Subgroups — C₁₁₃×#22187 #43290207

Order 28928.

1038 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#469×C₁₁₃	C₂	1	1	yes	maximal, characteristic
2	C₄×#121×C₁₁₃	C₂	1	2	yes	maximal, complemented
3	C₁₁₃×#1926	C₂	1	2	yes	maximal, complemented
4	C₁₁₃×#1919	C₂	1	2	yes	maximal, complemented
5	C₁₁₃×#1895	C₂	1	2	yes	maximal, complemented
6	C₁₁₃×#1879	C₂	1	2	yes	maximal, complemented
7	C₁₁₃×#1862	C₂	1	2	yes	maximal, complemented
8	C₁₁₃×#1809	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₁₃×#1728	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×D₈×C₁₁₃	C₂²	1	1	yes	characteristic
11	C₂²×M₁₆×C₁₁₃	C₂²	1	1	yes	characteristic
12	C₂²×M₁₆×C₁₁₃	C₂²	1	1	yes	characteristic
13	C₂×#121×C₁₁₃	C₂²	1	2	yes	—
14	C₂×C₄×D₄×C₁₁₃	—	2	2	no	—
15	C₂×C₄×D₄×C₁₁₃	C₂²	1	2	yes	—
16	C₂²×C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
17	#469×C₁₁₃	C₄	1	8	yes	complemented
18	C₂×#107×C₁₁₃	C₂²	1	1	yes	characteristic
19	C₂×#106×C₁₁₃	C₂²	1	1	yes	characteristic
20	C₂×#106×C₁₁₃	C₂²	1	1	yes	characteristic
21	C₂×#102×C₁₁₃	C₂²	1	2	yes	—
22	C₂×#102×C₁₁₃	C₂²	1	2	yes	—
23	C₂×#102×C₁₁₃	C₂²	1	2	yes	—
24	#390×C₁₁₃	C₂²	1	2	yes	—
25	#389×C₁₁₃	C₂²	1	2	yes	—
26	#388×C₁₁₃	—	2	2	no	—
27	#386×C₁₁₃	—	2	2	no	—
28	#380×C₁₁₃	C₂²	1	2	yes	—
29	C₄×C₄⋊C₄×C₁₁₃	C₂²	1	2	yes	—
30	C₄×C₂²⋊C₄×C₁₁₃	—	2	2	no	—
31	#340×C₁₁₃	C₂²	1	2	yes	—
32	#340×C₁₁₃	C₂²	1	2	yes	—
33	#340×C₁₁₃	C₂²	1	2	yes	—
34	#336×C₁₁₃	C₂²	1	2	yes	—
35	#336×C₁₁₃	C₂²	1	2	yes	—
36	C₂²×D₄×C₁₁₃	—	2	2	no	—
37	C₂²×D₄×C₁₁₃	C₂³	1	2	yes	—
38	C₂³×C₄×C₁₁₃	—	2	2	no	—
39	C₂³×C₄×C₁₁₃	D₄	1	1	yes	characteristic
40	C₂³×C₄×C₁₁₃	D₄	1	1	yes	characteristic
41	C₂³×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
42	C₂×D₈×C₁₁₃	—	4	1	no	—
43	C₂×D₈×C₁₁₃	—	4	1	no	—
44	C₂×D₈×C₁₁₃	C₂×C₄	1	4	yes	complemented
45	C₂×M₁₆×C₁₁₃	—	2	2	no	—
46	C₂×M₁₆×C₁₁₃	—	4	1	no	—
47	C₂×M₁₆×C₁₁₃	—	4	1	no	—
48	C₂×M₁₆×C₁₁₃	—	4	1	no	—
49	C₂×M₁₆×C₁₁₃	—	4	1	no	—
50	C₂×M₁₆×C₁₁₃	C₂×C₄	1	4	yes	complemented
51	C₂²×C₈×C₁₁₃	C₂³	1	2	yes	—
52	C₂²×C₈×C₁₁₃	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₁₁₃	C₂³	1	1	yes	characteristic
54	#121×C₁₁₃	—	2	4	no	—
55	#121×C₁₁₃	C₂×C₄	1	8	yes	complemented
56	C₄×D₄×C₁₁₃	—	4	2	no	—
57	C₄×D₄×C₁₁₃	—	4	2	no	—
58	C₄×D₄×C₁₁₃	—	4	8	no	—
59	C₂×C₄⋊C₄×C₁₁₃	—	2	2	no	—
60	C₂×C₄⋊C₄×C₁₁₃	C₂³	1	2	yes	—
61	C₂×C₄⋊C₄×C₁₁₃	C₂³	1	2	yes	—
62	C₂×C₄⋊C₄×C₁₁₃	C₂³	1	2	yes	—
63	C₂×C₂²⋊C₄×C₁₁₃	—	2	2	no	—
64	C₂×C₂²⋊C₄×C₁₁₃	—	2	2	no	—
65	C₂×C₂²⋊C₄×C₁₁₃	—	2	2	no	—
66	C₂×C₂²⋊C₄×C₁₁₃	—	2	2	no	—
67	C₂×C₂²⋊C₄×C₁₁₃	—	2	2	no	—
68	C₂×C₄²×C₁₁₃	—	2	2	no	—
69	C₂×C₄²×C₁₁₃	—	4	1	no	—
70	C₂×C₄²×C₁₁₃	—	4	1	no	—
71	C₂×C₄²×C₁₁₃	D₄	1	2	yes	—
72	C₂×C₄²×C₁₁₃	C₂³	1	2	yes	—
73	#107×C₁₁₃	—	4	1	no	—
74	#106×C₁₁₃	—	4	1	no	—
75	#106×C₁₁₃	C₂×C₄	1	4	yes	complemented
76	#102×C₁₁₃	—	4	2	no	—
77	#102×C₁₁₃	—	4	2	no	—
78	#102×C₁₁₃	C₂×C₄	1	8	yes	complemented
79	#95×C₁₁₃	—	2	2	no	—
80	#95×C₁₁₃	—	2	2	no	—
81	#95×C₁₁₃	—	2	2	no	—
82	C₂⁴×C₁₁₃	—	2	2	no	—
83	C₂⁴×C₁₁₃	C₂×D₄	1	1	yes	characteristic
84	C₂×D₄×C₁₁₃	—	2	8	no	—
85	C₂×D₄×C₁₁₃	—	2	2	no	—
86	C₂×D₄×C₁₁₃	—	2	2	no	—
87	C₂×D₄×C₁₁₃	—	2	2	no	—
88	C₂×D₄×C₁₁₃	—	2	2	no	—
89	C₂×D₄×C₁₁₃	—	2	2	no	—
90	C₂×D₄×C₁₁₃	—	4	4	no	—
91	C₂×D₄×C₁₁₃	—	4	4	no	—
92	C₂×D₄×C₁₁₃	C₂²×C₄	1	4	yes	—
93	C₂²×C₄×C₁₁₃	—	2	1	no	—
94	C₂²×C₄×C₁₁₃	—	2	1	no	—
95	C₂²×C₄×C₁₁₃	—	2	1	no	—
96	C₂²×C₄×C₁₁₃	—	2	1	no	—
97	C₂²×C₄×C₁₁₃	—	2	2	no	—
98	C₂²×C₄×C₁₁₃	—	2	4	no	—
99	C₂²×C₄×C₁₁₃	—	2	2	no	—
100	C₂²×C₄×C₁₁₃	—	2	2	no	—
101	C₂²×C₄×C₁₁₃	—	4	1	no	—
102	C₂²×C₄×C₁₁₃	—	4	1	no	—
103	C₂²×C₄×C₁₁₃	—	4	2	no	—
104	C₂²×C₄×C₁₁₃	—	4	4	no	—
105	C₂²×C₄×C₁₁₃	—	4	4	no	—
106	C₂²×C₄×C₁₁₃	—	4	4	no	—
107	C₂²×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
108	C₂²×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
109	C₂²×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
111	C₂²×C₄×C₁₁₃	#41	1	2	yes	—
112	C₂²×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
113	D₈×C₁₁₃	—	4	4	no	—
114	M₁₆×C₁₁₃	—	4	4	no	—
115	M₁₆×C₁₁₃	—	4	4	no	—
116	C₂×C₈×C₁₁₃	—	2	1	no	—
117	C₂×C₈×C₁₁₃	—	2	1	no	—
118	C₂×C₈×C₁₁₃	—	2	1	no	—
119	C₂×C₈×C₁₁₃	—	2	1	no	—
120	C₂×C₈×C₁₁₃	—	2	2	no	—
121	C₂×C₈×C₁₁₃	—	2	2	no	—
122	C₂×C₈×C₁₁₃	—	2	2	no	—
123	C₂×C₈×C₁₁₃	C₂²×C₄	1	2	yes	—
124	C₂×C₈×C₁₁₃	C₂²×C₄	1	2	yes	—
125	C₄⋊C₄×C₁₁₃	—	2	2	no	—
126	C₄⋊C₄×C₁₁₃	—	2	2	no	—
127	C₄⋊C₄×C₁₁₃	—	2	2	no	—
128	C₄⋊C₄×C₁₁₃	—	2	2	no	—
129	C₄⋊C₄×C₁₁₃	—	2	2	no	—
130	C₄⋊C₄×C₁₁₃	—	4	4	no	—
131	C₄⋊C₄×C₁₁₃	C₂²×C₄	1	4	yes	—
132	C₂²⋊C₄×C₁₁₃	—	2	8	no	—
133	C₂²⋊C₄×C₁₁₃	—	4	4	no	—
134	C₂²⋊C₄×C₁₁₃	—	4	4	no	—
135	C₂²⋊C₄×C₁₁₃	—	4	4	no	—
136	C₂²⋊C₄×C₁₁₃	—	4	4	no	—
137	C₄²×C₁₁₃	—	2	2	no	—
138	C₄²×C₁₁₃	—	2	2	no	—
139	C₄²×C₁₁₃	—	4	2	no	—
140	C₄²×C₁₁₃	—	4	4	no	—
141	C₂³×C₁₁₃	—	2	1	no	—
142	C₂³×C₁₁₃	—	2	1	no	—
143	C₂³×C₁₁₃	—	2	2	no	—
144	C₂³×C₁₁₃	—	2	2	no	—
145	C₂³×C₁₁₃	—	2	4	no	—
146	C₂³×C₁₁₃	—	4	2	no	—
147	C₂³×C₁₁₃	—	4	2	no	—
148	C₂³×C₁₁₃	—	4	2	no	—
149	C₂³×C₁₁₃	—	4	2	no	—
150	C₂³×C₁₁₃	—	4	2	no	—
151	C₂³×C₁₁₃	—	4	2	no	—
152	C₂³×C₁₁₃	C₄×D₄	1	2	yes	—
153	C₂³×C₁₁₃	C₂²×D₄	1	1	yes	characteristic
154	D₄×C₁₁₃	—	2	4	no	—
155	D₄×C₁₁₃	—	2	4	no	—
156	D₄×C₁₁₃	—	4	16	no	—
157	D₄×C₁₁₃	—	4	2	no	—
158	D₄×C₁₁₃	—	4	2	no	—
159	C₂×C₄×C₁₁₃	—	2	2	no	—
160	C₂×C₄×C₁₁₃	—	2	1	no	—
161	C₂×C₄×C₁₁₃	—	2	1	no	—
162	C₂×C₄×C₁₁₃	—	2	2	no	—
163	C₂×C₄×C₁₁₃	—	2	2	no	—
164	C₂×C₄×C₁₁₃	—	2	2	no	—
165	C₂×C₄×C₁₁₃	—	2	2	no	—
166	C₂×C₄×C₁₁₃	—	2	2	no	—
167	C₂×C₄×C₁₁₃	—	2	4	no	—
168	C₂×C₄×C₁₁₃	—	2	2	no	—
169	C₂×C₄×C₁₁₃	—	2	2	no	—
170	C₂×C₄×C₁₁₃	—	2	2	no	—
171	C₂×C₄×C₁₁₃	—	4	4	no	—
172	C₂×C₄×C₁₁₃	—	4	4	no	—
173	C₂×C₄×C₁₁₃	—	4	2	no	—
174	C₂×C₄×C₁₁₃	—	4	2	no	—
175	C₂×C₄×C₁₁₃	—	4	2	no	—
176	C₂×C₄×C₁₁₃	—	4	2	no	—
177	C₂×C₄×C₁₁₃	—	4	2	no	—
178	C₂×C₄×C₁₁₃	—	4	2	no	—
179	C₂×C₄×C₁₁₃	—	4	2	no	—
180	C₂×C₄×C₁₁₃	—	4	2	no	—
181	C₂×C₄×C₁₁₃	—	8	8	no	—
182	C₂×C₄×C₁₁₃	C₄×D₄	1	2	yes	—
183	C₂×C₄×C₁₁₃	AGL(1,ℤ/8)	1	4	yes	—
184	C₂×C₄×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
185	C₂×C₄×C₁₁₃	C₂×#41	1	1	yes	characteristic
186	C₂×C₄×C₁₁₃	H(5,𝔽₂)	1	1	yes	characteristic
187	C₂×C₄×C₁₁₃	H(5,𝔽₂)	1	1	yes	characteristic
188	C₂×C₄×C₁₁₃	H(5,𝔽₂)	1	1	yes	characteristic
189	C₂×C₄×C₁₁₃	2₋¹⁺⁴	1	1	yes	characteristic
190	C₈×C₁₁₃	—	2	2	no	—
191	C₈×C₁₁₃	—	2	2	no	—
192	C₈×C₁₁₃	—	4	2	no	—
193	C₂²×C₁₁₃	—	2	2	no	—
194	C₂²×C₁₁₃	—	2	4	no	—
195	C₂²×C₁₁₃	—	4	1	no	—
196	C₂²×C₁₁₃	—	4	1	no	—
197	C₂²×C₁₁₃	—	4	2	no	—
198	C₂²×C₁₁₃	—	4	4	no	—
199	C₂²×C₁₁₃	—	4	4	no	—
200	C₂²×C₁₁₃	—	4	4	no	—
201	C₂²×C₁₁₃	—	8	2	no	—
202	C₂²×C₁₁₃	—	8	2	no	—
203	C₂²×C₁₁₃	—	8	2	no	—
204	C₂²×C₁₁₃	—	8	2	no	—
205	C₂²×C₁₁₃	C₂×C₄×D₄	1	1	yes	characteristic
206	C₂²×C₁₁₃	#534	1	1	yes	characteristic
207	C₂²×C₁₁₃	#537	1	1	yes	characteristic
208	C₂²×C₁₁₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
209	C₂²×C₁₁₃	#575	1	2	yes	—
210	C₄×C₁₁₃	—	2	2	no	—
211	C₄×C₁₁₃	—	2	4	no	—
212	C₄×C₁₁₃	—	4	2	no	—
213	C₄×C₁₁₃	—	4	4	no	—
214	C₄×C₁₁₃	—	8	2	no	—
215	C₄×C₁₁₃	—	8	2	no	—
216	C₄×C₁₁₃	#518	1	1	yes	characteristic
217	C₄×C₁₁₃	#520	1	1	yes	characteristic
218	C₄×C₁₁₃	#538	1	1	yes	characteristic
219	C₄×C₁₁₃	#539	1	1	yes	characteristic
220	#22187	C₁₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
221	C₂×C₁₁₃	—	4	2	no	—
222	C₂×C₁₁₃	—	8	4	no	—
223	C₂×C₁₁₃	#2325	1	1	yes	characteristic
224	C₂×C₁₁₃	C₄×AGL(1,ℤ/8)	1	2	yes	—
225	C₂×C₁₁₃	#3192	1	1	yes	characteristic
226	C₂×C₁₁₃	#3197	1	1	yes	characteristic
227	C₂×C₁₁₃	#3251	1	2	yes	—
228	C₂×#469	C₂×C₁₁₃	1	1	yes	characteristic
229	C₄×#121	C₂×C₁₁₃	1	2	yes	complemented
230	#1926	C₂×C₁₁₃	1	2	yes	complemented
231	#1919	C₂×C₁₁₃	1	2	yes	complemented
232	#1895	C₂×C₁₁₃	1	2	yes	complemented
233	#1879	C₂×C₁₁₃	1	2	yes	complemented
234	#1862	C₂×C₁₁₃	1	2	yes	complemented
235	#1809	C₂×C₁₁₃	1	1	yes	characteristic, complemented
236	#1728	C₂×C₁₁₃	1	1	yes	characteristic, complemented
237	C₁₁₃	#22187	1	1	yes	characteristic, direct product factor, Sylow
238	C₂²×D₈	C₂²×C₁₁₃	1	1	yes	characteristic
239	C₂²×M₁₆	C₂²×C₁₁₃	1	1	yes	characteristic
240	C₂²×M₁₆	C₂²×C₁₁₃	1	1	yes	characteristic
241	C₂×#121	C₂²×C₁₁₃	1	2	yes	—
242	C₂×C₄×D₄	—	2	2	no	—
243	C₂×C₄×D₄	C₂²×C₁₁₃	1	2	yes	—
244	C₂²×C₄²	C₂²×C₁₁₃	1	1	yes	characteristic
245	#469	C₄×C₁₁₃	1	8	yes	complemented
246	C₂×#107	C₂²×C₁₁₃	1	1	yes	characteristic
247	C₂×#106	C₂²×C₁₁₃	1	1	yes	characteristic
248	C₂×#106	C₂²×C₁₁₃	1	1	yes	characteristic
249	C₂×#102	C₂²×C₁₁₃	1	2	yes	—
250	C₂×#102	C₂²×C₁₁₃	1	2	yes	—
251	C₂×#102	C₂²×C₁₁₃	1	2	yes	—
252	#390	C₂²×C₁₁₃	1	2	yes	—
253	#389	C₂²×C₁₁₃	1	2	yes	—
254	#388	—	2	2	no	—
255	#386	—	2	2	no	—
256	#380	C₂²×C₁₁₃	1	2	yes	—
257	C₄×C₄⋊C₄	C₂²×C₁₁₃	1	2	yes	—
258	C₄×C₂²⋊C₄	—	2	2	no	—
259	#340	C₂²×C₁₁₃	1	2	yes	—
260	#340	C₂²×C₁₁₃	1	2	yes	—
261	#340	C₂²×C₁₁₃	1	2	yes	—
262	#336	C₂²×C₁₁₃	1	2	yes	—
263	#336	C₂²×C₁₁₃	1	2	yes	—
264	C₂²×D₄	—	2	2	no	—
265	C₂²×D₄	C₂³×C₁₁₃	1	2	yes	—
266	C₂³×C₄	—	2	2	no	—
267	C₂³×C₄	D₄×C₁₁₃	1	1	yes	characteristic
268	C₂³×C₄	D₄×C₁₁₃	1	1	yes	characteristic
269	C₂³×C₄	C₂³×C₁₁₃	1	1	yes	characteristic
270	C₂×D₈	—	4	1	no	—
271	C₂×D₈	—	4	1	no	—
272	C₂×D₈	C₂×C₄×C₁₁₃	1	4	yes	complemented
273	C₂×M₁₆	—	2	2	no	—
274	C₂×M₁₆	—	4	1	no	—
275	C₂×M₁₆	—	4	1	no	—
276	C₂×M₁₆	—	4	1	no	—
277	C₂×M₁₆	—	4	1	no	—
278	C₂×M₁₆	C₂×C₄×C₁₁₃	1	4	yes	complemented
279	C₂²×C₈	C₂³×C₁₁₃	1	2	yes	—
280	C₂²×C₈	C₂³×C₁₁₃	1	1	yes	characteristic
281	C₂²×C₈	C₂³×C₁₁₃	1	1	yes	characteristic
282	#121	—	2	4	no	—
283	#121	C₂×C₄×C₁₁₃	1	8	yes	complemented
284	C₄×D₄	—	4	2	no	—
285	C₄×D₄	—	4	2	no	—
286	C₄×D₄	—	4	8	no	—
287	C₂×C₄⋊C₄	—	2	2	no	—
288	C₂×C₄⋊C₄	C₂³×C₁₁₃	1	2	yes	—
289	C₂×C₄⋊C₄	C₂³×C₁₁₃	1	2	yes	—
290	C₂×C₄⋊C₄	C₂³×C₁₁₃	1	2	yes	—
291	C₂×C₂²⋊C₄	—	2	2	no	—
292	C₂×C₂²⋊C₄	—	2	2	no	—
293	C₂×C₂²⋊C₄	—	2	2	no	—
294	C₂×C₂²⋊C₄	—	2	2	no	—
295	C₂×C₂²⋊C₄	—	2	2	no	—
296	C₂×C₄²	—	2	2	no	—
297	C₂×C₄²	—	4	1	no	—
298	C₂×C₄²	—	4	1	no	—
299	C₂×C₄²	D₄×C₁₁₃	1	2	yes	—
300	C₂×C₄²	C₂³×C₁₁₃	1	2	yes	—
301	#107	—	4	1	no	—
302	#106	—	4	1	no	—
303	#106	C₂×C₄×C₁₁₃	1	4	yes	complemented
304	#102	—	4	2	no	—
305	#102	—	4	2	no	—
306	#102	C₂×C₄×C₁₁₃	1	8	yes	complemented
307	#95	—	2	2	no	—
308	#95	—	2	2	no	—
309	#95	—	2	2	no	—
310	C₂⁴	—	2	2	no	—
311	C₂⁴	C₂×D₄×C₁₁₃	1	1	yes	characteristic
312	C₂×D₄	—	2	8	no	—
313	C₂×D₄	—	2	2	no	—
314	C₂×D₄	—	2	2	no	—
315	C₂×D₄	—	2	2	no	—
316	C₂×D₄	—	2	2	no	—
317	C₂×D₄	—	2	2	no	—
318	C₂×D₄	—	4	4	no	—
319	C₂×D₄	—	4	4	no	—
320	C₂×D₄	C₂²×C₄×C₁₁₃	1	4	yes	—
321	C₂²×C₄	—	2	1	no	—
322	C₂²×C₄	—	2	1	no	—
323	C₂²×C₄	—	2	1	no	—
324	C₂²×C₄	—	2	1	no	—
325	C₂²×C₄	—	2	2	no	—
326	C₂²×C₄	—	2	4	no	—
327	C₂²×C₄	—	2	2	no	—
328	C₂²×C₄	—	2	2	no	—
329	C₂²×C₄	—	4	1	no	—
330	C₂²×C₄	—	4	1	no	—
331	C₂²×C₄	—	4	2	no	—
332	C₂²×C₄	—	4	4	no	—
333	C₂²×C₄	—	4	4	no	—
334	C₂²×C₄	—	4	4	no	—
335	C₂²×C₄	C₂²×C₄×C₁₁₃	1	2	yes	—
336	C₂²×C₄	C₂×D₄×C₁₁₃	1	2	yes	—
337	C₂²×C₄	C₂×D₄×C₁₁₃	1	1	yes	characteristic
338	C₂²×C₄	C₂×D₄×C₁₁₃	1	2	yes	—
339	C₂²×C₄	#41×C₁₁₃	1	2	yes	—
340	C₂²×C₄	C₂⁴×C₁₁₃	1	1	yes	characteristic
341	D₈	—	4	4	no	—
342	M₁₆	—	4	4	no	—
343	M₁₆	—	4	4	no	—
344	C₂×C₈	—	2	1	no	—
345	C₂×C₈	—	2	1	no	—
346	C₂×C₈	—	2	1	no	—
347	C₂×C₈	—	2	1	no	—
348	C₂×C₈	—	2	2	no	—
349	C₂×C₈	—	2	2	no	—
350	C₂×C₈	—	2	2	no	—
351	C₂×C₈	C₂²×C₄×C₁₁₃	1	2	yes	—
352	C₂×C₈	C₂²×C₄×C₁₁₃	1	2	yes	—
353	C₄⋊C₄	—	2	2	no	—
354	C₄⋊C₄	—	2	2	no	—
355	C₄⋊C₄	—	2	2	no	—
356	C₄⋊C₄	—	2	2	no	—
357	C₄⋊C₄	—	2	2	no	—
358	C₄⋊C₄	—	4	4	no	—
359	C₄⋊C₄	C₂²×C₄×C₁₁₃	1	4	yes	—
360	C₂²⋊C₄	—	2	8	no	—
361	C₂²⋊C₄	—	4	4	no	—
362	C₂²⋊C₄	—	4	4	no	—
363	C₂²⋊C₄	—	4	4	no	—
364	C₂²⋊C₄	—	4	4	no	—
365	C₄²	—	2	2	no	—
366	C₄²	—	2	2	no	—
367	C₄²	—	4	2	no	—
368	C₄²	—	4	4	no	—
369	C₂³	—	2	1	no	—
370	C₂³	—	2	1	no	—
371	C₂³	—	2	2	no	—
372	C₂³	—	2	2	no	—
373	C₂³	—	2	4	no	—
374	C₂³	—	4	2	no	—
375	C₂³	—	4	2	no	—
376	C₂³	—	4	2	no	—
377	C₂³	—	4	2	no	—
378	C₂³	—	4	2	no	—
379	C₂³	—	4	2	no	—
380	C₂³	C₄×D₄×C₁₁₃	1	2	yes	—
381	C₂³	C₂²×D₄×C₁₁₃	1	1	yes	characteristic
382	D₄	—	2	4	no	—
383	D₄	—	2	4	no	—
384	D₄	—	4	16	no	—
385	D₄	—	4	2	no	—
386	D₄	—	4	2	no	—
387	C₂×C₄	—	2	2	no	—
388	C₂×C₄	—	2	1	no	—
389	C₂×C₄	—	2	1	no	—
390	C₂×C₄	—	2	2	no	—
391	C₂×C₄	—	2	2	no	—
392	C₂×C₄	—	2	2	no	—
393	C₂×C₄	—	2	2	no	—
394	C₂×C₄	—	2	2	no	—
395	C₂×C₄	—	2	4	no	—
396	C₂×C₄	—	2	2	no	—
397	C₂×C₄	—	2	2	no	—
398	C₂×C₄	—	2	2	no	—
399	C₂×C₄	—	4	4	no	—
400	C₂×C₄	—	4	4	no	—
401	C₂×C₄	—	4	2	no	—
402	C₂×C₄	—	4	2	no	—
403	C₂×C₄	—	4	2	no	—
404	C₂×C₄	—	4	2	no	—
405	C₂×C₄	—	4	2	no	—
406	C₂×C₄	—	4	2	no	—
407	C₂×C₄	—	4	2	no	—
408	C₂×C₄	—	4	2	no	—
409	C₂×C₄	—	8	8	no	—
410	C₂×C₄	C₄×D₄×C₁₁₃	1	2	yes	—
411	C₂×C₄	AGL(1,ℤ/8)×C₁₁₃	1	4	yes	—
412	C₂×C₄	C₂³×C₄×C₁₁₃	1	1	yes	characteristic
413	C₂×C₄	C₂×#41×C₁₁₃	1	1	yes	characteristic
414	C₂×C₄	H(5,𝔽₂)×C₁₁₃	1	1	yes	characteristic
415	C₂×C₄	H(5,𝔽₂)×C₁₁₃	1	1	yes	characteristic
416	C₂×C₄	H(5,𝔽₂)×C₁₁₃	1	1	yes	characteristic
417	C₂×C₄	2₋¹⁺⁴×C₁₁₃	1	1	yes	characteristic
418	C₈	—	2	2	no	—
419	C₈	—	2	2	no	—
420	C₈	—	4	2	no	—
421	C₂²	—	2	2	no	—
422	C₂²	—	2	4	no	—
423	C₂²	—	4	1	no	—
424	C₂²	—	4	1	no	—
425	C₂²	—	4	2	no	—
426	C₂²	—	4	4	no	—
427	C₂²	—	4	4	no	—
428	C₂²	—	4	4	no	—
429	C₂²	—	8	2	no	—
430	C₂²	—	8	2	no	—
431	C₂²	—	8	2	no	—
432	C₂²	—	8	2	no	—
433	C₂²	C₂×C₄×D₄×C₁₁₃	1	1	yes	characteristic
434	C₂²	#534×C₁₁₃	1	1	yes	characteristic
435	C₂²	#537×C₁₁₃	1	1	yes	characteristic
436	C₂²	C₂×AGL(1,ℤ/8)×C₁₁₃	1	2	yes	—
437	C₂²	#575×C₁₁₃	1	2	yes	—
438	C₄	—	2	2	no	—
439	C₄	—	2	4	no	—
440	C₄	—	4	2	no	—
441	C₄	—	4	4	no	—
442	C₄	—	8	2	no	—
443	C₄	—	8	2	no	—
444	C₄	#518×C₁₁₃	1	1	yes	characteristic
445	C₄	#520×C₁₁₃	1	1	yes	characteristic
446	C₄	#538×C₁₁₃	1	1	yes	characteristic
447	C₄	#539×C₁₁₃	1	1	yes	characteristic
448	C₂	—	4	2	no	—
449	C₂	—	8	4	no	—
450	C₂	C₁₁₃×#2325	1	1	yes	characteristic
451	C₂	C₄×AGL(1,ℤ/8)×C₁₁₃	1	2	yes	—
452	C₂	C₁₁₃×#3192	1	1	yes	characteristic
453	C₂	C₁₁₃×#3197	1	1	yes	characteristic
454	C₂	C₁₁₃×#3251	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	C₂×#469×C₁₁₃	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₁₃ ◁ C₂²×C₄²×C₁₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×#121×C₁₁₃	C₂	semidirect
3. C₁₁₃×#1926	C₂	semidirect
4. C₁₁₃×#1919	C₂	semidirect
5. C₁₁₃×#1895	C₂	semidirect
6. C₁₁₃×#1879	C₂	semidirect
7. C₁₁₃×#1862	C₂	semidirect
8. C₁₁₃×#1809	C₂	semidirect
9. C₁₁₃×#1728	C₂	semidirect
17. #469×C₁₁₃	C₄	semidirect
44. C₂×D₈×C₁₁₃	C₂×C₄	semidirect
50. C₂×M₁₆×C₁₁₃	C₂×C₄	semidirect
55. #121×C₁₁₃	C₂×C₄	semidirect
75. #106×C₁₁₃	C₂×C₄	semidirect
78. #102×C₁₁₃	C₂×C₄	semidirect
220. #22187	C₁₁₃	direct (N × H)
229. C₄×#121	C₂×C₁₁₃	semidirect
230. #1926	C₂×C₁₁₃	semidirect
231. #1919	C₂×C₁₁₃	semidirect
232. #1895	C₂×C₁₁₃	semidirect
233. #1879	C₂×C₁₁₃	semidirect
234. #1862	C₂×C₁₁₃	semidirect
235. #1809	C₂×C₁₁₃	semidirect
236. #1728	C₂×C₁₁₃	semidirect
237. C₁₁₃	#22187	direct (N × H)
245. #469	C₄×C₁₁₃	semidirect
272. C₂×D₈	C₂×C₄×C₁₁₃	semidirect
278. C₂×M₁₆	C₂×C₄×C₁₁₃	semidirect
283. #121	C₂×C₄×C₁₁₃	semidirect
303. #106	C₂×C₄×C₁₁₃	semidirect
306. #102	C₂×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×#469×C₁₁₃	2
C₄×#121×C₁₁₃	2
C₁₁₃×#1926	2
C₁₁₃×#1919	2
C₁₁₃×#1895	2
C₁₁₃×#1879	2
C₁₁₃×#1862	2
C₁₁₃×#1809	2
C₁₁₃×#1728	2
#22187	113

Subgroups — C113×#22187 #43290207

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁₃×#22187 #43290207