Subgroups — C₁₁₃×#23354 #43291374

Order 28928.

702 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₁₃×#2398	C₂	1	1	yes	maximal, characteristic
2	C₁₁₃×#1985	C₂	1	2	yes	maximal, complemented
3	C₁₁₃×#1924	C₂	1	2	yes	maximal, complemented
4	C₁₁₃×#1924	C₂	1	2	yes	maximal, complemented
5	C₁₁₃×#1870	C₂	1	1	yes	maximal, characteristic
6	C₁₁₃×#1869	C₂	1	1	yes	maximal, characteristic
7	C₁₁₃×#1851	C₂	1	2	yes	maximal, complemented
8	C₁₁₃×#1850	C₂	1	2	yes	maximal, complemented
9	C₁₁₃×#1832	C₂	1	2	yes	maximal, complemented
10	C₂²×M₁₆×C₁₁₃	C₂²	1	1	yes	characteristic
11	C₂³×C₈×C₁₁₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×Q₈×C₁₁₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₄×C₁₁₃	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₈×C₁₁₃	C₂²	1	2	yes	—
15	C₂×C₄⋊C₈×C₁₁₃	C₂²	1	2	yes	—
16	C₂×C₄⋊C₈×C₁₁₃	C₂²	1	2	yes	—
17	C₂×C₂²⋊C₈×C₁₁₃	C₂²	1	2	yes	—
18	C₂×C₂²⋊C₈×C₁₁₃	C₂²	1	2	yes	—
19	C₂×C₂²⋊C₈×C₁₁₃	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊C₈×C₁₁₃	C₂²	1	1	yes	characteristic
21	C₂×#97×C₁₁₃	C₂²	1	2	yes	—
22	#400×C₁₁₃	C₄	1	2	yes	—
23	#397×C₁₁₃	C₄	1	1	yes	characteristic
24	#397×C₁₁₃	C₄	1	1	yes	characteristic
25	#394×C₁₁₃	C₄	1	2	yes	—
26	#393×C₁₁₃	C₄	1	1	yes	characteristic
27	#393×C₁₁₃	C₄	1	1	yes	characteristic
28	C₄×C₂²⋊C₄×C₁₁₃	C₂²	1	2	yes	—
29	C₄×C₂²⋊C₄×C₁₁₃	C₂²	1	2	yes	—
30	#376×C₁₁₃	C₂²	1	1	yes	characteristic
31	#336×C₁₁₃	C₂²	1	2	yes	—
32	#336×C₁₁₃	C₂²	1	2	yes	—
33	#336×C₁₁₃	C₂²	1	2	yes	—
34	#336×C₁₁₃	C₂²	1	2	yes	—
35	#336×C₁₁₃	C₂²	1	2	yes	—
36	#336×C₁₁₃	C₂²	1	2	yes	—
37	C₂²×Q₈×C₁₁₃	C₂×C₄	1	1	yes	characteristic
38	C₂²×D₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
39	C₂³×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
40	C₂³×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
41	C₂×M₁₆×C₁₁₃	—	4	2	no	—
42	C₂×M₁₆×C₁₁₃	—	4	1	no	—
43	C₂²×C₈×C₁₁₃	—	4	2	no	—
44	C₂²×C₈×C₁₁₃	—	4	1	no	—
45	C₂²×C₈×C₁₁₃	C₂³	1	2	yes	—
46	C₂²×C₈×C₁₁₃	C₂³	1	2	yes	—
47	C₂²×C₈×C₁₁₃	C₂³	1	2	yes	—
48	C₂²×C₈×C₁₁₃	C₂³	1	2	yes	—
49	C₄×Q₈×C₁₁₃	—	4	1	no	—
50	C₄×Q₈×C₁₁₃	—	4	1	no	—
51	C₄×D₄×C₁₁₃	—	4	1	no	—
52	C₄×D₄×C₁₁₃	—	4	1	no	—
53	C₂×C₄⋊C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄⋊C₄×C₁₁₃	C₂×C₄	1	2	yes	—
55	C₂×C₄⋊C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
56	C₂×C₂²⋊C₄×C₁₁₃	C₂×C₄	1	2	yes	—
57	C₂×C₂²⋊C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
58	C₂×C₂²⋊C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
59	C₂×C₂²⋊C₄×C₁₁₃	C₂×C₄	1	2	yes	—
60	C₂×C₂²⋊C₄×C₁₁₃	C₂×C₄	1	2	yes	—
61	C₂×C₂²⋊C₄×C₁₁₃	C₂×C₄	1	2	yes	—
62	C₂×C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
63	C₂×C₄²×C₁₁₃	C₂³	1	2	yes	—
64	C₂×C₄²×C₁₁₃	C₂³	1	2	yes	—
65	C₄⋊C₈×C₁₁₃	—	4	2	no	—
66	C₄⋊C₈×C₁₁₃	—	4	2	no	—
67	C₄⋊C₈×C₁₁₃	—	4	2	no	—
68	C₂²⋊C₈×C₁₁₃	—	4	1	no	—
69	C₂²⋊C₈×C₁₁₃	—	4	1	no	—
70	C₂²⋊C₈×C₁₁₃	—	4	2	no	—
71	C₂²⋊C₈×C₁₁₃	—	4	2	no	—
72	#97×C₁₁₃	—	4	2	no	—
73	#95×C₁₁₃	C₂×C₄	1	1	yes	characteristic
74	#95×C₁₁₃	C₂×C₄	1	1	yes	characteristic
75	#95×C₁₁₃	C₂×C₄	1	2	yes	—
76	#95×C₁₁₃	C₂×C₄	1	2	yes	—
77	#95×C₁₁₃	C₂×C₄	1	2	yes	—
78	#95×C₁₁₃	C₂×C₄	1	2	yes	—
79	#95×C₁₁₃	C₂×C₄	1	2	yes	—
80	C₂⁴×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
81	C₂⁴×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
82	C₂×Q₈×C₁₁₃	—	4	1	no	—
83	C₂×Q₈×C₁₁₃	—	4	2	no	—
84	C₂×D₄×C₁₁₃	—	2	2	no	—
85	C₂×D₄×C₁₁₃	—	4	1	no	—
86	C₂×D₄×C₁₁₃	—	4	1	no	—
87	C₂²×C₄×C₁₁₃	—	2	1	no	—
88	C₂²×C₄×C₁₁₃	—	2	1	no	—
89	C₂²×C₄×C₁₁₃	—	2	2	no	—
90	C₂²×C₄×C₁₁₃	—	2	2	no	—
91	C₂²×C₄×C₁₁₃	—	2	1	no	—
92	C₂²×C₄×C₁₁₃	—	2	1	no	—
93	C₂²×C₄×C₁₁₃	—	2	2	no	—
94	C₂²×C₄×C₁₁₃	—	2	2	no	—
95	C₂²×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
96	C₂²×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
97	C₂²×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
98	C₂²×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
99	C₂²×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
100	C₂²×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
101	C₂²×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
102	C₂²×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
103	C₂²×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
104	M₁₆×C₁₁₃	—	4	4	no	—
105	C₂×C₈×C₁₁₃	—	2	2	no	—
106	C₂×C₈×C₁₁₃	—	2	2	no	—
107	C₂×C₈×C₁₁₃	—	2	2	no	—
108	C₂×C₈×C₁₁₃	—	2	2	no	—
109	C₂×C₈×C₁₁₃	—	2	2	no	—
110	C₂×C₈×C₁₁₃	—	2	2	no	—
111	C₂×C₈×C₁₁₃	—	2	2	no	—
112	C₂×C₈×C₁₁₃	—	2	2	no	—
113	C₂×C₈×C₁₁₃	—	2	2	no	—
114	C₂×C₈×C₁₁₃	—	2	2	no	—
115	C₂×C₈×C₁₁₃	—	2	2	no	—
116	C₂×C₈×C₁₁₃	—	2	2	no	—
117	C₂×C₈×C₁₁₃	—	4	4	no	—
118	C₄⋊C₄×C₁₁₃	—	4	1	no	—
119	C₄⋊C₄×C₁₁₃	—	4	1	no	—
120	C₄⋊C₄×C₁₁₃	—	4	2	no	—
121	C₂²⋊C₄×C₁₁₃	—	2	4	no	—
122	C₂²⋊C₄×C₁₁₃	—	4	2	no	—
123	C₂²⋊C₄×C₁₁₃	—	4	2	no	—
124	C₂²⋊C₄×C₁₁₃	—	4	2	no	—
125	C₂²⋊C₄×C₁₁₃	—	4	1	no	—
126	C₂²⋊C₄×C₁₁₃	—	4	1	no	—
127	C₄²×C₁₁₃	—	2	2	no	—
128	C₄²×C₁₁₃	—	2	2	no	—
129	C₄²×C₁₁₃	—	2	2	no	—
130	C₄²×C₁₁₃	—	2	1	no	—
131	C₄²×C₁₁₃	—	2	1	no	—
132	C₄²×C₁₁₃	—	2	2	no	—
133	C₂³×C₁₁₃	—	2	1	no	—
134	C₂³×C₁₁₃	—	2	2	no	—
135	C₂³×C₁₁₃	—	2	2	no	—
136	C₂³×C₁₁₃	—	2	1	no	—
137	C₂³×C₁₁₃	—	2	1	no	—
138	C₂³×C₁₁₃	—	2	2	no	—
139	C₂³×C₁₁₃	—	2	1	no	—
140	C₂³×C₁₁₃	—	2	1	no	—
141	C₂³×C₁₁₃	—	2	2	no	—
142	C₂³×C₁₁₃	#131	1	2	yes	—
143	C₂³×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
144	Q₈×C₁₁₃	—	4	4	no	—
145	D₄×C₁₁₃	—	4	4	no	—
146	C₂×C₄×C₁₁₃	—	2	1	no	—
147	C₂×C₄×C₁₁₃	—	2	1	no	—
148	C₂×C₄×C₁₁₃	—	2	2	no	—
149	C₂×C₄×C₁₁₃	—	2	2	no	—
150	C₂×C₄×C₁₁₃	—	2	1	no	—
151	C₂×C₄×C₁₁₃	—	2	1	no	—
152	C₂×C₄×C₁₁₃	—	2	2	no	—
153	C₂×C₄×C₁₁₃	—	2	2	no	—
154	C₂×C₄×C₁₁₃	—	2	2	no	—
155	C₂×C₄×C₁₁₃	—	2	2	no	—
156	C₂×C₄×C₁₁₃	—	2	2	no	—
157	C₂×C₄×C₁₁₃	—	2	2	no	—
158	C₂×C₄×C₁₁₃	—	2	1	no	—
159	C₂×C₄×C₁₁₃	—	2	2	no	—
160	C₂×C₄×C₁₁₃	—	2	2	no	—
161	C₂×C₄×C₁₁₃	—	2	2	no	—
162	C₂×C₄×C₁₁₃	—	2	2	no	—
163	C₂×C₄×C₁₁₃	—	2	2	no	—
164	C₂×C₄×C₁₁₃	—	2	1	no	—
165	C₂×C₄×C₁₁₃	—	2	2	no	—
166	C₂×C₄×C₁₁₃	—	2	2	no	—
167	C₂×C₄×C₁₁₃	—	4	1	no	—
168	C₂×C₄×C₁₁₃	—	4	1	no	—
169	C₂×C₄×C₁₁₃	—	4	2	no	—
170	C₂×C₄×C₁₁₃	—	4	2	no	—
171	C₂×C₄×C₁₁₃	—	4	1	no	—
172	C₂×C₄×C₁₁₃	—	4	1	no	—
173	C₂×C₄×C₁₁₃	H(5,𝔽₂)	1	2	yes	—
174	C₂×C₄×C₁₁₃	H(5,𝔽₂)	1	1	yes	characteristic
175	C₂×C₄×C₁₁₃	H(5,𝔽₂)	1	1	yes	characteristic
176	C₂×C₄×C₁₁₃	2₋¹⁺⁴	1	2	yes	—
177	C₈×C₁₁₃	—	4	2	no	—
178	C₈×C₁₁₃	—	4	2	no	—
179	C₈×C₁₁₃	—	4	2	no	—
180	C₈×C₁₁₃	—	4	2	no	—
181	C₂²×C₁₁₃	—	2	2	no	—
182	C₂²×C₁₁₃	—	2	4	no	—
183	C₂²×C₁₁₃	—	4	1	no	—
184	C₂²×C₁₁₃	—	4	1	no	—
185	C₂²×C₁₁₃	—	4	2	no	—
186	C₂²×C₁₁₃	—	4	1	no	—
187	C₂²×C₁₁₃	—	4	1	no	—
188	C₂²×C₁₁₃	—	4	1	no	—
189	C₂²×C₁₁₃	—	4	2	no	—
190	C₂²×C₁₁₃	—	4	2	no	—
191	C₂²×C₁₁₃	#518	1	1	yes	characteristic
192	C₂²×C₁₁₃	#520	1	2	yes	—
193	C₂²×C₁₁₃	C₂×#131	1	1	yes	characteristic
194	C₂²×C₁₁₃	#568	1	1	yes	characteristic
195	C₂²×C₁₁₃	#568	1	2	yes	—
196	C₄×C₁₁₃	—	4	1	no	—
197	C₄×C₁₁₃	—	4	1	no	—
198	C₄×C₁₁₃	—	4	2	no	—
199	C₄×C₁₁₃	—	4	1	no	—
200	C₄×C₁₁₃	—	4	1	no	—
201	C₄×C₁₁₃	—	4	2	no	—
202	C₄×C₁₁₃	—	4	2	no	—
203	C₄×C₁₁₃	—	4	2	no	—
204	C₄×C₁₁₃	H(3,𝔽₄)	1	1	yes	characteristic
205	C₄×C₁₁₃	#562	1	2	yes	—
206	C₄×C₁₁₃	#563	1	1	yes	characteristic
207	#23354	C₁₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
208	C₂×C₁₁₃	—	4	2	no	—
209	C₂×C₁₁₃	—	8	1	no	—
210	C₂×C₁₁₃	#2379	1	1	yes	characteristic
211	C₂×C₁₁₃	#2976	1	1	yes	characteristic
212	C₂×C₁₁₃	#2978	1	1	yes	characteristic
213	C₂×C₁₁₃	#2992	1	2	yes	—
214	C₂×C₁₁₃	#2995	1	2	yes	—
215	#2398	C₂×C₁₁₃	1	1	yes	characteristic
216	#1985	C₂×C₁₁₃	1	2	yes	complemented
217	#1924	C₂×C₁₁₃	1	2	yes	complemented
218	#1924	C₂×C₁₁₃	1	2	yes	complemented
219	#1870	C₂×C₁₁₃	1	1	yes	characteristic
220	#1869	C₂×C₁₁₃	1	1	yes	characteristic
221	#1851	C₂×C₁₁₃	1	2	yes	complemented
222	#1850	C₂×C₁₁₃	1	2	yes	complemented
223	#1832	C₂×C₁₁₃	1	2	yes	complemented
224	C₁₁₃	#23354	1	1	yes	characteristic, direct product factor, Sylow
225	C₂²×M₁₆	C₂²×C₁₁₃	1	1	yes	characteristic
226	C₂³×C₈	C₂²×C₁₁₃	1	1	yes	characteristic
227	C₂×C₄×Q₈	C₂²×C₁₁₃	1	1	yes	characteristic
228	C₂×C₄×D₄	C₂²×C₁₁₃	1	1	yes	characteristic
229	C₂×C₄⋊C₈	C₂²×C₁₁₃	1	2	yes	—
230	C₂×C₄⋊C₈	C₂²×C₁₁₃	1	2	yes	—
231	C₂×C₄⋊C₈	C₂²×C₁₁₃	1	2	yes	—
232	C₂×C₂²⋊C₈	C₂²×C₁₁₃	1	2	yes	—
233	C₂×C₂²⋊C₈	C₂²×C₁₁₃	1	2	yes	—
234	C₂×C₂²⋊C₈	C₂²×C₁₁₃	1	1	yes	characteristic
235	C₂×C₂²⋊C₈	C₂²×C₁₁₃	1	1	yes	characteristic
236	C₂×#97	C₂²×C₁₁₃	1	2	yes	—
237	#400	C₄×C₁₁₃	1	2	yes	—
238	#397	C₄×C₁₁₃	1	1	yes	characteristic
239	#397	C₄×C₁₁₃	1	1	yes	characteristic
240	#394	C₄×C₁₁₃	1	2	yes	—
241	#393	C₄×C₁₁₃	1	1	yes	characteristic
242	#393	C₄×C₁₁₃	1	1	yes	characteristic
243	C₄×C₂²⋊C₄	C₂²×C₁₁₃	1	2	yes	—
244	C₄×C₂²⋊C₄	C₂²×C₁₁₃	1	2	yes	—
245	#376	C₂²×C₁₁₃	1	1	yes	characteristic
246	#336	C₂²×C₁₁₃	1	2	yes	—
247	#336	C₂²×C₁₁₃	1	2	yes	—
248	#336	C₂²×C₁₁₃	1	2	yes	—
249	#336	C₂²×C₁₁₃	1	2	yes	—
250	#336	C₂²×C₁₁₃	1	2	yes	—
251	#336	C₂²×C₁₁₃	1	2	yes	—
252	C₂²×Q₈	C₂×C₄×C₁₁₃	1	1	yes	characteristic
253	C₂²×D₄	C₂×C₄×C₁₁₃	1	1	yes	characteristic
254	C₂³×C₄	C₂³×C₁₁₃	1	1	yes	characteristic
255	C₂³×C₄	C₂³×C₁₁₃	1	1	yes	characteristic
256	C₂×M₁₆	—	4	2	no	—
257	C₂×M₁₆	—	4	1	no	—
258	C₂²×C₈	—	4	2	no	—
259	C₂²×C₈	—	4	1	no	—
260	C₂²×C₈	C₂³×C₁₁₃	1	2	yes	—
261	C₂²×C₈	C₂³×C₁₁₃	1	2	yes	—
262	C₂²×C₈	C₂³×C₁₁₃	1	2	yes	—
263	C₂²×C₈	C₂³×C₁₁₃	1	2	yes	—
264	C₄×Q₈	—	4	1	no	—
265	C₄×Q₈	—	4	1	no	—
266	C₄×D₄	—	4	1	no	—
267	C₄×D₄	—	4	1	no	—
268	C₂×C₄⋊C₄	C₂×C₄×C₁₁₃	1	1	yes	characteristic
269	C₂×C₄⋊C₄	C₂×C₄×C₁₁₃	1	2	yes	—
270	C₂×C₄⋊C₄	C₂×C₄×C₁₁₃	1	1	yes	characteristic
271	C₂×C₂²⋊C₄	C₂×C₄×C₁₁₃	1	2	yes	—
272	C₂×C₂²⋊C₄	C₂×C₄×C₁₁₃	1	1	yes	characteristic
273	C₂×C₂²⋊C₄	C₂×C₄×C₁₁₃	1	1	yes	characteristic
274	C₂×C₂²⋊C₄	C₂×C₄×C₁₁₃	1	2	yes	—
275	C₂×C₂²⋊C₄	C₂×C₄×C₁₁₃	1	2	yes	—
276	C₂×C₂²⋊C₄	C₂×C₄×C₁₁₃	1	2	yes	—
277	C₂×C₄²	C₂³×C₁₁₃	1	1	yes	characteristic
278	C₂×C₄²	C₂³×C₁₁₃	1	2	yes	—
279	C₂×C₄²	C₂³×C₁₁₃	1	2	yes	—
280	C₄⋊C₈	—	4	2	no	—
281	C₄⋊C₈	—	4	2	no	—
282	C₄⋊C₈	—	4	2	no	—
283	C₂²⋊C₈	—	4	1	no	—
284	C₂²⋊C₈	—	4	1	no	—
285	C₂²⋊C₈	—	4	2	no	—
286	C₂²⋊C₈	—	4	2	no	—
287	#97	—	4	2	no	—
288	#95	C₂×C₄×C₁₁₃	1	1	yes	characteristic
289	#95	C₂×C₄×C₁₁₃	1	1	yes	characteristic
290	#95	C₂×C₄×C₁₁₃	1	2	yes	—
291	#95	C₂×C₄×C₁₁₃	1	2	yes	—
292	#95	C₂×C₄×C₁₁₃	1	2	yes	—
293	#95	C₂×C₄×C₁₁₃	1	2	yes	—
294	#95	C₂×C₄×C₁₁₃	1	2	yes	—
295	C₂⁴	C₂²×C₄×C₁₁₃	1	1	yes	characteristic
296	C₂⁴	C₂²×C₄×C₁₁₃	1	1	yes	characteristic
297	C₂×Q₈	—	4	1	no	—
298	C₂×Q₈	—	4	2	no	—
299	C₂×D₄	—	2	2	no	—
300	C₂×D₄	—	4	1	no	—
301	C₂×D₄	—	4	1	no	—
302	C₂²×C₄	—	2	1	no	—
303	C₂²×C₄	—	2	1	no	—
304	C₂²×C₄	—	2	2	no	—
305	C₂²×C₄	—	2	2	no	—
306	C₂²×C₄	—	2	1	no	—
307	C₂²×C₄	—	2	1	no	—
308	C₂²×C₄	—	2	2	no	—
309	C₂²×C₄	—	2	2	no	—
310	C₂²×C₄	C₂²×C₄×C₁₁₃	1	2	yes	—
311	C₂²×C₄	C₂²×C₄×C₁₁₃	1	2	yes	—
312	C₂²×C₄	C₂²×C₄×C₁₁₃	1	1	yes	characteristic
313	C₂²×C₄	C₂²×C₄×C₁₁₃	1	1	yes	characteristic
314	C₂²×C₄	C₂²×C₄×C₁₁₃	1	2	yes	—
315	C₂²×C₄	C₂²×C₄×C₁₁₃	1	1	yes	characteristic
316	C₂²×C₄	C₂²×C₄×C₁₁₃	1	2	yes	—
317	C₂²×C₄	C₂²×C₄×C₁₁₃	1	1	yes	characteristic
318	C₂²×C₄	C₂⁴×C₁₁₃	1	1	yes	characteristic
319	M₁₆	—	4	4	no	—
320	C₂×C₈	—	2	2	no	—
321	C₂×C₈	—	2	2	no	—
322	C₂×C₈	—	2	2	no	—
323	C₂×C₈	—	2	2	no	—
324	C₂×C₈	—	2	2	no	—
325	C₂×C₈	—	2	2	no	—
326	C₂×C₈	—	2	2	no	—
327	C₂×C₈	—	2	2	no	—
328	C₂×C₈	—	2	2	no	—
329	C₂×C₈	—	2	2	no	—
330	C₂×C₈	—	2	2	no	—
331	C₂×C₈	—	2	2	no	—
332	C₂×C₈	—	4	4	no	—
333	C₄⋊C₄	—	4	1	no	—
334	C₄⋊C₄	—	4	1	no	—
335	C₄⋊C₄	—	4	2	no	—
336	C₂²⋊C₄	—	2	4	no	—
337	C₂²⋊C₄	—	4	1	no	—
338	C₂²⋊C₄	—	4	1	no	—
339	C₂²⋊C₄	—	4	2	no	—
340	C₂²⋊C₄	—	4	2	no	—
341	C₂²⋊C₄	—	4	2	no	—
342	C₄²	—	2	2	no	—
343	C₄²	—	2	2	no	—
344	C₄²	—	2	2	no	—
345	C₄²	—	2	1	no	—
346	C₄²	—	2	1	no	—
347	C₄²	—	2	2	no	—
348	C₂³	—	2	1	no	—
349	C₂³	—	2	2	no	—
350	C₂³	—	2	2	no	—
351	C₂³	—	2	1	no	—
352	C₂³	—	2	1	no	—
353	C₂³	—	2	2	no	—
354	C₂³	—	2	1	no	—
355	C₂³	—	2	1	no	—
356	C₂³	—	2	2	no	—
357	C₂³	#131×C₁₁₃	1	2	yes	—
358	C₂³	C₂³×C₄×C₁₁₃	1	1	yes	characteristic
359	Q₈	—	4	4	no	—
360	D₄	—	4	4	no	—
361	C₂×C₄	—	2	1	no	—
362	C₂×C₄	—	2	1	no	—
363	C₂×C₄	—	2	2	no	—
364	C₂×C₄	—	2	2	no	—
365	C₂×C₄	—	2	1	no	—
366	C₂×C₄	—	2	1	no	—
367	C₂×C₄	—	2	2	no	—
368	C₂×C₄	—	2	2	no	—
369	C₂×C₄	—	2	2	no	—
370	C₂×C₄	—	2	2	no	—
371	C₂×C₄	—	2	2	no	—
372	C₂×C₄	—	2	2	no	—
373	C₂×C₄	—	2	1	no	—
374	C₂×C₄	—	2	2	no	—
375	C₂×C₄	—	2	2	no	—
376	C₂×C₄	—	2	2	no	—
377	C₂×C₄	—	2	2	no	—
378	C₂×C₄	—	2	2	no	—
379	C₂×C₄	—	2	1	no	—
380	C₂×C₄	—	2	2	no	—
381	C₂×C₄	—	2	2	no	—
382	C₂×C₄	—	4	1	no	—
383	C₂×C₄	—	4	1	no	—
384	C₂×C₄	—	4	2	no	—
385	C₂×C₄	—	4	2	no	—
386	C₂×C₄	—	4	1	no	—
387	C₂×C₄	—	4	1	no	—
388	C₂×C₄	H(5,𝔽₂)×C₁₁₃	1	2	yes	—
389	C₂×C₄	H(5,𝔽₂)×C₁₁₃	1	1	yes	characteristic
390	C₂×C₄	H(5,𝔽₂)×C₁₁₃	1	1	yes	characteristic
391	C₂×C₄	2₋¹⁺⁴×C₁₁₃	1	2	yes	—
392	C₈	—	4	2	no	—
393	C₈	—	4	2	no	—
394	C₈	—	4	2	no	—
395	C₈	—	4	2	no	—
396	C₂²	—	2	2	no	—
397	C₂²	—	2	4	no	—
398	C₂²	—	4	1	no	—
399	C₂²	—	4	1	no	—
400	C₂²	—	4	2	no	—
401	C₂²	—	4	1	no	—
402	C₂²	—	4	1	no	—
403	C₂²	—	4	1	no	—
404	C₂²	—	4	2	no	—
405	C₂²	—	4	2	no	—
406	C₂²	#518×C₁₁₃	1	1	yes	characteristic
407	C₂²	#520×C₁₁₃	1	2	yes	—
408	C₂²	C₂×#131×C₁₁₃	1	1	yes	characteristic
409	C₂²	#568×C₁₁₃	1	1	yes	characteristic
410	C₂²	#568×C₁₁₃	1	2	yes	—
411	C₄	—	4	1	no	—
412	C₄	—	4	1	no	—
413	C₄	—	4	1	no	—
414	C₄	—	4	2	no	—
415	C₄	—	4	1	no	—
416	C₄	—	4	2	no	—
417	C₄	—	4	2	no	—
418	C₄	—	4	2	no	—
419	C₄	H(3,𝔽₄)×C₁₁₃	1	1	yes	characteristic
420	C₄	#562×C₁₁₃	1	2	yes	—
421	C₄	#563×C₁₁₃	1	1	yes	characteristic
422	C₂	—	4	2	no	—
423	C₂	—	8	1	no	—
424	C₂	C₁₁₃×#2379	1	1	yes	characteristic
425	C₂	C₁₁₃×#2976	1	1	yes	characteristic
426	C₂	C₁₁₃×#2978	1	1	yes	characteristic
427	C₂	C₁₁₃×#2992	1	2	yes	—
428	C₂	C₁₁₃×#2995	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₁₁₃×#1985	C₂	semidirect
3. C₁₁₃×#1924	C₂	semidirect
4. C₁₁₃×#1924	C₂	semidirect
7. C₁₁₃×#1851	C₂	semidirect
8. C₁₁₃×#1850	C₂	semidirect
9. C₁₁₃×#1832	C₂	semidirect
207. #23354	C₁₁₃	direct (N × H)
216. #1985	C₂×C₁₁₃	semidirect
217. #1924	C₂×C₁₁₃	semidirect
218. #1924	C₂×C₁₁₃	semidirect
221. #1851	C₂×C₁₁₃	semidirect
222. #1850	C₂×C₁₁₃	semidirect
223. #1832	C₂×C₁₁₃	semidirect
224. C₁₁₃	#23354	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₁₃×#2398	2
C₁₁₃×#1985	2
C₁₁₃×#1924	2
C₁₁₃×#1870	2
C₁₁₃×#1869	2
C₁₁₃×#1851	2
C₁₁₃×#1850	2
C₁₁₃×#1832	2
#23354	113

Subgroups — C113×#23354 #43291374

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁₃×#23354 #43291374