Subgroups — C₁₁₃×#20646 #43288666

Order 28928.

1882 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×D₄×C₁₁₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#423×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#418×C₁₁₃	C₂	1	2	yes	maximal
4	C₁₁₃×#2285	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₁₃×#1804	C₂	1	4	yes	maximal, complemented
6	C₁₁₃×#1802	C₂	1	4	yes	maximal, complemented
7	C₁₁₃×#1736	C₂	1	2	yes	maximal, complemented
8	C₂³×D₄×C₁₁₃	C₂²	1	1	yes	characteristic
9	C₂⁴×C₄×C₁₁₃	—	2	1	no	—
10	C₂²×M₁₆×C₁₁₃	C₂²	1	2	yes	—
11	C₂×C₄×D₄×C₁₁₃	—	2	1	no	—
12	C₂×C₄×D₄×C₁₁₃	—	2	1	no	—
13	C₂×C₄×D₄×C₁₁₃	—	2	4	no	—
14	C₂×C₄×D₄×C₁₁₃	C₄	1	8	yes	complemented
15	C₂×C₄×D₄×C₁₁₃	C₂²	1	2	yes	—
16	C₂×C₄×D₄×C₁₁₃	C₂²	1	2	yes	—
17	C₂×#117×C₁₁₃	C₂²	1	2	yes	—
18	C₂²×C₄⋊C₄×C₁₁₃	C₂²	1	1	yes	characteristic
19	C₂²×C₂²⋊C₄×C₁₁₃	—	2	1	no	—
20	C₂²×C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
21	#423×C₁₁₃	—	2	4	no	—
22	C₂×C₄⋊C₈×C₁₁₃	C₂²	1	2	yes	complemented
23	C₂×C₄⋊C₈×C₁₁₃	C₂²	1	2	yes	complemented
24	#418×C₁₁₃	—	2	8	no	—
25	C₂×#102×C₁₁₃	C₂²	1	8	yes	—
26	#384×C₁₁₃	C₂²	1	2	yes	complemented
27	#382×C₁₁₃	C₂²	1	2	yes	complemented
28	#380×C₁₁₃	—	2	1	no	—
29	#380×C₁₁₃	—	2	1	no	—
30	C₄×C₄⋊C₄×C₁₁₃	—	2	1	no	—
31	C₄×C₄⋊C₄×C₁₁₃	—	2	1	no	—
32	#340×C₁₁₃	C₂²	1	8	yes	complemented
33	C₂⁵×C₁₁₃	—	2	1	no	—
34	C₂²×D₄×C₁₁₃	—	2	1	no	—
35	C₂²×D₄×C₁₁₃	—	2	2	no	—
36	C₂²×D₄×C₁₁₃	—	2	2	no	—
37	C₂²×D₄×C₁₁₃	—	2	2	no	—
38	C₂²×D₄×C₁₁₃	C₂×C₄	1	4	yes	—
39	C₂²×D₄×C₁₁₃	D₄	1	4	yes	—
40	C₂²×D₄×C₁₁₃	Q₈	1	4	yes	—
41	C₂²×D₄×C₁₁₃	C₂³	1	2	yes	—
42	C₂³×C₄×C₁₁₃	—	2	1	no	—
43	C₂³×C₄×C₁₁₃	—	2	8	no	—
44	C₂³×C₄×C₁₁₃	—	2	2	no	—
45	C₂³×C₄×C₁₁₃	—	2	2	no	—
46	C₂³×C₄×C₁₁₃	—	4	8	no	—
47	C₂³×C₄×C₁₁₃	D₄	1	1	yes	characteristic
48	C₂³×C₄×C₁₁₃	D₄	1	1	yes	characteristic
49	C₂³×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
50	C₂×M₁₆×C₁₁₃	—	2	4	no	—
51	C₂×M₁₆×C₁₁₃	—	2	4	no	—
52	C₂×M₁₆×C₁₁₃	—	2	4	no	—
53	C₂²×C₈×C₁₁₃	C₂³	1	4	yes	—
54	C₄×D₄×C₁₁₃	—	2	8	no	—
55	C₄×D₄×C₁₁₃	—	2	8	no	—
56	C₄×D₄×C₁₁₃	—	2	4	no	—
57	C₄×D₄×C₁₁₃	—	2	4	no	—
58	C₄×D₄×C₁₁₃	C₂×C₄	1	8	yes	—
59	C₄×D₄×C₁₁₃	C₂×C₄	1	8	yes	—
60	#117×C₁₁₃	—	2	4	no	—
61	#117×C₁₁₃	—	2	4	no	—
62	C₂×C₄⋊C₄×C₁₁₃	—	2	1	no	—
63	C₂×C₄⋊C₄×C₁₁₃	—	2	2	no	—
64	C₂×C₄⋊C₄×C₁₁₃	—	2	2	no	—
65	C₂×C₄⋊C₄×C₁₁₃	—	2	2	no	—
66	C₂×C₄⋊C₄×C₁₁₃	C₂×C₄	1	4	yes	—
67	C₂×C₄⋊C₄×C₁₁₃	C₂³	1	4	yes	—
68	C₂×C₄⋊C₄×C₁₁₃	C₂³	1	2	yes	—
69	C₂×C₂²⋊C₄×C₁₁₃	—	2	8	no	—
70	C₂×C₂²⋊C₄×C₁₁₃	—	2	2	no	—
71	C₂×C₂²⋊C₄×C₁₁₃	—	2	2	no	—
72	C₂×C₂²⋊C₄×C₁₁₃	—	2	2	no	—
73	C₂×C₄²×C₁₁₃	—	2	2	no	—
74	C₂×C₄²×C₁₁₃	C₂×C₄	1	4	yes	—
75	C₂×C₄²×C₁₁₃	D₄	1	1	yes	characteristic
76	C₂×C₄²×C₁₁₃	D₄	1	1	yes	characteristic
77	C₂×C₄²×C₁₁₃	D₄	1	4	yes	—
78	C₂×C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
79	C₂×C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
80	C₄⋊C₈×C₁₁₃	—	2	4	no	—
81	C₄⋊C₈×C₁₁₃	—	2	4	no	—
82	#102×C₁₁₃	—	2	16	no	—
83	#95×C₁₁₃	—	2	2	no	—
84	#95×C₁₁₃	—	2	2	no	—
85	C₂⁴×C₁₁₃	—	2	4	no	—
86	C₂⁴×C₁₁₃	—	2	4	no	—
87	C₂⁴×C₁₁₃	—	2	4	no	—
88	C₂⁴×C₁₁₃	—	2	2	no	—
89	C₂⁴×C₁₁₃	—	4	4	no	—
90	C₂⁴×C₁₁₃	—	4	4	no	—
91	C₂⁴×C₁₁₃	C₂×D₄	1	1	yes	characteristic
92	C₂×D₄×C₁₁₃	—	2	4	no	—
93	C₂×D₄×C₁₁₃	—	2	8	no	—
94	C₂×D₄×C₁₁₃	—	2	8	no	—
95	C₂×D₄×C₁₁₃	—	2	4	no	—
96	C₂×D₄×C₁₁₃	—	2	8	no	—
97	C₂×D₄×C₁₁₃	—	2	4	no	—
98	C₂×D₄×C₁₁₃	—	2	2	no	—
99	C₂×D₄×C₁₁₃	—	2	2	no	—
100	C₂×D₄×C₁₁₃	—	2	2	no	—
101	C₂×D₄×C₁₁₃	C₄⋊C₄	1	16	yes	complemented
102	C₂×D₄×C₁₁₃	C₂²×C₄	1	4	yes	—
103	C₂×D₄×C₁₁₃	C₂×D₄	1	4	yes	—
104	C₂×D₄×C₁₁₃	C₂×Q₈	1	4	yes	—
105	C₂²×C₄×C₁₁₃	—	2	4	no	—
106	C₂²×C₄×C₁₁₃	—	2	4	no	—
107	C₂²×C₄×C₁₁₃	—	2	2	no	—
108	C₂²×C₄×C₁₁₃	—	2	8	no	—
109	C₂²×C₄×C₁₁₃	—	2	8	no	—
110	C₂²×C₄×C₁₁₃	—	2	4	no	—
111	C₂²×C₄×C₁₁₃	—	4	2	no	—
112	C₂²×C₄×C₁₁₃	—	4	8	no	—
113	C₂²×C₄×C₁₁₃	—	4	32	no	—
114	C₂²×C₄×C₁₁₃	—	4	4	no	—
115	C₂²×C₄×C₁₁₃	—	4	8	no	—
116	C₂²×C₄×C₁₁₃	C₂²⋊C₄	1	4	yes	—
117	C₂²×C₄×C₁₁₃	C₂²⋊C₄	1	4	yes	—
118	C₂²×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
119	C₂²×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
120	C₂²×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
121	C₂²×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
122	C₂²×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
123	C₂²×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
124	C₂²×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
125	C₂²×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
126	C₂²×C₄×C₁₁₃	C₂×Q₈	1	2	yes	—
127	C₂²×C₄×C₁₁₃	#41	1	2	yes	—
128	C₂²×C₄×C₁₁₃	#41	1	2	yes	—
129	C₂²×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
130	M₁₆×C₁₁₃	—	4	8	no	—
131	C₂×C₈×C₁₁₃	—	2	4	no	—
132	C₂×C₈×C₁₁₃	—	2	4	no	—
133	C₂×C₈×C₁₁₃	—	2	4	no	—
134	C₄⋊C₄×C₁₁₃	—	2	4	no	—
135	C₄⋊C₄×C₁₁₃	—	2	2	no	—
136	C₄⋊C₄×C₁₁₃	—	2	4	no	—
137	C₄⋊C₄×C₁₁₃	—	2	4	no	—
138	C₄⋊C₄×C₁₁₃	—	4	4	no	—
139	C₄⋊C₄×C₁₁₃	—	4	4	no	—
140	C₄⋊C₄×C₁₁₃	C₂²×C₄	1	4	yes	—
141	C₂²⋊C₄×C₁₁₃	—	2	8	no	—
142	C₂²⋊C₄×C₁₁₃	—	2	8	no	—
143	C₂²⋊C₄×C₁₁₃	—	4	4	no	—
144	C₄²×C₁₁₃	—	4	4	no	—
145	C₄²×C₁₁₃	C₂²⋊C₄	1	4	yes	—
146	C₄²×C₁₁₃	C₂²⋊C₄	1	4	yes	—
147	C₄²×C₁₁₃	C₂²×C₄	1	2	yes	—
148	C₄²×C₁₁₃	C₂²×C₄	1	2	yes	—
149	C₄²×C₁₁₃	C₂×D₄	1	2	yes	—
150	C₄²×C₁₁₃	C₂×D₄	1	2	yes	—
151	C₂³×C₁₁₃	—	2	2	no	—
152	C₂³×C₁₁₃	—	2	2	no	—
153	C₂³×C₁₁₃	—	2	4	no	—
154	C₂³×C₁₁₃	—	2	4	no	—
155	C₂³×C₁₁₃	—	2	4	no	—
156	C₂³×C₁₁₃	—	2	16	no	—
157	C₂³×C₁₁₃	—	4	8	no	—
158	C₂³×C₁₁₃	—	4	16	no	—
159	C₂³×C₁₁₃	—	4	8	no	—
160	C₂³×C₁₁₃	—	4	16	no	—
161	C₂³×C₁₁₃	—	4	4	no	—
162	C₂³×C₁₁₃	—	4	4	no	—
163	C₂³×C₁₁₃	C₂×C₂²⋊C₄	1	2	yes	—
164	C₂³×C₁₁₃	#121	1	2	yes	—
165	C₂³×C₁₁₃	C₂²⋊Q₈	1	2	yes	—
166	C₂³×C₁₁₃	C₂²×D₄	1	1	yes	characteristic
167	D₄×C₁₁₃	—	2	4	no	—
168	D₄×C₁₁₃	—	2	4	no	—
169	D₄×C₁₁₃	—	2	4	no	—
170	D₄×C₁₁₃	—	2	16	no	—
171	D₄×C₁₁₃	C₂×C₄⋊C₄	1	8	yes	—
172	C₂×C₄×C₁₁₃	—	2	2	no	—
173	C₂×C₄×C₁₁₃	—	2	2	no	—
174	C₂×C₄×C₁₁₃	—	2	2	no	—
175	C₂×C₄×C₁₁₃	—	2	8	no	—
176	C₂×C₄×C₁₁₃	—	2	4	no	—
177	C₂×C₄×C₁₁₃	—	4	4	no	—
178	C₂×C₄×C₁₁₃	—	4	16	no	—
179	C₂×C₄×C₁₁₃	—	4	8	no	—
180	C₂×C₄×C₁₁₃	—	4	4	no	—
181	C₂×C₄×C₁₁₃	—	4	16	no	—
182	C₂×C₄×C₁₁₃	—	4	4	no	—
183	C₂×C₄×C₁₁₃	—	4	4	no	—
184	C₂×C₄×C₁₁₃	—	4	4	no	—
185	C₂×C₄×C₁₁₃	C₂×C₂²⋊C₄	1	2	yes	—
186	C₂×C₄×C₁₁₃	C₂×C₂²⋊C₄	1	2	yes	—
187	C₂×C₄×C₁₁₃	C₂×C₂²⋊C₄	1	2	yes	—
188	C₂×C₄×C₁₁₃	C₂×C₂²⋊C₄	1	2	yes	—
189	C₂×C₄×C₁₁₃	C₂×C₂²⋊C₄	1	2	yes	—
190	C₂×C₄×C₁₁₃	C₂×C₄⋊C₄	1	4	yes	—
191	C₂×C₄×C₁₁₃	#117	1	4	yes	—
192	C₂×C₄×C₁₁₃	#121	1	4	yes	—
193	C₂×C₄×C₁₁₃	#121	1	2	yes	—
194	C₂×C₄×C₁₁₃	C₂²⋊Q₈	1	4	yes	—
195	C₂×C₄×C₁₁₃	C₂²⋊Q₈	1	2	yes	—
196	C₂×C₄×C₁₁₃	AGL(1,ℤ/8)	1	4	yes	—
197	C₂×C₄×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
198	C₂×C₄×C₁₁₃	C₂²×D₄	1	1	yes	characteristic
199	C₂×C₄×C₁₁₃	C₂²×D₄	1	1	yes	characteristic
200	C₂×C₄×C₁₁₃	C₂²×Q₈	1	1	yes	characteristic
201	C₂×C₄×C₁₁₃	C₂×#41	1	1	yes	characteristic
202	C₂×C₄×C₁₁₃	C₂×#41	1	1	yes	characteristic
203	C₈×C₁₁₃	—	4	4	no	—
204	C₂²×C₁₁₃	—	2	4	no	—
205	C₂²×C₁₁₃	—	2	2	no	—
206	C₂²×C₁₁₃	—	2	2	no	—
207	C₂²×C₁₁₃	—	2	4	no	—
208	C₂²×C₁₁₃	—	2	8	no	—
209	C₂²×C₁₁₃	—	4	32	no	—
210	C₂²×C₁₁₃	—	4	8	no	—
211	C₂²×C₁₁₃	—	4	4	no	—
212	C₂²×C₁₁₃	—	4	4	no	—
213	C₂²×C₁₁₃	—	4	8	no	—
214	C₂²×C₁₁₃	#380	1	4	yes	—
215	C₂²×C₁₁₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
216	C₂²×C₁₁₃	C₂×#121	1	1	yes	characteristic
217	C₂²×C₁₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
218	C₂²×C₁₁₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
219	C₂²×C₁₁₃	#575	1	2	yes	—
220	C₄×C₁₁₃	—	2	2	no	—
221	C₄×C₁₁₃	—	2	2	no	—
222	C₄×C₁₁₃	—	2	2	no	—
223	C₄×C₁₁₃	—	4	4	no	—
224	C₄×C₁₁₃	—	8	4	no	—
225	C₄×C₁₁₃	#380	1	4	yes	—
226	C₄×C₁₁₃	#418	1	4	yes	complemented
227	C₄×C₁₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
228	C₄×C₁₁₃	C₂×#117	1	1	yes	characteristic
229	C₄×C₁₁₃	C₂×#121	1	1	yes	characteristic
230	C₄×C₁₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
231	#20646	C₁₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
232	C₂×C₁₁₃	—	2	4	no	—
233	C₂×C₁₁₃	—	4	8	no	—
234	C₂×C₁₁₃	C₂×#380	1	1	yes	characteristic
235	C₂×C₁₁₃	C₂×#418	1	1	yes	characteristic
236	C₂×C₁₁₃	#2896	1	1	yes	characteristic
237	C₂×C₁₁₃	#3036	1	2	yes	—
238	C₂×C₁₁₃	#3077	1	2	yes	—
239	C₂²×C₄×D₄	C₂×C₁₁₃	1	1	yes	characteristic
240	C₂×#423	C₂×C₁₁₃	1	1	yes	characteristic, complemented
241	C₂×#418	C₂×C₁₁₃	1	2	yes	—
242	#2285	C₂×C₁₁₃	1	1	yes	characteristic, complemented
243	#1804	C₂×C₁₁₃	1	4	yes	complemented
244	#1802	C₂×C₁₁₃	1	4	yes	complemented
245	#1736	C₂×C₁₁₃	1	2	yes	complemented
246	C₁₁₃	#20646	1	1	yes	characteristic, direct product factor, Sylow
247	C₂³×D₄	C₂²×C₁₁₃	1	1	yes	characteristic
248	C₂⁴×C₄	—	2	1	no	—
249	C₂²×M₁₆	C₂²×C₁₁₃	1	2	yes	—
250	C₂×C₄×D₄	—	2	1	no	—
251	C₂×C₄×D₄	—	2	1	no	—
252	C₂×C₄×D₄	—	2	4	no	—
253	C₂×C₄×D₄	C₄×C₁₁₃	1	8	yes	complemented
254	C₂×C₄×D₄	C₂²×C₁₁₃	1	2	yes	—
255	C₂×C₄×D₄	C₂²×C₁₁₃	1	2	yes	—
256	C₂×#117	C₂²×C₁₁₃	1	2	yes	—
257	C₂²×C₄⋊C₄	C₂²×C₁₁₃	1	1	yes	characteristic
258	C₂²×C₂²⋊C₄	—	2	1	no	—
259	C₂²×C₄²	C₂²×C₁₁₃	1	1	yes	characteristic
260	#423	—	2	4	no	—
261	C₂×C₄⋊C₈	C₂²×C₁₁₃	1	2	yes	complemented
262	C₂×C₄⋊C₈	C₂²×C₁₁₃	1	2	yes	complemented
263	#418	—	2	8	no	—
264	C₂×#102	C₂²×C₁₁₃	1	8	yes	—
265	#384	C₂²×C₁₁₃	1	2	yes	complemented
266	#382	C₂²×C₁₁₃	1	2	yes	complemented
267	#380	—	2	1	no	—
268	#380	—	2	1	no	—
269	C₄×C₄⋊C₄	—	2	1	no	—
270	C₄×C₄⋊C₄	—	2	1	no	—
271	#340	C₂²×C₁₁₃	1	8	yes	complemented
272	C₂⁵	—	2	1	no	—
273	C₂²×D₄	—	2	1	no	—
274	C₂²×D₄	—	2	2	no	—
275	C₂²×D₄	—	2	2	no	—
276	C₂²×D₄	—	2	2	no	—
277	C₂²×D₄	C₂×C₄×C₁₁₃	1	4	yes	—
278	C₂²×D₄	D₄×C₁₁₃	1	4	yes	—
279	C₂²×D₄	Q₈×C₁₁₃	1	4	yes	—
280	C₂²×D₄	C₂³×C₁₁₃	1	2	yes	—
281	C₂³×C₄	—	2	1	no	—
282	C₂³×C₄	—	2	8	no	—
283	C₂³×C₄	—	2	2	no	—
284	C₂³×C₄	—	2	2	no	—
285	C₂³×C₄	—	4	8	no	—
286	C₂³×C₄	D₄×C₁₁₃	1	1	yes	characteristic
287	C₂³×C₄	D₄×C₁₁₃	1	1	yes	characteristic
288	C₂³×C₄	C₂³×C₁₁₃	1	1	yes	characteristic
289	C₂×M₁₆	—	2	4	no	—
290	C₂×M₁₆	—	2	4	no	—
291	C₂×M₁₆	—	2	4	no	—
292	C₂²×C₈	C₂³×C₁₁₃	1	4	yes	—
293	C₄×D₄	—	2	8	no	—
294	C₄×D₄	—	2	8	no	—
295	C₄×D₄	—	2	4	no	—
296	C₄×D₄	—	2	4	no	—
297	C₄×D₄	C₂×C₄×C₁₁₃	1	8	yes	—
298	C₄×D₄	C₂×C₄×C₁₁₃	1	8	yes	—
299	#117	—	2	4	no	—
300	#117	—	2	4	no	—
301	C₂×C₄⋊C₄	—	2	1	no	—
302	C₂×C₄⋊C₄	—	2	2	no	—
303	C₂×C₄⋊C₄	—	2	2	no	—
304	C₂×C₄⋊C₄	—	2	2	no	—
305	C₂×C₄⋊C₄	C₂×C₄×C₁₁₃	1	4	yes	—
306	C₂×C₄⋊C₄	C₂³×C₁₁₃	1	4	yes	—
307	C₂×C₄⋊C₄	C₂³×C₁₁₃	1	2	yes	—
308	C₂×C₂²⋊C₄	—	2	8	no	—
309	C₂×C₂²⋊C₄	—	2	2	no	—
310	C₂×C₂²⋊C₄	—	2	2	no	—
311	C₂×C₂²⋊C₄	—	2	2	no	—
312	C₂×C₄²	—	2	2	no	—
313	C₂×C₄²	C₂×C₄×C₁₁₃	1	4	yes	—
314	C₂×C₄²	D₄×C₁₁₃	1	1	yes	characteristic
315	C₂×C₄²	D₄×C₁₁₃	1	1	yes	characteristic
316	C₂×C₄²	D₄×C₁₁₃	1	4	yes	—
317	C₂×C₄²	C₂³×C₁₁₃	1	1	yes	characteristic
318	C₂×C₄²	C₂³×C₁₁₃	1	1	yes	characteristic
319	C₄⋊C₈	—	2	4	no	—
320	C₄⋊C₈	—	2	4	no	—
321	#102	—	2	16	no	—
322	#95	—	2	2	no	—
323	#95	—	2	2	no	—
324	C₂⁴	—	2	4	no	—
325	C₂⁴	—	2	4	no	—
326	C₂⁴	—	2	4	no	—
327	C₂⁴	—	2	2	no	—
328	C₂⁴	—	4	4	no	—
329	C₂⁴	—	4	4	no	—
330	C₂⁴	C₂×D₄×C₁₁₃	1	1	yes	characteristic
331	C₂×D₄	—	2	4	no	—
332	C₂×D₄	—	2	8	no	—
333	C₂×D₄	—	2	8	no	—
334	C₂×D₄	—	2	4	no	—
335	C₂×D₄	—	2	8	no	—
336	C₂×D₄	—	2	4	no	—
337	C₂×D₄	—	2	2	no	—
338	C₂×D₄	—	2	2	no	—
339	C₂×D₄	—	2	2	no	—
340	C₂×D₄	C₄⋊C₄×C₁₁₃	1	16	yes	complemented
341	C₂×D₄	C₂²×C₄×C₁₁₃	1	4	yes	—
342	C₂×D₄	C₂×D₄×C₁₁₃	1	4	yes	—
343	C₂×D₄	C₂×Q₈×C₁₁₃	1	4	yes	—
344	C₂²×C₄	—	2	4	no	—
345	C₂²×C₄	—	2	4	no	—
346	C₂²×C₄	—	2	2	no	—
347	C₂²×C₄	—	2	8	no	—
348	C₂²×C₄	—	2	8	no	—
349	C₂²×C₄	—	2	4	no	—
350	C₂²×C₄	—	4	2	no	—
351	C₂²×C₄	—	4	8	no	—
352	C₂²×C₄	—	4	32	no	—
353	C₂²×C₄	—	4	4	no	—
354	C₂²×C₄	—	4	8	no	—
355	C₂²×C₄	C₂²⋊C₄×C₁₁₃	1	4	yes	—
356	C₂²×C₄	C₂²⋊C₄×C₁₁₃	1	4	yes	—
357	C₂²×C₄	C₂²×C₄×C₁₁₃	1	2	yes	—
358	C₂²×C₄	C₂×D₄×C₁₁₃	1	1	yes	characteristic
359	C₂²×C₄	C₂×D₄×C₁₁₃	1	1	yes	characteristic
360	C₂²×C₄	C₂×D₄×C₁₁₃	1	1	yes	characteristic
361	C₂²×C₄	C₂×D₄×C₁₁₃	1	1	yes	characteristic
362	C₂²×C₄	C₂×D₄×C₁₁₃	1	1	yes	characteristic
363	C₂²×C₄	C₂×D₄×C₁₁₃	1	2	yes	—
364	C₂²×C₄	C₂×D₄×C₁₁₃	1	2	yes	—
365	C₂²×C₄	C₂×Q₈×C₁₁₃	1	2	yes	—
366	C₂²×C₄	#41×C₁₁₃	1	2	yes	—
367	C₂²×C₄	#41×C₁₁₃	1	2	yes	—
368	C₂²×C₄	C₂⁴×C₁₁₃	1	1	yes	characteristic
369	M₁₆	—	4	8	no	—
370	C₂×C₈	—	2	4	no	—
371	C₂×C₈	—	2	4	no	—
372	C₂×C₈	—	2	4	no	—
373	C₄⋊C₄	—	2	4	no	—
374	C₄⋊C₄	—	2	2	no	—
375	C₄⋊C₄	—	2	4	no	—
376	C₄⋊C₄	—	2	4	no	—
377	C₄⋊C₄	—	4	4	no	—
378	C₄⋊C₄	—	4	4	no	—
379	C₄⋊C₄	C₂²×C₄×C₁₁₃	1	4	yes	—
380	C₂²⋊C₄	—	2	8	no	—
381	C₂²⋊C₄	—	2	8	no	—
382	C₂²⋊C₄	—	4	4	no	—
383	C₄²	—	4	4	no	—
384	C₄²	C₂²⋊C₄×C₁₁₃	1	4	yes	—
385	C₄²	C₂²⋊C₄×C₁₁₃	1	4	yes	—
386	C₄²	C₂²×C₄×C₁₁₃	1	2	yes	—
387	C₄²	C₂²×C₄×C₁₁₃	1	2	yes	—
388	C₄²	C₂×D₄×C₁₁₃	1	2	yes	—
389	C₄²	C₂×D₄×C₁₁₃	1	2	yes	—
390	C₂³	—	2	2	no	—
391	C₂³	—	2	2	no	—
392	C₂³	—	2	4	no	—
393	C₂³	—	2	4	no	—
394	C₂³	—	2	4	no	—
395	C₂³	—	2	16	no	—
396	C₂³	—	4	8	no	—
397	C₂³	—	4	16	no	—
398	C₂³	—	4	8	no	—
399	C₂³	—	4	16	no	—
400	C₂³	—	4	4	no	—
401	C₂³	—	4	4	no	—
402	C₂³	C₂×C₂²⋊C₄×C₁₁₃	1	2	yes	—
403	C₂³	#121×C₁₁₃	1	2	yes	—
404	C₂³	C₂²⋊Q₈×C₁₁₃	1	2	yes	—
405	C₂³	C₂²×D₄×C₁₁₃	1	1	yes	characteristic
406	D₄	—	2	4	no	—
407	D₄	—	2	4	no	—
408	D₄	—	2	4	no	—
409	D₄	—	2	16	no	—
410	D₄	C₂×C₄⋊C₄×C₁₁₃	1	8	yes	—
411	C₂×C₄	—	2	2	no	—
412	C₂×C₄	—	2	2	no	—
413	C₂×C₄	—	2	2	no	—
414	C₂×C₄	—	2	8	no	—
415	C₂×C₄	—	2	4	no	—
416	C₂×C₄	—	4	4	no	—
417	C₂×C₄	—	4	16	no	—
418	C₂×C₄	—	4	8	no	—
419	C₂×C₄	—	4	4	no	—
420	C₂×C₄	—	4	16	no	—
421	C₂×C₄	—	4	4	no	—
422	C₂×C₄	—	4	4	no	—
423	C₂×C₄	—	4	4	no	—
424	C₂×C₄	C₂×C₂²⋊C₄×C₁₁₃	1	2	yes	—
425	C₂×C₄	C₂×C₂²⋊C₄×C₁₁₃	1	2	yes	—
426	C₂×C₄	C₂×C₂²⋊C₄×C₁₁₃	1	2	yes	—
427	C₂×C₄	C₂×C₂²⋊C₄×C₁₁₃	1	2	yes	—
428	C₂×C₄	C₂×C₂²⋊C₄×C₁₁₃	1	2	yes	—
429	C₂×C₄	C₂×C₄⋊C₄×C₁₁₃	1	4	yes	—
430	C₂×C₄	#117×C₁₁₃	1	4	yes	—
431	C₂×C₄	#121×C₁₁₃	1	4	yes	—
432	C₂×C₄	#121×C₁₁₃	1	2	yes	—
433	C₂×C₄	C₂²⋊Q₈×C₁₁₃	1	4	yes	—
434	C₂×C₄	C₂²⋊Q₈×C₁₁₃	1	2	yes	—
435	C₂×C₄	AGL(1,ℤ/8)×C₁₁₃	1	4	yes	—
436	C₂×C₄	C₂³×C₄×C₁₁₃	1	1	yes	characteristic
437	C₂×C₄	C₂²×D₄×C₁₁₃	1	1	yes	characteristic
438	C₂×C₄	C₂²×D₄×C₁₁₃	1	1	yes	characteristic
439	C₂×C₄	C₂²×Q₈×C₁₁₃	1	1	yes	characteristic
440	C₂×C₄	C₂×#41×C₁₁₃	1	1	yes	characteristic
441	C₂×C₄	C₂×#41×C₁₁₃	1	1	yes	characteristic
442	C₈	—	4	4	no	—
443	C₂²	—	2	4	no	—
444	C₂²	—	2	2	no	—
445	C₂²	—	2	2	no	—
446	C₂²	—	2	4	no	—
447	C₂²	—	2	8	no	—
448	C₂²	—	4	32	no	—
449	C₂²	—	4	8	no	—
450	C₂²	—	4	4	no	—
451	C₂²	—	4	4	no	—
452	C₂²	—	4	8	no	—
453	C₂²	#380×C₁₁₃	1	4	yes	—
454	C₂²	C₂²×C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
455	C₂²	C₂×#121×C₁₁₃	1	1	yes	characteristic
456	C₂²	C₂×C₂²⋊Q₈×C₁₁₃	1	1	yes	characteristic
457	C₂²	C₂×AGL(1,ℤ/8)×C₁₁₃	1	2	yes	—
458	C₂²	#575×C₁₁₃	1	2	yes	—
459	C₄	—	2	2	no	—
460	C₄	—	2	2	no	—
461	C₄	—	2	2	no	—
462	C₄	—	4	4	no	—
463	C₄	—	8	4	no	—
464	C₄	#380×C₁₁₃	1	4	yes	—
465	C₄	#418×C₁₁₃	1	4	yes	complemented
466	C₄	C₂²×C₄⋊C₄×C₁₁₃	1	1	yes	characteristic
467	C₄	C₂×#117×C₁₁₃	1	1	yes	characteristic
468	C₄	C₂×#121×C₁₁₃	1	1	yes	characteristic
469	C₄	C₂×C₂²⋊Q₈×C₁₁₃	1	1	yes	characteristic
470	C₂	—	2	4	no	—
471	C₂	—	4	8	no	—
472	C₂	C₂×#380×C₁₁₃	1	1	yes	characteristic
473	C₂	C₂×#418×C₁₁₃	1	1	yes	characteristic
474	C₂	C₁₁₃×#2896	1	1	yes	characteristic
475	C₂	C₁₁₃×#3036	1	2	yes	—
476	C₂	C₁₁₃×#3077	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	C₂³×D₄×C₁₁₃	C₂²
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. C₂×#423×C₁₁₃	C₂	semidirect
4. C₁₁₃×#2285	C₂	semidirect
5. C₁₁₃×#1804	C₂	semidirect
6. C₁₁₃×#1802	C₂	semidirect
7. C₁₁₃×#1736	C₂	semidirect
14. C₂×C₄×D₄×C₁₁₃	C₄	semidirect
22. C₂×C₄⋊C₈×C₁₁₃	C₂²	semidirect
23. C₂×C₄⋊C₈×C₁₁₃	C₂²	semidirect
26. #384×C₁₁₃	C₂²	semidirect
27. #382×C₁₁₃	C₂²	semidirect
32. #340×C₁₁₃	C₂²	semidirect
101. C₂×D₄×C₁₁₃	C₄⋊C₄	semidirect
226. C₄×C₁₁₃	#418	semidirect
231. #20646	C₁₁₃	direct (N × H)
240. C₂×#423	C₂×C₁₁₃	semidirect
242. #2285	C₂×C₁₁₃	semidirect
243. #1804	C₂×C₁₁₃	semidirect
244. #1802	C₂×C₁₁₃	semidirect
245. #1736	C₂×C₁₁₃	semidirect
246. C₁₁₃	#20646	direct (N × H)
253. C₂×C₄×D₄	C₄×C₁₁₃	semidirect
261. C₂×C₄⋊C₈	C₂²×C₁₁₃	semidirect
262. C₂×C₄⋊C₈	C₂²×C₁₁₃	semidirect
265. #384	C₂²×C₁₁₃	semidirect
266. #382	C₂²×C₁₁₃	semidirect
271. #340	C₂²×C₁₁₃	semidirect
340. C₂×D₄	C₄⋊C₄×C₁₁₃	semidirect
465. C₄	#418×C₁₁₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×D₄×C₁₁₃	2
C₂×#423×C₁₁₃	2
C₂×#418×C₁₁₃	2
C₁₁₃×#2285	2
C₁₁₃×#1804	2
C₁₁₃×#1802	2
C₁₁₃×#1736	2
#20646	113

Subgroups — C113×#20646 #43288666

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁₃×#20646 #43288666