Subgroups — #26109067

Order 12032.

684 conjugacy classes of subgroups.

The subgroup lattice has 270 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, complemented
2	C₂×#15688867	C₂	1	2	yes	maximal
3	#20241271	C₂	1	2	yes	maximal, complemented
4	C₄×#127×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
6	C₂×C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
7	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
8	#15688867	—	2	4	no	—
9	C₂×#127×C₄₇	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₄×C₄₇	—	2	2	no	—
11	C₂×C₄×D₄×C₄₇	C₄	1	2	yes	—
12	C₄₇×#390	—	2	2	no	—
13	C₄₇×#390	C₄	1	2	yes	—
14	C₄₇×#384	C₂²	1	1	yes	characteristic
15	C₄³×C₄₇	C₂²	1	1	yes	characteristic
16	C₂²×C₄₇⋊C₈	—	2	4	no	—
17	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
18	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
19	C₄⋊(C₄₇⋊C₈)	—	4	4	no	—
20	C₄⋊(C₄₇⋊C₈)	—	4	4	no	—
21	C₂²×D₄×C₄₇	—	2	2	no	—
22	C₂²×D₄×C₄₇	C₂×C₄	1	2	yes	—
23	C₂³×C₄×C₄₇	—	2	2	no	—
24	#127×C₄₇	—	2	2	no	—
25	#127×C₄₇	D₄	1	2	yes	—
26	#127×C₄₇	Q₈	1	2	yes	—
27	C₄×D₄×C₄₇	—	4	4	no	—
28	C₄×D₄×C₄₇	—	4	4	no	—
29	C₄×D₄×C₄₇	—	4	4	no	—
30	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
31	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
32	C₂×C₂²⋊C₄×C₄₇	—	2	4	no	—
33	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
34	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₄₇	D₄	1	2	yes	—
37	C₂×C₄²×C₄₇	D₄	1	2	yes	—
38	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
39	C₂⁴×C₄₇	—	2	2	no	—
40	C₂×D₄×C₄₇	—	2	2	no	—
41	C₂×D₄×C₄₇	—	2	4	no	—
42	C₂×D₄×C₄₇	—	2	4	no	—
43	C₂×D₄×C₄₇	—	4	2	no	—
44	C₂×D₄×C₄₇	—	4	2	no	—
45	C₂×D₄×C₄₇	—	4	2	no	—
46	C₂×D₄×C₄₇	—	4	2	no	—
47	C₂×D₄×C₄₇	—	4	2	no	—
48	C₂×D₄×C₄₇	—	4	2	no	—
49	C₂×D₄×C₄₇	C₄⋊C₄	1	4	yes	—
50	C₂²×C₄×C₄₇	—	2	2	no	—
51	C₂²×C₄×C₄₇	—	4	4	no	—
52	C₂²×C₄×C₄₇	—	4	8	no	—
53	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
55	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
58	C₄⋊C₄×C₄₇	—	4	2	no	—
59	C₄⋊C₄×C₄₇	—	4	2	no	—
60	C₄⋊C₄×C₄₇	—	4	2	no	—
61	C₂²⋊C₄×C₄₇	—	4	4	no	—
62	C₂²⋊C₄×C₄₇	—	4	8	no	—
63	C₄²×C₄₇	—	2	2	no	—
64	C₄²×C₄₇	—	2	2	no	—
65	C₄²×C₄₇	—	2	2	no	—
66	C₄²×C₄₇	—	4	1	no	—
67	C₄²×C₄₇	D₈	1	4	yes	—
68	C₄²×C₄₇	SD₁₆	1	4	yes	—
69	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
70	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
71	C₄²×C₄₇	#41	1	2	yes	—
72	C₂×C₄₇⋊C₈	—	4	4	no	—
73	C₂×C₄₇⋊C₈	—	4	4	no	—
74	C₂×C₄₇⋊C₈	—	4	4	no	—
75	C₂³×C₄₇	—	2	4	no	—
76	C₂³×C₄₇	—	4	2	no	—
77	C₂³×C₄₇	—	4	2	no	—
78	C₂³×C₄₇	—	4	4	no	—
79	C₂³×C₄₇	—	4	4	no	—
80	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	D₄×C₄₇	—	4	4	no	—
82	D₄×C₄₇	—	4	4	no	—
83	D₄×C₄₇	—	4	4	no	—
84	D₄×C₄₇	—	4	4	no	—
85	D₄×C₄₇	—	4	4	no	—
86	D₄×C₄₇	—	4	4	no	—
87	C₂×C₄×C₄₇	—	2	1	no	—
88	C₂×C₄×C₄₇	—	2	2	no	—
89	C₂×C₄×C₄₇	—	2	1	no	—
90	C₂×C₄×C₄₇	—	2	2	no	—
91	C₂×C₄×C₄₇	—	2	1	no	—
92	C₂×C₄×C₄₇	—	2	2	no	—
93	C₂×C₄×C₄₇	—	4	2	no	—
94	C₂×C₄×C₄₇	—	4	4	no	—
95	C₂×C₄×C₄₇	—	8	8	no	—
96	C₂×C₄×C₄₇	#100	1	2	yes	—
97	C₂×C₄×C₄₇	#102	1	4	yes	—
98	C₂×C₄×C₄₇	#102	1	4	yes	—
99	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
101	C₂×C₄×C₄₇	#121	1	2	yes	—
102	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
103	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
104	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
105	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
106	C₂×C₄×C₄₇	#137	1	2	yes	—
107	C₄₇⋊C₈	—	8	4	no	—
108	#8585	—	47	1	no	maximal, Sylow
109	C₂²×C₄₇	—	4	4	no	—
110	C₂²×C₄₇	—	4	8	no	—
111	C₂²×C₄₇	—	8	2	no	—
112	C₂²×C₄₇	—	8	4	no	—
113	C₂²×C₄₇	—	8	2	no	—
114	C₂²×C₄₇	#380	1	1	yes	characteristic
115	C₂²×C₄₇	C₂×#100	1	1	yes	characteristic
116	C₂²×C₄₇	#413	1	1	yes	characteristic
117	C₂²×C₄₇	C₂×#102	1	2	yes	—
118	C₂²×C₄₇	#417	1	2	yes	—
119	C₄×C₄₇	—	2	1	no	—
120	C₄×C₄₇	—	2	1	no	—
121	C₄×C₄₇	—	2	4	no	—
122	C₄×C₄₇	—	4	1	no	—
123	C₄×C₄₇	—	8	2	no	—
124	C₄×C₄₇	#331	1	4	yes	complemented
125	C₄×C₄₇	#459	1	2	yes	—
126	C₄×C₄₇	#461	1	2	yes	—
127	C₄×C₄₇	#474	1	2	yes	—
128	C₄×C₄₇	#476	1	2	yes	—
129	C₄×#127	D₄₇	1	1	yes	characteristic
130	#1842	—	47	2	no	—
131	C₂×#331	—	47	2	no	—
132	#1296	—	47	2	no	—
133	C₂×C₄₇	—	8	4	no	—
134	C₂×C₄₇	C₂×#331	1	1	yes	characteristic
135	C₂×C₄₇	#1547	1	2	yes	—
136	C₂×C₄₇	#1560	1	1	yes	characteristic
137	C₂×C₄₇	#1800	1	1	yes	characteristic
138	C₂×C₄₇	#1802	1	2	yes	—
139	C₂×#127	C₂×D₄₇	1	1	yes	characteristic
140	C₂×C₄×D₄	—	2	2	no	—
141	C₂×C₄×D₄	Dic₄₇	1	2	yes	—
142	C₂×C₄⋊C₈	—	47	4	no	—
143	C₂×C₄⋊C₈	—	94	2	no	—
144	C₂×C₄⋊C₈	—	94	2	no	—
145	#390	—	2	2	no	—
146	#390	Dic₄₇	1	2	yes	—
147	#384	C₂×D₄₇	1	1	yes	characteristic
148	C₄³	C₂×D₄₇	1	1	yes	characteristic
149	#331	—	94	4	no	—
150	C₄₇	#8585	1	1	yes	characteristic, complemented, Sylow
151	C₂²×D₄	—	2	2	no	—
152	C₂²×D₄	C₂×Dic₄₇	1	2	yes	—
153	C₂³×C₄	—	2	2	no	—
154	C₂²×C₈	—	94	4	no	—
155	#127	—	2	2	no	—
156	#127	Dic₉₄	1	2	yes	—
157	#127	D₁₈₈	1	2	yes	—
158	C₄×D₄	—	4	4	no	—
159	C₄×D₄	—	4	4	no	—
160	C₄×D₄	—	4	4	no	—
161	C₂×C₄⋊C₄	—	2	2	no	—
162	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
163	C₂×C₂²⋊C₄	—	2	4	no	—
164	C₂×C₂²⋊C₄	—	2	2	no	—
165	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
166	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
167	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
168	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
169	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
170	C₄⋊C₈	—	94	4	no	—
171	C₄⋊C₈	—	94	4	no	—
172	C₄⋊C₈	—	188	4	no	—
173	C₄⋊C₈	—	188	4	no	—
174	C₂⁴	—	2	2	no	—
175	C₂×D₄	—	2	2	no	—
176	C₂×D₄	—	2	4	no	—
177	C₂×D₄	—	2	4	no	—
178	C₂×D₄	—	4	2	no	—
179	C₂×D₄	—	4	2	no	—
180	C₂×D₄	—	4	2	no	—
181	C₂×D₄	—	4	2	no	—
182	C₂×D₄	—	4	2	no	—
183	C₂×D₄	—	4	2	no	—
184	C₂×D₄	C₄⋊Dic₄₇	1	4	yes	—
185	C₂²×C₄	—	2	2	no	—
186	C₂²×C₄	—	4	4	no	—
187	C₂²×C₄	—	4	8	no	—
188	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
189	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
190	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
191	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
192	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
193	C₂×C₈	—	188	4	no	—
194	C₂×C₈	—	188	4	no	—
195	C₂×C₈	—	188	4	no	—
196	C₄⋊C₄	—	4	2	no	—
197	C₄⋊C₄	—	4	2	no	—
198	C₄⋊C₄	—	4	2	no	—
199	C₂²⋊C₄	—	4	4	no	—
200	C₂²⋊C₄	—	4	8	no	—
201	C₄²	—	2	2	no	—
202	C₄²	—	2	2	no	—
203	C₄²	—	2	2	no	—
204	C₄²	—	4	1	no	—
205	C₄²	#10625334	1	4	yes	—
206	C₄²	#10625335	1	4	yes	—
207	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
208	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
209	C₄²	#10625350	1	2	yes	—
210	C₂³	—	2	4	no	—
211	C₂³	—	4	2	no	—
212	C₂³	—	4	2	no	—
213	C₂³	—	4	4	no	—
214	C₂³	—	4	4	no	—
215	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
216	D₄	—	4	4	no	—
217	D₄	—	4	4	no	—
218	D₄	—	4	4	no	—
219	D₄	—	4	4	no	—
220	D₄	—	4	4	no	—
221	D₄	—	4	4	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	4	4	no	—
230	C₂×C₄	—	8	8	no	—
231	C₂×C₄	#13093941	1	2	yes	—
232	C₂×C₄	#13093945	1	4	yes	—
233	C₂×C₄	#13093945	1	4	yes	—
234	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
235	C₂×C₄	#13093983	1	1	yes	characteristic
236	C₂×C₄	#13093998	1	2	yes	—
237	C₂×C₄	#13094003	1	2	yes	—
238	C₂×C₄	C₂×#10625334	1	2	yes	—
239	C₂×C₄	C₂×#10625335	1	2	yes	—
240	C₂×C₄	#13094049	1	2	yes	—
241	C₂×C₄	#13094054	1	2	yes	—
242	C₈	—	376	4	no	—
243	C₂²	—	4	4	no	—
244	C₂²	—	4	8	no	—
245	C₂²	—	8	2	no	—
246	C₂²	—	8	4	no	—
247	C₂²	—	8	2	no	—
248	C₂²	#15688983	1	1	yes	characteristic
249	C₂²	C₂×#13093941	1	1	yes	characteristic
250	C₂²	#15689107	1	1	yes	characteristic
251	C₂²	C₂×#13093945	1	2	yes	—
252	C₂²	#15689124	1	2	yes	—
253	C₄	—	2	4	no	—
254	C₄	—	2	1	no	—
255	C₄	—	2	1	no	—
256	C₄	—	4	1	no	—
257	C₄	—	8	2	no	—
258	C₄	#15688867	1	4	yes	complemented
259	C₄	#15689318	1	2	yes	—
260	C₄	#15689328	1	2	yes	—
261	C₄	#15689403	1	2	yes	—
262	C₄	#15689413	1	2	yes	—
263	C₂	—	8	4	no	—
264	C₂	C₂×#15688867	1	1	yes	characteristic
265	C₂	#20242065	1	2	yes	—
266	C₂	#20242123	1	1	yes	characteristic
267	C₂	#20243380	1	1	yes	characteristic
268	C₂	#20243388	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₄₇
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄×#127×C₄₇	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂×#127×C₄₇	C₂²
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₄×#127	D₄₇
hypocenter	C₄₇	#8585
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×#127

Complements

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

N	H ≅ G/N	type
1. #20243540	C₂	semidirect
3. #20241271	C₂	semidirect
124. C₄×C₄₇	#331	semidirect
150. C₄₇	#8585	semidirect
258. C₄	#15688867	semidirect

Maximal subgroups

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

subgroup	index
#20243540	2
C₂×#15688867	2
#20241271	2
C₄×#127×C₄₇	2
#8585	47