Subgroups — C₁₁₃×#15224 #43283244

Order 28928.

1678 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#390×C₁₁₃	C₂	1	2	yes	maximal
2	C₂×#390×C₁₁₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#382×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₁₃×#1440	C₂	1	8	yes	maximal, complemented
5	C₁₁₃×C₄⋊#95	C₂	1	2	yes	maximal, complemented
6	C₄×#95×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₄×C₁₁₃	C₂²	1	1	yes	characteristic
8	C₂²×C₄⋊C₄×C₁₁₃	C₂²	1	2	yes	—
9	C₂²×C₄⋊C₄×C₁₁₃	C₂²	1	1	yes	characteristic
10	C₂²×C₂²⋊C₄×C₁₁₃	C₂²	1	8	yes	—
11	C₂²×C₂²⋊C₄×C₁₁₃	C₂²	1	4	yes	—
12	C₂²×C₄²×C₁₁₃	C₂²	1	2	yes	—
13	C₂²×C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
14	#390×C₁₁₃	—	2	8	no	—
15	#390×C₁₁₃	—	2	4	no	—
16	#382×C₁₁₃	—	2	4	no	—
17	C₂×#95×C₁₁₃	C₂²	1	4	yes	—
18	C₂×#95×C₁₁₃	C₂²	1	4	yes	—
19	C₂×#95×C₁₁₃	C₂²	1	8	yes	—
20	C₂⁵×C₁₁₃	C₂³	1	2	yes	—
21	C₂²×D₄×C₁₁₃	—	2	2	no	—
22	C₂²×D₄×C₁₁₃	—	2	2	no	—
23	C₂²×D₄×C₁₁₃	—	2	4	no	—
24	C₂²×D₄×C₁₁₃	—	2	4	no	—
25	C₂²×D₄×C₁₁₃	D₄	1	4	yes	—
26	C₂³×C₄×C₁₁₃	C₂³	1	4	yes	—
27	C₂³×C₄×C₁₁₃	C₂³	1	2	yes	—
28	C₂³×C₄×C₁₁₃	C₂³	1	2	yes	—
29	C₂³×C₄×C₁₁₃	C₂³	1	4	yes	—
30	C₂³×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₁₁₃	—	2	4	no	—
32	C₂×C₄⋊C₄×C₁₁₃	—	2	4	no	—
33	C₂×C₄⋊C₄×C₁₁₃	—	2	2	no	—
34	C₂×C₄⋊C₄×C₁₁₃	—	2	4	no	—
35	C₂×C₄⋊C₄×C₁₁₃	—	2	4	no	—
36	C₂×C₂²⋊C₄×C₁₁₃	—	2	16	no	—
37	C₂×C₂²⋊C₄×C₁₁₃	—	2	8	no	—
38	C₂×C₂²⋊C₄×C₁₁₃	—	2	16	no	—
39	C₂×C₂²⋊C₄×C₁₁₃	—	4	8	no	—
40	C₂×C₂²⋊C₄×C₁₁₃	—	4	8	no	—
41	C₂×C₄²×C₁₁₃	—	2	4	no	—
42	C₂×C₄²×C₁₁₃	—	2	4	no	—
43	C₂×C₄²×C₁₁₃	—	2	4	no	—
44	C₂×C₄²×C₁₁₃	—	2	4	no	—
45	C₂×C₄²×C₁₁₃	D₄	1	4	yes	—
46	#95×C₁₁₃	—	2	8	no	—
47	#95×C₁₁₃	—	2	8	no	—
48	#95×C₁₁₃	—	4	8	no	—
49	C₂⁴×C₁₁₃	—	2	2	no	—
50	C₂⁴×C₁₁₃	—	2	8	no	—
51	C₂⁴×C₁₁₃	—	2	2	no	—
52	C₂⁴×C₁₁₃	—	2	4	no	—
53	C₂⁴×C₁₁₃	—	2	4	no	—
54	C₂⁴×C₁₁₃	—	2	8	no	—
55	C₂⁴×C₁₁₃	C₂×D₄	1	4	yes	—
56	C₂⁴×C₁₁₃	C₂⁴	1	1	yes	characteristic
57	C₂×D₄×C₁₁₃	—	2	8	no	—
58	C₂×D₄×C₁₁₃	—	2	16	no	—
59	C₂×D₄×C₁₁₃	—	4	4	no	—
60	C₂×D₄×C₁₁₃	—	4	4	no	—
61	C₂×D₄×C₁₁₃	—	4	8	no	—
62	C₂²×C₄×C₁₁₃	—	2	4	no	—
63	C₂²×C₄×C₁₁₃	—	2	4	no	—
64	C₂²×C₄×C₁₁₃	—	2	4	no	—
65	C₂²×C₄×C₁₁₃	—	2	2	no	—
66	C₂²×C₄×C₁₁₃	—	2	2	no	—
67	C₂²×C₄×C₁₁₃	—	2	4	no	—
68	C₂²×C₄×C₁₁₃	—	2	4	no	—
69	C₂²×C₄×C₁₁₃	—	2	4	no	—
70	C₂²×C₄×C₁₁₃	—	2	4	no	—
71	C₂²×C₄×C₁₁₃	—	2	4	no	—
72	C₂²×C₄×C₁₁₃	—	2	4	no	—
73	C₂²×C₄×C₁₁₃	—	2	4	no	—
74	C₂²×C₄×C₁₁₃	—	2	2	no	—
75	C₂²×C₄×C₁₁₃	—	2	8	no	—
76	C₂²×C₄×C₁₁₃	—	2	4	no	—
77	C₂²×C₄×C₁₁₃	—	2	4	no	—
78	C₂²×C₄×C₁₁₃	—	2	8	no	—
79	C₂²×C₄×C₁₁₃	—	2	8	no	—
80	C₂²×C₄×C₁₁₃	C₂×D₄	1	4	yes	—
81	C₂²×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
82	C₂²×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₁₁₃	#41	1	8	yes	—
84	C₂²×C₄×C₁₁₃	#41	1	2	yes	—
85	C₂²×C₄×C₁₁₃	#41	1	4	yes	—
86	C₂²×C₄×C₁₁₃	#41	1	4	yes	—
87	C₄⋊C₄×C₁₁₃	—	4	8	no	—
88	C₄⋊C₄×C₁₁₃	—	4	4	no	—
89	C₂²⋊C₄×C₁₁₃	—	4	32	no	—
90	C₂²⋊C₄×C₁₁₃	—	4	16	no	—
91	C₄²×C₁₁₃	—	2	8	no	—
92	C₄²×C₁₁₃	—	4	8	no	—
93	C₂³×C₁₁₃	—	2	4	no	—
94	C₂³×C₁₁₃	—	2	8	no	—
95	C₂³×C₁₁₃	—	2	16	no	—
96	C₂³×C₁₁₃	—	4	2	no	—
97	C₂³×C₁₁₃	—	4	2	no	—
98	C₂³×C₁₁₃	—	4	4	no	—
99	C₂³×C₁₁₃	—	4	8	no	—
100	C₂³×C₁₁₃	—	4	4	no	—
101	C₂³×C₁₁₃	—	4	4	no	—
102	C₂³×C₁₁₃	—	4	8	no	—
103	C₂³×C₁₁₃	—	4	8	no	—
104	C₂³×C₁₁₃	—	4	8	no	—
105	C₂³×C₁₁₃	—	4	8	no	—
106	C₂³×C₁₁₃	C₂²×D₄	1	1	yes	characteristic
107	C₂³×C₁₁₃	C₂²×D₄	1	1	yes	characteristic
108	C₂³×C₁₁₃	C₂×#41	1	1	yes	characteristic
109	C₂³×C₁₁₃	C₂×#41	1	2	yes	—
110	C₂³×C₁₁₃	C₂×#41	1	2	yes	—
111	C₂³×C₁₁₃	C₂×#41	1	4	yes	—
112	C₂³×C₁₁₃	H(5,𝔽₂)	1	4	yes	—
113	D₄×C₁₁₃	—	4	16	no	—
114	C₂×C₄×C₁₁₃	—	2	4	no	—
115	C₂×C₄×C₁₁₃	—	2	8	no	—
116	C₂×C₄×C₁₁₃	—	2	4	no	—
117	C₂×C₄×C₁₁₃	—	2	4	no	—
118	C₂×C₄×C₁₁₃	—	2	8	no	—
119	C₂×C₄×C₁₁₃	—	2	8	no	—
120	C₂×C₄×C₁₁₃	—	4	4	no	—
121	C₂×C₄×C₁₁₃	—	4	4	no	—
122	C₂×C₄×C₁₁₃	—	4	4	no	—
123	C₂×C₄×C₁₁₃	—	4	4	no	—
124	C₂×C₄×C₁₁₃	—	4	4	no	—
125	C₂×C₄×C₁₁₃	—	4	4	no	—
126	C₂×C₄×C₁₁₃	—	4	4	no	—
127	C₂×C₄×C₁₁₃	—	4	2	no	—
128	C₂×C₄×C₁₁₃	—	4	4	no	—
129	C₂×C₄×C₁₁₃	—	4	4	no	—
130	C₂×C₄×C₁₁₃	—	4	4	no	—
131	C₂×C₄×C₁₁₃	—	4	8	no	—
132	C₂×C₄×C₁₁₃	—	4	8	no	—
133	C₂×C₄×C₁₁₃	—	4	8	no	—
134	C₂×C₄×C₁₁₃	#121	1	4	yes	—
135	C₂×C₄×C₁₁₃	#123	1	4	yes	—
136	C₂×C₄×C₁₁₃	#123	1	8	yes	—
137	C₂×C₄×C₁₁₃	#124	1	4	yes	—
138	C₂×C₄×C₁₁₃	#126	1	8	yes	—
139	C₂²×C₁₁₃	—	4	4	no	—
140	C₂²×C₁₁₃	—	4	16	no	—
141	C₂²×C₁₁₃	—	4	8	no	—
142	C₂²×C₁₁₃	—	8	2	no	—
143	C₂²×C₁₁₃	—	8	2	no	—
144	C₂²×C₁₁₃	—	8	4	no	—
145	C₂²×C₁₁₃	—	8	8	no	—
146	C₂²×C₁₁₃	C₂×#121	1	1	yes	characteristic
147	C₂²×C₁₁₃	C₂×#123	1	2	yes	—
148	C₂²×C₁₁₃	C₂×#123	1	1	yes	characteristic
149	C₂²×C₁₁₃	C₂×#124	1	1	yes	characteristic
150	C₂²×C₁₁₃	C₂×#126	1	2	yes	—
151	C₂²×C₁₁₃	#529	1	4	yes	—
152	C₂²×C₁₁₃	#532	1	2	yes	—
153	C₂²×C₁₁₃	#535	1	2	yes	—
154	C₂²×C₁₁₃	#540	1	4	yes	—
155	C₂²×C₁₁₃	#546	1	4	yes	—
156	C₂²×C₁₁₃	#553	1	4	yes	—
157	C₂²×C₁₁₃	#555	1	4	yes	—
158	C₂²×C₁₁₃	#559	1	4	yes	—
159	C₄×C₁₁₃	—	4	8	no	—
160	C₄×C₁₁₃	—	4	4	no	—
161	C₄×C₁₁₃	—	8	2	no	—
162	C₄×C₁₁₃	—	8	4	no	—
163	C₄×C₁₁₃	#397	1	8	yes	—
164	#15224	C₁₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
165	C₂×C₁₁₃	—	8	4	no	—
166	C₂×C₁₁₃	C₂×#397	1	1	yes	characteristic
167	C₂×C₁₁₃	#2420	1	4	yes	—
168	C₂×C₁₁₃	#2500	1	2	yes	—
169	C₂×C₁₁₃	#2513	1	4	yes	—
170	C₂×C₁₁₃	#2544	1	2	yes	—
171	C₂×C₁₁₃	#2641	1	1	yes	characteristic
172	C₂×C₁₁₃	#2861	1	1	yes	characteristic
173	C₂×#390	C₂×C₁₁₃	1	2	yes	—
174	C₂×#390	C₂×C₁₁₃	1	1	yes	characteristic
175	C₂×#382	C₂×C₁₁₃	1	1	yes	characteristic, complemented
176	#1440	C₂×C₁₁₃	1	8	yes	complemented
177	C₄⋊#95	C₂×C₁₁₃	1	2	yes	complemented
178	C₄×#95	C₂×C₁₁₃	1	1	yes	characteristic, complemented
179	C₁₁₃	#15224	1	1	yes	characteristic, direct product factor, Sylow
180	C₂³×D₄	C₂²×C₁₁₃	1	1	yes	characteristic
181	C₂²×C₄⋊C₄	C₂²×C₁₁₃	1	1	yes	characteristic
182	C₂²×C₄⋊C₄	C₂²×C₁₁₃	1	2	yes	—
183	C₂²×C₂²⋊C₄	C₂²×C₁₁₃	1	8	yes	—
184	C₂²×C₂²⋊C₄	C₂²×C₁₁₃	1	4	yes	—
185	C₂²×C₄²	C₂²×C₁₁₃	1	2	yes	—
186	C₂²×C₄²	C₂²×C₁₁₃	1	1	yes	characteristic
187	#390	—	2	8	no	—
188	#390	—	2	4	no	—
189	#382	—	2	4	no	—
190	C₂×#95	C₂²×C₁₁₃	1	4	yes	—
191	C₂×#95	C₂²×C₁₁₃	1	4	yes	—
192	C₂×#95	C₂²×C₁₁₃	1	8	yes	—
193	C₂⁵	C₂³×C₁₁₃	1	2	yes	—
194	C₂²×D₄	—	2	2	no	—
195	C₂²×D₄	—	2	2	no	—
196	C₂²×D₄	—	2	4	no	—
197	C₂²×D₄	—	2	4	no	—
198	C₂²×D₄	D₄×C₁₁₃	1	4	yes	—
199	C₂³×C₄	C₂³×C₁₁₃	1	4	yes	—
200	C₂³×C₄	C₂³×C₁₁₃	1	2	yes	—
201	C₂³×C₄	C₂³×C₁₁₃	1	2	yes	—
202	C₂³×C₄	C₂³×C₁₁₃	1	4	yes	—
203	C₂³×C₄	C₂³×C₁₁₃	1	1	yes	characteristic
204	C₂×C₄⋊C₄	—	2	4	no	—
205	C₂×C₄⋊C₄	—	2	4	no	—
206	C₂×C₄⋊C₄	—	2	2	no	—
207	C₂×C₄⋊C₄	—	2	4	no	—
208	C₂×C₄⋊C₄	—	2	4	no	—
209	C₂×C₂²⋊C₄	—	2	16	no	—
210	C₂×C₂²⋊C₄	—	2	8	no	—
211	C₂×C₂²⋊C₄	—	2	16	no	—
212	C₂×C₂²⋊C₄	—	4	8	no	—
213	C₂×C₂²⋊C₄	—	4	8	no	—
214	C₂×C₄²	—	2	4	no	—
215	C₂×C₄²	—	2	4	no	—
216	C₂×C₄²	—	2	4	no	—
217	C₂×C₄²	—	2	4	no	—
218	C₂×C₄²	D₄×C₁₁₃	1	4	yes	—
219	#95	—	2	8	no	—
220	#95	—	2	8	no	—
221	#95	—	4	8	no	—
222	C₂⁴	—	2	2	no	—
223	C₂⁴	—	2	8	no	—
224	C₂⁴	—	2	2	no	—
225	C₂⁴	—	2	4	no	—
226	C₂⁴	—	2	4	no	—
227	C₂⁴	—	2	8	no	—
228	C₂⁴	C₂×D₄×C₁₁₃	1	4	yes	—
229	C₂⁴	C₂⁴×C₁₁₃	1	1	yes	characteristic
230	C₂×D₄	—	2	8	no	—
231	C₂×D₄	—	2	16	no	—
232	C₂×D₄	—	4	4	no	—
233	C₂×D₄	—	4	4	no	—
234	C₂×D₄	—	4	8	no	—
235	C₂²×C₄	—	2	4	no	—
236	C₂²×C₄	—	2	4	no	—
237	C₂²×C₄	—	2	4	no	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	2	2	no	—
240	C₂²×C₄	—	2	4	no	—
241	C₂²×C₄	—	2	4	no	—
242	C₂²×C₄	—	2	4	no	—
243	C₂²×C₄	—	2	4	no	—
244	C₂²×C₄	—	2	4	no	—
245	C₂²×C₄	—	2	4	no	—
246	C₂²×C₄	—	2	4	no	—
247	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	—	2	8	no	—
249	C₂²×C₄	—	2	4	no	—
250	C₂²×C₄	—	2	4	no	—
251	C₂²×C₄	—	2	8	no	—
252	C₂²×C₄	—	2	8	no	—
253	C₂²×C₄	C₂×D₄×C₁₁₃	1	4	yes	—
254	C₂²×C₄	C₂×D₄×C₁₁₃	1	2	yes	—
255	C₂²×C₄	C₂×D₄×C₁₁₃	1	2	yes	—
256	C₂²×C₄	#41×C₁₁₃	1	8	yes	—
257	C₂²×C₄	#41×C₁₁₃	1	2	yes	—
258	C₂²×C₄	#41×C₁₁₃	1	4	yes	—
259	C₂²×C₄	#41×C₁₁₃	1	4	yes	—
260	C₄⋊C₄	—	4	8	no	—
261	C₄⋊C₄	—	4	4	no	—
262	C₂²⋊C₄	—	4	32	no	—
263	C₂²⋊C₄	—	4	16	no	—
264	C₄²	—	2	8	no	—
265	C₄²	—	4	8	no	—
266	C₂³	—	2	4	no	—
267	C₂³	—	2	8	no	—
268	C₂³	—	2	16	no	—
269	C₂³	—	4	2	no	—
270	C₂³	—	4	2	no	—
271	C₂³	—	4	4	no	—
272	C₂³	—	4	8	no	—
273	C₂³	—	4	4	no	—
274	C₂³	—	4	4	no	—
275	C₂³	—	4	8	no	—
276	C₂³	—	4	8	no	—
277	C₂³	—	4	8	no	—
278	C₂³	—	4	8	no	—
279	C₂³	C₂²×D₄×C₁₁₃	1	1	yes	characteristic
280	C₂³	C₂²×D₄×C₁₁₃	1	1	yes	characteristic
281	C₂³	C₂×#41×C₁₁₃	1	1	yes	characteristic
282	C₂³	C₂×#41×C₁₁₃	1	2	yes	—
283	C₂³	C₂×#41×C₁₁₃	1	2	yes	—
284	C₂³	C₂×#41×C₁₁₃	1	4	yes	—
285	C₂³	H(5,𝔽₂)×C₁₁₃	1	4	yes	—
286	D₄	—	4	16	no	—
287	C₂×C₄	—	2	4	no	—
288	C₂×C₄	—	2	8	no	—
289	C₂×C₄	—	2	4	no	—
290	C₂×C₄	—	2	4	no	—
291	C₂×C₄	—	2	8	no	—
292	C₂×C₄	—	2	8	no	—
293	C₂×C₄	—	4	4	no	—
294	C₂×C₄	—	4	4	no	—
295	C₂×C₄	—	4	4	no	—
296	C₂×C₄	—	4	4	no	—
297	C₂×C₄	—	4	4	no	—
298	C₂×C₄	—	4	4	no	—
299	C₂×C₄	—	4	4	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	4	no	—
302	C₂×C₄	—	4	4	no	—
303	C₂×C₄	—	4	4	no	—
304	C₂×C₄	—	4	8	no	—
305	C₂×C₄	—	4	8	no	—
306	C₂×C₄	—	4	8	no	—
307	C₂×C₄	#121×C₁₁₃	1	4	yes	—
308	C₂×C₄	#123×C₁₁₃	1	4	yes	—
309	C₂×C₄	#123×C₁₁₃	1	8	yes	—
310	C₂×C₄	#124×C₁₁₃	1	4	yes	—
311	C₂×C₄	#126×C₁₁₃	1	8	yes	—
312	C₂²	—	4	8	no	—
313	C₂²	—	4	4	no	—
314	C₂²	—	4	16	no	—
315	C₂²	—	8	2	no	—
316	C₂²	—	8	2	no	—
317	C₂²	—	8	4	no	—
318	C₂²	—	8	8	no	—
319	C₂²	C₂×#121×C₁₁₃	1	1	yes	characteristic
320	C₂²	C₂×#123×C₁₁₃	1	2	yes	—
321	C₂²	C₂×#123×C₁₁₃	1	1	yes	characteristic
322	C₂²	C₂×#124×C₁₁₃	1	1	yes	characteristic
323	C₂²	C₂×#126×C₁₁₃	1	2	yes	—
324	C₂²	#529×C₁₁₃	1	4	yes	—
325	C₂²	#532×C₁₁₃	1	2	yes	—
326	C₂²	#535×C₁₁₃	1	2	yes	—
327	C₂²	#540×C₁₁₃	1	4	yes	—
328	C₂²	#546×C₁₁₃	1	4	yes	—
329	C₂²	#553×C₁₁₃	1	4	yes	—
330	C₂²	#555×C₁₁₃	1	4	yes	—
331	C₂²	#559×C₁₁₃	1	4	yes	—
332	C₄	—	4	8	no	—
333	C₄	—	4	4	no	—
334	C₄	—	8	4	no	—
335	C₄	—	8	2	no	—
336	C₄	#397×C₁₁₃	1	8	yes	—
337	C₂	—	8	4	no	—
338	C₂	C₂×#397×C₁₁₃	1	1	yes	characteristic
339	C₂	C₁₁₃×#2420	1	4	yes	—
340	C₂	C₁₁₃×#2500	1	2	yes	—
341	C₂	C₁₁₃×#2513	1	4	yes	—
342	C₂	C₁₁₃×#2544	1	2	yes	—
343	C₂	C₁₁₃×#2641	1	1	yes	characteristic
344	C₂	C₁₁₃×#2861	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₁₁₃	C₂⁴
commutator, Frattini, cosocle	C₂⁴	C₂⁴×C₁₁₃
prime radical	C₂³×D₄×C₁₁₃	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×#382×C₁₁₃	C₂	semidirect
4. C₁₁₃×#1440	C₂	semidirect
5. C₁₁₃×C₄⋊#95	C₂	semidirect
6. C₄×#95×C₁₁₃	C₂	semidirect
164. #15224	C₁₁₃	direct (N × H)
175. C₂×#382	C₂×C₁₁₃	semidirect
176. #1440	C₂×C₁₁₃	semidirect
177. C₄⋊#95	C₂×C₁₁₃	semidirect
178. C₄×#95	C₂×C₁₁₃	semidirect
179. C₁₁₃	#15224	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×#390×C₁₁₃	2
C₂×#382×C₁₁₃	2
C₁₁₃×#1440	2
C₁₁₃×C₄⋊#95	2
C₄×#95×C₁₁₃	2
#15224	113

Subgroups — C113×#15224 #43283244

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁₃×#15224 #43283244