Subgroups — #26121427

Order 12032.

1222 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093935	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄²×Dic₄₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688877	C₂	1	4	yes	maximal
4	C₂²×#97×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₄₇⋊C₈	C₂²	1	2	yes	—
6	C₂²×C₄×Dic₄₇	C₄	1	4	yes	—
7	C₂²×C₄×Dic₄₇	C₂²	1	2	yes	—
8	C₂×#13093935	—	2	2	no	—
9	C₂×#13093935	—	2	2	no	—
10	C₂×#13093935	C₄	1	4	yes	complemented
11	C₄²×Dic₄₇	—	2	2	no	—
12	C₄²×Dic₄₇	—	2	2	no	—
13	#15688877	C₄	1	16	yes	complemented
14	C₂³×C₈×C₄₇	C₂²	1	2	yes	—
15	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
16	C₂×#97×C₄₇	—	2	2	no	—
17	C₂×#97×C₄₇	—	2	2	no	—
18	C₂×#97×C₄₇	C₄	1	4	yes	complemented
19	C₂³×Dic₄₇	C₂×C₄	1	4	yes	—
20	C₂²×C₄₇⋊C₈	—	2	2	no	—
21	C₂²×C₄₇⋊C₈	—	2	2	no	—
22	C₂²×C₄₇⋊C₈	—	2	4	no	—
23	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
24	C₂²×C₄₇⋊C₈	C₂×C₄	1	8	yes	—
25	C₂×C₄×Dic₄₇	—	2	8	no	—
26	C₂×C₄×Dic₄₇	—	2	2	no	—
27	C₂×C₄×Dic₄₇	—	2	2	no	—
28	C₂×C₄×Dic₄₇	—	2	4	no	—
29	C₂×C₄×Dic₄₇	—	2	16	no	—
30	C₂×C₄×Dic₄₇	C₂×C₄	1	8	yes	—
31	#13093935	—	2	8	no	—
32	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
33	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₄₇	—	2	2	no	—
35	C₂²×C₈×C₄₇	—	2	2	no	—
36	C₂²×C₈×C₄₇	—	2	4	no	—
37	C₂²×C₈×C₄₇	C₂×C₄	1	8	yes	—
38	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
39	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₄₇	D₄	1	2	yes	—
42	C₂×C₄²×C₄₇	D₄	1	2	yes	—
43	C₂×C₄²×C₄₇	D₄	1	2	yes	—
44	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
45	#97×C₄₇	—	2	8	no	—
46	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₄₇	—	2	8	no	—
48	C₂²×C₄×C₄₇	C₄²	1	4	yes	—
49	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
50	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
51	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
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	1	yes	characteristic
57	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	#41	1	4	yes	—
59	C₂²×Dic₄₇	—	2	4	no	—
60	C₂²×Dic₄₇	—	2	4	no	—
61	C₂²×Dic₄₇	—	2	8	no	—
62	C₂²×Dic₄₇	C₄²	1	8	yes	—
63	C₂²×Dic₄₇	M₁₆	1	16	yes	—
64	C₂×C₈×C₄₇	—	2	4	no	—
65	C₂×C₈×C₄₇	—	2	4	no	—
66	C₂×C₈×C₄₇	—	2	8	no	—
67	C₂×C₈×C₄₇	—	2	8	no	—
68	C₂×C₈×C₄₇	C₄²	1	8	yes	complemented
69	C₄²×C₄₇	C₂²⋊C₄	1	4	yes	—
70	C₄²×C₄₇	C₂²⋊C₄	1	4	yes	—
71	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
72	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
73	C₄×Dic₄₇	—	2	32	no	—
74	C₄×Dic₄₇	—	2	8	no	—
75	C₄×Dic₄₇	—	2	8	no	—
76	C₂×C₄₇⋊C₈	—	2	4	no	—
77	C₂×C₄₇⋊C₈	—	2	4	no	—
78	C₂×C₄₇⋊C₈	—	2	8	no	—
79	C₂×C₄₇⋊C₈	—	2	8	no	—
80	C₂×C₄₇⋊C₈	C₄²	1	8	yes	complemented
81	C₂³×C₄₇	C₂×C₄²	1	2	yes	—
82	C₂³×C₄₇	C₂×C₄²	1	1	yes	characteristic
83	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₂³×C₄₇	#117	1	2	yes	—
86	C₂³×C₄₇	C₂×M₁₆	1	8	yes	—
87	C₂×C₄×C₄₇	—	2	8	no	—
88	C₂×C₄×C₄₇	—	2	8	no	—
89	C₂×C₄×C₄₇	—	2	8	no	—
90	C₂×C₄×C₄₇	#95	1	8	yes	—
91	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
92	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
93	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₄₇	#117	1	2	yes	—
98	C₂×C₄×C₄₇	#117	1	2	yes	—
99	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
100	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
101	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
102	C₂×C₄×C₄₇	C₄×Q₈	1	4	yes	—
103	C₂×Dic₄₇	—	2	8	no	—
104	C₂×Dic₄₇	—	2	8	no	—
105	C₂×Dic₄₇	—	2	16	no	—
106	C₂×Dic₄₇	—	2	16	no	—
107	C₂×Dic₄₇	#97	1	16	yes	complemented
108	C₈×C₄₇	—	2	8	no	—
109	C₄₇⋊C₈	—	2	8	no	—
110	#10693	—	47	1	no	maximal, Sylow
111	C₂²×C₄₇	C₄³	1	1	yes	characteristic
112	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
113	C₂²×C₄₇	#376	1	1	yes	characteristic
114	C₂²×C₄₇	C₄×C₂²⋊C₄	1	2	yes	—
115	C₂²×C₄₇	C₄×C₄⋊C₄	1	2	yes	—
116	C₂²×C₄₇	C₂×#97	1	4	yes	—
117	C₂²×C₄₇	C₄×M₁₆	1	4	yes	—
118	C₂²×C₄₇	C₄×M₁₆	1	4	yes	—
119	C₂²×C₄₇	#407	1	4	yes	—
120	C₂²×C₄₇	#423	1	4	yes	—
121	C₂²×C₄₇	#432	1	8	yes	—
122	C₄×C₄₇	—	2	8	no	—
123	C₄×C₄₇	C₄×C₂²⋊C₄	1	2	yes	—
124	C₄×C₄₇	C₄×C₂²⋊C₄	1	2	yes	—
125	C₄×C₄₇	C₄×C₄⋊C₄	1	2	yes	—
126	C₄×C₄₇	C₄×C₄⋊C₄	1	2	yes	—
127	Dic₄₇	—	2	16	no	—
128	C₂²×#97	—	47	1	no	—
129	C₂²×#97	D₄₇	1	1	yes	characteristic
130	C₂×C₄³	—	47	1	no	—
131	C₂×#336	—	47	4	no	—
132	C₂×C₄₇	C₄×#95	1	1	yes	characteristic
133	C₂×C₄₇	C₄×#97	1	2	yes	—
134	C₂×C₄₇	#1728	1	2	yes	—
135	C₂×C₄₇	#1745	1	2	yes	—
136	C₂×C₄₇	#1749	1	4	yes	—
137	C₂×C₄₇	#1767	1	4	yes	—
138	C₂³×C₈	—	47	2	no	—
139	C₂³×C₈	C₂×D₄₇	1	2	yes	—
140	C₂²×C₄²	—	47	2	no	—
141	C₂²×C₄²	—	47	4	no	—
142	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
143	C₂×#97	—	2	2	no	—
144	C₂×#97	—	2	2	no	—
145	C₂×#97	—	47	4	no	—
146	C₂×#97	—	94	2	no	—
147	C₂×#97	—	94	2	no	—
148	C₂×#97	Dic₄₇	1	4	yes	complemented
149	C₄³	—	94	2	no	—
150	C₄³	—	94	2	no	—
151	#336	—	47	16	no	—
152	C₄₇	#10693	1	1	yes	characteristic, complemented, Sylow
153	C₂³×C₄	—	47	4	no	—
154	C₂³×C₄	C₄×D₄₇	1	2	yes	—
155	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
156	C₂²×C₈	—	2	4	no	—
157	C₂²×C₈	—	2	2	no	—
158	C₂²×C₈	—	2	2	no	—
159	C₂²×C₈	—	47	8	no	—
160	C₂²×C₈	—	47	4	no	—
161	C₂²×C₈	—	94	2	no	—
162	C₂²×C₈	—	94	4	no	—
163	C₂²×C₈	—	94	2	no	—
164	C₂²×C₈	C₄×D₄₇	1	8	yes	—
165	C₂²×C₈	C₂×Dic₄₇	1	4	yes	—
166	C₂×C₄²	—	47	8	no	—
167	C₂×C₄²	—	94	16	no	—
168	C₂×C₄²	—	94	8	no	—
169	C₂×C₄²	—	94	2	no	—
170	C₂×C₄²	—	94	4	no	—
171	C₂×C₄²	—	94	2	no	—
172	C₂×C₄²	Dic₉₄	1	2	yes	—
173	C₂×C₄²	C₄×D₄₇	1	2	yes	—
174	C₂×C₄²	D₁₈₈	1	2	yes	—
175	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
176	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
177	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
178	#97	—	2	8	no	—
179	#97	—	94	8	no	—
180	C₂⁴	C₂×C₄×D₄₇	1	1	yes	characteristic
181	C₂²×C₄	—	2	8	no	—
182	C₂²×C₄	—	47	8	no	—
183	C₂²×C₄	—	47	16	no	—
184	C₂²×C₄	—	94	4	no	—
185	C₂²×C₄	—	94	4	no	—
186	C₂²×C₄	—	94	8	no	—
187	C₂²×C₄	C₄×Dic₄₇	1	4	yes	—
188	C₂²×C₄	Dic₄₇⋊C₄	1	4	yes	—
189	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	4	yes	—
190	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
191	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
192	C₂²×C₄	C₂×C₄×D₄₇	1	4	yes	—
193	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
194	C₂²×C₄	#10625350	1	4	yes	—
195	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
196	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
197	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
198	C₂×C₈	—	2	4	no	—
199	C₂×C₈	—	2	4	no	—
200	C₂×C₈	—	2	8	no	—
201	C₂×C₈	—	2	8	no	—
202	C₂×C₈	—	47	8	no	—
203	C₂×C₈	—	94	4	no	—
204	C₂×C₈	—	94	4	no	—
205	C₂×C₈	—	94	8	no	—
206	C₂×C₈	—	94	8	no	—
207	C₂×C₈	C₄×Dic₄₇	1	8	yes	complemented
208	C₄²	—	94	32	no	—
209	C₄²	—	94	8	no	—
210	C₄²	—	94	8	no	—
211	C₄²	Dic₄₇⋊C₄	1	4	yes	—
212	C₄²	C₄⋊Dic₄₇	1	4	yes	—
213	C₄²	(C₂×D₄₇)⋊C₄	1	4	yes	—
214	C₄²	C₂²⋊Dic₄₇	1	4	yes	—
215	C₂³	C₄²×D₄₇	1	2	yes	—
216	C₂³	C₂×C₄×Dic₄₇	1	1	yes	characteristic
217	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
218	C₂³	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
219	C₂³	#13093981	1	2	yes	—
220	C₂³	M₁₆×D₄₇	1	8	yes	—
221	C₂×C₄	—	2	8	no	—
222	C₂×C₄	—	2	8	no	—
223	C₂×C₄	—	2	8	no	—
224	C₂×C₄	—	47	16	no	—
225	C₂×C₄	—	94	8	no	—
226	C₂×C₄	—	94	8	no	—
227	C₂×C₄	—	94	16	no	—
228	C₂×C₄	—	94	16	no	—
229	C₂×C₄	C₄₇⋊#95	1	8	yes	—
230	C₂×C₄	C₄²×D₄₇	1	2	yes	—
231	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
232	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
233	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
234	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
235	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
236	C₂×C₄	#13093981	1	2	yes	—
237	C₂×C₄	#13093983	1	2	yes	—
238	C₂×C₄	C₄×C₂²⋊D₄₇	1	4	yes	—
239	C₂×C₄	C₄×C₂²⋊D₄₇	1	4	yes	—
240	C₂×C₄	C₄×D₁₈₈	1	4	yes	—
241	C₂×C₄	C₄×Dic₉₄	1	4	yes	—
242	C₈	—	2	8	no	—
243	C₈	—	94	8	no	—
244	C₂²	C₄²×Dic₄₇	1	1	yes	characteristic
245	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
246	C₂²	#15688969	1	1	yes	characteristic
247	C₂²	C₄×(C₂×D₄₇)⋊C₄	1	2	yes	—
248	C₂²	C₄×Dic₄₇⋊C₄	1	2	yes	—
249	C₂²	#97×D₄₇	1	4	yes	—
250	C₂²	M₁₆×Dic₄₇	1	4	yes	—
251	C₂²	#15689075	1	4	yes	—
252	C₂²	#15689086	1	4	yes	—
253	C₂²	#15689146	1	4	yes	—
254	C₂²	#15689178	1	8	yes	—
255	C₄	—	2	8	no	—
256	C₄	—	94	16	no	—
257	C₄	C₄×C₂²⋊Dic₄₇	1	2	yes	—
258	C₄	C₄×(C₂×D₄₇)⋊C₄	1	2	yes	—
259	C₄	C₄×C₄⋊Dic₄₇	1	2	yes	—
260	C₄	C₄×Dic₄₇⋊C₄	1	2	yes	—
261	C₂	C₄×C₄₇⋊#95	1	1	yes	characteristic
262	C₂	#97×Dic₄₇	1	2	yes	—
263	C₂	#20243081	1	2	yes	—
264	C₂	#20243136	1	2	yes	—
265	C₂	#20243151	1	4	yes	—
266	C₂	#20243242	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂³×C₄	C₂²×D₄₇
commutator	C₂²×C₄₇	C₄³
Fitting	C₂²×#97×C₄₇	C₂
socle, prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂²×#97	D₄₇
hypocenter	C₄₇	#10693
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×#97

Complements

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

N	H ≅ G/N	type
10. C₂×#13093935	C₄	semidirect
13. #15688877	C₄	semidirect
18. C₂×#97×C₄₇	C₄	semidirect
68. C₂×C₈×C₄₇	C₄²	semidirect
80. C₂×C₄₇⋊C₈	C₄²	semidirect
107. C₂×Dic₄₇	#97	semidirect
148. C₂×#97	Dic₄₇	semidirect
152. C₄₇	#10693	semidirect
207. C₂×C₈	C₄×Dic₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13093935	2
C₂×C₄²×Dic₄₇	2
C₂×#15688877	2
C₂²×#97×C₄₇	2
#10693	47