Subgroups — C₁₁₃×#22708 #43290728

Order 28928.

910 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₄×C₁₁₃	C₂	1	1	yes	maximal, characteristic
2	C₁₁₃×#2354	C₂	1	2	yes	maximal
3	C₁₁₃×#1948	C₂	1	2	yes	maximal, complemented
4	C₁₁₃×#1854	C₂	1	4	yes	maximal, complemented
5	C₁₁₃×#1834	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₁₃×#1807	C₂	1	2	yes	maximal, complemented
7	C₄×#107×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₁₃×#1735	C₂	1	2	yes	maximal, complemented
9	C₂³×C₈×C₁₁₃	C₂²	1	2	yes	—
10	C₂×C₄×D₄×C₁₁₃	C₂²	1	1	yes	characteristic
11	C₂×#117×C₁₁₃	C₂²	1	4	yes	—
12	C₈×D₄×C₁₁₃	C₄	1	8	yes	complemented
13	C₂×#107×C₁₁₃	C₂²	1	2	yes	—
14	C₂×#106×C₁₁₃	C₂²	1	4	yes	—
15	C₂×C₄⋊C₈×C₁₁₃	C₂²	1	1	yes	characteristic
16	C₂×C₂²⋊C₈×C₁₁₃	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₁₁₃	C₂²	1	1	yes	characteristic
18	#390×C₁₁₃	—	2	2	no	—
19	#388×C₁₁₃	—	2	4	no	—
20	#384×C₁₁₃	C₂²	1	2	yes	—
21	#380×C₁₁₃	C₂²	1	4	yes	—
22	C₄×C₄⋊C₄×C₁₁₃	—	2	2	no	—
23	C₄×C₄⋊C₄×C₁₁₃	C₂²	1	2	yes	—
24	#340×C₁₁₃	C₂²	1	1	yes	characteristic
25	#340×C₁₁₃	C₂²	1	1	yes	characteristic
26	#340×C₁₁₃	C₂²	1	2	yes	—
27	#340×C₁₁₃	C₂²	1	2	yes	—
28	#340×C₁₁₃	C₂²	1	4	yes	—
29	C₂²×D₄×C₁₁₃	D₄	1	1	yes	characteristic
30	C₂³×C₄×C₁₁₃	C₂³	1	2	yes	—
31	C₂²×C₈×C₁₁₃	—	4	4	no	—
32	C₂²×C₈×C₁₁₃	C₂×C₄	1	8	yes	—
33	C₂²×C₈×C₁₁₃	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₁₁₃	C₂³	1	2	yes	—
35	C₂²×C₈×C₁₁₃	C₂³	1	1	yes	characteristic
36	C₄×D₄×C₁₁₃	C₂×C₄	1	4	yes	—
37	C₄×D₄×C₁₁₃	Q₈	1	4	yes	—
38	#117×C₁₁₃	—	2	8	no	—
39	#117×C₁₁₃	—	4	4	no	—
40	C₂×C₄⋊C₄×C₁₁₃	—	2	2	no	—
41	C₂×C₄⋊C₄×C₁₁₃	—	2	2	no	—
42	C₂×C₄⋊C₄×C₁₁₃	D₄	1	1	yes	characteristic
43	C₂×C₄⋊C₄×C₁₁₃	C₂³	1	4	yes	—
44	C₂×C₄⋊C₄×C₁₁₃	C₂³	1	2	yes	—
45	C₂×C₄⋊C₄×C₁₁₃	C₂³	1	2	yes	—
46	C₂×C₂²⋊C₄×C₁₁₃	—	2	4	no	—
47	C₂×C₂²⋊C₄×C₁₁₃	—	2	4	no	—
48	C₂×C₂²⋊C₄×C₁₁₃	D₄	1	2	yes	—
49	C₂×C₄²×C₁₁₃	—	2	4	no	—
50	C₂×C₄²×C₁₁₃	—	2	2	no	—
51	C₂×C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
52	#107×C₁₁₃	—	2	4	no	—
53	#106×C₁₁₃	—	4	4	no	—
54	C₄⋊C₈×C₁₁₃	C₂×C₄	1	4	yes	—
55	C₂²⋊C₈×C₁₁₃	C₂×C₄	1	8	yes	—
56	C₄×C₈×C₁₁₃	C₂×C₄	1	4	yes	—
57	#95×C₁₁₃	—	2	4	no	—
58	#95×C₁₁₃	—	2	2	no	—
59	C₂⁴×C₁₁₃	C₂×D₄	1	2	yes	—
60	C₂×D₄×C₁₁₃	—	2	2	no	—
61	C₂×D₄×C₁₁₃	—	2	2	no	—
62	C₂×D₄×C₁₁₃	C₄⋊C₄	1	4	yes	—
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	C₂²×C₄	1	4	yes	—
69	C₂²×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₁₃	C₂×Q₈	1	4	yes	—
73	C₂²×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
74	C₂×C₈×C₁₁₃	—	2	2	no	—
75	C₂×C₈×C₁₁₃	—	2	4	no	—
76	C₂×C₈×C₁₁₃	—	2	2	no	—
77	C₂×C₈×C₁₁₃	—	4	8	no	—
78	C₂×C₈×C₁₁₃	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₁₁₃	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₁₁₃	C₂²×C₄	1	4	yes	—
81	C₄⋊C₄×C₁₁₃	—	2	2	no	—
82	C₄⋊C₄×C₁₁₃	—	2	2	no	—
83	C₄⋊C₄×C₁₁₃	—	2	4	no	—
84	C₄⋊C₄×C₁₁₃	—	2	2	no	—
85	C₄⋊C₄×C₁₁₃	—	2	4	no	—
86	C₄⋊C₄×C₁₁₃	—	4	4	no	—
87	C₄⋊C₄×C₁₁₃	—	8	2	no	—
88	C₄⋊C₄×C₁₁₃	C₄⋊C₄	1	4	yes	—
89	C₄⋊C₄×C₁₁₃	#41	1	4	yes	—
90	C₂²⋊C₄×C₁₁₃	—	4	8	no	—
91	C₂²⋊C₄×C₁₁₃	—	8	4	no	—
92	C₂²⋊C₄×C₁₁₃	C₄⋊C₄	1	8	yes	—
93	C₄²×C₁₁₃	—	4	8	no	—
94	C₄²×C₁₁₃	—	4	4	no	—
95	C₄²×C₁₁₃	C₂²×C₄	1	2	yes	—
96	C₄²×C₁₁₃	C₂×Q₈	1	2	yes	—
97	C₂³×C₁₁₃	—	2	4	no	—
98	C₂³×C₁₁₃	—	2	2	no	—
99	C₂³×C₁₁₃	—	2	2	no	—
100	C₂³×C₁₁₃	—	2	4	no	—
101	C₂³×C₁₁₃	C₂×C₄⋊C₄	1	4	yes	—
102	C₂³×C₁₁₃	C₂²×D₄	1	1	yes	characteristic
103	D₄×C₁₁₃	—	2	8	no	—
104	C₂×C₄×C₁₁₃	—	2	4	no	—
105	C₂×C₄×C₁₁₃	—	2	2	no	—
106	C₂×C₄×C₁₁₃	—	2	4	no	—
107	C₂×C₄×C₁₁₃	—	2	4	no	—
108	C₂×C₄×C₁₁₃	—	2	8	no	—
109	C₂×C₄×C₁₁₃	—	2	4	no	—
110	C₂×C₄×C₁₁₃	—	4	2	no	—
111	C₂×C₄×C₁₁₃	—	4	2	no	—
112	C₂×C₄×C₁₁₃	—	4	4	no	—
113	C₂×C₄×C₁₁₃	—	4	2	no	—
114	C₂×C₄×C₁₁₃	—	4	2	no	—
115	C₂×C₄×C₁₁₃	—	4	4	no	—
116	C₂×C₄×C₁₁₃	—	4	2	no	—
117	C₂×C₄×C₁₁₃	—	4	4	no	—
118	C₂×C₄×C₁₁₃	C₂×C₄⋊C₄	1	2	yes	—
119	C₂×C₄×C₁₁₃	C₂×C₄⋊C₄	1	4	yes	—
120	C₂×C₄×C₁₁₃	C₂×C₄⋊C₄	1	2	yes	—
121	C₂×C₄×C₁₁₃	#135	1	2	yes	—
122	C₂×C₄×C₁₁₃	#135	1	2	yes	—
123	C₂×C₄×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
124	C₂×C₄×C₁₁₃	C₂²×Q₈	1	1	yes	characteristic
125	C₂×C₄×C₁₁₃	C₂×#41	1	2	yes	—
126	C₂×C₄×C₁₁₃	H(5,𝔽₂)	1	2	yes	—
127	C₈×C₁₁₃	—	2	2	no	—
128	C₈×C₁₁₃	—	2	2	no	—
129	C₈×C₁₁₃	—	2	4	no	—
130	C₂²×C₁₁₃	—	2	4	no	—
131	C₂²×C₁₁₃	—	2	4	no	—
132	C₂²×C₁₁₃	—	2	4	no	—
133	C₂²×C₁₁₃	—	4	4	no	—
134	C₂²×C₁₁₃	—	4	4	no	—
135	C₂²×C₁₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
136	C₂²×C₁₁₃	#546	1	2	yes	—
137	C₂²×C₁₁₃	C₂×#135	1	1	yes	characteristic
138	C₂²×C₁₁₃	C₂×#135	1	1	yes	characteristic
139	C₂²×C₁₁₃	#577	1	2	yes	—
140	C₄×C₁₁₃	—	2	4	no	—
141	C₄×C₁₁₃	—	2	2	no	—
142	C₄×C₁₁₃	—	4	4	no	—
143	C₄×C₁₁₃	—	8	2	no	—
144	C₄×C₁₁₃	—	8	4	no	—
145	C₄×C₁₁₃	#427	1	4	yes	—
146	C₄×C₁₁₃	C₄×#41	1	1	yes	characteristic
147	C₄×C₁₁₃	#518	1	1	yes	characteristic
148	C₄×C₁₁₃	#554	1	2	yes	—
149	#22708	C₁₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
150	C₂×C₁₁₃	—	4	4	no	—
151	C₂×C₁₁₃	#2349	1	1	yes	characteristic
152	C₂×C₁₁₃	C₂×#427	1	1	yes	characteristic
153	C₂×C₁₁₃	#2912	1	1	yes	characteristic
154	C₂×C₁₁₃	#3303	1	2	yes	—
155	C₂×C₁₁₃	#3304	1	2	yes	—
156	C₂×C₈×D₄	C₂×C₁₁₃	1	1	yes	characteristic
157	#2354	C₂×C₁₁₃	1	2	yes	—
158	#1948	C₂×C₁₁₃	1	2	yes	complemented
159	#1854	C₂×C₁₁₃	1	4	yes	complemented
160	#1834	C₂×C₁₁₃	1	1	yes	characteristic, complemented
161	#1807	C₂×C₁₁₃	1	2	yes	complemented
162	C₄×#107	C₂×C₁₁₃	1	1	yes	characteristic, complemented
163	#1735	C₂×C₁₁₃	1	2	yes	complemented
164	C₁₁₃	#22708	1	1	yes	characteristic, direct product factor, Sylow
165	C₂³×C₈	C₂²×C₁₁₃	1	2	yes	—
166	C₂×C₄×D₄	C₂²×C₁₁₃	1	1	yes	characteristic
167	C₂×#117	C₂²×C₁₁₃	1	4	yes	—
168	C₈×D₄	C₄×C₁₁₃	1	8	yes	complemented
169	C₂×#107	C₂²×C₁₁₃	1	2	yes	—
170	C₂×#106	C₂²×C₁₁₃	1	4	yes	—
171	C₂×C₄⋊C₈	C₂²×C₁₁₃	1	1	yes	characteristic
172	C₂×C₂²⋊C₈	C₂²×C₁₁₃	1	2	yes	—
173	C₂×C₄×C₈	C₂²×C₁₁₃	1	1	yes	characteristic
174	#390	—	2	2	no	—
175	#388	—	2	4	no	—
176	#384	C₂²×C₁₁₃	1	2	yes	—
177	#380	C₂²×C₁₁₃	1	4	yes	—
178	C₄×C₄⋊C₄	—	2	2	no	—
179	C₄×C₄⋊C₄	C₂²×C₁₁₃	1	2	yes	—
180	#340	C₂²×C₁₁₃	1	1	yes	characteristic
181	#340	C₂²×C₁₁₃	1	2	yes	—
182	#340	C₂²×C₁₁₃	1	2	yes	—
183	#340	C₂²×C₁₁₃	1	4	yes	—
184	#340	C₂²×C₁₁₃	1	1	yes	characteristic
185	C₂²×D₄	D₄×C₁₁₃	1	1	yes	characteristic
186	C₂³×C₄	C₂³×C₁₁₃	1	2	yes	—
187	C₂²×C₈	—	4	4	no	—
188	C₂²×C₈	C₂×C₄×C₁₁₃	1	8	yes	—
189	C₂²×C₈	C₂³×C₁₁₃	1	1	yes	characteristic
190	C₂²×C₈	C₂³×C₁₁₃	1	2	yes	—
191	C₂²×C₈	C₂³×C₁₁₃	1	1	yes	characteristic
192	C₄×D₄	C₂×C₄×C₁₁₃	1	4	yes	—
193	C₄×D₄	Q₈×C₁₁₃	1	4	yes	—
194	#117	—	2	8	no	—
195	#117	—	4	4	no	—
196	C₂×C₄⋊C₄	—	2	2	no	—
197	C₂×C₄⋊C₄	—	2	2	no	—
198	C₂×C₄⋊C₄	D₄×C₁₁₃	1	1	yes	characteristic
199	C₂×C₄⋊C₄	C₂³×C₁₁₃	1	4	yes	—
200	C₂×C₄⋊C₄	C₂³×C₁₁₃	1	2	yes	—
201	C₂×C₄⋊C₄	C₂³×C₁₁₃	1	2	yes	—
202	C₂×C₂²⋊C₄	—	2	4	no	—
203	C₂×C₂²⋊C₄	—	2	4	no	—
204	C₂×C₂²⋊C₄	D₄×C₁₁₃	1	2	yes	—
205	C₂×C₄²	—	2	4	no	—
206	C₂×C₄²	—	2	2	no	—
207	C₂×C₄²	C₂³×C₁₁₃	1	1	yes	characteristic
208	#107	—	2	4	no	—
209	#106	—	4	4	no	—
210	C₄⋊C₈	C₂×C₄×C₁₁₃	1	4	yes	—
211	C₂²⋊C₈	C₂×C₄×C₁₁₃	1	8	yes	—
212	C₄×C₈	C₂×C₄×C₁₁₃	1	4	yes	—
213	#95	—	2	4	no	—
214	#95	—	2	2	no	—
215	C₂⁴	C₂×D₄×C₁₁₃	1	2	yes	—
216	C₂×D₄	—	2	2	no	—
217	C₂×D₄	—	2	2	no	—
218	C₂×D₄	C₄⋊C₄×C₁₁₃	1	4	yes	—
219	C₂²×C₄	—	2	4	no	—
220	C₂²×C₄	—	2	4	no	—
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	4	no	—
224	C₂²×C₄	C₂²×C₄×C₁₁₃	1	4	yes	—
225	C₂²×C₄	C₂×D₄×C₁₁₃	1	2	yes	—
226	C₂²×C₄	C₂×D₄×C₁₁₃	1	1	yes	characteristic
227	C₂²×C₄	C₂×D₄×C₁₁₃	1	1	yes	characteristic
228	C₂²×C₄	C₂×Q₈×C₁₁₃	1	4	yes	—
229	C₂²×C₄	C₂⁴×C₁₁₃	1	1	yes	characteristic
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	4	8	no	—
234	C₂×C₈	C₂²×C₄×C₁₁₃	1	2	yes	—
235	C₂×C₈	C₂²×C₄×C₁₁₃	1	2	yes	—
236	C₂×C₈	C₂²×C₄×C₁₁₃	1	4	yes	—
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	—	2	2	no	—
239	C₄⋊C₄	—	2	4	no	—
240	C₄⋊C₄	—	2	2	no	—
241	C₄⋊C₄	—	2	4	no	—
242	C₄⋊C₄	—	4	4	no	—
243	C₄⋊C₄	—	8	2	no	—
244	C₄⋊C₄	C₄⋊C₄×C₁₁₃	1	4	yes	—
245	C₄⋊C₄	#41×C₁₁₃	1	4	yes	—
246	C₂²⋊C₄	—	4	8	no	—
247	C₂²⋊C₄	—	8	4	no	—
248	C₂²⋊C₄	C₄⋊C₄×C₁₁₃	1	8	yes	—
249	C₄²	—	4	8	no	—
250	C₄²	—	4	4	no	—
251	C₄²	C₂²×C₄×C₁₁₃	1	2	yes	—
252	C₄²	C₂×Q₈×C₁₁₃	1	2	yes	—
253	C₂³	—	2	4	no	—
254	C₂³	—	2	2	no	—
255	C₂³	—	2	2	no	—
256	C₂³	—	2	4	no	—
257	C₂³	C₂×C₄⋊C₄×C₁₁₃	1	4	yes	—
258	C₂³	C₂²×D₄×C₁₁₃	1	1	yes	characteristic
259	D₄	—	2	8	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	2	8	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	4	4	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	4	no	—
274	C₂×C₄	C₂×C₄⋊C₄×C₁₁₃	1	2	yes	—
275	C₂×C₄	C₂×C₄⋊C₄×C₁₁₃	1	4	yes	—
276	C₂×C₄	C₂×C₄⋊C₄×C₁₁₃	1	2	yes	—
277	C₂×C₄	#135×C₁₁₃	1	2	yes	—
278	C₂×C₄	#135×C₁₁₃	1	2	yes	—
279	C₂×C₄	C₂³×C₄×C₁₁₃	1	1	yes	characteristic
280	C₂×C₄	C₂²×Q₈×C₁₁₃	1	1	yes	characteristic
281	C₂×C₄	C₂×#41×C₁₁₃	1	2	yes	—
282	C₂×C₄	H(5,𝔽₂)×C₁₁₃	1	2	yes	—
283	C₈	—	2	2	no	—
284	C₈	—	2	2	no	—
285	C₈	—	2	4	no	—
286	C₂²	—	2	4	no	—
287	C₂²	—	2	4	no	—
288	C₂²	—	2	4	no	—
289	C₂²	—	4	4	no	—
290	C₂²	—	4	4	no	—
291	C₂²	C₂²×C₄⋊C₄×C₁₁₃	1	1	yes	characteristic
292	C₂²	#546×C₁₁₃	1	2	yes	—
293	C₂²	C₂×#135×C₁₁₃	1	1	yes	characteristic
294	C₂²	C₂×#135×C₁₁₃	1	1	yes	characteristic
295	C₂²	#577×C₁₁₃	1	2	yes	—
296	C₄	—	2	4	no	—
297	C₄	—	2	2	no	—
298	C₄	—	4	4	no	—
299	C₄	—	8	2	no	—
300	C₄	—	8	4	no	—
301	C₄	#427×C₁₁₃	1	4	yes	—
302	C₄	C₄×#41×C₁₁₃	1	1	yes	characteristic
303	C₄	#518×C₁₁₃	1	1	yes	characteristic
304	C₄	#554×C₁₁₃	1	2	yes	—
305	C₂	—	4	4	no	—
306	C₂	C₁₁₃×#2349	1	1	yes	characteristic
307	C₂	C₂×#427×C₁₁₃	1	1	yes	characteristic
308	C₂	C₁₁₃×#2912	1	1	yes	characteristic
309	C₂	C₁₁₃×#3303	1	2	yes	—
310	C₂	C₁₁₃×#3304	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₁₁₃×#1948	C₂	semidirect
4. C₁₁₃×#1854	C₂	semidirect
5. C₁₁₃×#1834	C₂	semidirect
6. C₁₁₃×#1807	C₂	semidirect
7. C₄×#107×C₁₁₃	C₂	semidirect
8. C₁₁₃×#1735	C₂	semidirect
12. C₈×D₄×C₁₁₃	C₄	semidirect
149. #22708	C₁₁₃	direct (N × H)
158. #1948	C₂×C₁₁₃	semidirect
159. #1854	C₂×C₁₁₃	semidirect
160. #1834	C₂×C₁₁₃	semidirect
161. #1807	C₂×C₁₁₃	semidirect
162. C₄×#107	C₂×C₁₁₃	semidirect
163. #1735	C₂×C₁₁₃	semidirect
164. C₁₁₃	#22708	direct (N × H)
168. C₈×D₄	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×C₈×D₄×C₁₁₃	2
C₁₁₃×#2354	2
C₁₁₃×#1948	2
C₁₁₃×#1854	2
C₁₁₃×#1834	2
C₁₁₃×#1807	2
C₄×#107×C₁₁₃	2
C₁₁₃×#1735	2
#22708	113

Subgroups — C113×#22708 #43290728

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁₃×#22708 #43290728