Subgroups — #26219763

Order 12032.

1022 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689166	C₂	1	2	yes	maximal
2	C₂×#15689147	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688989	C₂	1	1	yes	maximal, characteristic
4	#20244156	C₂	1	8	yes	maximal, complemented
5	#20243109	C₂	1	2	yes	maximal
6	C₄×#117×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×#10625329	C₂²	1	2	yes	—
8	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
9	#15689166	—	2	8	no	—
10	C₂×#13093957	C₂²	1	4	yes	—
11	C₂×#13093955	C₂²	1	4	yes	—
12	#15689147	C₄	1	8	yes	complemented
13	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
14	#15688989	—	2	1	no	—
15	#15688989	—	2	1	no	—
16	#15688989	C₂²	1	4	yes	—
17	C₂×C₄₇⋊#95	—	2	2	no	—
18	#15688885	C₂²	1	8	yes	—
19	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
20	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
21	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
22	C₄×C₄⋊C₄×C₄₇	C₂²	1	4	yes	—
23	C₄×C₂²⋊C₄×C₄₇	—	2	2	no	—
24	C₄₇×#376	—	2	1	no	—
25	C₄³×C₄₇	—	2	1	no	—
26	C₂³×Dic₄₇	—	2	2	no	—
27	C₂×#10625329	—	2	4	no	—
28	C₂×#10625329	—	2	4	no	—
29	C₂×#10625329	C₂×C₄	1	8	yes	—
30	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
31	C₂×C₄⋊Dic₄₇	—	2	4	no	—
32	C₂×C₄⋊Dic₄₇	—	2	4	no	—
33	C₂×C₄⋊Dic₄₇	—	2	2	no	—
34	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
35	#13093957	—	2	8	no	—
36	#13093955	—	2	8	no	—
37	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	16	yes	complemented
38	C₄₇⋊#95	—	2	4	no	—
39	C₄₇⋊#95	—	2	4	no	—
40	C₂³×C₄×C₄₇	D₄	1	2	yes	—
41	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
42	#117×C₄₇	—	2	2	no	—
43	#117×C₄₇	—	2	2	no	—
44	#117×C₄₇	D₄	1	4	yes	—
45	#117×C₄₇	Q₈	1	4	yes	—
46	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
47	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
48	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
49	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
50	C₂×C₄²×C₄₇	—	2	1	no	—
51	C₂×C₄²×C₄₇	—	2	1	no	—
52	C₂×C₄²×C₄₇	—	2	4	no	—
53	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
54	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
55	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
56	C₂×C₄²×C₄₇	Q₈	1	4	yes	—
57	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
58	#95×C₄₇	—	2	4	no	—
59	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	—	2	2	no	—
61	C₂²×C₄×C₄₇	—	2	2	no	—
62	C₂²×C₄×C₄₇	—	4	4	no	—
63	C₂²×C₄×C₄₇	C₄⋊C₄	1	8	yes	—
64	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
70	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
71	C₂²×C₄×C₄₇	#41	1	2	yes	—
72	C₂²×C₄×C₄₇	#41	1	2	yes	—
73	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
74	C₂²×Dic₄₇	—	2	4	no	—
75	C₂²×Dic₄₇	—	4	8	no	—
76	C₂²×Dic₄₇	—	4	4	no	—
77	C₄⋊C₄×C₄₇	—	2	2	no	—
78	C₄⋊C₄×C₄₇	—	2	2	no	—
79	C₄⋊C₄×C₄₇	C₂×D₄	1	4	yes	—
80	C₄⋊C₄×C₄₇	C₂×Q₈	1	4	yes	—
81	C₂²⋊C₄×C₄₇	—	2	4	no	—
82	C₂²⋊C₄×C₄₇	—	2	4	no	—
83	C₄²×C₄₇	—	2	4	no	—
84	C₄²×C₄₇	—	2	4	no	—
85	C₄²×C₄₇	—	4	8	no	—
86	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
87	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
88	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
89	C₄²×C₄₇	C₂×Q₈	1	4	yes	—
90	C₄⋊Dic₄₇	—	2	4	no	—
91	C₄⋊Dic₄₇	—	2	4	no	—
92	C₄⋊Dic₄₇	—	4	4	no	—
93	#10625329	—	2	16	no	—
94	C₂×C₄₇⋊C₈	—	2	4	no	—
95	C₂×C₄₇⋊C₈	—	2	4	no	—
96	C₂×C₄₇⋊C₈	C₂²×C₄	1	8	yes	—
97	C₂³×C₄₇	—	2	4	no	—
98	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
99	C₂³×C₄₇	#121	1	2	yes	—
100	C₂³×C₄₇	C₂²⋊Q₈	1	2	yes	—
101	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
102	C₂×C₄×C₄₇	—	2	4	no	—
103	C₂×C₄×C₄₇	—	2	4	no	—
104	C₂×C₄×C₄₇	—	2	4	no	—
105	C₂×C₄×C₄₇	—	2	4	no	—
106	C₂×C₄×C₄₇	—	4	8	no	—
107	C₂×C₄×C₄₇	—	4	4	no	—
108	C₂×C₄×C₄₇	—	4	4	no	—
109	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
110	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
111	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
112	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
113	C₂×C₄×C₄₇	C₄×Q₈	1	4	yes	—
114	C₂×C₄×C₄₇	#121	1	2	yes	—
115	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
116	C₂×C₄×C₄₇	#125	1	4	yes	—
117	C₂×C₄×C₄₇	C₄⋊Q₈	1	4	yes	—
118	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
119	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
121	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
122	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
123	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
124	C₂×Dic₄₇	—	4	4	no	—
125	C₂×Dic₄₇	—	4	8	no	—
126	C₂×Dic₄₇	—	8	8	no	—
127	C₄₇⋊C₈	—	2	8	no	—
128	#21172	—	47	1	no	maximal, Sylow
129	C₂²×C₄₇	—	2	4	no	—
130	C₂²×C₄₇	—	2	4	no	—
131	C₂²×C₄₇	—	2	4	no	—
132	C₂²×C₄₇	#389	1	4	yes	—
133	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
134	C₂²×C₄₇	C₂×#121	1	1	yes	characteristic
135	C₂²×C₄₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
136	C₂²×C₄₇	#575	1	4	yes	—
137	C₄×C₄₇	—	2	4	no	—
138	C₄×C₄₇	—	2	4	no	—
139	C₄×C₄₇	—	4	4	no	—
140	C₄×C₄₇	—	4	4	no	—
141	C₄×C₄₇	#389	1	4	yes	—
142	C₄×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
143	C₄×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
144	C₄×C₄₇	C₂×#125	1	1	yes	characteristic
145	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
146	Dic₄₇	—	8	4	no	—
147	C₂×#428	—	47	2	no	—
148	C₂×#423	—	47	1	no	—
149	C₂×#382	—	47	1	no	—
150	C₄×#117	D₄₇	1	1	yes	characteristic
151	#1949	—	47	8	no	—
152	#1736	—	47	2	no	—
153	C₂×C₄₇	—	2	4	no	—
154	C₂×C₄₇	C₂×#389	1	1	yes	characteristic
155	C₂×C₄₇	#2911	1	2	yes	—
156	C₂×C₄₇	#3040	1	2	yes	—
157	C₂×C₄₇	#3080	1	2	yes	—
158	C₂²×M₁₆	—	47	2	no	—
159	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
160	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
161	C₂²×C₄⋊C₄	—	47	2	no	—
162	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
163	#428	—	94	8	no	—
164	C₂×#107	—	47	4	no	—
165	C₂×#106	—	47	4	no	—
166	#423	—	47	8	no	—
167	C₂×C₄⋊C₈	—	47	4	no	—
168	#382	—	47	4	no	—
169	#382	—	94	1	no	—
170	#382	—	94	1	no	—
171	C₄×C₄⋊C₄	C₂×D₄₇	1	4	yes	—
172	C₄×C₂²⋊C₄	—	2	2	no	—
173	#376	—	2	1	no	—
174	C₂×#95	—	94	2	no	—
175	C₄³	—	2	1	no	—
176	#340	—	47	8	no	—
177	C₄₇	#21172	1	1	yes	characteristic, complemented, Sylow
178	C₂³×C₄	—	94	2	no	—
179	C₂³×C₄	C₂²⋊D₄₇	1	2	yes	—
180	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
181	C₂×M₁₆	—	47	8	no	—
182	C₂×M₁₆	—	94	4	no	—
183	C₂×M₁₆	—	94	4	no	—
184	C₂²×C₈	—	47	4	no	—
185	#117	—	2	2	no	—
186	#117	—	2	2	no	—
187	#117	Dic₉₄	1	4	yes	—
188	#117	D₁₈₈	1	4	yes	—
189	C₂×C₄⋊C₄	—	47	4	no	—
190	C₂×C₄⋊C₄	—	94	4	no	—
191	C₂×C₄⋊C₄	—	94	4	no	—
192	C₂×C₄⋊C₄	—	94	2	no	—
193	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
194	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
195	C₂×C₂²⋊C₄	—	2	1	no	—
196	C₂×C₂²⋊C₄	—	2	1	no	—
197	C₂×C₄²	—	2	1	no	—
198	C₂×C₄²	—	2	1	no	—
199	C₂×C₄²	—	2	4	no	—
200	C₂×C₄²	Dic₉₄	1	4	yes	—
201	C₂×C₄²	C₄×D₄₇	1	4	yes	—
202	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
203	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
204	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
205	#107	—	94	8	no	—
206	#106	—	94	8	no	—
207	C₄⋊C₈	—	47	16	no	—
208	#95	—	2	4	no	—
209	#95	—	94	4	no	—
210	#95	—	94	4	no	—
211	C₂⁴	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	4	4	no	—
215	C₂²×C₄	—	94	4	no	—
216	C₂²×C₄	—	188	8	no	—
217	C₂²×C₄	—	188	4	no	—
218	C₂²×C₄	Dic₄₇⋊C₄	1	8	yes	—
219	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
220	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
221	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
222	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
223	C₂²×C₄	#10625350	1	2	yes	—
224	C₂²×C₄	#10625350	1	2	yes	—
225	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
226	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
227	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
228	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
229	M₁₆	—	94	16	no	—
230	C₂×C₈	—	47	8	no	—
231	C₂×C₈	—	94	4	no	—
232	C₂×C₈	—	94	4	no	—
233	C₄⋊C₄	—	2	2	no	—
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	94	4	no	—
236	C₄⋊C₄	—	94	4	no	—
237	C₄⋊C₄	—	188	4	no	—
238	C₄⋊C₄	C₂×Dic₉₄	1	4	yes	—
239	C₄⋊C₄	C₂×D₁₈₈	1	4	yes	—
240	C₂²⋊C₄	—	2	4	no	—
241	C₂²⋊C₄	—	2	4	no	—
242	C₄²	—	2	4	no	—
243	C₄²	—	2	4	no	—
244	C₄²	—	4	8	no	—
245	C₄²	Dic₄₇⋊C₄	1	4	yes	—
246	C₄²	Dic₄₇⋊C₄	1	4	yes	—
247	C₄²	C₂×Dic₉₄	1	4	yes	—
248	C₄²	C₂×C₄×D₄₇	1	4	yes	—
249	C₂³	—	2	4	no	—
250	C₂³	C₂×Dic₄₇⋊C₄	1	2	yes	—
251	C₂³	#13093998	1	2	yes	—
252	C₂³	#13094003	1	2	yes	—
253	C₂³	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	4	8	no	—
259	C₂×C₄	—	4	4	no	—
260	C₂×C₄	—	4	4	no	—
261	C₂×C₄	—	376	8	no	—
262	C₂×C₄	—	188	4	no	—
263	C₂×C₄	—	188	8	no	—
264	C₂×C₄	C₂×Dic₄₇⋊C₄	1	4	yes	—
265	C₂×C₄	C₂×Dic₄₇⋊C₄	1	4	yes	—
266	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
267	C₂×C₄	C₄×D₁₈₈	1	4	yes	—
268	C₂×C₄	C₄×Dic₉₄	1	4	yes	—
269	C₂×C₄	#13093998	1	2	yes	—
270	C₂×C₄	#13094003	1	2	yes	—
271	C₂×C₄	#13094016	1	4	yes	—
272	C₂×C₄	#13094024	1	4	yes	—
273	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
274	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
275	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
276	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
277	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
278	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
279	C₈	—	94	8	no	—
280	C₂²	—	2	4	no	—
281	C₂²	—	2	4	no	—
282	C₂²	—	2	4	no	—
283	C₂²	#15689021	1	4	yes	—
284	C₂²	C₂²×Dic₄₇⋊C₄	1	1	yes	characteristic
285	C₂²	C₂×#13093998	1	1	yes	characteristic
286	C₂²	C₂×#13094003	1	1	yes	characteristic
287	C₂²	#15689921	1	4	yes	—
288	C₄	—	2	4	no	—
289	C₄	—	2	4	no	—
290	C₄	—	4	4	no	—
291	C₄	—	4	4	no	—
292	C₄	—	376	4	no	—
293	C₄	#15689021	1	4	yes	—
294	C₄	C₂×C₄×D₁₈₈	1	1	yes	characteristic
295	C₄	C₂×C₄×Dic₉₄	1	1	yes	characteristic
296	C₄	C₂×#13094016	1	1	yes	characteristic
297	C₄	C₂×#13094024	1	1	yes	characteristic
298	C₂	—	2	4	no	—
299	C₂	C₂×#15689021	1	1	yes	characteristic
300	C₂	#20251977	1	2	yes	—
301	C₂	#20252882	1	2	yes	—
302	C₂	#20253210	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄×#117×C₄₇	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₄×#117	D₄₇
hypocenter	C₄₇	#21172
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₄×#117

Complements

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

N	H ≅ G/N	type
4. #20244156	C₂	semidirect
12. #15689147	C₄	semidirect
37. C₄⋊(C₄₇⋊C₈)	C₂×C₄	semidirect
177. C₄₇	#21172	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689166	2
C₂×#15689147	2
C₂×#15688989	2
#20244156	2
#20243109	2
C₄×#117×C₄₇	2
#21172	47