Subgroups — C₁₁₃×#17666 #43285686

Order 28928.

1306 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#124×C₁₁₃	C₂	1	1	yes	maximal, characteristic
2	C₁₁₃×#2920	C₂	1	2	yes	maximal, complemented
3	C₁₁₃×#1533	C₂	1	4	yes	maximal, complemented
4	C₁₁₃×#1527	C₂	1	8	yes	maximal, complemented
5	C₂³×Q₈×C₁₁₃	C₄	1	1	yes	characteristic
6	C₂³×D₄×C₁₁₃	C₄	1	1	yes	characteristic
7	C₂×#124×C₁₁₃	—	2	2	no	—
8	C₂×#124×C₁₁₃	—	2	2	no	—
9	C₂×#124×C₁₁₃	—	2	2	no	—
10	C₂×#124×C₁₁₃	C₄	1	4	yes	—
11	C₂×#124×C₁₁₃	C₄	1	2	yes	—
12	C₂×#124×C₁₁₃	C₂²	1	4	yes	—
13	C₂×#124×C₁₁₃	C₂²	1	2	yes	—
14	C₂²×C₂²⋊C₄×C₁₁₃	—	2	2	no	—
15	C₂²×C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
16	#432×C₁₁₃	C₂²	1	4	yes	complemented
17	#432×C₁₁₃	C₂²	1	4	yes	complemented
18	#431×C₁₁₃	—	2	2	no	—
19	#431×C₁₁₃	—	2	2	no	—
20	#423×C₁₁₃	—	2	2	no	—
21	#407×C₁₁₃	—	2	2	no	—
22	C₄×M₁₆×C₁₁₃	C₂²	1	4	yes	complemented
23	#329×C₁₁₃	C₂²	1	16	yes	complemented
24	C₂⁵×C₁₁₃	—	2	1	no	—
25	C₂²×Q₈×C₁₁₃	—	2	2	no	—
26	C₂²×Q₈×C₁₁₃	—	2	1	no	—
27	C₂²×Q₈×C₁₁₃	—	4	4	no	—
28	C₂²×Q₈×C₁₁₃	C₂×C₄	1	4	yes	—
29	C₂²×Q₈×C₁₁₃	C₂×C₄	1	2	yes	—
30	C₂²×D₄×C₁₁₃	—	2	2	no	—
31	C₂²×D₄×C₁₁₃	—	2	1	no	—
32	C₂²×D₄×C₁₁₃	—	4	4	no	—
33	C₂²×D₄×C₁₁₃	C₂×C₄	1	2	yes	—
34	C₂²×D₄×C₁₁₃	C₂×C₄	1	4	yes	—
35	C₂³×C₄×C₁₁₃	—	2	1	no	—
36	C₂³×C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
37	C₂³×C₄×C₁₁₃	D₄	1	2	yes	—
38	C₂×M₁₆×C₁₁₃	—	2	4	no	—
39	#124×C₁₁₃	—	2	2	no	—
40	#124×C₁₁₃	—	2	2	no	—
41	#124×C₁₁₃	—	2	4	no	—
42	#124×C₁₁₃	—	2	4	no	—
43	#124×C₁₁₃	—	2	4	no	—
44	#124×C₁₁₃	—	2	4	no	—
45	#124×C₁₁₃	—	2	4	no	—
46	#124×C₁₁₃	C₂×C₄	1	8	yes	—
47	#124×C₁₁₃	C₂×C₄	1	4	yes	—
48	#124×C₁₁₃	C₂³	1	4	yes	—
49	C₂×C₂²⋊C₄×C₁₁₃	—	2	4	no	—
50	C₂×C₂²⋊C₄×C₁₁₃	—	2	4	no	—
51	C₂×C₂²⋊C₄×C₁₁₃	—	2	4	no	—
52	C₂×C₂²⋊C₄×C₁₁₃	—	2	4	no	—
53	C₂×C₂²⋊C₄×C₁₁₃	—	2	4	no	—
54	C₂×C₂²⋊C₄×C₁₁₃	—	2	4	no	—
55	C₂×C₄²×C₁₁₃	C₂×C₄	1	2	yes	—
56	C₂×C₄²×C₁₁₃	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄²×C₁₁₃	D₄	1	2	yes	—
58	C₂×C₄²×C₁₁₃	D₄	1	2	yes	—
59	C₂×C₄²×C₁₁₃	D₄	1	2	yes	—
60	C₂×C₄²×C₁₁₃	C₂³	1	2	yes	—
61	C₂×C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
62	C₄⋊C₈×C₁₁₃	—	2	4	no	—
63	C₄⋊C₈×C₁₁₃	—	2	4	no	—
64	C₂²⋊C₈×C₁₁₃	—	2	4	no	—
65	C₂²⋊C₈×C₁₁₃	—	2	4	no	—
66	#97×C₁₁₃	C₂³	1	8	yes	complemented
67	C₄×C₈×C₁₁₃	—	2	4	no	—
68	C₂⁴×C₁₁₃	—	2	4	no	—
69	C₂⁴×C₁₁₃	—	2	2	no	—
70	C₂⁴×C₁₁₃	—	4	4	no	—
71	C₂⁴×C₁₁₃	—	4	8	no	—
72	C₂⁴×C₁₁₃	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×Q₈×C₁₁₃	—	2	2	no	—
74	C₂×Q₈×C₁₁₃	—	2	4	no	—
75	C₂×Q₈×C₁₁₃	—	4	8	no	—
76	C₂×Q₈×C₁₁₃	—	4	8	no	—
77	C₂×Q₈×C₁₁₃	—	4	4	no	—
78	C₂×Q₈×C₁₁₃	—	4	4	no	—
79	C₂×Q₈×C₁₁₃	C₂²×C₄	1	4	yes	—
80	C₂×D₄×C₁₁₃	—	2	2	no	—
81	C₂×D₄×C₁₁₃	—	2	4	no	—
82	C₂×D₄×C₁₁₃	—	4	4	no	—
83	C₂×D₄×C₁₁₃	—	4	8	no	—
84	C₂×D₄×C₁₁₃	—	4	8	no	—
85	C₂×D₄×C₁₁₃	—	4	4	no	—
86	C₂×D₄×C₁₁₃	C₂²×C₄	1	4	yes	—
87	C₂²×C₄×C₁₁₃	—	2	4	no	—
88	C₂²×C₄×C₁₁₃	—	2	4	no	—
89	C₂²×C₄×C₁₁₃	—	2	2	no	—
90	C₂²×C₄×C₁₁₃	—	4	4	no	—
91	C₂²×C₄×C₁₁₃	—	4	4	no	—
92	C₂²×C₄×C₁₁₃	C₂²⋊C₄	1	2	yes	—
93	C₂²×C₄×C₁₁₃	C₂²⋊C₄	1	2	yes	—
94	C₂²×C₄×C₁₁₃	C₂²⋊C₄	1	2	yes	—
95	C₂²×C₄×C₁₁₃	C₂²⋊C₄	1	2	yes	—
96	C₂²×C₄×C₁₁₃	C₂²⋊C₄	1	1	yes	characteristic
97	C₂²×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
98	C₂²×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
99	C₂²×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
100	C₂²×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
101	C₂²×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
102	M₁₆×C₁₁₃	—	4	8	no	—
103	C₂×C₈×C₁₁₃	—	2	8	no	—
104	C₂²⋊C₄×C₁₁₃	—	2	8	no	—
105	C₂²⋊C₄×C₁₁₃	—	2	8	no	—
106	C₂²⋊C₄×C₁₁₃	—	2	8	no	—
107	C₂²⋊C₄×C₁₁₃	—	2	8	no	—
108	C₄²×C₁₁₃	C₂²⋊C₄	1	1	yes	characteristic
109	C₄²×C₁₁₃	C₂²⋊C₄	1	1	yes	characteristic
110	C₄²×C₁₁₃	C₂²⋊C₄	1	2	yes	—
111	C₄²×C₁₁₃	C₂²⋊C₄	1	2	yes	—
112	C₄²×C₁₁₃	C₂²×C₄	1	2	yes	—
113	C₄²×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
114	C₄²×C₁₁₃	C₂×D₄	1	2	yes	—
115	C₄²×C₁₁₃	C₂×D₄	1	2	yes	—
116	C₄²×C₁₁₃	C₂×D₄	1	2	yes	—
117	C₄²×C₁₁₃	C₂⁴	1	1	yes	characteristic
118	C₂³×C₁₁₃	—	2	2	no	—
119	C₂³×C₁₁₃	—	2	4	no	—
120	C₂³×C₁₁₃	—	2	4	no	—
121	C₂³×C₁₁₃	—	4	8	no	—
122	C₂³×C₁₁₃	—	4	8	no	—
123	C₂³×C₁₁₃	—	4	8	no	—
124	C₂³×C₁₁₃	—	4	4	no	—
125	C₂³×C₁₁₃	—	4	8	no	—
126	C₂³×C₁₁₃	—	8	16	no	—
127	C₂³×C₁₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	C₂³×C₁₁₃	C₂×C₂²⋊C₄	1	2	yes	—
129	Q₈×C₁₁₃	—	4	4	no	—
130	Q₈×C₁₁₃	—	4	4	no	—
131	Q₈×C₁₁₃	—	4	8	no	—
132	D₄×C₁₁₃	—	4	4	no	—
133	D₄×C₁₁₃	—	4	4	no	—
134	D₄×C₁₁₃	—	4	8	no	—
135	C₂×C₄×C₁₁₃	—	2	2	no	—
136	C₂×C₄×C₁₁₃	—	2	4	no	—
137	C₂×C₄×C₁₁₃	—	2	4	no	—
138	C₂×C₄×C₁₁₃	—	2	4	no	—
139	C₂×C₄×C₁₁₃	—	2	4	no	—
140	C₂×C₄×C₁₁₃	—	2	4	no	—
141	C₂×C₄×C₁₁₃	—	4	2	no	—
142	C₂×C₄×C₁₁₃	—	4	2	no	—
143	C₂×C₄×C₁₁₃	—	4	2	no	—
144	C₂×C₄×C₁₁₃	—	4	2	no	—
145	C₂×C₄×C₁₁₃	—	4	8	no	—
146	C₂×C₄×C₁₁₃	—	4	4	no	—
147	C₂×C₄×C₁₁₃	#100	1	4	yes	—
148	C₂×C₄×C₁₁₃	C₂×C₂²⋊C₄	1	2	yes	—
149	C₂×C₄×C₁₁₃	C₂×C₂²⋊C₄	1	2	yes	—
150	C₂×C₄×C₁₁₃	C₂×C₂²⋊C₄	1	2	yes	—
151	C₂×C₄×C₁₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
152	C₂×C₄×C₁₁₃	C₂×C₂²⋊C₄	1	2	yes	—
153	C₂×C₄×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
154	C₂×C₄×C₁₁₃	C₂²×D₄	1	2	yes	—
155	C₈×C₁₁₃	—	4	8	no	—
156	C₂²×C₁₁₃	—	2	8	no	—
157	C₂²×C₁₁₃	—	2	2	no	—
158	C₂²×C₁₁₃	—	2	2	no	—
159	C₂²×C₁₁₃	—	2	2	no	—
160	C₂²×C₁₁₃	—	4	4	no	—
161	C₂²×C₁₁₃	—	4	8	no	—
162	C₂²×C₁₁₃	—	8	8	no	—
163	C₂²×C₁₁₃	—	8	16	no	—
164	C₂²×C₁₁₃	C₂×#100	1	2	yes	—
165	C₂²×C₁₁₃	#421	1	4	yes	—
166	C₂²×C₁₁₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
167	C₄×C₁₁₃	—	2	1	no	—
168	C₄×C₁₁₃	—	2	1	no	—
169	C₄×C₁₁₃	—	2	2	no	—
170	C₄×C₁₁₃	—	2	2	no	—
171	C₄×C₁₁₃	—	2	2	no	—
172	C₄×C₁₁₃	—	2	2	no	—
173	C₄×C₁₁₃	—	4	4	no	—
174	C₄×C₁₁₃	C₂×#100	1	4	yes	—
175	#17666	C₁₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
176	C₂×C₁₁₃	—	2	2	no	—
177	C₂×C₁₁₃	—	2	4	no	—
178	C₂×C₁₁₃	—	8	4	no	—
179	C₂×C₁₁₃	C₂²×#100	1	1	yes	characteristic
180	C₂×C₁₁₃	C₂×#421	1	2	yes	—
181	C₂²×#124	C₂×C₁₁₃	1	1	yes	characteristic
182	#2920	C₂×C₁₁₃	1	2	yes	complemented
183	#1533	C₂×C₁₁₃	1	4	yes	complemented
184	#1527	C₂×C₁₁₃	1	8	yes	complemented
185	C₁₁₃	#17666	1	1	yes	characteristic, direct product factor, Sylow
186	C₂³×Q₈	C₄×C₁₁₃	1	1	yes	characteristic
187	C₂³×D₄	C₄×C₁₁₃	1	1	yes	characteristic
188	C₂×#124	—	2	2	no	—
189	C₂×#124	—	2	2	no	—
190	C₂×#124	—	2	2	no	—
191	C₂×#124	C₄×C₁₁₃	1	4	yes	—
192	C₂×#124	C₄×C₁₁₃	1	2	yes	—
193	C₂×#124	C₂²×C₁₁₃	1	4	yes	—
194	C₂×#124	C₂²×C₁₁₃	1	2	yes	—
195	C₂²×C₂²⋊C₄	—	2	2	no	—
196	C₂²×C₄²	C₂²×C₁₁₃	1	1	yes	characteristic
197	#432	C₂²×C₁₁₃	1	4	yes	complemented
198	#432	C₂²×C₁₁₃	1	4	yes	complemented
199	#431	—	2	2	no	—
200	#431	—	2	2	no	—
201	#423	—	2	2	no	—
202	#407	—	2	2	no	—
203	C₄×M₁₆	C₂²×C₁₁₃	1	4	yes	complemented
204	#329	C₂²×C₁₁₃	1	16	yes	complemented
205	C₂⁵	—	2	1	no	—
206	C₂²×Q₈	—	2	2	no	—
207	C₂²×Q₈	—	2	1	no	—
208	C₂²×Q₈	—	4	4	no	—
209	C₂²×Q₈	C₂×C₄×C₁₁₃	1	4	yes	—
210	C₂²×Q₈	C₂×C₄×C₁₁₃	1	2	yes	—
211	C₂²×D₄	—	2	2	no	—
212	C₂²×D₄	—	2	1	no	—
213	C₂²×D₄	—	4	4	no	—
214	C₂²×D₄	C₂×C₄×C₁₁₃	1	2	yes	—
215	C₂²×D₄	C₂×C₄×C₁₁₃	1	4	yes	—
216	C₂³×C₄	—	2	1	no	—
217	C₂³×C₄	C₂×C₄×C₁₁₃	1	1	yes	characteristic
218	C₂³×C₄	D₄×C₁₁₃	1	2	yes	—
219	C₂×M₁₆	—	2	4	no	—
220	#124	—	2	2	no	—
221	#124	—	2	2	no	—
222	#124	—	2	4	no	—
223	#124	—	2	4	no	—
224	#124	—	2	4	no	—
225	#124	—	2	4	no	—
226	#124	—	2	4	no	—
227	#124	C₂×C₄×C₁₁₃	1	8	yes	—
228	#124	C₂×C₄×C₁₁₃	1	4	yes	—
229	#124	C₂³×C₁₁₃	1	4	yes	—
230	C₂×C₂²⋊C₄	—	2	4	no	—
231	C₂×C₂²⋊C₄	—	2	4	no	—
232	C₂×C₂²⋊C₄	—	2	4	no	—
233	C₂×C₂²⋊C₄	—	2	4	no	—
234	C₂×C₂²⋊C₄	—	2	4	no	—
235	C₂×C₂²⋊C₄	—	2	4	no	—
236	C₂×C₄²	C₂×C₄×C₁₁₃	1	2	yes	—
237	C₂×C₄²	C₂×C₄×C₁₁₃	1	1	yes	characteristic
238	C₂×C₄²	D₄×C₁₁₃	1	2	yes	—
239	C₂×C₄²	D₄×C₁₁₃	1	2	yes	—
240	C₂×C₄²	D₄×C₁₁₃	1	2	yes	—
241	C₂×C₄²	C₂³×C₁₁₃	1	2	yes	—
242	C₂×C₄²	C₂³×C₁₁₃	1	1	yes	characteristic
243	C₄⋊C₈	—	2	4	no	—
244	C₄⋊C₈	—	2	4	no	—
245	C₂²⋊C₈	—	2	4	no	—
246	C₂²⋊C₈	—	2	4	no	—
247	#97	C₂³×C₁₁₃	1	8	yes	complemented
248	C₄×C₈	—	2	4	no	—
249	C₂⁴	—	2	4	no	—
250	C₂⁴	—	2	2	no	—
251	C₂⁴	—	4	4	no	—
252	C₂⁴	—	4	8	no	—
253	C₂⁴	C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
254	C₂×Q₈	—	2	2	no	—
255	C₂×Q₈	—	2	4	no	—
256	C₂×Q₈	—	4	8	no	—
257	C₂×Q₈	—	4	8	no	—
258	C₂×Q₈	—	4	4	no	—
259	C₂×Q₈	—	4	4	no	—
260	C₂×Q₈	C₂²×C₄×C₁₁₃	1	4	yes	—
261	C₂×D₄	—	2	2	no	—
262	C₂×D₄	—	2	4	no	—
263	C₂×D₄	—	4	4	no	—
264	C₂×D₄	—	4	8	no	—
265	C₂×D₄	—	4	8	no	—
266	C₂×D₄	—	4	4	no	—
267	C₂×D₄	C₂²×C₄×C₁₁₃	1	4	yes	—
268	C₂²×C₄	—	2	4	no	—
269	C₂²×C₄	—	2	4	no	—
270	C₂²×C₄	—	2	2	no	—
271	C₂²×C₄	—	4	4	no	—
272	C₂²×C₄	—	4	4	no	—
273	C₂²×C₄	C₂²⋊C₄×C₁₁₃	1	2	yes	—
274	C₂²×C₄	C₂²⋊C₄×C₁₁₃	1	2	yes	—
275	C₂²×C₄	C₂²⋊C₄×C₁₁₃	1	2	yes	—
276	C₂²×C₄	C₂²⋊C₄×C₁₁₃	1	2	yes	—
277	C₂²×C₄	C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
278	C₂²×C₄	C₂²×C₄×C₁₁₃	1	1	yes	characteristic
279	C₂²×C₄	C₂²×C₄×C₁₁₃	1	2	yes	—
280	C₂²×C₄	C₂×D₄×C₁₁₃	1	2	yes	—
281	C₂²×C₄	C₂×D₄×C₁₁₃	1	2	yes	—
282	C₂²×C₄	C₂×D₄×C₁₁₃	1	2	yes	—
283	M₁₆	—	4	8	no	—
284	C₂×C₈	—	2	8	no	—
285	C₂²⋊C₄	—	2	8	no	—
286	C₂²⋊C₄	—	2	8	no	—
287	C₂²⋊C₄	—	2	8	no	—
288	C₂²⋊C₄	—	2	8	no	—
289	C₄²	C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
290	C₄²	C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
291	C₄²	C₂²⋊C₄×C₁₁₃	1	2	yes	—
292	C₄²	C₂²⋊C₄×C₁₁₃	1	2	yes	—
293	C₄²	C₂²×C₄×C₁₁₃	1	2	yes	—
294	C₄²	C₂²×C₄×C₁₁₃	1	1	yes	characteristic
295	C₄²	C₂×D₄×C₁₁₃	1	2	yes	—
296	C₄²	C₂×D₄×C₁₁₃	1	2	yes	—
297	C₄²	C₂×D₄×C₁₁₃	1	2	yes	—
298	C₄²	C₂⁴×C₁₁₃	1	1	yes	characteristic
299	C₂³	—	2	2	no	—
300	C₂³	—	2	4	no	—
301	C₂³	—	2	4	no	—
302	C₂³	—	4	8	no	—
303	C₂³	—	4	8	no	—
304	C₂³	—	4	8	no	—
305	C₂³	—	4	4	no	—
306	C₂³	—	4	8	no	—
307	C₂³	—	8	16	no	—
308	C₂³	C₂×C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
309	C₂³	C₂×C₂²⋊C₄×C₁₁₃	1	2	yes	—
310	Q₈	—	4	4	no	—
311	Q₈	—	4	4	no	—
312	Q₈	—	4	8	no	—
313	D₄	—	4	4	no	—
314	D₄	—	4	4	no	—
315	D₄	—	4	8	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	4	no	—
318	C₂×C₄	—	2	4	no	—
319	C₂×C₄	—	2	4	no	—
320	C₂×C₄	—	2	4	no	—
321	C₂×C₄	—	2	4	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	4	8	no	—
327	C₂×C₄	—	4	4	no	—
328	C₂×C₄	#100×C₁₁₃	1	4	yes	—
329	C₂×C₄	C₂×C₂²⋊C₄×C₁₁₃	1	2	yes	—
330	C₂×C₄	C₂×C₂²⋊C₄×C₁₁₃	1	2	yes	—
331	C₂×C₄	C₂×C₂²⋊C₄×C₁₁₃	1	2	yes	—
332	C₂×C₄	C₂×C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
333	C₂×C₄	C₂×C₂²⋊C₄×C₁₁₃	1	2	yes	—
334	C₂×C₄	C₂³×C₄×C₁₁₃	1	1	yes	characteristic
335	C₂×C₄	C₂²×D₄×C₁₁₃	1	2	yes	—
336	C₈	—	4	8	no	—
337	C₂²	—	2	8	no	—
338	C₂²	—	2	2	no	—
339	C₂²	—	2	2	no	—
340	C₂²	—	2	2	no	—
341	C₂²	—	4	4	no	—
342	C₂²	—	4	8	no	—
343	C₂²	—	8	8	no	—
344	C₂²	—	8	16	no	—
345	C₂²	C₂×#100×C₁₁₃	1	2	yes	—
346	C₂²	#421×C₁₁₃	1	4	yes	—
347	C₂²	C₂²×C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
348	C₄	—	2	1	no	—
349	C₄	—	2	1	no	—
350	C₄	—	2	2	no	—
351	C₄	—	2	2	no	—
352	C₄	—	2	2	no	—
353	C₄	—	2	2	no	—
354	C₄	—	4	4	no	—
355	C₄	C₂×#100×C₁₁₃	1	4	yes	—
356	C₂	—	2	2	no	—
357	C₂	—	2	4	no	—
358	C₂	—	8	4	no	—
359	C₂	C₂²×#100×C₁₁₃	1	1	yes	characteristic
360	C₂	C₂×#421×C₁₁₃	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₁₃	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₁₁₃
Frattini, cosocle	C₄²	C₂⁴×C₁₁₃
prime radical	C₂³×D₄×C₁₁₃	C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₁₃ ◁ C₂²×C₄²×C₁₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₁₃×#2920	C₂	semidirect
3. C₁₁₃×#1533	C₂	semidirect
4. C₁₁₃×#1527	C₂	semidirect
16. #432×C₁₁₃	C₂²	semidirect
17. #432×C₁₁₃	C₂²	semidirect
22. C₄×M₁₆×C₁₁₃	C₂²	semidirect
23. #329×C₁₁₃	C₂²	semidirect
66. #97×C₁₁₃	C₂³	semidirect
175. #17666	C₁₁₃	direct (N × H)
182. #2920	C₂×C₁₁₃	semidirect
183. #1533	C₂×C₁₁₃	semidirect
184. #1527	C₂×C₁₁₃	semidirect
185. C₁₁₃	#17666	direct (N × H)
197. #432	C₂²×C₁₁₃	semidirect
198. #432	C₂²×C₁₁₃	semidirect
203. C₄×M₁₆	C₂²×C₁₁₃	semidirect
204. #329	C₂²×C₁₁₃	semidirect
247. #97	C₂³×C₁₁₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#124×C₁₁₃	2
C₁₁₃×#2920	2
C₁₁₃×#1533	2
C₁₁₃×#1527	2
#17666	113

Subgroups — C113×#17666 #43285686

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁₃×#17666 #43285686