Subgroups — C₁₁₃×#20608 #43288628

Order 28928.

3798 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#585×C₁₁₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#410×C₁₁₃	C₂	1	3	yes	maximal, complemented
3	C₁₁₃×#2297	C₂	1	3	yes	maximal, complemented
4	C₁₁₃×#1790	C₂	1	8	yes	maximal, complemented
5	#585×C₁₁₃	—	2	12	no	—
6	#585×C₁₁₃	C₄	1	8	yes	complemented
7	C₂×2₋¹⁺⁴×C₁₁₃	—	2	3	no	—
8	C₂×H(5,𝔽₂)×C₁₁₃	—	2	3	no	—
9	C₂×H(5,𝔽₂)×C₁₁₃	C₂²	1	4	yes	—
10	C₂²×#41×C₁₁₃	—	2	6	no	—
11	C₂²×#41×C₁₁₃	C₂²	1	3	yes	—
12	C₂×#117×C₁₁₃	C₂²	1	3	yes	complemented
13	C₂²×C₄²×C₁₁₃	C₂²	1	1	yes	characteristic, complemented
14	#410×C₁₁₃	—	2	12	no	—
15	C₂×#99×C₁₁₃	C₂²	1	12	yes	complemented
16	#391×C₁₁₃	—	2	3	no	—
17	#390×C₁₁₃	—	2	3	no	—
18	#386×C₁₁₃	C₂²	1	12	yes	complemented
19	C₄×C₂²⋊C₄×C₁₁₃	—	2	6	no	—
20	2₋¹⁺⁴×C₁₁₃	—	2	24	no	—
21	2₋¹⁺⁴×C₁₁₃	—	2	24	no	—
22	H(5,𝔽₂)×C₁₁₃	—	2	24	no	—
23	H(5,𝔽₂)×C₁₁₃	—	2	24	no	—
24	H(5,𝔽₂)×C₁₁₃	—	2	24	no	—
25	H(5,𝔽₂)×C₁₁₃	C₂×C₄	1	16	yes	—
26	C₂×#41×C₁₁₃	—	2	4	no	—
27	C₂×#41×C₁₁₃	—	2	12	no	—
28	C₂×#41×C₁₁₃	—	2	48	no	—
29	C₂×#41×C₁₁₃	—	2	48	no	—
30	C₂×#41×C₁₁₃	—	2	6	no	—
31	C₂×#41×C₁₁₃	—	2	12	no	—
32	C₂×#41×C₁₁₃	—	2	6	no	—
33	C₂×#41×C₁₁₃	—	2	12	no	—
34	C₂×#41×C₁₁₃	C₂×C₄	1	12	yes	—
35	C₂×#41×C₁₁₃	D₄	1	12	yes	—
36	C₂²×Q₈×C₁₁₃	—	2	6	no	—
37	C₂²×Q₈×C₁₁₃	D₄	1	3	yes	—
38	C₂²×D₄×C₁₁₃	—	2	6	no	—
39	C₂²×D₄×C₁₁₃	—	2	12	no	—
40	C₂²×D₄×C₁₁₃	D₄	1	3	yes	—
41	C₂²×D₄×C₁₁₃	C₂³	1	6	yes	—
42	C₂³×C₄×C₁₁₃	—	2	4	no	—
43	C₂³×C₄×C₁₁₃	D₄	1	6	yes	—
44	C₂³×C₄×C₁₁₃	C₂³	1	2	yes	—
45	C₂³×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
46	#117×C₁₁₃	—	2	12	no	—
47	C₂×C₄⋊C₄×C₁₁₃	—	2	3	no	—
48	C₂×C₂²⋊C₄×C₁₁₃	—	2	12	no	—
49	C₂×C₂²⋊C₄×C₁₁₃	C₂³	1	6	yes	—
50	C₂×C₄²×C₁₁₃	—	2	3	no	—
51	C₂×C₄²×C₁₁₃	—	2	3	no	—
52	C₂×C₄²×C₁₁₃	—	2	3	no	—
53	#99×C₁₁₃	—	2	24	no	—
54	#95×C₁₁₃	—	2	12	no	—
55	C₂⁴×C₁₁₃	—	2	4	no	—
56	C₂⁴×C₁₁₃	C₂×D₄	1	6	yes	—
57	C₂⁴×C₁₁₃	C₂⁴	1	1	yes	characteristic
58	#41×C₁₁₃	—	2	32	no	—
59	#41×C₁₁₃	—	2	48	no	—
60	#41×C₁₁₃	—	2	96	no	—
61	#41×C₁₁₃	—	2	24	no	—
62	#41×C₁₁₃	—	2	48	no	—
63	#41×C₁₁₃	—	2	24	no	—
64	#41×C₁₁₃	—	2	24	no	—
65	#41×C₁₁₃	—	2	24	no	—
66	C₂×Q₈×C₁₁₃	—	2	4	no	—
67	C₂×Q₈×C₁₁₃	—	2	48	no	—
68	C₂×Q₈×C₁₁₃	—	2	6	no	—
69	C₂×Q₈×C₁₁₃	—	2	6	no	—
70	C₂×Q₈×C₁₁₃	C₂²⋊C₄	1	12	yes	complemented
71	C₂×D₄×C₁₁₃	—	2	12	no	—
72	C₂×D₄×C₁₁₃	—	2	48	no	—
73	C₂×D₄×C₁₁₃	—	2	48	no	—
74	C₂×D₄×C₁₁₃	—	2	48	no	—
75	C₂×D₄×C₁₁₃	—	2	12	no	—
76	C₂×D₄×C₁₁₃	—	2	12	no	—
77	C₂×D₄×C₁₁₃	—	2	6	no	—
78	C₂×D₄×C₁₁₃	—	2	6	no	—
79	C₂×D₄×C₁₁₃	C₂²⋊C₄	1	12	yes	complemented
80	C₂×D₄×C₁₁₃	C₂²×C₄	1	12	yes	—
81	C₂×D₄×C₁₁₃	C₂×D₄	1	12	yes	—
82	C₂²×C₄×C₁₁₃	—	2	12	no	—
83	C₂²×C₄×C₁₁₃	—	2	32	no	—
84	C₂²×C₄×C₁₁₃	—	2	6	no	—
85	C₂²×C₄×C₁₁₃	—	2	12	no	—
86	C₂²×C₄×C₁₁₃	—	2	6	no	—
87	C₂²×C₄×C₁₁₃	—	2	6	no	—
88	C₂²×C₄×C₁₁₃	—	2	6	no	—
89	C₂²×C₄×C₁₁₃	—	4	4	no	—
90	C₂²×C₄×C₁₁₃	C₂²⋊C₄	1	24	yes	complemented
91	C₂²×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
92	C₂²×C₄×C₁₁₃	C₂×D₄	1	6	yes	—
93	C₂²×C₄×C₁₁₃	C₂×D₄	1	3	yes	—
94	C₂²×C₄×C₁₁₃	C₂×D₄	1	6	yes	—
95	C₂²×C₄×C₁₁₃	C₂×D₄	1	3	yes	—
96	C₄⋊C₄×C₁₁₃	—	4	6	no	—
97	C₂²⋊C₄×C₁₁₃	—	2	12	no	—
98	C₂²⋊C₄×C₁₁₃	—	4	24	no	—
99	C₄²×C₁₁₃	—	4	3	no	—
100	C₄²×C₁₁₃	—	4	6	no	—
101	C₄²×C₁₁₃	—	4	1	no	—
102	C₂³×C₁₁₃	—	2	16	no	—
103	C₂³×C₁₁₃	—	2	6	no	—
104	C₂³×C₁₁₃	—	2	6	no	—
105	C₂³×C₁₁₃	—	4	2	no	—
106	C₂³×C₁₁₃	—	4	12	no	—
107	C₂³×C₁₁₃	—	8	8	no	—
108	C₂³×C₁₁₃	C₂×C₂²⋊C₄	1	12	yes	—
109	C₂³×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
110	C₂³×C₁₁₃	C₂²×D₄	1	3	yes	—
111	C₂³×C₁₁₃	C₂²×D₄	1	3	yes	—
112	Q₈×C₁₁₃	—	2	16	no	—
113	Q₈×C₁₁₃	—	2	24	no	—
114	D₄×C₁₁₃	—	2	48	no	—
115	D₄×C₁₁₃	—	2	24	no	—
116	D₄×C₁₁₃	—	2	24	no	—
117	D₄×C₁₁₃	—	2	24	no	—
118	C₂×C₄×C₁₁₃	—	2	48	no	—
119	C₂×C₄×C₁₁₃	—	2	6	no	—
120	C₂×C₄×C₁₁₃	—	2	6	no	—
121	C₂×C₄×C₁₁₃	—	2	24	no	—
122	C₂×C₄×C₁₁₃	—	2	24	no	—
123	C₂×C₄×C₁₁₃	—	2	12	no	—
124	C₂×C₄×C₁₁₃	—	4	6	no	—
125	C₂×C₄×C₁₁₃	—	4	2	no	—
126	C₂×C₄×C₁₁₃	—	4	12	no	—
127	C₂×C₄×C₁₁₃	—	4	6	no	—
128	C₂×C₄×C₁₁₃	C₂×C₂²⋊C₄	1	12	yes	—
129	C₂×C₄×C₁₁₃	C₂×C₂²⋊C₄	1	6	yes	—
130	C₂×C₄×C₁₁₃	C₂×C₂²⋊C₄	1	6	yes	—
131	C₂×C₄×C₁₁₃	#120	1	3	yes	—
132	C₂×C₄×C₁₁₃	#120	1	6	yes	—
133	C₂×C₄×C₁₁₃	#120	1	6	yes	—
134	C₂×C₄×C₁₁₃	#120	1	1	yes	characteristic
135	C₂²×C₁₁₃	—	2	6	no	—
136	C₂²×C₁₁₃	—	2	12	no	—
137	C₂²×C₁₁₃	—	4	1	no	—
138	C₂²×C₁₁₃	—	4	3	no	—
139	C₂²×C₁₁₃	—	4	12	no	—
140	C₂²×C₁₁₃	—	4	12	no	—
141	C₂²×C₁₁₃	—	8	6	no	—
142	C₂²×C₁₁₃	—	8	16	no	—
143	C₂²×C₁₁₃	C₂²×C₂²⋊C₄	1	3	yes	—
144	C₂²×C₁₁₃	C₂×#120	1	3	yes	—
145	C₂²×C₁₁₃	C₂×#120	1	1	yes	characteristic
146	C₄×C₁₁₃	—	2	12	no	—
147	C₄×C₁₁₃	—	4	4	no	—
148	C₄×C₁₁₃	—	8	6	no	—
149	C₄×C₁₁₃	#379	1	2	yes	—
150	C₄×C₁₁₃	#379	1	6	yes	—
151	#20608	C₁₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
152	C₂×C₁₁₃	—	2	3	no	—
153	C₂×C₁₁₃	—	2	3	no	—
154	C₂×C₁₁₃	—	4	12	no	—
155	C₂×C₁₁₃	C₂×#379	1	1	yes	characteristic
156	C₂×C₁₁₃	#3025	1	2	yes	—
157	C₂×#585	C₂×C₁₁₃	1	1	yes	characteristic
158	C₂×#410	C₂×C₁₁₃	1	3	yes	complemented
159	#2297	C₂×C₁₁₃	1	3	yes	complemented
160	#1790	C₂×C₁₁₃	1	8	yes	complemented
161	C₁₁₃	#20608	1	1	yes	characteristic, direct product factor, Sylow
162	#585	—	2	12	no	—
163	#585	C₄×C₁₁₃	1	8	yes	complemented
164	C₂×2₋¹⁺⁴	—	2	3	no	—
165	C₂×H(5,𝔽₂)	—	2	3	no	—
166	C₂×H(5,𝔽₂)	C₂²×C₁₁₃	1	4	yes	—
167	C₂²×#41	—	2	6	no	—
168	C₂²×#41	C₂²×C₁₁₃	1	3	yes	—
169	C₂×#117	C₂²×C₁₁₃	1	3	yes	complemented
170	C₂²×C₄²	C₂²×C₁₁₃	1	1	yes	characteristic, complemented
171	#410	—	2	12	no	—
172	C₂×#99	C₂²×C₁₁₃	1	12	yes	complemented
173	#391	—	2	3	no	—
174	#390	—	2	3	no	—
175	#386	C₂²×C₁₁₃	1	12	yes	complemented
176	C₄×C₂²⋊C₄	—	2	6	no	—
177	2₋¹⁺⁴	—	2	24	no	—
178	2₋¹⁺⁴	—	2	24	no	—
179	H(5,𝔽₂)	—	2	24	no	—
180	H(5,𝔽₂)	—	2	24	no	—
181	H(5,𝔽₂)	—	2	24	no	—
182	H(5,𝔽₂)	C₂×C₄×C₁₁₃	1	16	yes	—
183	C₂×#41	—	2	4	no	—
184	C₂×#41	—	2	12	no	—
185	C₂×#41	—	2	48	no	—
186	C₂×#41	—	2	48	no	—
187	C₂×#41	—	2	6	no	—
188	C₂×#41	—	2	12	no	—
189	C₂×#41	—	2	6	no	—
190	C₂×#41	—	2	12	no	—
191	C₂×#41	C₂×C₄×C₁₁₃	1	12	yes	—
192	C₂×#41	D₄×C₁₁₃	1	12	yes	—
193	C₂²×Q₈	—	2	6	no	—
194	C₂²×Q₈	D₄×C₁₁₃	1	3	yes	—
195	C₂²×D₄	—	2	6	no	—
196	C₂²×D₄	—	2	12	no	—
197	C₂²×D₄	D₄×C₁₁₃	1	3	yes	—
198	C₂²×D₄	C₂³×C₁₁₃	1	6	yes	—
199	C₂³×C₄	—	2	4	no	—
200	C₂³×C₄	D₄×C₁₁₃	1	6	yes	—
201	C₂³×C₄	C₂³×C₁₁₃	1	2	yes	—
202	C₂³×C₄	C₂³×C₁₁₃	1	1	yes	characteristic
203	#117	—	2	12	no	—
204	C₂×C₄⋊C₄	—	2	3	no	—
205	C₂×C₂²⋊C₄	—	2	12	no	—
206	C₂×C₂²⋊C₄	C₂³×C₁₁₃	1	6	yes	—
207	C₂×C₄²	—	2	3	no	—
208	C₂×C₄²	—	2	3	no	—
209	C₂×C₄²	—	2	3	no	—
210	#99	—	2	24	no	—
211	#95	—	2	12	no	—
212	C₂⁴	—	2	4	no	—
213	C₂⁴	C₂×D₄×C₁₁₃	1	6	yes	—
214	C₂⁴	C₂⁴×C₁₁₃	1	1	yes	characteristic
215	#41	—	2	32	no	—
216	#41	—	2	48	no	—
217	#41	—	2	96	no	—
218	#41	—	2	24	no	—
219	#41	—	2	48	no	—
220	#41	—	2	24	no	—
221	#41	—	2	24	no	—
222	#41	—	2	24	no	—
223	C₂×Q₈	—	2	4	no	—
224	C₂×Q₈	—	2	48	no	—
225	C₂×Q₈	—	2	6	no	—
226	C₂×Q₈	—	2	6	no	—
227	C₂×Q₈	C₂²⋊C₄×C₁₁₃	1	12	yes	complemented
228	C₂×D₄	—	2	12	no	—
229	C₂×D₄	—	2	48	no	—
230	C₂×D₄	—	2	48	no	—
231	C₂×D₄	—	2	48	no	—
232	C₂×D₄	—	2	12	no	—
233	C₂×D₄	—	2	12	no	—
234	C₂×D₄	—	2	6	no	—
235	C₂×D₄	—	2	6	no	—
236	C₂×D₄	C₂²⋊C₄×C₁₁₃	1	12	yes	complemented
237	C₂×D₄	C₂²×C₄×C₁₁₃	1	12	yes	—
238	C₂×D₄	C₂×D₄×C₁₁₃	1	12	yes	—
239	C₂²×C₄	—	2	12	no	—
240	C₂²×C₄	—	2	32	no	—
241	C₂²×C₄	—	2	6	no	—
242	C₂²×C₄	—	2	6	no	—
243	C₂²×C₄	—	2	12	no	—
244	C₂²×C₄	—	2	6	no	—
245	C₂²×C₄	—	2	6	no	—
246	C₂²×C₄	—	4	4	no	—
247	C₂²×C₄	C₂²⋊C₄×C₁₁₃	1	24	yes	complemented
248	C₂²×C₄	C₂²×C₄×C₁₁₃	1	2	yes	—
249	C₂²×C₄	C₂×D₄×C₁₁₃	1	6	yes	—
250	C₂²×C₄	C₂×D₄×C₁₁₃	1	3	yes	—
251	C₂²×C₄	C₂×D₄×C₁₁₃	1	3	yes	—
252	C₂²×C₄	C₂×D₄×C₁₁₃	1	6	yes	—
253	C₄⋊C₄	—	4	6	no	—
254	C₂²⋊C₄	—	2	12	no	—
255	C₂²⋊C₄	—	4	24	no	—
256	C₄²	—	4	1	no	—
257	C₄²	—	4	3	no	—
258	C₄²	—	4	6	no	—
259	C₂³	—	2	16	no	—
260	C₂³	—	2	6	no	—
261	C₂³	—	2	6	no	—
262	C₂³	—	4	2	no	—
263	C₂³	—	4	12	no	—
264	C₂³	—	8	8	no	—
265	C₂³	C₂×C₂²⋊C₄×C₁₁₃	1	12	yes	—
266	C₂³	C₂³×C₄×C₁₁₃	1	1	yes	characteristic
267	C₂³	C₂²×D₄×C₁₁₃	1	3	yes	—
268	C₂³	C₂²×D₄×C₁₁₃	1	3	yes	—
269	Q₈	—	2	16	no	—
270	Q₈	—	2	24	no	—
271	D₄	—	2	48	no	—
272	D₄	—	2	24	no	—
273	D₄	—	2	24	no	—
274	D₄	—	2	24	no	—
275	C₂×C₄	—	2	48	no	—
276	C₂×C₄	—	2	6	no	—
277	C₂×C₄	—	2	6	no	—
278	C₂×C₄	—	2	24	no	—
279	C₂×C₄	—	2	24	no	—
280	C₂×C₄	—	2	12	no	—
281	C₂×C₄	—	4	6	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	12	no	—
284	C₂×C₄	—	4	6	no	—
285	C₂×C₄	C₂×C₂²⋊C₄×C₁₁₃	1	12	yes	—
286	C₂×C₄	C₂×C₂²⋊C₄×C₁₁₃	1	6	yes	—
287	C₂×C₄	C₂×C₂²⋊C₄×C₁₁₃	1	6	yes	—
288	C₂×C₄	#120×C₁₁₃	1	1	yes	characteristic
289	C₂×C₄	#120×C₁₁₃	1	3	yes	—
290	C₂×C₄	#120×C₁₁₃	1	6	yes	—
291	C₂×C₄	#120×C₁₁₃	1	6	yes	—
292	C₂²	—	2	6	no	—
293	C₂²	—	2	12	no	—
294	C₂²	—	4	1	no	—
295	C₂²	—	4	3	no	—
296	C₂²	—	4	12	no	—
297	C₂²	—	4	12	no	—
298	C₂²	—	8	6	no	—
299	C₂²	—	8	16	no	—
300	C₂²	C₂²×C₂²⋊C₄×C₁₁₃	1	3	yes	—
301	C₂²	C₂×#120×C₁₁₃	1	3	yes	—
302	C₂²	C₂×#120×C₁₁₃	1	1	yes	characteristic
303	C₄	—	2	12	no	—
304	C₄	—	4	4	no	—
305	C₄	—	8	6	no	—
306	C₄	#379×C₁₁₃	1	2	yes	—
307	C₄	#379×C₁₁₃	1	6	yes	—
308	C₂	—	2	3	no	—
309	C₂	—	2	3	no	—
310	C₂	—	4	12	no	—
311	C₂	C₂×#379×C₁₁₃	1	1	yes	characteristic
312	C₂	C₁₁₃×#3025	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₁₃	#120
commutator	C₂³	C₂³×C₄×C₁₁₃
Frattini, cosocle	C₂⁴	C₂⁴×C₁₁₃
socle	C₂²×C₁₁₃	C₂×#120
prime radical	C₂×#585×C₁₁₃	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₁₃ ◁ C₂³×C₄×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×#2297	C₂	semidirect
4. C₁₁₃×#1790	C₂	semidirect
6. #585×C₁₁₃	C₄	semidirect
12. C₂×#117×C₁₁₃	C₂²	semidirect
13. C₂²×C₄²×C₁₁₃	C₂²	semidirect
15. C₂×#99×C₁₁₃	C₂²	semidirect
18. #386×C₁₁₃	C₂²	semidirect
70. C₂×Q₈×C₁₁₃	C₂²⋊C₄	semidirect
79. C₂×D₄×C₁₁₃	C₂²⋊C₄	semidirect
90. C₂²×C₄×C₁₁₃	C₂²⋊C₄	semidirect
151. #20608	C₁₁₃	direct (N × H)
158. C₂×#410	C₂×C₁₁₃	semidirect
159. #2297	C₂×C₁₁₃	semidirect
160. #1790	C₂×C₁₁₃	semidirect
161. C₁₁₃	#20608	direct (N × H)
163. #585	C₄×C₁₁₃	semidirect
169. C₂×#117	C₂²×C₁₁₃	semidirect
170. C₂²×C₄²	C₂²×C₁₁₃	semidirect
172. C₂×#99	C₂²×C₁₁₃	semidirect
175. #386	C₂²×C₁₁₃	semidirect
227. C₂×Q₈	C₂²⋊C₄×C₁₁₃	semidirect
236. C₂×D₄	C₂²⋊C₄×C₁₁₃	semidirect
247. 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
C₂×#585×C₁₁₃	2
C₂×#410×C₁₁₃	2
C₁₁₃×#2297	2
C₁₁₃×#1790	2
#20608	113

Subgroups — C113×#20608 #43288628

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁₃×#20608 #43288628