	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
183	C₂×C₈	—	47	8	no	—
184	C₂×C₈	—	47	8	no	—
185	C₂×C₈	—	47	8	no	—
186	C₄⋊C₄	—	2	4	no	—
187	C₄⋊C₄	—	2	4	no	—
188	C₄⋊C₄	—	4	4	no	—
189	C₄⋊C₄	—	4	4	no	—
190	C₄⋊C₄	—	4	4	no	—
191	C₄⋊C₄	—	4	4	no	—
192	C₄⋊C₄	—	94	8	no	—
193	C₄⋊C₄	—	188	8	no	—
194	C₄⋊C₄	—	188	4	no	—
195	C₄⋊C₄	—	188	8	no	—
196	C₄²	—	2	4	no	—
197	C₄²	—	2	4	no	—
198	C₄²	D₄×D₄₇	1	4	yes	—
199	C₄²	Q₈×D₄₇	1	4	yes	—
200	C₄²	Q₈×D₄₇	1	4	yes	—
201	C₂³	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
202	Q₈	—	376	8	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	4	4	no	—
208	C₂×C₄	—	4	4	no	—
209	C₂×C₄	—	4	2	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	—	188	8	no	—
212	C₂×C₄	—	188	4	no	—
213	C₂×C₄	#13094020	1	2	yes	—
214	C₂×C₄	#13094020	1	2	yes	—
215	C₂×C₄	#13094022	1	4	yes	—
216	C₂×C₄	#13094022	1	4	yes	—
217	C₂×C₄	#13094049	1	4	yes	—
218	C₂×C₄	#13094054	1	4	yes	—
219	C₂×C₄	C₂×D₄×D₄₇	1	2	yes	—
220	C₂×C₄	C₂×Q₈×D₄₇	1	2	yes	—
221	C₂×C₄	C₂×Q₈×D₄₇	1	2	yes	—
222	C₈	—	47	16	no	—
223	C₂²	C₂×#13094020	1	1	yes	characteristic
224	C₂²	C₂×#13094022	1	1	yes	characteristic
225	C₂²	C₂×#13094022	1	1	yes	characteristic
226	C₂²	C₂×#13094049	1	2	yes	—
227	C₂²	C₂×#13094054	1	2	yes	—
228	C₄	—	2	2	no	—
229	C₄	—	2	2	no	—
230	C₄	—	2	4	no	—
231	C₄	—	8	2	no	—
232	C₄	—	8	2	no	—
233	C₄	—	376	4	no	—
234	C₄	#15689533	1	8	yes	complemented
235	C₄	C₄⋊Q₈×D₄₇	1	4	yes	—
236	C₂	#20251784	1	1	yes	characteristic
237	C₂	#20253841	1	1	yes	characteristic
238	C₂	#20253848	1	1	yes	characteristic
239	C₂	C₂×#15689533	1	2	yes	—
240	C₂	#20253935	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₄₇
commutator, cosocle	C₂²×C₄×C₄₇	C₂⁴
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#2859	C₂
socle, prime radical	C₂³×C₄₇	C₂²×D₄
hypercenter	#2859	D₄₇
hypocenter	C₄₇	#33316
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2859

Complements

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

N	H ≅ G/N	type
37. #13093935	Q₈	semidirect
38. C₄×C₄₇⋊C₈	Q₈	semidirect
101. C₄₇⋊C₈	C₄⋊Q₈	semidirect
113. C₄×C₄₇	#501	semidirect
146. C₄₇	#33316	semidirect
234. C₄	#15689533	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689533	2
#20251786	2
#20243557	2
#20243547	2
#20243544	2
C₄×#13093935	2
C₄₇×#2859	2
#33316	47