Subgroups — #26121455

Order 12032.

998 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#10625329	C₂	1	1	yes	maximal, characteristic
2	C₄×#13093983	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688883	C₂	1	4	yes	maximal
4	C₂×C₄₇×#376	C₂	1	1	yes	maximal, characteristic
5	C₂²×#10625329	C₂²	1	2	yes	—
6	C₂×#13093983	C₂²	1	2	yes	—
7	C₄×#10625329	—	2	2	no	—
8	C₄×#10625329	C₄	1	4	yes	complemented
9	C₂×#13093935	—	2	1	no	—
10	C₂×C₄×C₄₇⋊C₈	—	2	1	no	—
11	C₄×C₄⋊Dic₄₇	C₄	1	4	yes	complemented
12	C₄×C₂²⋊Dic₄₇	—	2	2	no	—
13	#15688969	—	2	1	no	—
14	C₄²×Dic₄₇	—	2	1	no	—
15	#15688883	C₄	1	16	yes	complemented
16	C₂²×C₄²×C₄₇	C₂²	1	2	yes	—
17	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
18	C₄₇×#376	—	2	2	no	—
19	C₄₇×#376	C₄	1	4	yes	complemented
20	C₂×#95×C₄₇	—	2	2	no	—
21	C₂×#10625329	—	2	2	no	—
22	C₂×#10625329	—	2	4	no	—
23	C₂×#10625329	C₂×C₄	1	4	yes	—
24	C₂×#10625329	C₂×C₄	1	8	yes	—
25	C₂²×C₄₇⋊C₈	—	2	2	no	—
26	#13093983	—	2	4	no	—
27	#13093983	C₂×C₄	1	8	yes	—
28	C₂×C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
29	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
30	C₂×C₄×Dic₄₇	—	2	2	no	—
31	C₂×C₄×Dic₄₇	—	2	4	no	—
32	#13093935	—	2	4	no	—
33	C₄×C₄₇⋊C₈	—	2	4	no	—
34	C₄₇⋊#95	—	2	4	no	—
35	C₂³×C₄×C₄₇	—	2	2	no	—
36	C₂³×C₄×C₄₇	D₄	1	2	yes	—
37	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
38	C₂×C₄²×C₄₇	—	2	2	no	—
39	C₂×C₄²×C₄₇	—	2	4	no	—
40	C₂×C₄²×C₄₇	C₂×C₄	1	8	yes	—
41	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
42	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₄₇	D₄	1	2	yes	—
48	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
49	#95×C₄₇	—	2	8	no	—
50	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₄₇	—	2	4	no	—
52	C₂²×C₄×C₄₇	—	2	8	no	—
53	C₂²×C₄×C₄₇	—	4	4	no	—
54	C₂²×C₄×C₄₇	—	4	8	no	—
55	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
58	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
59	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
63	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	#41	1	4	yes	—
67	C₂²×Dic₄₇	—	2	4	no	—
68	C₄²×C₄₇	—	2	4	no	—
69	C₄²×C₄₇	—	2	8	no	—
70	C₄²×C₄₇	C₄²	1	4	yes	—
71	C₄²×C₄₇	C₄²	1	8	yes	complemented
72	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
73	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
74	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
75	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
76	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
77	C₂²⋊Dic₄₇	—	2	8	no	—
78	C₄⋊Dic₄₇	—	2	4	no	—
79	C₄⋊Dic₄₇	C₄²	1	8	yes	—
80	C₄×Dic₄₇	—	2	8	no	—
81	C₄×Dic₄₇	—	4	8	no	—
82	#10625329	—	2	4	no	—
83	#10625329	—	2	8	no	—
84	#10625329	C₄²	1	8	yes	complemented
85	C₂×C₄₇⋊C₈	—	2	4	no	—
86	C₂×C₄₇⋊C₈	—	2	8	no	—
87	C₂³×C₄₇	—	2	4	no	—
88	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂³×C₄₇	C₄×D₄	1	4	yes	—
92	C₂×C₄×C₄₇	—	2	4	no	—
93	C₂×C₄×C₄₇	—	2	4	no	—
94	C₂×C₄×C₄₇	—	2	8	no	—
95	C₂×C₄×C₄₇	—	4	4	no	—
96	C₂×C₄×C₄₇	—	4	4	no	—
97	C₂×C₄×C₄₇	—	4	8	no	—
98	C₂×C₄×C₄₇	—	4	16	no	—
99	C₂×C₄×C₄₇	#95	1	4	yes	—
100	C₂×C₄×C₄₇	#95	1	4	yes	—
101	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
102	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
103	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
104	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
105	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₄₇	#117	1	2	yes	—
109	C₂×C₄×C₄₇	#117	1	2	yes	—
110	C₂×C₄×C₄₇	#117	1	2	yes	—
111	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
112	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
113	C₂×C₄×C₄₇	C₄×Q₈	1	4	yes	—
114	C₂×Dic₄₇	—	2	8	no	—
115	C₂×Dic₄₇	—	4	8	no	—
116	C₄₇⋊C₈	—	2	8	no	—
117	#10701	—	47	1	no	maximal, Sylow
118	C₂²×C₄₇	—	2	2	no	—
119	C₂²×C₄₇	—	2	2	no	—
120	C₂²×C₄₇	—	2	8	no	—
121	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
122	C₂²×C₄₇	C₄×C₂²⋊C₄	1	2	yes	—
123	C₂²×C₄₇	C₄×C₂²⋊C₄	1	2	yes	—
124	C₂²×C₄₇	C₄×C₄⋊C₄	1	2	yes	—
125	C₂²×C₄₇	#421	1	4	yes	—
126	C₄×C₄₇	—	2	4	no	—
127	C₄×C₄₇	—	2	4	no	—
128	C₄×C₄₇	—	4	8	no	—
129	C₄×C₄₇	C₄³	1	1	yes	characteristic
130	C₄×C₄₇	#376	1	1	yes	characteristic
131	C₄×C₄₇	C₄×C₂²⋊C₄	1	2	yes	—
132	C₄×C₄₇	C₄×C₄⋊C₄	1	2	yes	—
133	C₄×C₄₇	C₄×C₄⋊C₄	1	2	yes	—
134	Dic₄₇	—	4	8	no	—
135	C₂×C₄×M₁₆	—	47	1	no	—
136	C₄×#117	—	47	1	no	—
137	C₂×#376	D₄₇	1	1	yes	characteristic
138	C₂×#339	—	47	4	no	—
139	C₂×C₄₇	—	2	2	no	—
140	C₂×C₄₇	—	2	2	no	—
141	C₂×C₄₇	C₄×#95	1	1	yes	characteristic
142	C₂×C₄₇	#1733	1	2	yes	—
143	C₂×C₄₇	#1759	1	4	yes	—
144	C₂²×M₁₆	—	47	2	no	—
145	C₂×#117	—	47	2	no	—
146	C₂²×C₄²	C₂×D₄₇	1	2	yes	—
147	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
148	C₄×M₁₆	—	47	4	no	—
149	C₄×M₁₆	—	94	2	no	—
150	C₂×#97	—	94	1	no	—
151	C₂×C₄×C₈	—	94	1	no	—
152	C₄×C₄⋊C₄	—	47	4	no	—
153	C₄×C₂²⋊C₄	—	94	2	no	—
154	#376	—	2	2	no	—
155	#376	—	94	1	no	—
156	#376	Dic₄₇	1	4	yes	complemented
157	C₂×#95	—	2	2	no	—
158	C₄³	—	94	1	no	—
159	#339	—	47	16	no	—
160	C₄₇	#10701	1	1	yes	characteristic, complemented, Sylow
161	C₂³×C₄	—	2	2	no	—
162	C₂³×C₄	C₂²⋊D₄₇	1	2	yes	—
163	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
164	C₂×M₁₆	—	47	8	no	—
165	C₂×M₁₆	—	47	4	no	—
166	C₂×M₁₆	—	94	4	no	—
167	C₂×M₁₆	—	94	2	no	—
168	C₂²×C₈	—	94	2	no	—
169	#117	—	47	8	no	—
170	#117	—	94	4	no	—
171	C₂×C₄⋊C₄	—	47	4	no	—
172	C₂×C₂²⋊C₄	—	94	2	no	—
173	C₂×C₄²	—	2	4	no	—
174	C₂×C₄²	—	2	2	no	—
175	C₂×C₄²	—	94	2	no	—
176	C₂×C₄²	—	94	4	no	—
177	C₂×C₄²	Dic₉₄	1	2	yes	—
178	C₂×C₄²	C₄×D₄₇	1	8	yes	—
179	C₂×C₄²	C₄×D₄₇	1	2	yes	—
180	C₂×C₄²	C₄×D₄₇	1	2	yes	—
181	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
182	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
183	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
184	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
185	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
186	#97	—	94	4	no	—
187	C₄×C₈	—	94	4	no	—
188	#95	—	2	8	no	—
189	#95	—	94	4	no	—
190	C₂⁴	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
191	C₂²×C₄	—	2	4	no	—
192	C₂²×C₄	—	2	8	no	—
193	C₂²×C₄	—	4	4	no	—
194	C₂²×C₄	—	4	8	no	—
195	C₂²×C₄	—	94	4	no	—
196	C₂²×C₄	Dic₄₇⋊C₄	1	4	yes	—
197	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
198	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
199	C₂²×C₄	C₂²⋊Dic₄₇	1	4	yes	—
200	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
201	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
202	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
203	C₂²×C₄	C₂×C₄×D₄₇	1	4	yes	—
204	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
205	C₂²×C₄	#10625350	1	4	yes	—
206	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
207	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
208	M₁₆	—	47	8	no	—
209	M₁₆	—	94	4	no	—
210	M₁₆	—	94	8	no	—
211	C₂×C₈	—	94	4	no	—
212	C₂×C₈	—	94	8	no	—
213	C₄⋊C₄	—	47	8	no	—
214	C₄⋊C₄	—	94	4	no	—
215	C₂²⋊C₄	—	94	8	no	—
216	C₄²	—	2	4	no	—
217	C₄²	—	2	8	no	—
218	C₄²	—	94	8	no	—
219	C₄²	—	188	8	no	—
220	C₄²	C₄×Dic₄₇	1	4	yes	—
221	C₄²	C₄×Dic₄₇	1	8	yes	complemented
222	C₄²	Dic₄₇⋊C₄	1	4	yes	—
223	C₄²	C₄⋊Dic₄₇	1	2	yes	—
224	C₄²	C₄⋊Dic₄₇	1	2	yes	—
225	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
226	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
227	C₂³	—	2	4	no	—
228	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
229	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
230	C₂³	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
231	C₂³	C₄×C₂²⋊D₄₇	1	4	yes	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	8	no	—
235	C₂×C₄	—	4	4	no	—
236	C₂×C₄	—	4	4	no	—
237	C₂×C₄	—	4	8	no	—
238	C₂×C₄	—	4	16	no	—
239	C₂×C₄	—	94	8	no	—
240	C₂×C₄	—	188	8	no	—
241	C₂×C₄	C₄₇⋊#95	1	4	yes	—
242	C₂×C₄	C₄₇⋊#95	1	4	yes	—
243	C₂×C₄	C₄²×D₄₇	1	2	yes	—
244	C₂×C₄	C₄²×D₄₇	1	2	yes	—
245	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
246	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
247	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
248	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
249	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
250	C₂×C₄	#13093981	1	2	yes	—
251	C₂×C₄	#13093981	1	2	yes	—
252	C₂×C₄	#13093983	1	2	yes	—
253	C₂×C₄	C₄×C₂²⋊D₄₇	1	4	yes	—
254	C₂×C₄	C₄×D₁₈₈	1	4	yes	—
255	C₂×C₄	C₄×Dic₉₄	1	4	yes	—
256	C₈	—	94	8	no	—
257	C₂²	—	2	2	no	—
258	C₂²	—	2	2	no	—
259	C₂²	—	2	8	no	—
260	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
261	C₂²	C₄×C₂²⋊Dic₄₇	1	2	yes	—
262	C₂²	C₄×(C₂×D₄₇)⋊C₄	1	2	yes	—
263	C₂²	C₄×Dic₄₇⋊C₄	1	2	yes	—
264	C₂²	#15689141	1	4	yes	—
265	C₄	—	2	4	no	—
266	C₄	—	2	4	no	—
267	C₄	—	4	8	no	—
268	C₄	—	188	8	no	—
269	C₄	C₄²×Dic₄₇	1	1	yes	characteristic
270	C₄	#15688969	1	1	yes	characteristic
271	C₄	C₄×(C₂×D₄₇)⋊C₄	1	2	yes	—
272	C₄	C₄×C₄⋊Dic₄₇	1	2	yes	—
273	C₄	C₄×Dic₄₇⋊C₄	1	2	yes	—
274	C₂	—	2	2	no	—
275	C₂	—	2	2	no	—
276	C₂	C₄×C₄₇⋊#95	1	1	yes	characteristic
277	C₂	#20243098	1	2	yes	—
278	C₂	#20243199	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₈₈
commutator	C₄×C₄₇	C₄³
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₂×C₄₇×#376	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂×#376	D₄₇
hypocenter	C₄₇	#10701
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#376

Complements

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

N	H ≅ G/N	type
8. C₄×#10625329	C₄	semidirect
11. C₄×C₄⋊Dic₄₇	C₄	semidirect
15. #15688883	C₄	semidirect
19. C₄₇×#376	C₄	semidirect
71. C₄²×C₄₇	C₄²	semidirect
84. #10625329	C₄²	semidirect
156. #376	Dic₄₇	semidirect
160. C₄₇	#10701	semidirect
221. C₄²	C₄×Dic₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#10625329	2
C₄×#13093983	2
C₂×#15688883	2
C₂×C₄₇×#376	2
#10701	47