Subgroups — C₁₁₃×#23430 #43291450

Order 28928.

2706 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×AGL(1,ℤ/8)×C₁₁₃	C₂	1	2	yes	maximal, complemented
2	C₂²×#127×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#418×C₁₁₃	C₂	1	2	yes	maximal, complemented
4	C₁₁₃×#2297	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₁₃×#2014	C₂	1	4	yes	maximal, complemented
6	C₁₁₃×#2011	C₂	1	4	yes	maximal, complemented
7	C₁₁₃×#1728	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#41×C₁₁₃	C₂²	1	1	yes	characteristic
9	C₂³×D₄×C₁₁₃	—	2	2	no	—
10	C₂³×D₄×C₁₁₃	C₂²	1	2	yes	—
11	C₂×AGL(1,ℤ/8)×C₁₁₃	—	2	8	no	—
12	C₂×AGL(1,ℤ/8)×C₁₁₃	—	2	8	no	—
13	C₂×AGL(1,ℤ/8)×C₁₁₃	—	2	8	no	—
14	C₂²×SD₁₆×C₁₁₃	C₂²	1	4	yes	complemented
15	C₂²×D₈×C₁₁₃	C₂²	1	4	yes	complemented
16	C₂²×M₁₆×C₁₁₃	C₂²	1	2	yes	—
17	C₂×#127×C₁₁₃	—	2	8	no	—
18	C₂×#127×C₁₁₃	—	2	2	no	—
19	C₂×#127×C₁₁₃	C₂²	1	4	yes	—
20	C₂×#117×C₁₁₃	C₂²	1	1	yes	characteristic
21	C₂²×C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
22	#418×C₁₁₃	—	2	8	no	—
23	C₂×#102×C₁₁₃	C₂²	1	8	yes	complemented
24	#391×C₁₁₃	C₂²	1	2	yes	complemented
25	#390×C₁₁₃	C₂²	1	2	yes	complemented
26	#386×C₁₁₃	—	2	2	no	—
27	C₄×C₂²⋊C₄×C₁₁₃	—	2	2	no	—
28	#336×C₁₁₃	C₂²	1	4	yes	complemented
29	C₂⁵×C₁₁₃	—	2	2	no	—
30	C₂×#41×C₁₁₃	—	2	2	no	—
31	C₂×#41×C₁₁₃	—	2	4	no	—
32	C₂×#41×C₁₁₃	—	2	4	no	—
33	C₂×#41×C₁₁₃	D₄	1	4	yes	complemented
34	C₂²×Q₈×C₁₁₃	C₂³	1	1	yes	characteristic
35	C₂²×D₄×C₁₁₃	—	2	16	no	—
36	C₂²×D₄×C₁₁₃	—	2	1	no	—
37	C₂²×D₄×C₁₁₃	—	2	4	no	—
38	C₂²×D₄×C₁₁₃	—	2	4	no	—
39	C₂²×D₄×C₁₁₃	—	2	4	no	—
40	C₂²×D₄×C₁₁₃	—	2	4	no	—
41	C₂²×D₄×C₁₁₃	—	2	2	no	—
42	C₂²×D₄×C₁₁₃	—	2	8	no	—
43	C₂²×D₄×C₁₁₃	—	2	4	no	—
44	C₂²×D₄×C₁₁₃	—	4	16	no	—
45	C₂²×D₄×C₁₁₃	D₄	1	8	yes	—
46	C₂²×D₄×C₁₁₃	C₂³	1	4	yes	—
47	C₂²×D₄×C₁₁₃	C₂³	1	1	yes	characteristic
48	C₂³×C₄×C₁₁₃	—	2	1	no	—
49	C₂³×C₄×C₁₁₃	D₄	1	2	yes	—
50	C₂³×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
51	AGL(1,ℤ/8)×C₁₁₃	—	2	64	no	—
52	C₂×SD₁₆×C₁₁₃	—	2	8	no	—
53	C₂×SD₁₆×C₁₁₃	—	2	8	no	—
54	C₂×SD₁₆×C₁₁₃	—	2	8	no	—
55	C₂×D₈×C₁₁₃	—	2	8	no	—
56	C₂×D₈×C₁₁₃	—	2	8	no	—
57	C₂×D₈×C₁₁₃	—	2	8	no	—
58	C₂×M₁₆×C₁₁₃	—	2	4	no	—
59	C₂×M₁₆×C₁₁₃	—	2	4	no	—
60	C₂×M₁₆×C₁₁₃	D₄	1	8	yes	complemented
61	C₂²×C₈×C₁₁₃	C₂³	1	4	yes	—
62	#127×C₁₁₃	—	2	16	no	—
63	#127×C₁₁₃	—	2	16	no	—
64	#117×C₁₁₃	—	2	4	no	—
65	C₂×C₄⋊C₄×C₁₁₃	C₂³	1	2	yes	—
66	C₂×C₂²⋊C₄×C₁₁₃	—	2	1	no	—
67	C₂×C₂²⋊C₄×C₁₁₃	—	2	4	no	—
68	C₂×C₄²×C₁₁₃	—	2	4	no	—
69	C₂×C₄²×C₁₁₃	—	2	1	no	—
70	C₂×C₄²×C₁₁₃	D₄	1	2	yes	—
71	C₂×C₄²×C₁₁₃	C₂³	1	2	yes	—
72	#102×C₁₁₃	—	2	16	no	—
73	#95×C₁₁₃	—	2	4	no	—
74	C₂⁴×C₁₁₃	—	2	8	no	—
75	C₂⁴×C₁₁₃	—	2	1	no	—
76	C₂⁴×C₁₁₃	—	2	4	no	—
77	C₂⁴×C₁₁₃	—	4	16	no	—
78	C₂⁴×C₁₁₃	—	4	4	no	—
79	C₂⁴×C₁₁₃	—	4	4	no	—
80	C₂⁴×C₁₁₃	C₂×D₄	1	1	yes	characteristic
81	#41×C₁₁₃	—	2	16	no	—
82	#41×C₁₁₃	—	4	8	no	—
83	C₂×Q₈×C₁₁₃	—	2	1	no	—
84	C₂×Q₈×C₁₁₃	—	2	2	no	—
85	C₂×Q₈×C₁₁₃	—	2	2	no	—
86	C₂×Q₈×C₁₁₃	C₂×D₄	1	2	yes	complemented
87	C₂×D₄×C₁₁₃	—	2	32	no	—
88	C₂×D₄×C₁₁₃	—	2	4	no	—
89	C₂×D₄×C₁₁₃	—	2	16	no	—
90	C₂×D₄×C₁₁₃	—	2	8	no	—
91	C₂×D₄×C₁₁₃	—	2	16	no	—
92	C₂×D₄×C₁₁₃	—	2	4	no	—
93	C₂×D₄×C₁₁₃	—	2	2	no	—
94	C₂×D₄×C₁₁₃	—	2	2	no	—
95	C₂×D₄×C₁₁₃	—	2	4	no	—
96	C₂×D₄×C₁₁₃	—	2	4	no	—
97	C₂×D₄×C₁₁₃	—	2	4	no	—
98	C₂×D₄×C₁₁₃	—	2	4	no	—
99	C₂×D₄×C₁₁₃	—	2	1	no	—
100	C₂×D₄×C₁₁₃	—	4	64	no	—
101	C₂×D₄×C₁₁₃	—	4	16	no	—
102	C₂×D₄×C₁₁₃	—	4	2	no	—
103	C₂×D₄×C₁₁₃	—	4	2	no	—
104	C₂×D₄×C₁₁₃	—	4	8	no	—
105	C₂×D₄×C₁₁₃	—	4	8	no	—
106	C₂×D₄×C₁₁₃	—	4	4	no	—
107	C₂×D₄×C₁₁₃	—	4	4	no	—
108	C₂×D₄×C₁₁₃	—	4	16	no	—
109	C₂×D₄×C₁₁₃	C₂×D₄	1	8	yes	—
110	C₂×D₄×C₁₁₃	C₂×D₄	1	2	yes	complemented
111	C₂²×C₄×C₁₁₃	—	2	4	no	—
112	C₂²×C₄×C₁₁₃	—	2	8	no	—
113	C₂²×C₄×C₁₁₃	—	2	1	no	—
114	C₂²×C₄×C₁₁₃	—	2	4	no	—
115	C₂²×C₄×C₁₁₃	—	2	2	no	—
116	C₂²×C₄×C₁₁₃	—	4	2	no	—
117	C₂²×C₄×C₁₁₃	—	4	2	no	—
118	C₂²×C₄×C₁₁₃	C₂×D₄	1	4	yes	—
119	C₂²×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
120	C₂²×C₄×C₁₁₃	C₂×D₄	1	4	yes	—
121	C₂²×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
122	C₂²×C₄×C₁₁₃	C₂×D₄	1	4	yes	—
123	C₂²×C₄×C₁₁₃	#41	1	2	yes	—
124	C₂²×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
125	SD₁₆×C₁₁₃	—	2	32	no	—
126	D₈×C₁₁₃	—	2	32	no	—
127	M₁₆×C₁₁₃	—	2	16	no	—
128	C₂×C₈×C₁₁₃	—	2	4	no	—
129	C₂×C₈×C₁₁₃	—	2	4	no	—
130	C₂×C₈×C₁₁₃	C₂×D₄	1	8	yes	complemented
131	C₄⋊C₄×C₁₁₃	—	2	4	no	—
132	C₂²⋊C₄×C₁₁₃	—	4	2	no	—
133	C₂²⋊C₄×C₁₁₃	—	4	8	no	—
134	C₄²×C₁₁₃	—	2	4	no	—
135	C₄²×C₁₁₃	—	2	4	no	—
136	C₄²×C₁₁₃	—	4	2	no	—
137	C₂³×C₁₁₃	—	2	4	no	—
138	C₂³×C₁₁₃	—	2	8	no	—
139	C₂³×C₁₁₃	—	2	2	no	—
140	C₂³×C₁₁₃	—	4	32	no	—
141	C₂³×C₁₁₃	—	4	1	no	—
142	C₂³×C₁₁₃	—	4	4	no	—
143	C₂³×C₁₁₃	—	4	1	no	—
144	C₂³×C₁₁₃	—	4	4	no	—
145	C₂³×C₁₁₃	—	4	4	no	—
146	C₂³×C₁₁₃	—	4	16	no	—
147	C₂³×C₁₁₃	—	4	16	no	—
148	C₂³×C₁₁₃	—	8	16	no	—
149	C₂³×C₁₁₃	—	8	16	no	—
150	C₂³×C₁₁₃	#120	1	2	yes	—
151	C₂³×C₁₁₃	#121	1	2	yes	—
152	C₂³×C₁₁₃	#127	1	2	yes	—
153	C₂³×C₁₁₃	C₂²×D₄	1	1	yes	characteristic
154	Q₈×C₁₁₃	—	2	4	no	—
155	Q₈×C₁₁₃	—	4	2	no	—
156	D₄×C₁₁₃	—	2	4	no	—
157	D₄×C₁₁₃	—	2	8	no	—
158	D₄×C₁₁₃	—	2	8	no	—
159	D₄×C₁₁₃	—	4	64	no	—
160	D₄×C₁₁₃	—	4	8	no	—
161	D₄×C₁₁₃	—	4	4	no	—
162	D₄×C₁₁₃	—	4	4	no	—
163	D₄×C₁₁₃	—	4	2	no	—
164	D₄×C₁₁₃	—	4	16	no	—
165	C₂×C₄×C₁₁₃	—	2	16	no	—
166	C₂×C₄×C₁₁₃	—	2	4	no	—
167	C₂×C₄×C₁₁₃	—	2	4	no	—
168	C₂×C₄×C₁₁₃	—	2	2	no	—
169	C₂×C₄×C₁₁₃	—	4	1	no	—
170	C₂×C₄×C₁₁₃	—	4	1	no	—
171	C₂×C₄×C₁₁₃	—	4	8	no	—
172	C₂×C₄×C₁₁₃	—	4	2	no	—
173	C₂×C₄×C₁₁₃	—	4	4	no	—
174	C₂×C₄×C₁₁₃	#120	1	8	yes	—
175	C₂×C₄×C₁₁₃	#120	1	2	yes	—
176	C₂×C₄×C₁₁₃	#121	1	8	yes	—
177	C₂×C₄×C₁₁₃	#121	1	2	yes	—
178	C₂×C₄×C₁₁₃	#127	1	2	yes	—
179	C₂×C₄×C₁₁₃	AGL(1,ℤ/8)	1	8	yes	complemented
180	C₂×C₄×C₁₁₃	C₂²×D₄	1	1	yes	characteristic
181	C₂×C₄×C₁₁₃	C₂²×D₄	1	2	yes	—
182	C₂×C₄×C₁₁₃	C₂²×D₄	1	2	yes	—
183	C₂×C₄×C₁₁₃	C₂×#41	1	1	yes	characteristic
184	C₈×C₁₁₃	—	2	8	no	—
185	C₂²×C₁₁₃	—	2	4	no	—
186	C₂²×C₁₁₃	—	2	4	no	—
187	C₂²×C₁₁₃	—	2	4	no	—
188	C₂²×C₁₁₃	—	4	2	no	—
189	C₂²×C₁₁₃	—	4	8	no	—
190	C₂²×C₁₁₃	—	4	16	no	—
191	C₂²×C₁₁₃	—	4	4	no	—
192	C₂²×C₁₁₃	—	8	32	no	—
193	C₂²×C₁₁₃	—	8	2	no	—
194	C₂²×C₁₁₃	—	8	8	no	—
195	C₂²×C₁₁₃	—	8	2	no	—
196	C₂²×C₁₁₃	—	8	8	no	—
197	C₂²×C₁₁₃	#392	1	4	yes	—
198	C₂²×C₁₁₃	C₂×#120	1	1	yes	characteristic
199	C₂²×C₁₁₃	C₂×#121	1	1	yes	characteristic
200	C₂²×C₁₁₃	C₂×#127	1	1	yes	characteristic
201	C₂²×C₁₁₃	C₂×AGL(1,ℤ/8)	1	4	yes	—
202	C₄×C₁₁₃	—	2	8	no	—
203	C₄×C₁₁₃	—	4	2	no	—
204	C₄×C₁₁₃	—	8	2	no	—
205	C₄×C₁₁₃	#392	1	4	yes	—
206	C₄×C₁₁₃	C₂×#120	1	2	yes	—
207	C₄×C₁₁₃	C₂×#121	1	2	yes	—
208	#23430	C₁₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
209	C₂×C₁₁₃	—	2	4	no	—
210	C₂×C₁₁₃	—	8	2	no	—
211	C₂×C₁₁₃	—	8	8	no	—
212	C₂×C₁₁₃	C₂×#392	1	1	yes	characteristic
213	C₂×C₁₁₃	#3003	1	2	yes	—
214	C₂×C₁₁₃	#3038	1	2	yes	—
215	C₂×C₁₁₃	#3151	1	2	yes	—
216	C₂²×AGL(1,ℤ/8)	C₂×C₁₁₃	1	2	yes	complemented
217	C₂²×#127	C₂×C₁₁₃	1	1	yes	characteristic, complemented
218	C₂×#418	C₂×C₁₁₃	1	2	yes	complemented
219	#2297	C₂×C₁₁₃	1	1	yes	characteristic, complemented
220	#2014	C₂×C₁₁₃	1	4	yes	complemented
221	#2011	C₂×C₁₁₃	1	4	yes	complemented
222	#1728	C₂×C₁₁₃	1	1	yes	characteristic, complemented
223	C₁₁₃	#23430	1	1	yes	characteristic, direct product factor, Sylow
224	C₂²×#41	C₂²×C₁₁₃	1	1	yes	characteristic
225	C₂³×D₄	—	2	2	no	—
226	C₂³×D₄	C₂²×C₁₁₃	1	2	yes	—
227	C₂×AGL(1,ℤ/8)	—	2	8	no	—
228	C₂×AGL(1,ℤ/8)	—	2	8	no	—
229	C₂×AGL(1,ℤ/8)	—	2	8	no	—
230	C₂²×SD₁₆	C₂²×C₁₁₃	1	4	yes	complemented
231	C₂²×D₈	C₂²×C₁₁₃	1	4	yes	complemented
232	C₂²×M₁₆	C₂²×C₁₁₃	1	2	yes	—
233	C₂×#127	—	2	8	no	—
234	C₂×#127	—	2	2	no	—
235	C₂×#127	C₂²×C₁₁₃	1	4	yes	—
236	C₂×#117	C₂²×C₁₁₃	1	1	yes	characteristic
237	C₂²×C₄²	C₂²×C₁₁₃	1	1	yes	characteristic
238	#418	—	2	8	no	—
239	C₂×#102	C₂²×C₁₁₃	1	8	yes	complemented
240	#391	C₂²×C₁₁₃	1	2	yes	complemented
241	#390	C₂²×C₁₁₃	1	2	yes	complemented
242	#386	—	2	2	no	—
243	C₄×C₂²⋊C₄	—	2	2	no	—
244	#336	C₂²×C₁₁₃	1	4	yes	complemented
245	C₂⁵	—	2	2	no	—
246	C₂×#41	—	2	2	no	—
247	C₂×#41	—	2	4	no	—
248	C₂×#41	—	2	4	no	—
249	C₂×#41	D₄×C₁₁₃	1	4	yes	complemented
250	C₂²×Q₈	C₂³×C₁₁₃	1	1	yes	characteristic
251	C₂²×D₄	—	2	16	no	—
252	C₂²×D₄	—	2	1	no	—
253	C₂²×D₄	—	2	4	no	—
254	C₂²×D₄	—	2	4	no	—
255	C₂²×D₄	—	2	4	no	—
256	C₂²×D₄	—	2	4	no	—
257	C₂²×D₄	—	2	2	no	—
258	C₂²×D₄	—	2	8	no	—
259	C₂²×D₄	—	2	4	no	—
260	C₂²×D₄	—	4	16	no	—
261	C₂²×D₄	D₄×C₁₁₃	1	8	yes	—
262	C₂²×D₄	C₂³×C₁₁₃	1	4	yes	—
263	C₂²×D₄	C₂³×C₁₁₃	1	1	yes	characteristic
264	C₂³×C₄	—	2	1	no	—
265	C₂³×C₄	D₄×C₁₁₃	1	2	yes	—
266	C₂³×C₄	C₂³×C₁₁₃	1	1	yes	characteristic
267	AGL(1,ℤ/8)	—	2	64	no	—
268	C₂×SD₁₆	—	2	8	no	—
269	C₂×SD₁₆	—	2	8	no	—
270	C₂×SD₁₆	—	2	8	no	—
271	C₂×D₈	—	2	8	no	—
272	C₂×D₈	—	2	8	no	—
273	C₂×D₈	—	2	8	no	—
274	C₂×M₁₆	—	2	4	no	—
275	C₂×M₁₆	—	2	4	no	—
276	C₂×M₁₆	D₄×C₁₁₃	1	8	yes	complemented
277	C₂²×C₈	C₂³×C₁₁₃	1	4	yes	—
278	#127	—	2	16	no	—
279	#127	—	2	16	no	—
280	#117	—	2	4	no	—
281	C₂×C₄⋊C₄	C₂³×C₁₁₃	1	2	yes	—
282	C₂×C₂²⋊C₄	—	2	1	no	—
283	C₂×C₂²⋊C₄	—	2	4	no	—
284	C₂×C₄²	—	2	4	no	—
285	C₂×C₄²	—	2	1	no	—
286	C₂×C₄²	D₄×C₁₁₃	1	2	yes	—
287	C₂×C₄²	C₂³×C₁₁₃	1	2	yes	—
288	#102	—	2	16	no	—
289	#95	—	2	4	no	—
290	C₂⁴	—	2	8	no	—
291	C₂⁴	—	2	1	no	—
292	C₂⁴	—	2	4	no	—
293	C₂⁴	—	4	16	no	—
294	C₂⁴	—	4	4	no	—
295	C₂⁴	—	4	4	no	—
296	C₂⁴	C₂×D₄×C₁₁₃	1	1	yes	characteristic
297	#41	—	2	16	no	—
298	#41	—	4	8	no	—
299	C₂×Q₈	—	2	1	no	—
300	C₂×Q₈	—	2	2	no	—
301	C₂×Q₈	—	2	2	no	—
302	C₂×Q₈	C₂×D₄×C₁₁₃	1	2	yes	complemented
303	C₂×D₄	—	2	32	no	—
304	C₂×D₄	—	2	4	no	—
305	C₂×D₄	—	2	16	no	—
306	C₂×D₄	—	2	8	no	—
307	C₂×D₄	—	2	16	no	—
308	C₂×D₄	—	2	4	no	—
309	C₂×D₄	—	2	2	no	—
310	C₂×D₄	—	2	2	no	—
311	C₂×D₄	—	2	4	no	—
312	C₂×D₄	—	2	4	no	—
313	C₂×D₄	—	2	4	no	—
314	C₂×D₄	—	2	4	no	—
315	C₂×D₄	—	2	1	no	—
316	C₂×D₄	—	4	64	no	—
317	C₂×D₄	—	4	16	no	—
318	C₂×D₄	—	4	2	no	—
319	C₂×D₄	—	4	2	no	—
320	C₂×D₄	—	4	8	no	—
321	C₂×D₄	—	4	8	no	—
322	C₂×D₄	—	4	4	no	—
323	C₂×D₄	—	4	4	no	—
324	C₂×D₄	—	4	16	no	—
325	C₂×D₄	C₂×D₄×C₁₁₃	1	8	yes	—
326	C₂×D₄	C₂×D₄×C₁₁₃	1	2	yes	complemented
327	C₂²×C₄	—	2	4	no	—
328	C₂²×C₄	—	2	8	no	—
329	C₂²×C₄	—	2	1	no	—
330	C₂²×C₄	—	2	4	no	—
331	C₂²×C₄	—	2	2	no	—
332	C₂²×C₄	—	4	2	no	—
333	C₂²×C₄	—	4	2	no	—
334	C₂²×C₄	C₂×D₄×C₁₁₃	1	4	yes	—
335	C₂²×C₄	C₂×D₄×C₁₁₃	1	2	yes	—
336	C₂²×C₄	C₂×D₄×C₁₁₃	1	4	yes	—
337	C₂²×C₄	C₂×D₄×C₁₁₃	1	1	yes	characteristic
338	C₂²×C₄	C₂×D₄×C₁₁₃	1	4	yes	—
339	C₂²×C₄	#41×C₁₁₃	1	2	yes	—
340	C₂²×C₄	C₂⁴×C₁₁₃	1	1	yes	characteristic
341	SD₁₆	—	2	32	no	—
342	D₈	—	2	32	no	—
343	M₁₆	—	2	16	no	—
344	C₂×C₈	—	2	4	no	—
345	C₂×C₈	—	2	4	no	—
346	C₂×C₈	C₂×D₄×C₁₁₃	1	8	yes	complemented
347	C₄⋊C₄	—	2	4	no	—
348	C₂²⋊C₄	—	4	2	no	—
349	C₂²⋊C₄	—	4	8	no	—
350	C₄²	—	2	4	no	—
351	C₄²	—	2	4	no	—
352	C₄²	—	4	2	no	—
353	C₂³	—	2	4	no	—
354	C₂³	—	2	8	no	—
355	C₂³	—	2	2	no	—
356	C₂³	—	4	32	no	—
357	C₂³	—	4	1	no	—
358	C₂³	—	4	4	no	—
359	C₂³	—	4	1	no	—
360	C₂³	—	4	4	no	—
361	C₂³	—	4	4	no	—
362	C₂³	—	4	16	no	—
363	C₂³	—	4	16	no	—
364	C₂³	—	8	16	no	—
365	C₂³	—	8	16	no	—
366	C₂³	#120×C₁₁₃	1	2	yes	—
367	C₂³	#121×C₁₁₃	1	2	yes	—
368	C₂³	#127×C₁₁₃	1	2	yes	—
369	C₂³	C₂²×D₄×C₁₁₃	1	1	yes	characteristic
370	Q₈	—	2	4	no	—
371	Q₈	—	4	2	no	—
372	D₄	—	2	4	no	—
373	D₄	—	2	8	no	—
374	D₄	—	2	8	no	—
375	D₄	—	4	64	no	—
376	D₄	—	4	8	no	—
377	D₄	—	4	4	no	—
378	D₄	—	4	4	no	—
379	D₄	—	4	2	no	—
380	D₄	—	4	16	no	—
381	C₂×C₄	—	2	16	no	—
382	C₂×C₄	—	2	4	no	—
383	C₂×C₄	—	2	4	no	—
384	C₂×C₄	—	2	2	no	—
385	C₂×C₄	—	4	1	no	—
386	C₂×C₄	—	4	1	no	—
387	C₂×C₄	—	4	8	no	—
388	C₂×C₄	—	4	2	no	—
389	C₂×C₄	—	4	4	no	—
390	C₂×C₄	#120×C₁₁₃	1	8	yes	—
391	C₂×C₄	#120×C₁₁₃	1	2	yes	—
392	C₂×C₄	#121×C₁₁₃	1	8	yes	—
393	C₂×C₄	#121×C₁₁₃	1	2	yes	—
394	C₂×C₄	#127×C₁₁₃	1	2	yes	—
395	C₂×C₄	AGL(1,ℤ/8)×C₁₁₃	1	8	yes	complemented
396	C₂×C₄	C₂²×D₄×C₁₁₃	1	1	yes	characteristic
397	C₂×C₄	C₂²×D₄×C₁₁₃	1	2	yes	—
398	C₂×C₄	C₂²×D₄×C₁₁₃	1	2	yes	—
399	C₂×C₄	C₂×#41×C₁₁₃	1	1	yes	characteristic
400	C₈	—	2	8	no	—
401	C₂²	—	2	4	no	—
402	C₂²	—	2	4	no	—
403	C₂²	—	2	4	no	—
404	C₂²	—	4	2	no	—
405	C₂²	—	4	8	no	—
406	C₂²	—	4	16	no	—
407	C₂²	—	4	4	no	—
408	C₂²	—	8	32	no	—
409	C₂²	—	8	8	no	—
410	C₂²	—	8	2	no	—
411	C₂²	—	8	2	no	—
412	C₂²	—	8	8	no	—
413	C₂²	#392×C₁₁₃	1	4	yes	—
414	C₂²	C₂×#120×C₁₁₃	1	1	yes	characteristic
415	C₂²	C₂×#121×C₁₁₃	1	1	yes	characteristic
416	C₂²	C₂×#127×C₁₁₃	1	1	yes	characteristic
417	C₂²	C₂×AGL(1,ℤ/8)×C₁₁₃	1	4	yes	—
418	C₄	—	2	8	no	—
419	C₄	—	4	2	no	—
420	C₄	—	8	2	no	—
421	C₄	#392×C₁₁₃	1	4	yes	—
422	C₄	C₂×#120×C₁₁₃	1	2	yes	—
423	C₄	C₂×#121×C₁₁₃	1	2	yes	—
424	C₂	—	2	4	no	—
425	C₂	—	8	8	no	—
426	C₂	—	8	2	no	—
427	C₂	C₂×#392×C₁₁₃	1	1	yes	characteristic
428	C₂	C₁₁₃×#3003	1	2	yes	—
429	C₂	C₁₁₃×#3038	1	2	yes	—
430	C₂	C₁₁₃×#3151	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₁₃
LCS	G ▷ C₂²×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
1. C₂²×AGL(1,ℤ/8)×C₁₁₃	C₂	semidirect
2. C₂²×#127×C₁₁₃	C₂	semidirect
3. C₂×#418×C₁₁₃	C₂	semidirect
4. C₁₁₃×#2297	C₂	semidirect
5. C₁₁₃×#2014	C₂	semidirect
6. C₁₁₃×#2011	C₂	semidirect
7. C₁₁₃×#1728	C₂	semidirect
14. C₂²×SD₁₆×C₁₁₃	C₂²	semidirect
15. C₂²×D₈×C₁₁₃	C₂²	semidirect
23. C₂×#102×C₁₁₃	C₂²	semidirect
24. #391×C₁₁₃	C₂²	semidirect
25. #390×C₁₁₃	C₂²	semidirect
28. #336×C₁₁₃	C₂²	semidirect
33. C₂×#41×C₁₁₃	D₄	semidirect
60. C₂×M₁₆×C₁₁₃	D₄	semidirect
86. C₂×Q₈×C₁₁₃	C₂×D₄	semidirect
110. C₂×D₄×C₁₁₃	C₂×D₄	semidirect
130. C₂×C₈×C₁₁₃	C₂×D₄	semidirect
179. C₂×C₄×C₁₁₃	AGL(1,ℤ/8)	semidirect
208. #23430	C₁₁₃	direct (N × H)
216. C₂²×AGL(1,ℤ/8)	C₂×C₁₁₃	semidirect
217. C₂²×#127	C₂×C₁₁₃	semidirect
218. C₂×#418	C₂×C₁₁₃	semidirect
219. #2297	C₂×C₁₁₃	semidirect
220. #2014	C₂×C₁₁₃	semidirect
221. #2011	C₂×C₁₁₃	semidirect
222. #1728	C₂×C₁₁₃	semidirect
223. C₁₁₃	#23430	direct (N × H)
230. C₂²×SD₁₆	C₂²×C₁₁₃	semidirect
231. C₂²×D₈	C₂²×C₁₁₃	semidirect
239. C₂×#102	C₂²×C₁₁₃	semidirect
240. #391	C₂²×C₁₁₃	semidirect
241. #390	C₂²×C₁₁₃	semidirect
244. #336	C₂²×C₁₁₃	semidirect
249. C₂×#41	D₄×C₁₁₃	semidirect
276. C₂×M₁₆	D₄×C₁₁₃	semidirect
302. C₂×Q₈	C₂×D₄×C₁₁₃	semidirect
326. C₂×D₄	C₂×D₄×C₁₁₃	semidirect
346. C₂×C₈	C₂×D₄×C₁₁₃	semidirect
395. C₂×C₄	AGL(1,ℤ/8)×C₁₁₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×AGL(1,ℤ/8)×C₁₁₃	2
C₂²×#127×C₁₁₃	2
C₂×#418×C₁₁₃	2
C₁₁₃×#2297	2
C₁₁₃×#2014	2
C₁₁₃×#2011	2
C₁₁₃×#1728	2
#23430	113

Subgroups — C113×#23430 #43291450

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁₃×#23430 #43291450