Subgroups — #26122106

Order 12032.

814 conjugacy classes of subgroups.

The subgroup lattice has 294 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×C₄×C₄₇⋊C₈	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×C₄×C₄₇⋊C₈	—	2	2	no	—
11	C₂×C₄×C₄₇⋊C₈	—	2	2	no	—
12	C₂×C₄×C₄₇⋊C₈	C₄	1	2	yes	—
13	C₂×C₄×C₄₇⋊C₈	C₄	1	2	yes	—
14	#15688887	—	2	8	no	—
15	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
16	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
17	C₄₇×#423	—	2	2	no	—
18	C₄₇×#423	C₄	1	4	yes	—
19	C₂×C₄⋊C₈×C₄₇	—	2	2	no	—
20	C₂×#10625329	—	2	2	no	—
21	C₂×#10625329	—	2	4	no	—
22	C₂×#10625329	—	2	4	no	—
23	C₂×#10625329	C₂×C₄	1	4	yes	—
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	—	2	2	no	—
29	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
30	C₂²×C₄₇⋊C₈	D₄	1	4	yes	—
31	C₂²×C₄₇⋊C₈	Q₈	1	4	yes	—
32	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
33	C₄×C₄₇⋊C₈	—	2	4	no	—
34	C₄×C₄₇⋊C₈	—	2	4	no	—
35	C₂³×C₄×C₄₇	Q₈	1	2	yes	—
36	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
37	C₂×M₁₆×C₄₇	—	2	2	no	—
38	C₂×M₁₆×C₄₇	—	2	4	no	—
39	C₂×M₁₆×C₄₇	—	2	4	no	—
40	C₂×M₁₆×C₄₇	C₂×C₄	1	4	yes	—
41	C₂²×C₈×C₄₇	—	2	2	no	—
42	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₄₇	D₄	1	2	yes	—
47	C₂×C₄²×C₄₇	D₄	1	2	yes	—
48	C₂×C₄²×C₄₇	D₄	1	2	yes	—
49	C₄⋊C₈×C₄₇	—	2	8	no	—
50	C₂⁴×C₄₇	C₂×Q₈	1	1	yes	characteristic
51	C₂²×C₄×C₄₇	—	2	8	no	—
52	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
54	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
55	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
56	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
64	C₂²×C₄×C₄₇	#41	1	2	yes	—
65	C₂²×C₄×C₄₇	#41	1	2	yes	—
66	M₁₆×C₄₇	—	2	8	no	—
67	M₁₆×C₄₇	—	4	4	no	—
68	C₂×C₈×C₄₇	—	2	4	no	—
69	C₂×C₈×C₄₇	—	4	2	no	—
70	C₂×C₈×C₄₇	—	4	2	no	—
71	C₄²×C₄₇	C₄²	1	2	yes	—
72	C₄²×C₄₇	C₄²	1	2	yes	—
73	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
74	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
75	C₄²×C₄₇	C₂²⋊C₄	1	4	yes	—
76	C₄²×C₄₇	C₂²⋊C₄	1	4	yes	—
77	#10625329	—	2	8	no	—
78	#10625329	—	4	4	no	—
79	C₂×C₄₇⋊C₈	—	2	4	no	—
80	C₂×C₄₇⋊C₈	—	2	4	no	—
81	C₂×C₄₇⋊C₈	—	2	4	no	—
82	C₂×C₄₇⋊C₈	—	2	4	no	—
83	C₂×C₄₇⋊C₈	—	2	4	no	—
84	C₂×C₄₇⋊C₈	—	2	4	no	—
85	C₂×C₄₇⋊C₈	—	2	4	no	—
86	C₂×C₄₇⋊C₈	—	4	2	no	—
87	C₂×C₄₇⋊C₈	—	4	2	no	—
88	C₂×C₄₇⋊C₈	C₄⋊C₄	1	8	yes	—
89	C₂³×C₄₇	—	2	4	no	—
90	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂³×C₄₇	#125	1	2	yes	—
94	C₂³×C₄₇	C₄⋊Q₈	1	2	yes	—
95	C₂×C₄×C₄₇	—	2	4	no	—
96	C₂×C₄×C₄₇	—	2	8	no	—
97	C₂×C₄×C₄₇	—	2	8	no	—
98	C₂×C₄×C₄₇	#95	1	4	yes	—
99	C₂×C₄×C₄₇	#95	1	4	yes	—
100	C₂×C₄×C₄₇	#108	1	8	yes	—
101	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
102	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
106	C₂×C₄×C₄₇	#117	1	2	yes	—
107	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
108	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
109	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
110	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
111	C₂×C₄×C₄₇	#121	1	2	yes	—
112	C₂×C₄×C₄₇	#121	1	2	yes	—
113	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
114	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
115	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
116	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
117	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
118	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
119	C₈×C₄₇	—	4	4	no	—
120	C₄₇⋊C₈	—	2	8	no	—
121	C₄₇⋊C₈	—	4	4	no	—
122	#10844	—	47	1	no	maximal, Sylow
123	C₂²×C₄₇	—	2	4	no	—
124	C₂²×C₄₇	—	2	4	no	—
125	C₂²×C₄₇	—	2	2	no	—
126	C₂²×C₄₇	—	2	2	no	—
127	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
128	C₂²×C₄₇	#382	1	1	yes	characteristic
129	C₂²×C₄₇	#384	1	1	yes	characteristic
130	C₂²×C₄₇	#389	1	2	yes	—
131	C₂²×C₄₇	#389	1	2	yes	—
132	C₂²×C₄₇	C₂×#108	1	4	yes	—
133	C₄×C₄₇	—	2	4	no	—
134	C₄×C₄₇	#341	1	8	yes	complemented
135	C₄×C₄₇	C₄×C₄⋊C₄	1	2	yes	—
136	C₄×C₄₇	#380	1	2	yes	—
137	C₄×C₄₇	#390	1	1	yes	characteristic
138	C₄×C₄₇	#390	1	1	yes	characteristic
139	C₄×C₄₇	#391	1	1	yes	characteristic
140	C₄×C₄₇	#391	1	1	yes	characteristic
141	C₂×#423	—	47	1	no	—
142	C₂×#423	D₄₇	1	1	yes	characteristic
143	C₂²×C₄×C₈	—	47	1	no	—
144	C₂×#341	—	47	4	no	—
145	C₂×C₄₇	—	2	2	no	—
146	C₂×C₄₇	—	2	2	no	—
147	C₂×C₄₇	C₄⋊#95	1	1	yes	characteristic
148	C₂×C₄₇	C₂×#341	1	2	yes	—
149	C₂×C₄₇	#1837	1	2	yes	—
150	C₂×C₄₇	#1848	1	2	yes	—
151	C₂²×M₁₆	—	47	2	no	—
152	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
153	C₂³×C₈	—	47	2	no	—
154	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
155	#423	—	2	2	no	—
156	#423	—	47	4	no	—
157	#423	—	94	2	no	—
158	#423	Dic₄₇	1	4	yes	—
159	C₂×C₄⋊C₈	—	2	2	no	—
160	C₂×C₄⋊C₈	—	94	2	no	—
161	C₂×C₄×C₈	—	47	2	no	—
162	C₂×C₄×C₈	—	47	2	no	—
163	C₂×C₄×C₈	—	94	2	no	—
164	C₂×C₄×C₈	—	94	2	no	—
165	#341	—	94	8	no	—
166	C₄₇	#10844	1	1	yes	characteristic, complemented, Sylow
167	C₂³×C₄	Dic₉₄	1	2	yes	—
168	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
169	C₂×M₁₆	—	2	2	no	—
170	C₂×M₁₆	—	2	4	no	—
171	C₂×M₁₆	—	2	4	no	—
172	C₂×M₁₆	—	47	4	no	—
173	C₂×M₁₆	—	94	2	no	—
174	C₂×M₁₆	—	94	4	no	—
175	C₂×M₁₆	—	94	4	no	—
176	C₂×M₁₆	C₂×Dic₄₇	1	4	yes	—
177	C₂²×C₈	—	2	2	no	—
178	C₂²×C₈	—	47	4	no	—
179	C₂²×C₈	—	47	4	no	—
180	C₂²×C₈	—	47	4	no	—
181	C₂²×C₈	—	94	2	no	—
182	C₂²×C₈	—	94	2	no	—
183	C₂²×C₈	—	94	2	no	—
184	C₂²×C₈	—	94	2	no	—
185	C₂²×C₈	—	94	2	no	—
186	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
187	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
188	C₂×C₄²	C₄×D₄₇	1	2	yes	—
189	C₂×C₄²	D₁₈₈	1	2	yes	—
190	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
191	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
192	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
193	C₄⋊C₈	—	2	8	no	—
194	C₄⋊C₈	—	94	8	no	—
195	C₄×C₈	—	94	4	no	—
196	C₄×C₈	—	94	4	no	—
197	C₂⁴	C₂×Dic₉₄	1	1	yes	characteristic
198	C₂²×C₄	—	2	8	no	—
199	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
200	C₂²×C₄	Dic₄₇⋊C₄	1	4	yes	—
201	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
202	C₂²×C₄	C₄⋊Dic₄₇	1	4	yes	—
203	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
204	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
205	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
206	C₂²×C₄	D₄×D₄₇	1	2	yes	—
207	C₂²×C₄	#10625352	1	2	yes	—
208	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
209	C₂²×C₄	#10625354	1	2	yes	—
210	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
211	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
212	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
213	M₁₆	—	2	8	no	—
214	M₁₆	—	4	4	no	—
215	M₁₆	—	94	8	no	—
216	M₁₆	—	188	4	no	—
217	C₂×C₈	—	2	4	no	—
218	C₂×C₈	—	4	2	no	—
219	C₂×C₈	—	4	2	no	—
220	C₂×C₈	—	47	8	no	—
221	C₂×C₈	—	94	4	no	—
222	C₂×C₈	—	94	4	no	—
223	C₂×C₈	—	94	4	no	—
224	C₂×C₈	—	94	4	no	—
225	C₂×C₈	—	94	4	no	—
226	C₂×C₈	—	94	4	no	—
227	C₂×C₈	—	94	4	no	—
228	C₂×C₈	—	188	2	no	—
229	C₂×C₈	—	188	2	no	—
230	C₄²	C₄×Dic₄₇	1	2	yes	—
231	C₄²	C₄×Dic₄₇	1	2	yes	—
232	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
233	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
234	C₄²	(C₂×D₄₇)⋊C₄	1	4	yes	—
235	C₄²	C₂²⋊Dic₄₇	1	4	yes	—
236	C₂³	—	2	4	no	—
237	C₂³	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
238	C₂³	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
239	C₂³	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
240	C₂³	#13094015	1	2	yes	—
241	C₂³	#13094023	1	2	yes	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	8	no	—
244	C₂×C₄	—	2	8	no	—
245	C₂×C₄	C₄₇⋊#95	1	4	yes	—
246	C₂×C₄	C₄₇⋊#95	1	4	yes	—
247	C₂×C₄	#13093959	1	8	yes	—
248	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
249	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
250	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
251	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
252	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
253	C₂×C₄	#13093982	1	2	yes	—
254	C₂×C₄	#13093986	1	2	yes	—
255	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
256	C₂×C₄	#13093989	1	2	yes	—
257	C₂×C₄	Q₈×Dic₄₇	1	2	yes	—
258	C₂×C₄	#13093995	1	2	yes	—
259	C₂×C₄	#13093999	1	2	yes	—
260	C₂×C₄	#13094000	1	2	yes	—
261	C₂×C₄	#13094004	1	2	yes	—
262	C₂×C₄	#13094011	1	1	yes	characteristic
263	C₂×C₄	#13094013	1	1	yes	characteristic
264	C₂×C₄	#13094020	1	1	yes	characteristic
265	C₂×C₄	#13094022	1	1	yes	characteristic
266	C₈	—	4	4	no	—
267	C₈	—	94	8	no	—
268	C₈	—	188	4	no	—
269	C₂²	—	2	4	no	—
270	C₂²	—	2	4	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	2	2	no	—
273	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
274	C₂²	#15688988	1	1	yes	characteristic
275	C₂²	#15688992	1	1	yes	characteristic
276	C₂²	#15689019	1	2	yes	—
277	C₂²	#15689022	1	2	yes	—
278	C₂²	C₂×#13093959	1	4	yes	—
279	C₄	—	2	4	no	—
280	C₄	#15688887	1	8	yes	complemented
281	C₄	C₄⋊C₄×Dic₄₇	1	2	yes	—
282	C₄	#15688980	1	2	yes	—
283	C₄	#15689024	1	1	yes	characteristic
284	C₄	#15689026	1	1	yes	characteristic
285	C₄	#15689028	1	1	yes	characteristic
286	C₄	#15689030	1	1	yes	characteristic
287	C₂	—	2	2	no	—
288	C₂	—	2	2	no	—
289	C₂	#20241617	1	1	yes	characteristic
290	C₂	C₂×#15688887	1	2	yes	—
291	C₂	#20243525	1	2	yes	—
292	C₂	#20243559	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	#10844
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
134. C₄×C₄₇	#341	semidirect
166. C₄₇	#10844	semidirect
280. C₄	#15688887	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×C₄×C₄₇⋊C₈	2
C₂×#15688887	2
C₂×C₄₇×#423	2
#10844	47