Subgroups — #26127502

Order 12032.

414 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₄₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
2	#20242136	C₂	1	1	yes	maximal, characteristic, complemented
3	#20241431	C₂	1	4	yes	maximal, complemented
4	C₄₇×#3147	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄₇⋊C₁₆	—	2	1	no	—
6	#15689176	—	2	1	no	—
7	C₄⋊(C₄₇⋊C₁₆)	—	2	1	no	—
8	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	2	yes	—
9	#15688876	—	2	1	no	—
10	#15688874	C₂²	1	2	yes	—
11	C₂×#135×C₄₇	—	2	1	no	—
12	C₂×#135×C₄₇	C₄	1	2	yes	—
13	C₄₇×#532	—	2	1	no	—
14	C₄₇×#495	—	2	1	no	—
15	C₄₇×#495	C₂²	1	2	yes	—
16	C₄₇×#494	C₄	1	1	yes	characteristic
17	C₄₇×#493	C₄	1	1	yes	characteristic
18	C₄₇×#492	—	2	1	no	—
19	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
20	C₂×C₄₇⋊C₁₆	—	2	2	no	—
21	C₂×C₄₇⋊C₁₆	—	4	2	no	—
22	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
23	C₂²⋊(C₄₇⋊C₈)	—	4	1	no	—
24	C₄×C₄₇⋊C₈	—	4	1	no	—
25	C₂×#41×C₄₇	—	2	2	no	—
26	#135×C₄₇	—	4	4	no	—
27	#135×C₄₇	—	4	2	no	—
28	#135×C₄₇	—	4	2	no	—
29	C₂×Q₁₆×C₄₇	—	2	1	no	—
30	C₂×Q₁₆×C₄₇	C₂×C₄	1	2	yes	—
31	C₂×SD₁₆×C₄₇	—	2	2	no	—
32	C₂×SD₁₆×C₄₇	—	2	2	no	—
33	C₂×D₈×C₄₇	—	2	1	no	—
34	C₂×D₈×C₄₇	C₂×C₄	1	2	yes	—
35	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
37	C₄⋊Q₈×C₄₇	—	2	1	no	—
38	#127×C₄₇	—	2	1	no	—
39	#124×C₄₇	—	2	2	no	—
40	#121×C₄₇	—	4	1	no	—
41	#121×C₄₇	—	4	1	no	—
42	C₄×D₄×C₄₇	—	4	1	no	—
43	C₄×D₄×C₄₇	—	4	1	no	—
44	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
45	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
46	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
47	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
48	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
49	#41×C₄₇	—	4	4	no	—
50	#41×C₄₇	—	4	4	no	—
51	C₂×Q₈×C₄₇	—	2	2	no	—
52	C₂×D₄×C₄₇	—	2	2	no	—
53	C₂×D₄×C₄₇	—	4	2	no	—
54	C₂×D₄×C₄₇	—	4	1	no	—
55	C₂×D₄×C₄₇	—	4	1	no	—
56	C₂²×C₄×C₄₇	—	4	2	no	—
57	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	SD₁₆	1	1	yes	characteristic
60	Q₁₆×C₄₇	—	4	2	no	—
61	Q₁₆×C₄₇	—	4	2	no	—
62	SD₁₆×C₄₇	—	4	4	no	—
63	SD₁₆×C₄₇	—	4	2	no	—
64	SD₁₆×C₄₇	—	4	2	no	—
65	D₈×C₄₇	—	4	2	no	—
66	D₈×C₄₇	—	4	2	no	—
67	C₂×C₈×C₄₇	—	2	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	D₈	1	2	yes	—
71	C₂×C₈×C₄₇	SD₁₆	1	2	yes	—
72	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
74	C₄⋊C₄×C₄₇	—	4	1	no	—
75	C₄⋊C₄×C₄₇	—	4	1	no	—
76	C₂²⋊C₄×C₄₇	—	4	2	no	—
77	C₂²⋊C₄×C₄₇	—	4	2	no	—
78	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
79	C₄²×C₄₇	D₈	1	1	yes	characteristic
80	C₄²×C₄₇	SD₁₆	1	1	yes	characteristic
81	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
82	C₂×C₄₇⋊C₈	—	4	2	no	—
83	C₄₇⋊C₁₆	—	4	2	no	—
84	C₂³×C₄₇	—	4	2	no	—
85	C₂³×C₄₇	#102	1	1	yes	characteristic
86	Q₈×C₄₇	—	4	2	no	—
87	Q₈×C₄₇	—	4	2	no	—
88	D₄×C₄₇	—	4	2	no	—
89	D₄×C₄₇	—	4	2	no	—
90	D₄×C₄₇	—	8	2	no	—
91	D₄×C₄₇	—	8	4	no	—
92	D₄×C₄₇	—	8	2	no	—
93	C₂×C₄×C₄₇	—	2	2	no	—
94	C₂×C₄×C₄₇	—	4	1	no	—
95	C₂×C₄×C₄₇	—	4	2	no	—
96	C₂×C₄×C₄₇	—	8	4	no	—
97	C₂×C₄×C₄₇	#100	1	2	yes	—
98	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
99	C₂×C₄×C₄₇	#102	1	2	yes	—
100	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
101	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
102	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
104	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
105	C₈×C₄₇	—	2	1	no	—
106	C₈×C₄₇	—	2	1	no	—
107	C₈×C₄₇	#102	1	2	yes	—
108	C₈×C₄₇	C₂×D₈	1	1	yes	characteristic
109	C₈×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
110	C₄₇⋊C₈	—	8	2	no	—
111	#11924	—	47	1	no	maximal, Sylow
112	C₂²×C₄₇	—	2	1	no	—
113	C₂²×C₄₇	—	2	1	no	—
114	C₂²×C₄₇	—	8	2	no	—
115	C₂²×C₄₇	—	8	2	no	—
116	C₂²×C₄₇	—	8	2	no	—
117	C₂²×C₄₇	#331	1	2	yes	—
118	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
119	C₄×C₄₇	—	2	1	no	—
120	C₄×C₄₇	—	2	1	no	—
121	C₄×C₄₇	—	2	1	no	—
122	C₄×C₄₇	—	8	2	no	—
123	C₄×C₄₇	#331	1	2	yes	—
124	C₄×C₄₇	#359	1	2	yes	—
125	C₄×C₄₇	C₂×#100	1	1	yes	characteristic
126	C₄×C₄₇	C₂×#102	1	1	yes	characteristic
127	#3147	D₄₇	1	1	yes	characteristic
128	C₂×C₄⋊C₁₆	—	47	1	no	—
129	#1564	—	47	1	no	—
130	#1360	—	47	4	no	—
131	C₂×C₄₇	—	2	2	no	—
132	C₂×C₄₇	—	16	2	no	—
133	C₂×C₄₇	C₂×#331	1	1	yes	characteristic
134	C₂×C₄₇	C₂×#359	1	1	yes	characteristic
135	C₂×C₄₇	#2145	1	1	yes	characteristic
136	C₂×#135	—	2	1	no	—
137	C₂×#135	Dic₄₇	1	2	yes	—
138	#532	—	2	1	no	—
139	C₂²×C₁₆	—	94	1	no	—
140	#495	—	2	1	no	—
141	#495	C₂×D₄₇	1	2	yes	—
142	#494	Dic₄₇	1	1	yes	characteristic
143	#493	Dic₄₇	1	1	yes	characteristic
144	#492	—	2	1	no	—
145	#431	—	94	1	no	—
146	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
147	C₄⋊C₁₆	—	47	2	no	—
148	C₄⋊C₁₆	—	94	1	no	—
149	#335	—	94	1	no	—
150	#334	—	47	2	no	—
151	C₄₇	#11924	1	1	yes	characteristic, complemented, Sylow
152	C₂×#41	—	2	2	no	—
153	#135	—	4	4	no	—
154	#135	—	4	2	no	—
155	#135	—	4	2	no	—
156	C₂×Q₁₆	—	2	1	no	—
157	C₂×Q₁₆	C₂×Dic₄₇	1	2	yes	—
158	C₂×SD₁₆	—	2	2	no	—
159	C₂×SD₁₆	—	2	2	no	—
160	C₂×D₈	—	2	1	no	—
161	C₂×D₈	C₂×Dic₄₇	1	2	yes	—
162	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
163	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
164	C₄⋊Q₈	—	2	1	no	—
165	#127	—	2	1	no	—
166	#124	—	2	2	no	—
167	#121	—	4	1	no	—
168	#121	—	4	1	no	—
169	C₄×D₄	—	4	1	no	—
170	C₄×D₄	—	4	1	no	—
171	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
172	C₂×C₁₆	—	94	2	no	—
173	C₂×C₁₆	—	188	2	no	—
174	C₄⋊C₈	—	94	2	no	—
175	C₂²⋊C₈	—	188	1	no	—
176	C₄×C₈	—	188	1	no	—
177	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
178	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
179	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
180	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
181	#41	—	4	4	no	—
182	#41	—	4	4	no	—
183	C₂×Q₈	—	2	2	no	—
184	C₂×D₄	—	2	2	no	—
185	C₂×D₄	—	4	2	no	—
186	C₂×D₄	—	4	1	no	—
187	C₂×D₄	—	4	1	no	—
188	C₂²×C₄	—	4	2	no	—
189	C₂²×C₄	#10625334	1	1	yes	characteristic
190	C₂²×C₄	#10625335	1	1	yes	characteristic
191	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
192	Q₁₆	—	4	2	no	—
193	Q₁₆	—	4	2	no	—
194	SD₁₆	—	4	4	no	—
195	SD₁₆	—	4	2	no	—
196	SD₁₆	—	4	2	no	—
197	D₈	—	4	2	no	—
198	D₈	—	4	2	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	188	2	no	—
201	C₂×C₈	#10625334	1	2	yes	—
202	C₂×C₈	#10625335	1	2	yes	—
203	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
204	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
205	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
206	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
207	C₄⋊C₄	—	4	1	no	—
208	C₄⋊C₄	—	4	1	no	—
209	C₂²⋊C₄	—	4	2	no	—
210	C₂²⋊C₄	—	4	2	no	—
211	C₄²	#10625334	1	1	yes	characteristic
212	C₄²	#10625335	1	1	yes	characteristic
213	C₄²	C₂²⋊Dic₄₇	1	1	yes	characteristic
214	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
215	C₁₆	—	188	2	no	—
216	C₂³	—	4	2	no	—
217	C₂³	#13093945	1	1	yes	characteristic
218	Q₈	—	4	2	no	—
219	Q₈	—	4	2	no	—
220	D₄	—	4	2	no	—
221	D₄	—	4	2	no	—
222	D₄	—	8	2	no	—
223	D₄	—	8	4	no	—
224	D₄	—	8	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	4	1	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	8	4	no	—
229	C₂×C₄	#13093941	1	2	yes	—
230	C₂×C₄	#13093945	1	1	yes	characteristic
231	C₂×C₄	#13093945	1	2	yes	—
232	C₂×C₄	#13093945	1	1	yes	characteristic
233	C₂×C₄	#13093945	1	1	yes	characteristic
234	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
235	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
236	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
237	C₈	—	2	1	no	—
238	C₈	—	2	1	no	—
239	C₈	—	376	2	no	—
240	C₈	#13093945	1	2	yes	—
241	C₈	C₂×#10625334	1	1	yes	characteristic
242	C₈	C₂×#10625335	1	1	yes	characteristic
243	C₂²	—	2	1	no	—
244	C₂²	—	2	1	no	—
245	C₂²	—	8	2	no	—
246	C₂²	—	8	2	no	—
247	C₂²	—	8	2	no	—
248	C₂²	#15688867	1	2	yes	—
249	C₂²	C₂×#13093945	1	1	yes	characteristic
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	8	2	no	—
254	C₄	#15688867	1	2	yes	—
255	C₄	#15688930	1	2	yes	—
256	C₄	C₂×#13093941	1	1	yes	characteristic
257	C₄	C₂×#13093945	1	1	yes	characteristic
258	C₂	—	2	2	no	—
259	C₂	—	16	2	no	—
260	C₂	C₂×#15688867	1	1	yes	characteristic
261	C₂	C₂×#15688930	1	1	yes	characteristic
262	C₂	#20245124	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13093945
commutator	C₂×C₈×C₄₇	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₄₇
Fitting, prime radical	C₄₇×#3147	C₂
socle	C₂²×C₄₇	C₂×#102
cosocle	C₄×C₈×C₄₇	C₂³
hypercenter	#3147	D₄₇
hypocenter	C₄₇	#11924
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3147

Complements

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

N	H ≅ G/N	type
1. C₂×C₄⋊(C₄₇⋊C₁₆)	C₂	semidirect
2. #20242136	C₂	semidirect
3. #20241431	C₂	semidirect
151. C₄₇	#11924	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊(C₄₇⋊C₁₆)	2
#20242136	2
#20241431	2
C₄₇×#3147	2
#11924	47