Subgroups — C₁₁₃×#20586 #43288606

Order 28928.

4042 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	C₂²×H(5,𝔽₂)×C₁₁₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#410×C₁₁₃	C₂	1	4	yes	maximal, complemented
3	C₁₁₃×#2309	C₂	1	2	yes	maximal, complemented
4	C₁₁₃×#1781	C₂	1	8	yes	maximal, complemented
5	C₂×H(5,𝔽₂)×C₁₁₃	—	2	16	no	—
6	C₂×H(5,𝔽₂)×C₁₁₃	—	2	8	no	—
7	C₂²×#41×C₁₁₃	C₄	1	4	yes	complemented
8	C₂²×#41×C₁₁₃	C₂²	1	2	yes	—
9	C₂³×D₄×C₁₁₃	C₄	1	4	yes	complemented
10	C₂³×D₄×C₁₁₃	C₂²	1	4	yes	—
11	C₂³×D₄×C₁₁₃	C₂²	1	1	yes	characteristic
12	C₂×#117×C₁₁₃	C₂²	1	4	yes	complemented
13	#410×C₁₁₃	—	2	16	no	—
14	C₂×#99×C₁₁₃	C₂²	1	16	yes	complemented
15	#390×C₁₁₃	—	2	8	no	—
16	#379×C₁₁₃	C₂²	1	8	yes	complemented
17	C₂⁵×C₁₁₃	C₂×C₄	1	4	yes	—
18	C₂⁵×C₁₁₃	C₂³	1	2	yes	—
19	H(5,𝔽₂)×C₁₁₃	—	2	64	no	—
20	H(5,𝔽₂)×C₁₁₃	—	2	64	no	—
21	C₂×#41×C₁₁₃	—	2	16	no	—
22	C₂×#41×C₁₁₃	—	2	16	no	—
23	C₂×#41×C₁₁₃	—	2	16	no	—
24	C₂×#41×C₁₁₃	—	2	8	no	—
25	C₂×#41×C₁₁₃	—	2	8	no	—
26	C₂×#41×C₁₁₃	D₄	1	16	yes	—
27	C₂²×Q₈×C₁₁₃	C₂×C₄	1	2	yes	—
28	C₂²×D₄×C₁₁₃	—	2	16	no	—
29	C₂²×D₄×C₁₁₃	—	2	16	no	—
30	C₂²×D₄×C₁₁₃	—	2	16	no	—
31	C₂²×D₄×C₁₁₃	—	2	32	no	—
32	C₂²×D₄×C₁₁₃	—	2	16	no	—
33	C₂²×D₄×C₁₁₃	—	2	8	no	—
34	C₂²×D₄×C₁₁₃	C₂×C₄	1	8	yes	—
35	C₂²×D₄×C₁₁₃	C₂×C₄	1	4	yes	—
36	C₂²×D₄×C₁₁₃	C₂×C₄	1	2	yes	—
37	C₂²×D₄×C₁₁₃	D₄	1	8	yes	—
38	C₂²×D₄×C₁₁₃	C₂³	1	4	yes	—
39	C₂³×C₄×C₁₁₃	C₂×C₄	1	4	yes	—
40	C₂³×C₄×C₁₁₃	C₂×C₄	1	4	yes	—
41	C₂³×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
42	#117×C₁₁₃	—	2	16	no	—
43	C₂×C₄⋊C₄×C₁₁₃	—	2	4	no	—
44	C₂×C₂²⋊C₄×C₁₁₃	—	2	16	no	—
45	C₂×C₂²⋊C₄×C₁₁₃	C₂³	1	8	yes	—
46	C₂×C₄²×C₁₁₃	—	2	4	no	—
47	#99×C₁₁₃	—	2	32	no	—
48	C₂⁴×C₁₁₃	—	2	16	no	—
49	C₂⁴×C₁₁₃	—	2	8	no	—
50	C₂⁴×C₁₁₃	—	2	8	no	—
51	C₂⁴×C₁₁₃	—	8	4	no	—
52	C₂⁴×C₁₁₃	—	8	8	no	—
53	C₂⁴×C₁₁₃	C₂²×C₄	1	4	yes	—
54	C₂⁴×C₁₁₃	C₂²×C₄	1	4	yes	—
55	C₂⁴×C₁₁₃	C₂×D₄	1	8	yes	—
56	C₂⁴×C₁₁₃	C₂⁴	1	1	yes	characteristic
57	#41×C₁₁₃	—	2	128	no	—
58	#41×C₁₁₃	—	2	32	no	—
59	#41×C₁₁₃	—	2	32	no	—
60	C₂×Q₈×C₁₁₃	—	2	4	no	—
61	C₂×Q₈×C₁₁₃	—	2	4	no	—
62	C₂×Q₈×C₁₁₃	C₂²⋊C₄	1	8	yes	complemented
63	C₂×D₄×C₁₁₃	—	2	128	no	—
64	C₂×D₄×C₁₁₃	—	2	128	no	—
65	C₂×D₄×C₁₁₃	—	2	16	no	—
66	C₂×D₄×C₁₁₃	—	2	16	no	—
67	C₂×D₄×C₁₁₃	—	2	8	no	—
68	C₂×D₄×C₁₁₃	—	2	8	no	—
69	C₂×D₄×C₁₁₃	—	2	16	no	—
70	C₂×D₄×C₁₁₃	—	2	8	no	—
71	C₂×D₄×C₁₁₃	—	2	8	no	—
72	C₂×D₄×C₁₁₃	—	2	4	no	—
73	C₂×D₄×C₁₁₃	—	2	4	no	—
74	C₂×D₄×C₁₁₃	—	2	16	no	—
75	C₂×D₄×C₁₁₃	—	2	16	no	—
76	C₂×D₄×C₁₁₃	C₂²⋊C₄	1	16	yes	—
77	C₂×D₄×C₁₁₃	C₂²⋊C₄	1	8	yes	complemented
78	C₂×D₄×C₁₁₃	C₂×D₄	1	16	yes	—
79	C₂²×C₄×C₁₁₃	—	2	8	no	—
80	C₂²×C₄×C₁₁₃	—	2	16	no	—
81	C₂²×C₄×C₁₁₃	—	2	8	no	—
82	C₂²×C₄×C₁₁₃	—	2	8	no	—
83	C₂²×C₄×C₁₁₃	—	2	8	no	—
84	C₂²×C₄×C₁₁₃	C₂²⋊C₄	1	16	yes	complemented
85	C₂²×C₄×C₁₁₃	C₂²×C₄	1	4	yes	—
86	C₂²×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
87	C₂²×C₄×C₁₁₃	C₂×D₄	1	4	yes	—
88	C₂²×C₄×C₁₁₃	C₂×D₄	1	8	yes	—
89	C₄⋊C₄×C₁₁₃	—	4	8	no	—
90	C₂²⋊C₄×C₁₁₃	—	2	16	no	—
91	C₂²⋊C₄×C₁₁₃	—	8	16	no	—
92	C₄²×C₁₁₃	—	4	8	no	—
93	C₂³×C₁₁₃	—	2	32	no	—
94	C₂³×C₁₁₃	—	2	8	no	—
95	C₂³×C₁₁₃	—	2	4	no	—
96	C₂³×C₁₁₃	—	2	4	no	—
97	C₂³×C₁₁₃	—	2	16	no	—
98	C₂³×C₁₁₃	—	4	8	no	—
99	C₂³×C₁₁₃	—	4	2	no	—
100	C₂³×C₁₁₃	—	4	16	no	—
101	C₂³×C₁₁₃	—	4	8	no	—
102	C₂³×C₁₁₃	—	8	16	no	—
103	C₂³×C₁₁₃	—	8	16	no	—
104	C₂³×C₁₁₃	—	8	32	no	—
105	C₂³×C₁₁₃	C₂×C₂²⋊C₄	1	8	yes	—
106	C₂³×C₁₁₃	C₂×C₂²⋊C₄	1	8	yes	—
107	C₂³×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
108	C₂³×C₁₁₃	C₂²×D₄	1	2	yes	—
109	C₂³×C₁₁₃	C₂²×D₄	1	4	yes	—
110	Q₈×C₁₁₃	—	2	16	no	—
111	D₄×C₁₁₃	—	2	64	no	—
112	D₄×C₁₁₃	—	2	32	no	—
113	D₄×C₁₁₃	—	2	32	no	—
114	D₄×C₁₁₃	—	2	16	no	—
115	C₂×C₄×C₁₁₃	—	2	32	no	—
116	C₂×C₄×C₁₁₃	—	2	4	no	—
117	C₂×C₄×C₁₁₃	—	2	4	no	—
118	C₂×C₄×C₁₁₃	—	2	32	no	—
119	C₂×C₄×C₁₁₃	—	4	16	no	—
120	C₂×C₄×C₁₁₃	—	4	8	no	—
121	C₂×C₄×C₁₁₃	C₂×C₂²⋊C₄	1	8	yes	—
122	C₂×C₄×C₁₁₃	C₂×C₂²⋊C₄	1	4	yes	—
123	C₂×C₄×C₁₁₃	C₂×C₂²⋊C₄	1	4	yes	—
124	C₂×C₄×C₁₁₃	C₂×C₂²⋊C₄	1	4	yes	—
125	C₂×C₄×C₁₁₃	#120	1	8	yes	—
126	C₂×C₄×C₁₁₃	#120	1	8	yes	—
127	C₂²×C₁₁₃	—	2	4	no	—
128	C₂²×C₁₁₃	—	2	2	no	—
129	C₂²×C₁₁₃	—	2	16	no	—
130	C₂²×C₁₁₃	—	2	8	no	—
131	C₂²×C₁₁₃	—	2	8	no	—
132	C₂²×C₁₁₃	—	4	4	no	—
133	C₂²×C₁₁₃	—	4	8	no	—
134	C₂²×C₁₁₃	—	4	16	no	—
135	C₂²×C₁₁₃	—	4	8	no	—
136	C₂²×C₁₁₃	—	4	8	no	—
137	C₂²×C₁₁₃	—	8	16	no	—
138	C₂²×C₁₁₃	—	8	32	no	—
139	C₂²×C₁₁₃	C₂²×C₂²⋊C₄	1	2	yes	—
140	C₂²×C₁₁₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
141	C₂²×C₁₁₃	C₂×#120	1	4	yes	—
142	C₄×C₁₁₃	—	2	8	no	—
143	C₄×C₁₁₃	—	8	8	no	—
144	C₄×C₁₁₃	#379	1	8	yes	—
145	#20586	C₁₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
146	C₂×C₁₁₃	—	2	4	no	—
147	C₂×C₁₁₃	—	2	2	no	—
148	C₂×C₁₁₃	—	4	8	no	—
149	C₂×C₁₁₃	—	4	8	no	—
150	C₂×C₁₁₃	C₂×#379	1	1	yes	characteristic
151	C₂×C₁₁₃	#2883	1	2	yes	—
152	C₂²×H(5,𝔽₂)	C₂×C₁₁₃	1	1	yes	characteristic
153	C₂×#410	C₂×C₁₁₃	1	4	yes	complemented
154	#2309	C₂×C₁₁₃	1	2	yes	complemented
155	#1781	C₂×C₁₁₃	1	8	yes	complemented
156	C₁₁₃	#20586	1	1	yes	characteristic, direct product factor, Sylow
157	C₂×H(5,𝔽₂)	—	2	16	no	—
158	C₂×H(5,𝔽₂)	—	2	8	no	—
159	C₂²×#41	C₄×C₁₁₃	1	4	yes	complemented
160	C₂²×#41	C₂²×C₁₁₃	1	2	yes	—
161	C₂³×D₄	C₄×C₁₁₃	1	4	yes	complemented
162	C₂³×D₄	C₂²×C₁₁₃	1	4	yes	—
163	C₂³×D₄	C₂²×C₁₁₃	1	1	yes	characteristic
164	C₂×#117	C₂²×C₁₁₃	1	4	yes	complemented
165	#410	—	2	16	no	—
166	C₂×#99	C₂²×C₁₁₃	1	16	yes	complemented
167	#390	—	2	8	no	—
168	#379	C₂²×C₁₁₃	1	8	yes	complemented
169	C₂⁵	C₂×C₄×C₁₁₃	1	4	yes	—
170	C₂⁵	C₂³×C₁₁₃	1	2	yes	—
171	H(5,𝔽₂)	—	2	64	no	—
172	H(5,𝔽₂)	—	2	64	no	—
173	C₂×#41	—	2	16	no	—
174	C₂×#41	—	2	16	no	—
175	C₂×#41	—	2	16	no	—
176	C₂×#41	—	2	8	no	—
177	C₂×#41	—	2	8	no	—
178	C₂×#41	D₄×C₁₁₃	1	16	yes	—
179	C₂²×Q₈	C₂×C₄×C₁₁₃	1	2	yes	—
180	C₂²×D₄	—	2	16	no	—
181	C₂²×D₄	—	2	16	no	—
182	C₂²×D₄	—	2	16	no	—
183	C₂²×D₄	—	2	32	no	—
184	C₂²×D₄	—	2	16	no	—
185	C₂²×D₄	—	2	8	no	—
186	C₂²×D₄	C₂×C₄×C₁₁₃	1	8	yes	—
187	C₂²×D₄	C₂×C₄×C₁₁₃	1	4	yes	—
188	C₂²×D₄	C₂×C₄×C₁₁₃	1	2	yes	—
189	C₂²×D₄	D₄×C₁₁₃	1	8	yes	—
190	C₂²×D₄	C₂³×C₁₁₃	1	4	yes	—
191	C₂³×C₄	C₂×C₄×C₁₁₃	1	4	yes	—
192	C₂³×C₄	C₂×C₄×C₁₁₃	1	4	yes	—
193	C₂³×C₄	C₂³×C₁₁₃	1	1	yes	characteristic
194	#117	—	2	16	no	—
195	C₂×C₄⋊C₄	—	2	4	no	—
196	C₂×C₂²⋊C₄	—	2	16	no	—
197	C₂×C₂²⋊C₄	C₂³×C₁₁₃	1	8	yes	—
198	C₂×C₄²	—	2	4	no	—
199	#99	—	2	32	no	—
200	C₂⁴	—	2	8	no	—
201	C₂⁴	—	2	16	no	—
202	C₂⁴	—	2	8	no	—
203	C₂⁴	—	8	4	no	—
204	C₂⁴	—	8	8	no	—
205	C₂⁴	C₂²×C₄×C₁₁₃	1	4	yes	—
206	C₂⁴	C₂²×C₄×C₁₁₃	1	4	yes	—
207	C₂⁴	C₂×D₄×C₁₁₃	1	8	yes	—
208	C₂⁴	C₂⁴×C₁₁₃	1	1	yes	characteristic
209	#41	—	2	128	no	—
210	#41	—	2	32	no	—
211	#41	—	2	32	no	—
212	C₂×Q₈	—	2	4	no	—
213	C₂×Q₈	—	2	4	no	—
214	C₂×Q₈	C₂²⋊C₄×C₁₁₃	1	8	yes	complemented
215	C₂×D₄	—	2	128	no	—
216	C₂×D₄	—	2	128	no	—
217	C₂×D₄	—	2	16	no	—
218	C₂×D₄	—	2	16	no	—
219	C₂×D₄	—	2	8	no	—
220	C₂×D₄	—	2	8	no	—
221	C₂×D₄	—	2	16	no	—
222	C₂×D₄	—	2	8	no	—
223	C₂×D₄	—	2	4	no	—
224	C₂×D₄	—	2	8	no	—
225	C₂×D₄	—	2	4	no	—
226	C₂×D₄	—	2	16	no	—
227	C₂×D₄	—	2	16	no	—
228	C₂×D₄	C₂²⋊C₄×C₁₁₃	1	16	yes	—
229	C₂×D₄	C₂²⋊C₄×C₁₁₃	1	8	yes	complemented
230	C₂×D₄	C₂×D₄×C₁₁₃	1	16	yes	—
231	C₂²×C₄	—	2	8	no	—
232	C₂²×C₄	—	2	16	no	—
233	C₂²×C₄	—	2	8	no	—
234	C₂²×C₄	—	2	8	no	—
235	C₂²×C₄	—	2	8	no	—
236	C₂²×C₄	C₂²⋊C₄×C₁₁₃	1	16	yes	complemented
237	C₂²×C₄	C₂²×C₄×C₁₁₃	1	4	yes	—
238	C₂²×C₄	C₂²×C₄×C₁₁₃	1	2	yes	—
239	C₂²×C₄	C₂×D₄×C₁₁₃	1	4	yes	—
240	C₂²×C₄	C₂×D₄×C₁₁₃	1	8	yes	—
241	C₄⋊C₄	—	4	8	no	—
242	C₂²⋊C₄	—	2	16	no	—
243	C₂²⋊C₄	—	8	16	no	—
244	C₄²	—	4	8	no	—
245	C₂³	—	2	32	no	—
246	C₂³	—	2	8	no	—
247	C₂³	—	2	4	no	—
248	C₂³	—	2	4	no	—
249	C₂³	—	2	16	no	—
250	C₂³	—	4	8	no	—
251	C₂³	—	4	2	no	—
252	C₂³	—	4	16	no	—
253	C₂³	—	4	8	no	—
254	C₂³	—	8	16	no	—
255	C₂³	—	8	16	no	—
256	C₂³	—	8	32	no	—
257	C₂³	C₂×C₂²⋊C₄×C₁₁₃	1	8	yes	—
258	C₂³	C₂×C₂²⋊C₄×C₁₁₃	1	8	yes	—
259	C₂³	C₂³×C₄×C₁₁₃	1	1	yes	characteristic
260	C₂³	C₂²×D₄×C₁₁₃	1	2	yes	—
261	C₂³	C₂²×D₄×C₁₁₃	1	4	yes	—
262	Q₈	—	2	16	no	—
263	D₄	—	2	64	no	—
264	D₄	—	2	32	no	—
265	D₄	—	2	32	no	—
266	D₄	—	2	16	no	—
267	C₂×C₄	—	2	32	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	32	no	—
271	C₂×C₄	—	4	16	no	—
272	C₂×C₄	—	4	8	no	—
273	C₂×C₄	C₂×C₂²⋊C₄×C₁₁₃	1	8	yes	—
274	C₂×C₄	C₂×C₂²⋊C₄×C₁₁₃	1	4	yes	—
275	C₂×C₄	C₂×C₂²⋊C₄×C₁₁₃	1	4	yes	—
276	C₂×C₄	C₂×C₂²⋊C₄×C₁₁₃	1	4	yes	—
277	C₂×C₄	#120×C₁₁₃	1	8	yes	—
278	C₂×C₄	#120×C₁₁₃	1	8	yes	—
279	C₂²	—	2	4	no	—
280	C₂²	—	2	2	no	—
281	C₂²	—	2	8	no	—
282	C₂²	—	2	8	no	—
283	C₂²	—	2	16	no	—
284	C₂²	—	4	4	no	—
285	C₂²	—	4	8	no	—
286	C₂²	—	4	16	no	—
287	C₂²	—	4	8	no	—
288	C₂²	—	4	8	no	—
289	C₂²	—	8	16	no	—
290	C₂²	—	8	32	no	—
291	C₂²	C₂²×C₂²⋊C₄×C₁₁₃	1	2	yes	—
292	C₂²	C₂²×C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
293	C₂²	C₂×#120×C₁₁₃	1	4	yes	—
294	C₄	—	2	8	no	—
295	C₄	—	8	8	no	—
296	C₄	#379×C₁₁₃	1	8	yes	—
297	C₂	—	2	4	no	—
298	C₂	—	2	2	no	—
299	C₂	—	4	8	no	—
300	C₂	—	4	8	no	—
301	C₂	C₂×#379×C₁₁₃	1	1	yes	characteristic
302	C₂	C₁₁₃×#2883	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₁₃	C₂²×C₂²⋊C₄
commutator	C₂³	C₂³×C₄×C₁₁₃
Frattini, cosocle	C₂⁴	C₂⁴×C₁₁₃
prime radical	C₂²×H(5,𝔽₂)×C₁₁₃	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₁₃ ◁ C₂³×D₄×C₁₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#410×C₁₁₃	C₂	semidirect
3. C₁₁₃×#2309	C₂	semidirect
4. C₁₁₃×#1781	C₂	semidirect
7. C₂²×#41×C₁₁₃	C₄	semidirect
9. C₂³×D₄×C₁₁₃	C₄	semidirect
12. C₂×#117×C₁₁₃	C₂²	semidirect
14. C₂×#99×C₁₁₃	C₂²	semidirect
16. #379×C₁₁₃	C₂²	semidirect
62. C₂×Q₈×C₁₁₃	C₂²⋊C₄	semidirect
77. C₂×D₄×C₁₁₃	C₂²⋊C₄	semidirect
84. C₂²×C₄×C₁₁₃	C₂²⋊C₄	semidirect
145. #20586	C₁₁₃	direct (N × H)
153. C₂×#410	C₂×C₁₁₃	semidirect
154. #2309	C₂×C₁₁₃	semidirect
155. #1781	C₂×C₁₁₃	semidirect
156. C₁₁₃	#20586	direct (N × H)
159. C₂²×#41	C₄×C₁₁₃	semidirect
161. C₂³×D₄	C₄×C₁₁₃	semidirect
164. C₂×#117	C₂²×C₁₁₃	semidirect
166. C₂×#99	C₂²×C₁₁₃	semidirect
168. #379	C₂²×C₁₁₃	semidirect
214. C₂×Q₈	C₂²⋊C₄×C₁₁₃	semidirect
229. C₂×D₄	C₂²⋊C₄×C₁₁₃	semidirect
236. C₂²×C₄	C₂²⋊C₄×C₁₁₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×H(5,𝔽₂)×C₁₁₃	2
C₂×#410×C₁₁₃	2
C₁₁₃×#2309	2
C₁₁₃×#1781	2
#20586	113

Subgroups — C113×#20586 #43288606

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁₃×#20586 #43288606