Subgroups — C₂×C₁₁₃×#1274 #43275873

Order 28928.

758 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×#431×C₁₁₃	C₂	1	2	yes	maximal
2	C₂×#376×C₁₁₃	C₂	1	1	yes	maximal, characteristic
3	C₁₁₃×#1274	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₄²×C₁₁₃	C₄	1	2	yes	—
5	C₂²×C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
6	#431×C₁₁₃	—	2	2	no	—
7	#431×C₁₁₃	—	2	2	no	—
8	#431×C₁₁₃	C₄	1	4	yes	complemented
9	#431×C₁₁₃	C₂²	1	4	yes	—
10	C₂×C₄⋊C₈×C₁₁₃	—	2	2	no	—
11	C₂×C₂²⋊C₈×C₁₁₃	—	2	2	no	—
12	C₂×C₄×C₈×C₁₁₃	C₄	1	4	yes	complemented
13	#376×C₁₁₃	—	2	2	no	—
14	#376×C₁₁₃	C₄	1	2	yes	—
15	#376×C₁₁₃	C₂²	1	2	yes	—
16	C₂×#95×C₁₁₃	—	2	2	no	—
17	C₂³×C₄×C₁₁₃	—	2	2	no	—
18	C₂³×C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
19	C₂³×C₄×C₁₁₃	D₄	1	2	yes	—
20	C₂²×C₈×C₁₁₃	—	2	4	no	—
21	C₂×C₄²×C₁₁₃	—	2	2	no	—
22	C₂×C₄²×C₁₁₃	—	2	4	no	—
23	C₂×C₄²×C₁₁₃	C₈	1	8	yes	complemented
24	C₂×C₄²×C₁₁₃	C₂×C₄	1	4	yes	—
25	C₂×C₄²×C₁₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₁₁₃	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₁₁₃	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₁₁₃	D₄	1	2	yes	—
29	C₂×C₄²×C₁₁₃	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₁₁₃	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₁₁₃	Q₈	1	1	yes	characteristic
32	C₂×C₄²×C₁₁₃	Q₈	1	1	yes	characteristic
33	C₂×C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
34	C₄⋊C₈×C₁₁₃	—	2	4	no	—
35	C₄⋊C₈×C₁₁₃	—	2	4	no	—
36	C₂²⋊C₈×C₁₁₃	—	2	4	no	—
37	C₂²⋊C₈×C₁₁₃	—	2	4	no	—
38	C₄×C₈×C₁₁₃	—	2	4	no	—
39	C₄×C₈×C₁₁₃	C₂×C₄	1	8	yes	complemented
40	#95×C₁₁₃	—	2	4	no	—
41	#95×C₁₁₃	—	2	4	no	—
42	C₂⁴×C₁₁₃	C₂²⋊C₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₁₃	—	2	8	no	—
44	C₂²×C₄×C₁₁₃	—	2	4	no	—
45	C₂²×C₄×C₁₁₃	—	4	4	no	—
46	C₂²×C₄×C₁₁₃	—	4	8	no	—
47	C₂²×C₄×C₁₁₃	C₄²	1	1	yes	characteristic
48	C₂²×C₄×C₁₁₃	C₄²	1	1	yes	characteristic
49	C₂²×C₄×C₁₁₃	C₂²⋊C₄	1	2	yes	—
50	C₂²×C₄×C₁₁₃	C₂²⋊C₄	1	2	yes	—
51	C₂²×C₄×C₁₁₃	C₂²⋊C₄	1	2	yes	—
52	C₂²×C₄×C₁₁₃	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₁₃	C₄⋊C₄	1	2	yes	—
54	C₂²×C₄×C₁₁₃	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄×C₁₁₃	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₁₃	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₁₃	C₂×C₈	1	4	yes	—
58	C₂²×C₄×C₁₁₃	M₁₆	1	4	yes	—
59	C₂²×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
61	C₂×C₈×C₁₁₃	—	2	8	no	—
62	C₂×C₈×C₁₁₃	—	4	8	no	—
63	C₄²×C₁₁₃	—	2	8	no	—
64	C₄²×C₁₁₃	—	2	4	no	—
65	C₄²×C₁₁₃	C₄²	1	2	yes	—
66	C₄²×C₁₁₃	C₂²⋊C₄	1	2	yes	—
67	C₄²×C₁₁₃	C₂²⋊C₄	1	2	yes	—
68	C₄²×C₁₁₃	C₄⋊C₄	1	1	yes	characteristic
69	C₄²×C₁₁₃	C₄⋊C₄	1	1	yes	characteristic
70	C₄²×C₁₁₃	C₄⋊C₄	1	2	yes	—
71	C₄²×C₁₁₃	C₄⋊C₄	1	2	yes	—
72	C₄²×C₁₁₃	C₂×C₈	1	8	yes	complemented
73	C₄²×C₁₁₃	C₂²×C₄	1	2	yes	—
74	C₄²×C₁₁₃	C₂×D₄	1	1	yes	characteristic
75	C₄²×C₁₁₃	C₂×Q₈	1	1	yes	characteristic
76	C₂³×C₁₁₃	—	2	4	no	—
77	C₂³×C₁₁₃	#95	1	1	yes	characteristic
78	C₂³×C₁₁₃	#95	1	1	yes	characteristic
79	C₂³×C₁₁₃	C₂²⋊C₈	1	4	yes	—
80	C₂³×C₁₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₁₁₃	—	2	4	no	—
82	C₂×C₄×C₁₁₃	—	2	4	no	—
83	C₂×C₄×C₁₁₃	—	2	8	no	—
84	C₂×C₄×C₁₁₃	—	4	2	no	—
85	C₂×C₄×C₁₁₃	—	4	2	no	—
86	C₂×C₄×C₁₁₃	—	4	2	no	—
87	C₂×C₄×C₁₁₃	—	4	2	no	—
88	C₂×C₄×C₁₁₃	—	4	8	no	—
89	C₂×C₄×C₁₁₃	—	4	16	no	—
90	C₂×C₄×C₁₁₃	#95	1	2	yes	—
91	C₂×C₄×C₁₁₃	#95	1	2	yes	—
92	C₂×C₄×C₁₁₃	#95	1	1	yes	characteristic
93	C₂×C₄×C₁₁₃	#95	1	1	yes	characteristic
94	C₂×C₄×C₁₁₃	C₄×C₈	1	2	yes	—
95	C₂×C₄×C₁₁₃	C₄×C₈	1	2	yes	—
96	C₂×C₄×C₁₁₃	#97	1	2	yes	—
97	C₂×C₄×C₁₁₃	#97	1	2	yes	—
98	C₂×C₄×C₁₁₃	C₂²⋊C₈	1	4	yes	—
99	C₂×C₄×C₁₁₃	C₄⋊C₈	1	4	yes	—
100	C₂×C₄×C₁₁₃	C₄⋊C₈	1	4	yes	—
101	C₂×C₄×C₁₁₃	C₂×C₄²	1	1	yes	characteristic
102	C₂×C₄×C₁₁₃	C₂×C₂²⋊C₄	1	2	yes	—
103	C₂×C₄×C₁₁₃	C₂×C₄⋊C₄	1	2	yes	—
104	C₂×C₄×C₁₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₁₁₃	C₂²×C₈	1	2	yes	—
106	C₂×C₄×C₁₁₃	C₂×M₁₆	1	2	yes	—
107	C₈×C₁₁₃	—	4	8	no	—
108	C₂²×C₁₁₃	—	2	2	no	—
109	C₂²×C₁₁₃	—	2	2	no	—
110	C₂²×C₁₁₃	—	2	8	no	—
111	C₂²×C₁₁₃	#336	1	2	yes	—
112	C₂²×C₁₁₃	#336	1	2	yes	—
113	C₂²×C₁₁₃	H(3,ℤ/4)	1	4	yes	—
114	C₂²×C₁₁₃	C₂×#95	1	1	yes	characteristic
115	C₂²×C₁₁₃	C₂×C₂²⋊C₈	1	2	yes	—
116	C₄×C₁₁₃	—	2	2	no	—
117	C₄×C₁₁₃	—	2	2	no	—
118	C₄×C₁₁₃	—	2	2	no	—
119	C₄×C₁₁₃	—	2	2	no	—
120	C₄×C₁₁₃	—	4	8	no	—
121	C₄×C₁₁₃	#336	1	2	yes	—
122	C₄×C₁₁₃	#336	1	2	yes	—
123	C₄×C₁₁₃	C₂×C₄×C₈	1	1	yes	characteristic
124	C₄×C₁₁₃	C₂×#97	1	1	yes	characteristic
125	C₄×C₁₁₃	C₂×C₄⋊C₈	1	2	yes	—
126	C₂×#1274	C₁₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
127	C₂×C₁₁₃	—	2	2	no	—
128	C₂×C₁₁₃	—	2	2	no	—
129	C₂×C₁₁₃	#1274	1	4	yes	direct product factor
130	C₂×C₁₁₃	C₂×#336	1	1	yes	characteristic
131	C₂×C₁₁₃	C₂×H(3,ℤ/4)	1	2	yes	—
132	C₂×#431	C₂×C₁₁₃	1	2	yes	—
133	C₂×#376	C₂×C₁₁₃	1	1	yes	characteristic
134	#1274	C₂×C₁₁₃	1	4	yes	direct product factor
135	C₁₁₃	C₂×#1274	1	1	yes	characteristic, direct product factor, Sylow
136	C₂²×C₄²	C₄×C₁₁₃	1	2	yes	—
137	C₂²×C₄²	C₂²×C₁₁₃	1	1	yes	characteristic
138	#431	—	2	2	no	—
139	#431	—	2	2	no	—
140	#431	C₄×C₁₁₃	1	4	yes	complemented
141	#431	C₂²×C₁₁₃	1	4	yes	—
142	C₂×C₄⋊C₈	—	2	2	no	—
143	C₂×C₂²⋊C₈	—	2	2	no	—
144	C₂×C₄×C₈	C₄×C₁₁₃	1	4	yes	complemented
145	#376	—	2	2	no	—
146	#376	C₄×C₁₁₃	1	2	yes	—
147	#376	C₂²×C₁₁₃	1	2	yes	—
148	C₂×#95	—	2	2	no	—
149	C₂³×C₄	—	2	2	no	—
150	C₂³×C₄	C₂×C₄×C₁₁₃	1	1	yes	characteristic
151	C₂³×C₄	D₄×C₁₁₃	1	2	yes	—
152	C₂²×C₈	—	2	4	no	—
153	C₂×C₄²	—	2	2	no	—
154	C₂×C₄²	—	2	4	no	—
155	C₂×C₄²	C₈×C₁₁₃	1	8	yes	complemented
156	C₂×C₄²	C₂×C₄×C₁₁₃	1	4	yes	—
157	C₂×C₄²	C₂×C₄×C₁₁₃	1	1	yes	characteristic
158	C₂×C₄²	C₂×C₄×C₁₁₃	1	2	yes	—
159	C₂×C₄²	C₂×C₄×C₁₁₃	1	2	yes	—
160	C₂×C₄²	D₄×C₁₁₃	1	2	yes	—
161	C₂×C₄²	D₄×C₁₁₃	1	1	yes	characteristic
162	C₂×C₄²	D₄×C₁₁₃	1	1	yes	characteristic
163	C₂×C₄²	Q₈×C₁₁₃	1	1	yes	characteristic
164	C₂×C₄²	Q₈×C₁₁₃	1	1	yes	characteristic
165	C₂×C₄²	C₂³×C₁₁₃	1	1	yes	characteristic
166	C₄⋊C₈	—	2	4	no	—
167	C₄⋊C₈	—	2	4	no	—
168	C₂²⋊C₈	—	2	4	no	—
169	C₂²⋊C₈	—	2	4	no	—
170	C₄×C₈	—	2	4	no	—
171	C₄×C₈	C₂×C₄×C₁₁₃	1	8	yes	complemented
172	#95	—	2	4	no	—
173	#95	—	2	4	no	—
174	C₂⁴	C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
175	C₂²×C₄	—	2	8	no	—
176	C₂²×C₄	—	2	4	no	—
177	C₂²×C₄	—	4	4	no	—
178	C₂²×C₄	—	4	8	no	—
179	C₂²×C₄	C₄²×C₁₁₃	1	1	yes	characteristic
180	C₂²×C₄	C₄²×C₁₁₃	1	1	yes	characteristic
181	C₂²×C₄	C₂²⋊C₄×C₁₁₃	1	2	yes	—
182	C₂²×C₄	C₂²⋊C₄×C₁₁₃	1	2	yes	—
183	C₂²×C₄	C₂²⋊C₄×C₁₁₃	1	2	yes	—
184	C₂²×C₄	C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
185	C₂²×C₄	C₄⋊C₄×C₁₁₃	1	2	yes	—
186	C₂²×C₄	C₄⋊C₄×C₁₁₃	1	2	yes	—
187	C₂²×C₄	C₄⋊C₄×C₁₁₃	1	1	yes	characteristic
188	C₂²×C₄	C₄⋊C₄×C₁₁₃	1	1	yes	characteristic
189	C₂²×C₄	C₂×C₈×C₁₁₃	1	4	yes	—
190	C₂²×C₄	M₁₆×C₁₁₃	1	4	yes	—
191	C₂²×C₄	C₂²×C₄×C₁₁₃	1	1	yes	characteristic
192	C₂²×C₄	C₂×D₄×C₁₁₃	1	2	yes	—
193	C₂×C₈	—	2	8	no	—
194	C₂×C₈	—	4	8	no	—
195	C₄²	—	2	8	no	—
196	C₄²	—	2	4	no	—
197	C₄²	C₄²×C₁₁₃	1	2	yes	—
198	C₄²	C₂²⋊C₄×C₁₁₃	1	2	yes	—
199	C₄²	C₂²⋊C₄×C₁₁₃	1	2	yes	—
200	C₄²	C₄⋊C₄×C₁₁₃	1	1	yes	characteristic
201	C₄²	C₄⋊C₄×C₁₁₃	1	1	yes	characteristic
202	C₄²	C₄⋊C₄×C₁₁₃	1	2	yes	—
203	C₄²	C₄⋊C₄×C₁₁₃	1	2	yes	—
204	C₄²	C₂×C₈×C₁₁₃	1	8	yes	complemented
205	C₄²	C₂²×C₄×C₁₁₃	1	2	yes	—
206	C₄²	C₂×D₄×C₁₁₃	1	1	yes	characteristic
207	C₄²	C₂×Q₈×C₁₁₃	1	1	yes	characteristic
208	C₂³	—	2	4	no	—
209	C₂³	#95×C₁₁₃	1	1	yes	characteristic
210	C₂³	#95×C₁₁₃	1	1	yes	characteristic
211	C₂³	C₂²⋊C₈×C₁₁₃	1	4	yes	—
212	C₂³	C₂×C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	—	2	8	no	—
216	C₂×C₄	—	4	2	no	—
217	C₂×C₄	—	4	2	no	—
218	C₂×C₄	—	4	2	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	—	4	16	no	—
221	C₂×C₄	—	4	8	no	—
222	C₂×C₄	#95×C₁₁₃	1	2	yes	—
223	C₂×C₄	#95×C₁₁₃	1	2	yes	—
224	C₂×C₄	#95×C₁₁₃	1	1	yes	characteristic
225	C₂×C₄	#95×C₁₁₃	1	1	yes	characteristic
226	C₂×C₄	C₄×C₈×C₁₁₃	1	2	yes	—
227	C₂×C₄	C₄×C₈×C₁₁₃	1	2	yes	—
228	C₂×C₄	#97×C₁₁₃	1	2	yes	—
229	C₂×C₄	#97×C₁₁₃	1	2	yes	—
230	C₂×C₄	C₂²⋊C₈×C₁₁₃	1	4	yes	—
231	C₂×C₄	C₄⋊C₈×C₁₁₃	1	4	yes	—
232	C₂×C₄	C₄⋊C₈×C₁₁₃	1	4	yes	—
233	C₂×C₄	C₂×C₄²×C₁₁₃	1	1	yes	characteristic
234	C₂×C₄	C₂×C₂²⋊C₄×C₁₁₃	1	2	yes	—
235	C₂×C₄	C₂×C₄⋊C₄×C₁₁₃	1	2	yes	—
236	C₂×C₄	C₂×C₄⋊C₄×C₁₁₃	1	1	yes	characteristic
237	C₂×C₄	C₂²×C₈×C₁₁₃	1	2	yes	—
238	C₂×C₄	C₂×M₁₆×C₁₁₃	1	2	yes	—
239	C₈	—	4	8	no	—
240	C₂²	—	2	8	no	—
241	C₂²	—	2	2	no	—
242	C₂²	—	2	2	no	—
243	C₂²	#336×C₁₁₃	1	2	yes	—
244	C₂²	#336×C₁₁₃	1	2	yes	—
245	C₂²	H(3,ℤ/4)×C₁₁₃	1	4	yes	—
246	C₂²	C₂×#95×C₁₁₃	1	1	yes	characteristic
247	C₂²	C₂×C₂²⋊C₈×C₁₁₃	1	2	yes	—
248	C₄	—	2	2	no	—
249	C₄	—	2	2	no	—
250	C₄	—	2	2	no	—
251	C₄	—	2	2	no	—
252	C₄	—	4	8	no	—
253	C₄	#336×C₁₁₃	1	2	yes	—
254	C₄	#336×C₁₁₃	1	2	yes	—
255	C₄	C₂×C₄×C₈×C₁₁₃	1	1	yes	characteristic
256	C₄	C₂×#97×C₁₁₃	1	1	yes	characteristic
257	C₄	C₂×C₄⋊C₈×C₁₁₃	1	2	yes	—
258	C₂	—	2	2	no	—
259	C₂	—	2	2	no	—
260	C₂	C₁₁₃×#1274	1	4	yes	direct product factor
261	C₂	C₂×#336×C₁₁₃	1	1	yes	characteristic
262	C₂	C₂×H(3,ℤ/4)×C₁₁₃	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₁₃	C₄²
commutator	C₄	C₂×C₄×C₈×C₁₁₃
Frattini, cosocle	C₂×C₄²	C₂³×C₁₁₃
socle	C₂³×C₁₁₃	#95
prime radical	C₂⁴×C₁₁₃	C₂²⋊C₄
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₁₃×#1274	C₂	direct (N × H)
8. #431×C₁₁₃	C₄	semidirect
12. C₂×C₄×C₈×C₁₁₃	C₄	semidirect
23. C₂×C₄²×C₁₁₃	C₈	semidirect
39. C₄×C₈×C₁₁₃	C₂×C₄	semidirect
72. C₄²×C₁₁₃	C₂×C₈	semidirect
126. C₂×#1274	C₁₁₃	direct (N × H)
129. C₂×C₁₁₃	#1274	direct (N × H)
134. #1274	C₂×C₁₁₃	direct (N × H)
135. C₁₁₃	C₂×#1274	direct (N × H)
140. #431	C₄×C₁₁₃	semidirect
144. C₂×C₄×C₈	C₄×C₁₁₃	semidirect
155. C₂×C₄²	C₈×C₁₁₃	semidirect
171. C₄×C₈	C₂×C₄×C₁₁₃	semidirect
204. C₄²	C₂×C₈×C₁₁₃	semidirect
260. C₂	C₁₁₃×#1274	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#431×C₁₁₃	2
C₂×#376×C₁₁₃	2
C₁₁₃×#1274	2
C₂×#1274	113

Subgroups — C2×C113×#1274 #43275873

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₁₃×#1274 #43275873