Subgroups — #26109207

Order 12032.

398 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Dic₉₄	C₂	1	1	yes	maximal, characteristic
2	#20243381	C₂	1	2	yes	maximal
3	#20242147	C₂	1	1	yes	maximal, characteristic
4	#20241262	C₂	1	2	yes	maximal
5	C₄₇×#1567	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₉₄	C₂²	1	1	yes	characteristic
7	C₈×Dic₉₄	—	2	1	no	—
8	C₈×Dic₉₄	—	2	1	no	—
9	C₈×Dic₉₄	—	2	1	no	—
10	C₈×Dic₉₄	—	2	1	no	—
11	#15689147	C₂²	1	2	yes	—
12	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
13	C₂×Dic₄₇⋊C₈	C₄	1	2	yes	—
14	C₂×#13093942	—	2	2	no	—
15	C₂×C₈×Dic₄₇	C₄	1	2	yes	—
16	#15688891	—	2	2	no	—
17	#15688876	—	2	1	no	—
18	#15688876	—	2	1	no	—
19	C₄₇×#423	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
21	C₄₇×#335	—	2	1	no	—
22	C₄₇×#335	—	2	1	no	—
23	C₂²×Dic₉₄	D₄	1	1	yes	characteristic
24	C₂×#10625329	—	2	2	no	—
25	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
26	C₄×Dic₉₄	—	2	1	no	—
27	C₄×Dic₉₄	—	2	1	no	—
28	C₄×Dic₉₄	—	2	1	no	—
29	C₄×Dic₉₄	—	2	1	no	—
30	C₂×C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
31	C₂×Dic₄₇⋊C₄	Q₈	1	2	yes	—
32	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
33	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
34	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
35	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
36	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
37	Dic₄₇⋊C₈	—	2	2	no	—
38	Dic₄₇⋊C₈	—	2	2	no	—
39	#13093942	—	4	4	no	—
40	C₈×Dic₄₇	—	2	2	no	—
41	C₈×Dic₄₇	—	2	2	no	—
42	C₂×M₁₆×C₄₇	—	2	2	no	—
43	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
46	C₄⋊C₈×C₄₇	—	2	2	no	—
47	C₄⋊C₈×C₄₇	—	2	2	no	—
48	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
49	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
50	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
51	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
55	C₂²×Dic₄₇	C₄⋊C₄	1	2	yes	—
56	C₂²×Dic₄₇	C₄⋊C₄	1	2	yes	—
57	C₂×Dic₉₄	—	2	1	no	—
58	C₂×Dic₉₄	—	2	1	no	—
59	C₂×Dic₉₄	—	2	1	no	—
60	C₂×Dic₉₄	—	2	1	no	—
61	C₂×Dic₉₄	D₈	1	2	yes	—
62	C₂×Dic₉₄	Q₁₆	1	2	yes	—
63	M₁₆×C₄₇	—	4	2	no	—
64	M₁₆×C₄₇	—	4	2	no	—
65	C₂×C₈×C₄₇	—	2	2	no	—
66	C₂×C₈×C₄₇	—	4	1	no	—
67	C₂×C₈×C₄₇	—	4	2	no	—
68	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
72	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
73	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₄₇	#41	1	1	yes	characteristic
75	C₄²×C₄₇	#41	1	1	yes	characteristic
76	C₄⋊Dic₄₇	—	2	1	no	—
77	C₄⋊Dic₄₇	—	2	1	no	—
78	Dic₄₇⋊C₄	—	2	2	no	—
79	Dic₄₇⋊C₄	—	2	2	no	—
80	C₄×Dic₄₇	—	2	2	no	—
81	C₄×Dic₄₇	—	2	2	no	—
82	#10625329	—	4	2	no	—
83	#10625329	—	4	2	no	—
84	C₂×C₄₇⋊C₈	—	2	2	no	—
85	C₂×C₄₇⋊C₈	—	4	2	no	—
86	C₂×C₄₇⋊C₈	—	4	2	no	—
87	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₄₇	—	2	1	no	—
89	C₂×C₄×C₄₇	—	2	1	no	—
90	C₂×C₄×C₄₇	—	4	1	no	—
91	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
93	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
94	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
95	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
96	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
97	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
98	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
99	C₂×C₄×C₄₇	#135	1	1	yes	characteristic
100	C₂×C₄×C₄₇	#135	1	1	yes	characteristic
101	C₂×Dic₄₇	—	2	2	no	—
102	C₂×Dic₄₇	—	2	2	no	—
103	C₂×Dic₄₇	—	2	2	no	—
104	C₂×Dic₄₇	—	4	2	no	—
105	C₂×Dic₄₇	#107	1	4	yes	—
106	Dic₉₄	—	2	2	no	—
107	Dic₉₄	—	2	2	no	—
108	Dic₉₄	—	2	2	no	—
109	Dic₉₄	—	2	2	no	—
110	C₈×C₄₇	—	2	1	no	—
111	C₈×C₄₇	—	2	1	no	—
112	C₈×C₄₇	—	2	1	no	—
113	C₈×C₄₇	—	8	2	no	—
114	C₈×C₄₇	#101	1	2	yes	complemented
115	C₄₇⋊C₈	—	4	2	no	—
116	C₄₇⋊C₈	—	8	2	no	—
117	#8617	—	47	1	no	maximal, Sylow
118	C₂²×C₄₇	—	2	1	no	—
119	C₂²×C₄₇	—	2	1	no	—
120	C₂²×C₄₇	#380	1	1	yes	characteristic
121	C₂²×C₄₇	C₂×#107	1	1	yes	characteristic
122	C₂²×C₄₇	#427	1	1	yes	characteristic
123	C₄×C₄₇	—	2	1	no	—
124	C₄×C₄₇	—	2	1	no	—
125	C₄×C₄₇	—	2	1	no	—
126	C₄×C₄₇	C₂×#101	1	1	yes	characteristic
127	C₄×C₄₇	#413	1	1	yes	characteristic
128	C₄×C₄₇	#466	1	1	yes	characteristic
129	C₄×C₄₇	#466	1	1	yes	characteristic
130	C₄×C₄₇	#467	1	1	yes	characteristic
131	C₄×C₄₇	#467	1	1	yes	characteristic
132	Dic₄₇	—	2	4	no	—
133	Dic₄₇	—	4	2	no	—
134	C₂×C₈×Q₈	—	47	1	no	—
135	#1801	—	47	2	no	—
136	#1567	—	47	1	no	—
137	#1567	D₄₇	1	1	yes	characteristic
138	#1292	—	47	2	no	—
139	C₂×C₄₇	—	2	1	no	—
140	C₂×C₄₇	—	2	1	no	—
141	C₂×C₄₇	#1801	1	1	yes	characteristic
142	C₂×C₄₇	#1808	1	1	yes	characteristic
143	C₂×C₄₇	#1813	1	1	yes	characteristic
144	C₂×C₄×Q₈	—	47	1	no	—
145	C₈×Q₈	—	94	1	no	—
146	C₈×Q₈	—	94	1	no	—
147	C₈×Q₈	—	94	1	no	—
148	C₈×Q₈	—	94	1	no	—
149	#423	—	47	2	no	—
150	#423	C₂×D₄₇	1	2	yes	—
151	C₂×C₄⋊C₈	—	47	2	no	—
152	C₂×C₄⋊C₈	—	47	1	no	—
153	C₂×#101	—	94	2	no	—
154	C₂×C₄×C₈	—	47	2	no	—
155	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
156	#343	—	94	2	no	—
157	#335	—	2	1	no	—
158	#335	—	2	1	no	—
159	#335	—	94	1	no	—
160	#335	—	94	1	no	—
161	C₄₇	#8617	1	1	yes	characteristic, complemented, Sylow
162	C₂²×Q₈	—	47	1	no	—
163	C₂×M₁₆	—	2	2	no	—
164	C₂×M₁₆	—	94	2	no	—
165	C₂²×C₈	—	47	2	no	—
166	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
167	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
168	C₄×Q₈	—	94	1	no	—
169	C₄×Q₈	—	94	1	no	—
170	C₄×Q₈	—	94	1	no	—
171	C₄×Q₈	—	94	1	no	—
172	C₂×C₄⋊C₄	—	47	1	no	—
173	C₂×C₄⋊C₄	—	47	2	no	—
174	C₂×C₄²	—	47	2	no	—
175	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
176	C₄⋊C₈	—	2	2	no	—
177	C₄⋊C₈	—	2	2	no	—
178	C₄⋊C₈	—	94	2	no	—
179	C₄⋊C₈	—	94	2	no	—
180	C₄⋊C₈	—	94	1	no	—
181	C₄⋊C₈	—	94	2	no	—
182	C₄⋊C₈	—	94	2	no	—
183	C₄⋊C₈	—	94	1	no	—
184	#101	—	188	4	no	—
185	C₄×C₈	—	94	2	no	—
186	C₄×C₈	—	94	2	no	—
187	C₄×C₈	D₁₈₈	1	1	yes	characteristic
188	C₄×C₈	D₁₈₈	1	1	yes	characteristic
189	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
190	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
191	C₂×Q₈	—	47	2	no	—
192	C₂×Q₈	—	47	2	no	—
193	C₂×Q₈	—	94	1	no	—
194	C₂×Q₈	—	94	1	no	—
195	C₂×Q₈	—	94	1	no	—
196	C₂×Q₈	—	94	1	no	—
197	C₂²×C₄	—	47	2	no	—
198	C₂²×C₄	—	47	2	no	—
199	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
200	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
201	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
202	M₁₆	—	4	2	no	—
203	M₁₆	—	4	2	no	—
204	M₁₆	—	188	2	no	—
205	M₁₆	—	188	2	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	4	1	no	—
208	C₂×C₈	—	4	2	no	—
209	C₂×C₈	—	94	2	no	—
210	C₂×C₈	—	188	2	no	—
211	C₂×C₈	—	188	2	no	—
212	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
213	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
214	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
215	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
216	C₄⋊C₄	—	94	1	no	—
217	C₄⋊C₄	—	94	1	no	—
218	C₄⋊C₄	—	94	2	no	—
219	C₄⋊C₄	—	94	2	no	—
220	C₄²	—	94	2	no	—
221	C₄²	—	94	2	no	—
222	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
223	C₄²	#10625352	1	1	yes	characteristic
224	C₄²	#10625354	1	1	yes	characteristic
225	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
226	C₂³	C₄⋊C₄×D₄₇	1	1	yes	characteristic
227	Q₈	—	94	2	no	—
228	Q₈	—	94	2	no	—
229	Q₈	—	94	2	no	—
230	Q₈	—	94	2	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	4	1	no	—
234	C₂×C₄	—	47	4	no	—
235	C₂×C₄	—	94	2	no	—
236	C₂×C₄	—	94	2	no	—
237	C₂×C₄	—	94	2	no	—
238	C₂×C₄	—	188	2	no	—
239	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
240	C₂×C₄	#13093982	1	1	yes	characteristic
241	C₂×C₄	#13093995	1	1	yes	characteristic
242	C₂×C₄	#13093999	1	1	yes	characteristic
243	C₂×C₄	#13094000	1	1	yes	characteristic
244	C₂×C₄	#13094004	1	1	yes	characteristic
245	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
246	C₂×C₄	Q₁₆×D₄₇	1	1	yes	characteristic
247	C₂×C₄	#13094043	1	1	yes	characteristic
248	C₂×C₄	#13094047	1	1	yes	characteristic
249	C₈	—	2	1	no	—
250	C₈	—	2	1	no	—
251	C₈	—	2	1	no	—
252	C₈	—	8	2	no	—
253	C₈	—	376	2	no	—
254	C₈	—	188	2	no	—
255	C₈	#13093942	1	2	yes	complemented
256	C₂²	—	2	1	no	—
257	C₂²	—	2	1	no	—
258	C₂²	#15688980	1	1	yes	characteristic
259	C₂²	#107×D₄₇	1	1	yes	characteristic
260	C₂²	#15689160	1	1	yes	characteristic
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	2	1	no	—
264	C₄	—	94	4	no	—
265	C₄	—	188	2	no	—
266	C₄	C₂×#13093942	1	1	yes	characteristic
267	C₄	#15689105	1	1	yes	characteristic
268	C₄	#15689356	1	1	yes	characteristic
269	C₄	#15689360	1	1	yes	characteristic
270	C₄	#15689361	1	1	yes	characteristic
271	C₄	#15689365	1	1	yes	characteristic
272	C₂	—	2	1	no	—
273	C₂	—	2	1	no	—
274	C₂	#20243381	1	1	yes	characteristic
275	C₂	#20243414	1	1	yes	characteristic
276	C₂	#20243435	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15688980
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1567	C₂
socle	C₂²×C₄₇	#380
prime radical	C₂³×C₄₇	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1567	D₄₇
hypocenter	C₄₇	#8617
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1567

Complements

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

N	H ≅ G/N	type
114. C₈×C₄₇	#101	semidirect
161. C₄₇	#8617	semidirect
255. C₈	#13093942	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Dic₉₄	2
#20243381	2
#20242147	2
#20241262	2
C₄₇×#1567	2
#8617	47