Subgroups — #26122109

Order 12032.

774 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689147	C₂	1	1	yes	maximal, characteristic
2	C₂²×#13093935	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688887	C₂	1	4	yes	maximal
4	C₂×C₄₇×#423	C₂	1	1	yes	maximal, characteristic
5	C₂²×#10625329	C₂²	1	2	yes	—
6	C₂³×C₄₇⋊C₈	C₂²	1	2	yes	—
7	#15689147	—	2	2	no	—
8	#15689147	C₄	1	4	yes	—
9	C₂×C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
10	C₂×#13093935	—	2	2	no	—
11	C₂×#13093935	—	2	2	no	—
12	C₂×#13093935	C₄	1	4	yes	—
13	#15688887	—	2	8	no	—
14	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
15	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
16	C₄₇×#423	—	2	2	no	—
17	C₄₇×#423	C₄	1	4	yes	—
18	C₂×C₄⋊C₈×C₄₇	—	2	2	no	—
19	C₂×#10625329	—	2	2	no	—
20	C₂×#10625329	—	2	4	no	—
21	C₂×#10625329	—	2	4	no	—
22	C₂×#10625329	C₂×C₄	1	4	yes	—
23	C₂²×C₄₇⋊C₈	—	2	2	no	—
24	C₂²×C₄₇⋊C₈	—	2	2	no	—
25	C₂²×C₄₇⋊C₈	—	2	2	no	—
26	C₂²×C₄₇⋊C₈	—	2	2	no	—
27	C₂²×C₄₇⋊C₈	—	2	2	no	—
28	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
29	C₂²×C₄₇⋊C₈	D₄	1	4	yes	—
30	C₂²×C₄₇⋊C₈	Q₈	1	4	yes	—
31	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
32	#13093935	—	2	8	no	—
33	C₂³×C₄×C₄₇	Q₈	1	2	yes	—
34	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
35	C₂×M₁₆×C₄₇	—	2	2	no	—
36	C₂×M₁₆×C₄₇	—	2	4	no	—
37	C₂×M₁₆×C₄₇	—	2	4	no	—
38	C₂×M₁₆×C₄₇	C₂×C₄	1	4	yes	—
39	C₂²×C₈×C₄₇	—	2	2	no	—
40	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₄₇	D₄	1	2	yes	—
44	C₂×C₄²×C₄₇	D₄	1	2	yes	—
45	C₂×C₄²×C₄₇	D₄	1	2	yes	—
46	C₄⋊C₈×C₄₇	—	2	8	no	—
47	C₂⁴×C₄₇	C₂×Q₈	1	1	yes	characteristic
48	C₂²×C₄×C₄₇	—	4	4	no	—
49	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
50	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
51	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
52	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
59	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
60	C₂²×C₄×C₄₇	#41	1	2	yes	—
61	C₂²×C₄×C₄₇	#41	1	2	yes	—
62	M₁₆×C₄₇	—	2	8	no	—
63	M₁₆×C₄₇	—	4	4	no	—
64	C₂×C₈×C₄₇	—	2	4	no	—
65	C₂×C₈×C₄₇	—	4	2	no	—
66	C₂×C₈×C₄₇	—	4	2	no	—
67	C₄²×C₄₇	C₄²	1	4	yes	—
68	C₄²×C₄₇	C₂²⋊C₄	1	4	yes	—
69	C₄²×C₄₇	C₂²⋊C₄	1	4	yes	—
70	C₄²×C₄₇	C₂²⋊C₄	1	4	yes	—
71	#10625329	—	2	8	no	—
72	#10625329	—	4	4	no	—
73	C₂×C₄₇⋊C₈	—	2	4	no	—
74	C₂×C₄₇⋊C₈	—	2	4	no	—
75	C₂×C₄₇⋊C₈	—	2	4	no	—
76	C₂×C₄₇⋊C₈	—	2	4	no	—
77	C₂×C₄₇⋊C₈	—	2	4	no	—
78	C₂×C₄₇⋊C₈	—	2	4	no	—
79	C₂×C₄₇⋊C₈	—	2	4	no	—
80	C₂×C₄₇⋊C₈	—	4	2	no	—
81	C₂×C₄₇⋊C₈	—	4	2	no	—
82	C₂×C₄₇⋊C₈	C₄⋊C₄	1	8	yes	—
83	C₂³×C₄₇	—	2	4	no	—
84	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
86	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
87	C₂³×C₄₇	#125	1	2	yes	—
88	C₂³×C₄₇	C₄⋊Q₈	1	2	yes	—
89	C₂×C₄×C₄₇	—	2	8	no	—
90	C₂×C₄×C₄₇	—	4	4	no	—
91	C₂×C₄×C₄₇	—	4	4	no	—
92	C₂×C₄×C₄₇	#95	1	8	yes	—
93	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
94	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
98	C₂×C₄×C₄₇	#117	1	2	yes	—
99	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
100	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
101	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
102	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
103	C₂×C₄×C₄₇	#121	1	2	yes	—
104	C₂×C₄×C₄₇	#121	1	2	yes	—
105	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
106	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
107	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
108	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
109	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
110	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
111	C₈×C₄₇	—	4	4	no	—
112	C₄₇⋊C₈	—	2	8	no	—
113	C₄₇⋊C₈	—	4	4	no	—
114	#10845	—	47	1	no	maximal, Sylow
115	C₂²×C₄₇	—	2	4	no	—
116	C₂²×C₄₇	—	2	4	no	—
117	C₂²×C₄₇	—	2	2	no	—
118	C₂²×C₄₇	—	2	2	no	—
119	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
120	C₂²×C₄₇	#382	1	1	yes	characteristic
121	C₂²×C₄₇	#384	1	1	yes	characteristic
122	C₂²×C₄₇	#389	1	2	yes	—
123	C₂²×C₄₇	#389	1	2	yes	—
124	C₂²×C₄₇	#430	1	4	yes	—
125	C₄×C₄₇	—	2	8	no	—
126	C₄×C₄₇	C₄×C₄⋊C₄	1	2	yes	—
127	C₄×C₄₇	#380	1	2	yes	—
128	C₄×C₄₇	#390	1	1	yes	characteristic
129	C₄×C₄₇	#390	1	1	yes	characteristic
130	C₄×C₄₇	#391	1	1	yes	characteristic
131	C₄×C₄₇	#391	1	1	yes	characteristic
132	C₂×#423	—	47	1	no	—
133	C₂×#423	D₄₇	1	1	yes	characteristic
134	C₂²×#97	—	47	1	no	—
135	C₂×#341	—	47	4	no	—
136	C₂×C₄₇	—	2	2	no	—
137	C₂×C₄₇	—	2	2	no	—
138	C₂×C₄₇	C₄⋊#95	1	1	yes	characteristic
139	C₂×C₄₇	#1738	1	2	yes	—
140	C₂×C₄₇	#1838	1	2	yes	—
141	C₂×C₄₇	#1849	1	2	yes	—
142	C₂²×M₁₆	—	47	2	no	—
143	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
144	C₂³×C₈	—	47	2	no	—
145	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
146	#423	—	2	2	no	—
147	#423	—	47	4	no	—
148	#423	—	94	2	no	—
149	#423	Dic₄₇	1	4	yes	—
150	C₂×C₄⋊C₈	—	2	2	no	—
151	C₂×C₄⋊C₈	—	94	2	no	—
152	C₂×#97	—	47	4	no	—
153	C₂×#97	—	94	2	no	—
154	C₂×#97	—	94	2	no	—
155	#341	—	94	8	no	—
156	C₄₇	#10845	1	1	yes	characteristic, complemented, Sylow
157	C₂³×C₄	Dic₉₄	1	2	yes	—
158	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
159	C₂×M₁₆	—	2	2	no	—
160	C₂×M₁₆	—	2	4	no	—
161	C₂×M₁₆	—	2	4	no	—
162	C₂×M₁₆	—	47	4	no	—
163	C₂×M₁₆	—	94	2	no	—
164	C₂×M₁₆	—	94	4	no	—
165	C₂×M₁₆	—	94	4	no	—
166	C₂×M₁₆	C₂×Dic₄₇	1	4	yes	—
167	C₂²×C₈	—	2	2	no	—
168	C₂²×C₈	—	47	4	no	—
169	C₂²×C₈	—	47	4	no	—
170	C₂²×C₈	—	47	4	no	—
171	C₂²×C₈	—	94	2	no	—
172	C₂²×C₈	—	94	2	no	—
173	C₂²×C₈	—	94	2	no	—
174	C₂²×C₈	—	94	2	no	—
175	C₂²×C₈	—	94	2	no	—
176	C₂×C₄²	C₄×D₄₇	1	2	yes	—
177	C₂×C₄²	C₄×D₄₇	1	2	yes	—
178	C₂×C₄²	D₁₈₈	1	2	yes	—
179	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
180	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
181	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
182	C₄⋊C₈	—	2	8	no	—
183	C₄⋊C₈	—	94	8	no	—
184	#97	—	94	8	no	—
185	C₂⁴	C₂×Dic₉₄	1	1	yes	characteristic
186	C₂²×C₄	—	4	4	no	—
187	C₂²×C₄	Dic₄₇⋊C₄	1	4	yes	—
188	C₂²×C₄	Dic₄₇⋊C₄	1	4	yes	—
189	C₂²×C₄	C₄⋊Dic₄₇	1	4	yes	—
190	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
191	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
192	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
193	C₂²×C₄	D₄×D₄₇	1	2	yes	—
194	C₂²×C₄	#10625352	1	2	yes	—
195	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
196	C₂²×C₄	#10625354	1	2	yes	—
197	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
198	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
199	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
200	M₁₆	—	2	8	no	—
201	M₁₆	—	4	4	no	—
202	M₁₆	—	94	8	no	—
203	M₁₆	—	188	4	no	—
204	C₂×C₈	—	2	4	no	—
205	C₂×C₈	—	4	2	no	—
206	C₂×C₈	—	4	2	no	—
207	C₂×C₈	—	47	8	no	—
208	C₂×C₈	—	94	4	no	—
209	C₂×C₈	—	94	4	no	—
210	C₂×C₈	—	94	4	no	—
211	C₂×C₈	—	94	4	no	—
212	C₂×C₈	—	94	4	no	—
213	C₂×C₈	—	94	4	no	—
214	C₂×C₈	—	94	4	no	—
215	C₂×C₈	—	188	2	no	—
216	C₂×C₈	—	188	2	no	—
217	C₄²	C₄×Dic₄₇	1	4	yes	—
218	C₄²	(C₂×D₄₇)⋊C₄	1	4	yes	—
219	C₄²	(C₂×D₄₇)⋊C₄	1	4	yes	—
220	C₄²	C₂²⋊Dic₄₇	1	4	yes	—
221	C₂³	—	2	4	no	—
222	C₂³	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
223	C₂³	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
224	C₂³	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
225	C₂³	#13094015	1	2	yes	—
226	C₂³	#13094023	1	2	yes	—
227	C₂×C₄	—	2	8	no	—
228	C₂×C₄	—	4	4	no	—
229	C₂×C₄	—	4	4	no	—
230	C₂×C₄	C₄₇⋊#95	1	8	yes	—
231	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
232	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
233	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
234	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
235	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
236	C₂×C₄	#13093982	1	2	yes	—
237	C₂×C₄	#13093986	1	2	yes	—
238	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
239	C₂×C₄	#13093989	1	2	yes	—
240	C₂×C₄	Q₈×Dic₄₇	1	2	yes	—
241	C₂×C₄	#13093995	1	2	yes	—
242	C₂×C₄	#13093999	1	2	yes	—
243	C₂×C₄	#13094000	1	2	yes	—
244	C₂×C₄	#13094004	1	2	yes	—
245	C₂×C₄	#13094011	1	1	yes	characteristic
246	C₂×C₄	#13094013	1	1	yes	characteristic
247	C₂×C₄	#13094020	1	1	yes	characteristic
248	C₂×C₄	#13094022	1	1	yes	characteristic
249	C₈	—	4	4	no	—
250	C₈	—	94	8	no	—
251	C₈	—	188	4	no	—
252	C₂²	—	2	4	no	—
253	C₂²	—	2	4	no	—
254	C₂²	—	2	2	no	—
255	C₂²	—	2	2	no	—
256	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
257	C₂²	#15688988	1	1	yes	characteristic
258	C₂²	#15688992	1	1	yes	characteristic
259	C₂²	#15689019	1	2	yes	—
260	C₂²	#15689022	1	2	yes	—
261	C₂²	#15689171	1	4	yes	—
262	C₄	—	2	8	no	—
263	C₄	C₄⋊C₄×Dic₄₇	1	2	yes	—
264	C₄	#15688980	1	2	yes	—
265	C₄	#15689024	1	1	yes	characteristic
266	C₄	#15689026	1	1	yes	characteristic
267	C₄	#15689028	1	1	yes	characteristic
268	C₄	#15689030	1	1	yes	characteristic
269	C₂	—	2	2	no	—
270	C₂	—	2	2	no	—
271	C₂	#20241617	1	1	yes	characteristic
272	C₂	#20243114	1	2	yes	—
273	C₂	#20243528	1	2	yes	—
274	C₂	#20243562	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₂×C₄₇×#423	C₂
socle	C₂³×C₄₇	C₂×C₄⋊C₄
prime radical	C₂⁴×C₄₇	C₂×Q₈
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂×#423	D₄₇
hypocenter	C₄₇	#10845
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
156. C₄₇	#10845	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689147	2
C₂²×#13093935	2
C₂×#15688887	2
C₂×C₄₇×#423	2
#10845	47