Subgroups — C₁₁₃×#24254 #43292274

Order 28928.

1476 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#550×C₁₁₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#460×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#452×C₁₁₃	C₂	1	2	yes	maximal, complemented
4	C₂×#441×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₁₃×#2536	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₁₃×#2034	C₂	1	2	yes	maximal, complemented
7	C₁₁₃×#2000	C₂	1	2	yes	maximal, complemented
8	C₁₁₃×#1961	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₁₃×#1955	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₁₃×#1805	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₁₃×#1788	C₂	1	2	yes	maximal, complemented
12	C₂²×#41×C₁₁₃	C₂²	1	2	yes	—
13	C₂²×Q₁₆×C₁₁₃	C₂²	1	1	yes	characteristic
14	C₂²×SD₁₆×C₁₁₃	C₂²	1	2	yes	—
15	#550×C₁₁₃	—	2	4	no	—
16	#550×C₁₁₃	—	2	4	no	—
17	C₂×#127×C₁₁₃	—	2	1	no	—
18	C₂×#127×C₁₁₃	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊Q₈×C₁₁₃	C₂²	1	2	yes	—
20	C₂×#121×C₁₁₃	—	2	2	no	—
21	C₂×#121×C₁₁₃	C₂²	1	2	yes	—
22	C₂×C₄×Q₈×C₁₁₃	C₂²	1	1	yes	characteristic
23	C₂×C₄×Q₈×C₁₁₃	C₂²	1	1	yes	characteristic
24	C₂×C₄×D₄×C₁₁₃	—	2	1	no	—
25	C₂×C₄×D₄×C₁₁₃	C₂²	1	1	yes	characteristic
26	#460×C₁₁₃	—	2	4	no	—
27	#452×C₁₁₃	—	2	8	no	—
28	#441×C₁₁₃	—	2	4	no	—
29	C₂×#106×C₁₁₃	C₂²	1	1	yes	characteristic
30	C₂×C₄⋊C₈×C₁₁₃	C₂²	1	1	yes	characteristic
31	C₂×#103×C₁₁₃	C₂²	1	1	yes	characteristic
32	C₂×#103×C₁₁₃	C₂²	1	2	yes	—
33	C₂×#103×C₁₁₃	C₂²	1	1	yes	characteristic
34	C₂×#102×C₁₁₃	C₂²	1	2	yes	—
35	C₂×#102×C₁₁₃	C₂²	1	2	yes	—
36	C₂×#102×C₁₁₃	C₂²	1	2	yes	—
37	C₂×C₂²⋊C₈×C₁₁₃	C₂²	1	2	yes	—
38	C₂×C₂²⋊C₈×C₁₁₃	C₂²	1	2	yes	—
39	C₂×#97×C₁₁₃	C₂²	1	1	yes	characteristic
40	#398×C₁₁₃	—	2	1	no	—
41	#394×C₁₁₃	—	2	1	no	—
42	#388×C₁₁₃	—	2	2	no	—
43	#382×C₁₁₃	C₂²	1	1	yes	characteristic
44	#380×C₁₁₃	C₂²	1	2	yes	—
45	#340×C₁₁₃	C₂²	1	2	yes	—
46	C₂×#41×C₁₁₃	—	2	4	no	—
47	C₂×#41×C₁₁₃	—	2	4	no	—
48	C₂×#41×C₁₁₃	—	2	4	no	—
49	C₂×#41×C₁₁₃	—	2	4	no	—
50	C₂×#41×C₁₁₃	—	4	2	no	—
51	C₂×#41×C₁₁₃	—	4	2	no	—
52	C₂²×Q₈×C₁₁₃	C₂³	1	1	yes	characteristic
53	C₂²×Q₈×C₁₁₃	C₂³	1	1	yes	characteristic
54	C₂²×D₄×C₁₁₃	—	2	2	no	—
55	C₂²×D₄×C₁₁₃	—	2	2	no	—
56	C₂²×D₄×C₁₁₃	—	2	1	no	—
57	C₂²×D₄×C₁₁₃	—	2	1	no	—
58	C₂²×D₄×C₁₁₃	D₄	1	1	yes	characteristic
59	C₂²×D₄×C₁₁₃	C₂³	1	2	yes	—
60	C₂³×C₄×C₁₁₃	—	2	2	no	—
61	C₂³×C₄×C₁₁₃	C₂³	1	2	yes	—
62	C₂×Q₁₆×C₁₁₃	—	2	2	no	—
63	C₂×Q₁₆×C₁₁₃	—	2	2	no	—
64	C₂×Q₁₆×C₁₁₃	D₄	1	4	yes	complemented
65	C₂×SD₁₆×C₁₁₃	—	2	4	no	—
66	C₂×SD₁₆×C₁₁₃	—	2	4	no	—
67	C₂×SD₁₆×C₁₁₃	—	4	2	no	—
68	C₂²×C₈×C₁₁₃	C₂³	1	2	yes	—
69	C₂²×C₈×C₁₁₃	C₂³	1	1	yes	characteristic
70	C₂²×C₈×C₁₁₃	C₂³	1	1	yes	characteristic
71	#127×C₁₁₃	—	2	8	no	—
72	#127×C₁₁₃	—	2	2	no	—
73	#127×C₁₁₃	D₄	1	4	yes	—
74	C₂²⋊Q₈×C₁₁₃	—	2	4	no	—
75	C₂²⋊Q₈×C₁₁₃	—	4	2	no	—
76	#121×C₁₁₃	—	2	16	no	—
77	#121×C₁₁₃	—	2	4	no	—
78	#121×C₁₁₃	—	2	4	no	—
79	C₄×Q₈×C₁₁₃	—	2	2	no	—
80	C₄×Q₈×C₁₁₃	—	2	2	no	—
81	C₄×Q₈×C₁₁₃	—	2	2	no	—
82	C₄×Q₈×C₁₁₃	D₄	1	4	yes	—
83	C₄×D₄×C₁₁₃	—	2	8	no	—
84	C₄×D₄×C₁₁₃	—	2	2	no	—
85	C₄×D₄×C₁₁₃	—	2	2	no	—
86	C₂×C₄⋊C₄×C₁₁₃	—	2	1	no	—
87	C₂×C₄⋊C₄×C₁₁₃	—	2	1	no	—
88	C₂×C₄⋊C₄×C₁₁₃	—	2	2	no	—
89	C₂×C₄⋊C₄×C₁₁₃	D₄	1	1	yes	characteristic
90	C₂×C₄⋊C₄×C₁₁₃	C₂³	1	1	yes	characteristic
91	C₂×C₄⋊C₄×C₁₁₃	C₂³	1	2	yes	—
92	C₂×C₄⋊C₄×C₁₁₃	C₂³	1	1	yes	characteristic
93	C₂×C₂²⋊C₄×C₁₁₃	—	2	2	no	—
94	C₂×C₂²⋊C₄×C₁₁₃	—	2	2	no	—
95	C₂×C₂²⋊C₄×C₁₁₃	—	2	2	no	—
96	C₂×C₂²⋊C₄×C₁₁₃	D₄	1	2	yes	—
97	C₂×C₄²×C₁₁₃	—	2	1	no	—
98	C₂×C₄²×C₁₁₃	—	2	1	no	—
99	C₂×C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
100	#106×C₁₁₃	—	2	2	no	—
101	C₄⋊C₈×C₁₁₃	—	2	2	no	—
102	#103×C₁₁₃	—	2	2	no	—
103	#103×C₁₁₃	—	2	4	no	—
104	#103×C₁₁₃	—	2	2	no	—
105	#102×C₁₁₃	—	2	4	no	—
106	#102×C₁₁₃	—	2	4	no	—
107	#102×C₁₁₃	—	4	2	no	—
108	C₂²⋊C₈×C₁₁₃	—	2	4	no	—
109	C₂²⋊C₈×C₁₁₃	—	4	2	no	—
110	#97×C₁₁₃	—	2	2	no	—
111	#95×C₁₁₃	—	2	2	no	—
112	#95×C₁₁₃	—	2	1	no	—
113	#95×C₁₁₃	—	2	1	no	—
114	C₂⁴×C₁₁₃	—	2	2	no	—
115	C₂⁴×C₁₁₃	C₂×D₄	1	2	yes	—
116	#41×C₁₁₃	—	4	8	no	—
117	#41×C₁₁₃	—	4	8	no	—
118	#41×C₁₁₃	—	4	8	no	—
119	#41×C₁₁₃	—	4	8	no	—
120	C₂×Q₈×C₁₁₃	—	2	1	no	—
121	C₂×Q₈×C₁₁₃	—	2	1	no	—
122	C₂×Q₈×C₁₁₃	—	2	1	no	—
123	C₂×Q₈×C₁₁₃	—	2	1	no	—
124	C₂×Q₈×C₁₁₃	—	2	1	no	—
125	C₂×Q₈×C₁₁₃	—	2	1	no	—
126	C₂×Q₈×C₁₁₃	—	2	1	no	—
127	C₂×Q₈×C₁₁₃	—	2	1	no	—
128	C₂×Q₈×C₁₁₃	C₂×D₄	1	2	yes	—
129	C₂×Q₈×C₁₁₃	C₂×D₄	1	2	yes	—
130	C₂×Q₈×C₁₁₃	C₂×D₄	1	2	yes	—
131	C₂×Q₈×C₁₁₃	#41	1	2	yes	—
132	C₂×D₄×C₁₁₃	—	2	8	no	—
133	C₂×D₄×C₁₁₃	—	2	8	no	—
134	C₂×D₄×C₁₁₃	—	2	2	no	—
135	C₂×D₄×C₁₁₃	—	2	4	no	—
136	C₂×D₄×C₁₁₃	—	2	4	no	—
137	C₂×D₄×C₁₁₃	—	2	2	no	—
138	C₂×D₄×C₁₁₃	—	2	2	no	—
139	C₂×D₄×C₁₁₃	—	2	2	no	—
140	C₂×D₄×C₁₁₃	—	2	2	no	—
141	C₂×D₄×C₁₁₃	—	2	2	no	—
142	C₂×D₄×C₁₁₃	—	4	2	no	—
143	C₂×D₄×C₁₁₃	—	4	4	no	—
144	C₂×D₄×C₁₁₃	—	4	4	no	—
145	C₂×D₄×C₁₁₃	—	4	4	no	—
146	C₂×D₄×C₁₁₃	—	8	2	no	—
147	C₂×D₄×C₁₁₃	—	8	2	no	—
148	C₂×D₄×C₁₁₃	—	8	2	no	—
149	C₂×D₄×C₁₁₃	C₂×D₄	1	4	yes	—
150	C₂²×C₄×C₁₁₃	—	2	1	no	—
151	C₂²×C₄×C₁₁₃	—	2	8	no	—
152	C₂²×C₄×C₁₁₃	—	2	1	no	—
153	C₂²×C₄×C₁₁₃	—	2	1	no	—
154	C₂²×C₄×C₁₁₃	—	2	2	no	—
155	C₂²×C₄×C₁₁₃	—	2	2	no	—
156	C₂²×C₄×C₁₁₃	—	2	2	no	—
157	C₂²×C₄×C₁₁₃	—	2	2	no	—
158	C₂²×C₄×C₁₁₃	—	2	2	no	—
159	C₂²×C₄×C₁₁₃	—	2	2	no	—
160	C₂²×C₄×C₁₁₃	—	2	2	no	—
161	C₂²×C₄×C₁₁₃	—	2	1	no	—
162	C₂²×C₄×C₁₁₃	—	4	4	no	—
163	C₂²×C₄×C₁₁₃	—	4	4	no	—
164	C₂²×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
165	C₂²×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
166	C₂²×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
167	C₂²×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
168	Q₁₆×C₁₁₃	—	2	8	no	—
169	SD₁₆×C₁₁₃	—	4	8	no	—
170	C₂×C₈×C₁₁₃	—	2	1	no	—
171	C₂×C₈×C₁₁₃	—	2	1	no	—
172	C₂×C₈×C₁₁₃	—	2	1	no	—
173	C₂×C₈×C₁₁₃	—	2	2	no	—
174	C₂×C₈×C₁₁₃	—	2	2	no	—
175	C₂×C₈×C₁₁₃	—	2	1	no	—
176	C₂×C₈×C₁₁₃	—	2	2	no	—
177	C₂×C₈×C₁₁₃	—	2	1	no	—
178	C₂×C₈×C₁₁₃	C₂×D₄	1	2	yes	—
179	C₄⋊C₄×C₁₁₃	—	2	4	no	—
180	C₄⋊C₄×C₁₁₃	—	2	2	no	—
181	C₄⋊C₄×C₁₁₃	—	2	1	no	—
182	C₄⋊C₄×C₁₁₃	—	2	1	no	—
183	C₄⋊C₄×C₁₁₃	—	2	2	no	—
184	C₄⋊C₄×C₁₁₃	—	2	1	no	—
185	C₄⋊C₄×C₁₁₃	—	2	2	no	—
186	C₄⋊C₄×C₁₁₃	—	4	4	no	—
187	C₄⋊C₄×C₁₁₃	—	4	2	no	—
188	C₄⋊C₄×C₁₁₃	C₂×D₄	1	2	yes	—
189	C₂²⋊C₄×C₁₁₃	—	2	8	no	—
190	C₂²⋊C₄×C₁₁₃	—	2	4	no	—
191	C₂²⋊C₄×C₁₁₃	—	4	4	no	—
192	C₂²⋊C₄×C₁₁₃	—	4	4	no	—
193	C₄²×C₁₁₃	—	2	4	no	—
194	C₄²×C₁₁₃	—	4	2	no	—
195	C₄²×C₁₁₃	C₂×D₄	1	2	yes	—
196	C₄²×C₁₁₃	C₂×D₄	1	2	yes	—
197	C₂³×C₁₁₃	—	2	2	no	—
198	C₂³×C₁₁₃	—	2	4	no	—
199	C₂³×C₁₁₃	—	2	2	no	—
200	C₂³×C₁₁₃	—	2	2	no	—
201	C₂³×C₁₁₃	—	2	2	no	—
202	C₂³×C₁₁₃	—	2	2	no	—
203	C₂³×C₁₁₃	—	2	2	no	—
204	C₂³×C₁₁₃	—	2	2	no	—
205	C₂³×C₁₁₃	—	4	2	no	—
206	C₂³×C₁₁₃	—	4	2	no	—
207	C₂³×C₁₁₃	—	4	2	no	—
208	C₂³×C₁₁₃	—	4	2	no	—
209	C₂³×C₁₁₃	—	4	2	no	—
210	C₂³×C₁₁₃	—	4	2	no	—
211	C₂³×C₁₁₃	C₂²×D₄	1	1	yes	characteristic
212	Q₈×C₁₁₃	—	2	2	no	—
213	Q₈×C₁₁₃	—	2	2	no	—
214	Q₈×C₁₁₃	—	2	2	no	—
215	Q₈×C₁₁₃	—	2	2	no	—
216	Q₈×C₁₁₃	—	2	2	no	—
217	Q₈×C₁₁₃	—	4	1	no	—
218	Q₈×C₁₁₃	—	4	1	no	—
219	Q₈×C₁₁₃	#121	1	4	yes	—
220	D₄×C₁₁₃	—	2	4	no	—
221	D₄×C₁₁₃	—	2	4	no	—
222	D₄×C₁₁₃	—	4	4	no	—
223	D₄×C₁₁₃	—	4	8	no	—
224	D₄×C₁₁₃	—	4	16	no	—
225	D₄×C₁₁₃	—	4	2	no	—
226	D₄×C₁₁₃	—	4	2	no	—
227	D₄×C₁₁₃	—	8	4	no	—
228	D₄×C₁₁₃	—	8	8	no	—
229	C₂×C₄×C₁₁₃	—	2	2	no	—
230	C₂×C₄×C₁₁₃	—	2	2	no	—
231	C₂×C₄×C₁₁₃	—	2	2	no	—
232	C₂×C₄×C₁₁₃	—	2	2	no	—
233	C₂×C₄×C₁₁₃	—	2	2	no	—
234	C₂×C₄×C₁₁₃	—	2	2	no	—
235	C₂×C₄×C₁₁₃	—	2	2	no	—
236	C₂×C₄×C₁₁₃	—	2	2	no	—
237	C₂×C₄×C₁₁₃	—	2	2	no	—
238	C₂×C₄×C₁₁₃	—	2	2	no	—
239	C₂×C₄×C₁₁₃	—	4	16	no	—
240	C₂×C₄×C₁₁₃	—	4	1	no	—
241	C₂×C₄×C₁₁₃	—	4	1	no	—
242	C₂×C₄×C₁₁₃	—	4	2	no	—
243	C₂×C₄×C₁₁₃	—	4	2	no	—
244	C₂×C₄×C₁₁₃	—	4	2	no	—
245	C₂×C₄×C₁₁₃	—	4	2	no	—
246	C₂×C₄×C₁₁₃	—	4	2	no	—
247	C₂×C₄×C₁₁₃	—	4	1	no	—
248	C₂×C₄×C₁₁₃	—	4	2	no	—
249	C₂×C₄×C₁₁₃	—	4	2	no	—
250	C₂×C₄×C₁₁₃	—	4	2	no	—
251	C₂×C₄×C₁₁₃	—	4	2	no	—
252	C₂×C₄×C₁₁₃	#120	1	2	yes	—
253	C₂×C₄×C₁₁₃	#120	1	2	yes	—
254	C₂×C₄×C₁₁₃	C₂²×D₄	1	1	yes	characteristic
255	C₂×C₄×C₁₁₃	C₂²×D₄	1	1	yes	characteristic
256	C₂×C₄×C₁₁₃	C₂²×D₄	1	1	yes	characteristic
257	C₂×C₄×C₁₁₃	C₂×#41	1	1	yes	characteristic
258	C₂×C₄×C₁₁₃	H(5,𝔽₂)	1	1	yes	characteristic
259	C₂×C₄×C₁₁₃	H(5,𝔽₂)	1	1	yes	characteristic
260	C₈×C₁₁₃	—	2	2	no	—
261	C₈×C₁₁₃	—	4	1	no	—
262	C₈×C₁₁₃	—	4	2	no	—
263	C₂²×C₁₁₃	—	4	4	no	—
264	C₂²×C₁₁₃	—	4	2	no	—
265	C₂²×C₁₁₃	—	4	2	no	—
266	C₂²×C₁₁₃	—	4	4	no	—
267	C₂²×C₁₁₃	—	4	2	no	—
268	C₂²×C₁₁₃	—	4	4	no	—
269	C₂²×C₁₁₃	—	4	2	no	—
270	C₂²×C₁₁₃	—	4	2	no	—
271	C₂²×C₁₁₃	—	4	2	no	—
272	C₂²×C₁₁₃	—	4	2	no	—
273	C₂²×C₁₁₃	—	8	2	no	—
274	C₂²×C₁₁₃	—	8	2	no	—
275	C₂²×C₁₁₃	—	8	2	no	—
276	C₂²×C₁₁₃	—	8	2	no	—
277	C₂²×C₁₁₃	C₂×#120	1	1	yes	characteristic
278	C₂²×C₁₁₃	D₄²	1	1	yes	characteristic
279	C₂²×C₁₁₃	#546	1	1	yes	characteristic
280	C₂²×C₁₁₃	#575	1	2	yes	—
281	C₂²×C₁₁₃	#577	1	2	yes	—
282	C₄×C₁₁₃	—	2	4	no	—
283	C₄×C₁₁₃	—	2	2	no	—
284	C₄×C₁₁₃	—	2	2	no	—
285	C₄×C₁₁₃	—	4	2	no	—
286	C₄×C₁₁₃	—	4	2	no	—
287	C₄×C₁₁₃	—	4	2	no	—
288	C₄×C₁₁₃	—	8	1	no	—
289	C₄×C₁₁₃	—	8	2	no	—
290	C₄×C₁₁₃	C₂×#121	1	1	yes	characteristic
291	C₄×C₁₁₃	#535	1	1	yes	characteristic
292	C₄×C₁₁₃	D₄²	1	1	yes	characteristic
293	C₄×C₁₁₃	#546	1	1	yes	characteristic
294	#24254	C₁₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
295	C₂×C₁₁₃	—	8	2	no	—
296	C₂×C₁₁₃	—	8	4	no	—
297	C₂×C₁₁₃	#2408	1	1	yes	characteristic
298	C₂×C₁₁₃	#3005	1	1	yes	characteristic
299	C₂×C₁₁₃	#3009	1	1	yes	characteristic
300	C₂×C₁₁₃	#3271	1	2	yes	—
301	C₂×C₁₁₃	#3317	1	2	yes	—
302	C₂×#550	C₂×C₁₁₃	1	1	yes	characteristic
303	C₂×#460	C₂×C₁₁₃	1	1	yes	characteristic, complemented
304	C₂×#452	C₂×C₁₁₃	1	2	yes	complemented
305	C₂×#441	C₂×C₁₁₃	1	1	yes	characteristic, complemented
306	#2536	C₂×C₁₁₃	1	1	yes	characteristic, complemented
307	#2034	C₂×C₁₁₃	1	2	yes	complemented
308	#2000	C₂×C₁₁₃	1	2	yes	complemented
309	#1961	C₂×C₁₁₃	1	1	yes	characteristic, complemented
310	#1955	C₂×C₁₁₃	1	1	yes	characteristic, complemented
311	#1805	C₂×C₁₁₃	1	1	yes	characteristic, complemented
312	#1788	C₂×C₁₁₃	1	2	yes	complemented
313	C₁₁₃	#24254	1	1	yes	characteristic, direct product factor, Sylow
314	C₂²×#41	C₂²×C₁₁₃	1	2	yes	—
315	C₂²×Q₁₆	C₂²×C₁₁₃	1	1	yes	characteristic
316	C₂²×SD₁₆	C₂²×C₁₁₃	1	2	yes	—
317	#550	—	2	4	no	—
318	#550	—	2	4	no	—
319	C₂×#127	—	2	1	no	—
320	C₂×#127	C₂²×C₁₁₃	1	1	yes	characteristic
321	C₂×C₂²⋊Q₈	C₂²×C₁₁₃	1	2	yes	—
322	C₂×#121	—	2	2	no	—
323	C₂×#121	C₂²×C₁₁₃	1	2	yes	—
324	C₂×C₄×Q₈	C₂²×C₁₁₃	1	1	yes	characteristic
325	C₂×C₄×Q₈	C₂²×C₁₁₃	1	1	yes	characteristic
326	C₂×C₄×D₄	—	2	1	no	—
327	C₂×C₄×D₄	C₂²×C₁₁₃	1	1	yes	characteristic
328	#460	—	2	4	no	—
329	#452	—	2	8	no	—
330	#441	—	2	4	no	—
331	C₂×#106	C₂²×C₁₁₃	1	1	yes	characteristic
332	C₂×C₄⋊C₈	C₂²×C₁₁₃	1	1	yes	characteristic
333	C₂×#103	C₂²×C₁₁₃	1	2	yes	—
334	C₂×#103	C₂²×C₁₁₃	1	1	yes	characteristic
335	C₂×#103	C₂²×C₁₁₃	1	1	yes	characteristic
336	C₂×#102	C₂²×C₁₁₃	1	2	yes	—
337	C₂×#102	C₂²×C₁₁₃	1	2	yes	—
338	C₂×#102	C₂²×C₁₁₃	1	2	yes	—
339	C₂×C₂²⋊C₈	C₂²×C₁₁₃	1	2	yes	—
340	C₂×C₂²⋊C₈	C₂²×C₁₁₃	1	2	yes	—
341	C₂×#97	C₂²×C₁₁₃	1	1	yes	characteristic
342	#398	—	2	1	no	—
343	#394	—	2	1	no	—
344	#388	—	2	2	no	—
345	#382	C₂²×C₁₁₃	1	1	yes	characteristic
346	#380	C₂²×C₁₁₃	1	2	yes	—
347	#340	C₂²×C₁₁₃	1	2	yes	—
348	C₂×#41	—	2	4	no	—
349	C₂×#41	—	2	4	no	—
350	C₂×#41	—	2	4	no	—
351	C₂×#41	—	2	4	no	—
352	C₂×#41	—	4	2	no	—
353	C₂×#41	—	4	2	no	—
354	C₂²×Q₈	C₂³×C₁₁₃	1	1	yes	characteristic
355	C₂²×Q₈	C₂³×C₁₁₃	1	1	yes	characteristic
356	C₂²×D₄	—	2	2	no	—
357	C₂²×D₄	—	2	2	no	—
358	C₂²×D₄	—	2	1	no	—
359	C₂²×D₄	—	2	1	no	—
360	C₂²×D₄	D₄×C₁₁₃	1	1	yes	characteristic
361	C₂²×D₄	C₂³×C₁₁₃	1	2	yes	—
362	C₂³×C₄	—	2	2	no	—
363	C₂³×C₄	C₂³×C₁₁₃	1	2	yes	—
364	C₂×Q₁₆	—	2	2	no	—
365	C₂×Q₁₆	—	2	2	no	—
366	C₂×Q₁₆	D₄×C₁₁₃	1	4	yes	complemented
367	C₂×SD₁₆	—	2	4	no	—
368	C₂×SD₁₆	—	2	4	no	—
369	C₂×SD₁₆	—	4	2	no	—
370	C₂²×C₈	C₂³×C₁₁₃	1	2	yes	—
371	C₂²×C₈	C₂³×C₁₁₃	1	1	yes	characteristic
372	C₂²×C₈	C₂³×C₁₁₃	1	1	yes	characteristic
373	#127	—	2	8	no	—
374	#127	—	2	2	no	—
375	#127	D₄×C₁₁₃	1	4	yes	—
376	C₂²⋊Q₈	—	2	4	no	—
377	C₂²⋊Q₈	—	4	2	no	—
378	#121	—	2	16	no	—
379	#121	—	2	4	no	—
380	#121	—	2	4	no	—
381	C₄×Q₈	—	2	2	no	—
382	C₄×Q₈	—	2	2	no	—
383	C₄×Q₈	—	2	2	no	—
384	C₄×Q₈	D₄×C₁₁₃	1	4	yes	—
385	C₄×D₄	—	2	8	no	—
386	C₄×D₄	—	2	2	no	—
387	C₄×D₄	—	2	2	no	—
388	C₂×C₄⋊C₄	—	2	1	no	—
389	C₂×C₄⋊C₄	—	2	1	no	—
390	C₂×C₄⋊C₄	—	2	2	no	—
391	C₂×C₄⋊C₄	D₄×C₁₁₃	1	1	yes	characteristic
392	C₂×C₄⋊C₄	C₂³×C₁₁₃	1	1	yes	characteristic
393	C₂×C₄⋊C₄	C₂³×C₁₁₃	1	2	yes	—
394	C₂×C₄⋊C₄	C₂³×C₁₁₃	1	1	yes	characteristic
395	C₂×C₂²⋊C₄	—	2	2	no	—
396	C₂×C₂²⋊C₄	—	2	2	no	—
397	C₂×C₂²⋊C₄	—	2	2	no	—
398	C₂×C₂²⋊C₄	D₄×C₁₁₃	1	2	yes	—
399	C₂×C₄²	—	2	1	no	—
400	C₂×C₄²	—	2	1	no	—
401	C₂×C₄²	C₂³×C₁₁₃	1	1	yes	characteristic
402	#106	—	2	2	no	—
403	C₄⋊C₈	—	2	2	no	—
404	#103	—	2	2	no	—
405	#103	—	2	4	no	—
406	#103	—	2	2	no	—
407	#102	—	2	4	no	—
408	#102	—	2	4	no	—
409	#102	—	4	2	no	—
410	C₂²⋊C₈	—	2	4	no	—
411	C₂²⋊C₈	—	4	2	no	—
412	#97	—	2	2	no	—
413	#95	—	2	2	no	—
414	#95	—	2	1	no	—
415	#95	—	2	1	no	—
416	C₂⁴	—	2	2	no	—
417	C₂⁴	C₂×D₄×C₁₁₃	1	2	yes	—
418	#41	—	4	8	no	—
419	#41	—	4	8	no	—
420	#41	—	4	8	no	—
421	#41	—	4	8	no	—
422	C₂×Q₈	—	2	1	no	—
423	C₂×Q₈	—	2	1	no	—
424	C₂×Q₈	—	2	1	no	—
425	C₂×Q₈	—	2	1	no	—
426	C₂×Q₈	—	2	1	no	—
427	C₂×Q₈	—	2	1	no	—
428	C₂×Q₈	—	2	1	no	—
429	C₂×Q₈	—	2	1	no	—
430	C₂×Q₈	C₂×D₄×C₁₁₃	1	2	yes	—
431	C₂×Q₈	C₂×D₄×C₁₁₃	1	2	yes	—
432	C₂×Q₈	C₂×D₄×C₁₁₃	1	2	yes	—
433	C₂×Q₈	#41×C₁₁₃	1	2	yes	—
434	C₂×D₄	—	2	8	no	—
435	C₂×D₄	—	2	8	no	—
436	C₂×D₄	—	2	2	no	—
437	C₂×D₄	—	2	4	no	—
438	C₂×D₄	—	2	4	no	—
439	C₂×D₄	—	2	2	no	—
440	C₂×D₄	—	2	2	no	—
441	C₂×D₄	—	2	2	no	—
442	C₂×D₄	—	2	2	no	—
443	C₂×D₄	—	2	2	no	—
444	C₂×D₄	—	4	2	no	—
445	C₂×D₄	—	4	4	no	—
446	C₂×D₄	—	4	4	no	—
447	C₂×D₄	—	4	4	no	—
448	C₂×D₄	—	8	2	no	—
449	C₂×D₄	—	8	2	no	—
450	C₂×D₄	—	8	2	no	—
451	C₂×D₄	C₂×D₄×C₁₁₃	1	4	yes	—
452	C₂²×C₄	—	2	1	no	—
453	C₂²×C₄	—	2	8	no	—
454	C₂²×C₄	—	2	1	no	—
455	C₂²×C₄	—	2	1	no	—
456	C₂²×C₄	—	2	2	no	—
457	C₂²×C₄	—	2	2	no	—
458	C₂²×C₄	—	2	2	no	—
459	C₂²×C₄	—	2	2	no	—
460	C₂²×C₄	—	2	2	no	—
461	C₂²×C₄	—	2	2	no	—
462	C₂²×C₄	—	2	2	no	—
463	C₂²×C₄	—	2	1	no	—
464	C₂²×C₄	—	4	4	no	—
465	C₂²×C₄	—	4	4	no	—
466	C₂²×C₄	C₂×D₄×C₁₁₃	1	2	yes	—
467	C₂²×C₄	C₂×D₄×C₁₁₃	1	1	yes	characteristic
468	C₂²×C₄	C₂×D₄×C₁₁₃	1	1	yes	characteristic
469	C₂²×C₄	C₂⁴×C₁₁₃	1	1	yes	characteristic
470	Q₁₆	—	2	8	no	—
471	SD₁₆	—	4	8	no	—
472	C₂×C₈	—	2	1	no	—
473	C₂×C₈	—	2	1	no	—
474	C₂×C₈	—	2	1	no	—
475	C₂×C₈	—	2	2	no	—
476	C₂×C₈	—	2	2	no	—
477	C₂×C₈	—	2	1	no	—
478	C₂×C₈	—	2	2	no	—
479	C₂×C₈	—	2	1	no	—
480	C₂×C₈	C₂×D₄×C₁₁₃	1	2	yes	—
481	C₄⋊C₄	—	2	4	no	—
482	C₄⋊C₄	—	2	2	no	—
483	C₄⋊C₄	—	2	1	no	—
484	C₄⋊C₄	—	2	1	no	—
485	C₄⋊C₄	—	2	2	no	—
486	C₄⋊C₄	—	2	1	no	—
487	C₄⋊C₄	—	2	2	no	—
488	C₄⋊C₄	—	4	4	no	—
489	C₄⋊C₄	—	4	2	no	—
490	C₄⋊C₄	C₂×D₄×C₁₁₃	1	2	yes	—
491	C₂²⋊C₄	—	2	8	no	—
492	C₂²⋊C₄	—	2	4	no	—
493	C₂²⋊C₄	—	4	4	no	—
494	C₂²⋊C₄	—	4	4	no	—
495	C₄²	—	2	4	no	—
496	C₄²	—	4	2	no	—
497	C₄²	C₂×D₄×C₁₁₃	1	2	yes	—
498	C₄²	C₂×D₄×C₁₁₃	1	2	yes	—
499	C₂³	—	2	2	no	—
500	C₂³	—	2	4	no	—
501	C₂³	—	2	2	no	—
502	C₂³	—	2	2	no	—
503	C₂³	—	2	2	no	—
504	C₂³	—	2	2	no	—
505	C₂³	—	2	2	no	—
506	C₂³	—	2	2	no	—
507	C₂³	—	4	2	no	—
508	C₂³	—	4	2	no	—
509	C₂³	—	4	2	no	—
510	C₂³	—	4	2	no	—
511	C₂³	—	4	2	no	—
512	C₂³	—	4	2	no	—
513	C₂³	C₂²×D₄×C₁₁₃	1	1	yes	characteristic
514	Q₈	—	2	2	no	—
515	Q₈	—	2	2	no	—
516	Q₈	—	2	2	no	—
517	Q₈	—	2	2	no	—
518	Q₈	—	2	2	no	—
519	Q₈	—	4	1	no	—
520	Q₈	—	4	1	no	—
521	Q₈	#121×C₁₁₃	1	4	yes	—
522	D₄	—	2	4	no	—
523	D₄	—	2	4	no	—
524	D₄	—	4	16	no	—
525	D₄	—	4	4	no	—
526	D₄	—	4	8	no	—
527	D₄	—	4	2	no	—
528	D₄	—	4	2	no	—
529	D₄	—	8	4	no	—
530	D₄	—	8	8	no	—
531	C₂×C₄	—	2	2	no	—
532	C₂×C₄	—	2	2	no	—
533	C₂×C₄	—	2	2	no	—
534	C₂×C₄	—	2	2	no	—
535	C₂×C₄	—	2	2	no	—
536	C₂×C₄	—	2	2	no	—
537	C₂×C₄	—	2	2	no	—
538	C₂×C₄	—	2	2	no	—
539	C₂×C₄	—	2	2	no	—
540	C₂×C₄	—	2	2	no	—
541	C₂×C₄	—	4	16	no	—
542	C₂×C₄	—	4	1	no	—
543	C₂×C₄	—	4	1	no	—
544	C₂×C₄	—	4	2	no	—
545	C₂×C₄	—	4	2	no	—
546	C₂×C₄	—	4	2	no	—
547	C₂×C₄	—	4	2	no	—
548	C₂×C₄	—	4	2	no	—
549	C₂×C₄	—	4	1	no	—
550	C₂×C₄	—	4	2	no	—
551	C₂×C₄	—	4	2	no	—
552	C₂×C₄	—	4	2	no	—
553	C₂×C₄	—	4	2	no	—
554	C₂×C₄	#120×C₁₁₃	1	2	yes	—
555	C₂×C₄	#120×C₁₁₃	1	2	yes	—
556	C₂×C₄	C₂²×D₄×C₁₁₃	1	1	yes	characteristic
557	C₂×C₄	C₂²×D₄×C₁₁₃	1	1	yes	characteristic
558	C₂×C₄	C₂²×D₄×C₁₁₃	1	1	yes	characteristic
559	C₂×C₄	C₂×#41×C₁₁₃	1	1	yes	characteristic
560	C₂×C₄	H(5,𝔽₂)×C₁₁₃	1	1	yes	characteristic
561	C₂×C₄	H(5,𝔽₂)×C₁₁₃	1	1	yes	characteristic
562	C₈	—	2	2	no	—
563	C₈	—	4	1	no	—
564	C₈	—	4	2	no	—
565	C₂²	—	4	2	no	—
566	C₂²	—	4	2	no	—
567	C₂²	—	4	4	no	—
568	C₂²	—	4	2	no	—
569	C₂²	—	4	2	no	—
570	C₂²	—	4	4	no	—
571	C₂²	—	4	2	no	—
572	C₂²	—	4	4	no	—
573	C₂²	—	4	2	no	—
574	C₂²	—	4	2	no	—
575	C₂²	—	8	2	no	—
576	C₂²	—	8	2	no	—
577	C₂²	—	8	2	no	—
578	C₂²	—	8	2	no	—
579	C₂²	C₂×#120×C₁₁₃	1	1	yes	characteristic
580	C₂²	D₄²×C₁₁₃	1	1	yes	characteristic
581	C₂²	#546×C₁₁₃	1	1	yes	characteristic
582	C₂²	#575×C₁₁₃	1	2	yes	—
583	C₂²	#577×C₁₁₃	1	2	yes	—
584	C₄	—	2	4	no	—
585	C₄	—	2	2	no	—
586	C₄	—	2	2	no	—
587	C₄	—	4	2	no	—
588	C₄	—	4	2	no	—
589	C₄	—	4	2	no	—
590	C₄	—	8	1	no	—
591	C₄	—	8	2	no	—
592	C₄	C₂×#121×C₁₁₃	1	1	yes	characteristic
593	C₄	#535×C₁₁₃	1	1	yes	characteristic
594	C₄	D₄²×C₁₁₃	1	1	yes	characteristic
595	C₄	#546×C₁₁₃	1	1	yes	characteristic
596	C₂	—	8	2	no	—
597	C₂	—	8	4	no	—
598	C₂	C₁₁₃×#2408	1	1	yes	characteristic
599	C₂	C₁₁₃×#3005	1	1	yes	characteristic
600	C₂	C₁₁₃×#3009	1	1	yes	characteristic
601	C₂	C₁₁₃×#3271	1	2	yes	—
602	C₂	C₁₁₃×#3317	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₁₃
prime radical	C₂×#550×C₁₁₃	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₁₃ ◁ C₂×C₄×D₄×C₁₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#460×C₁₁₃	C₂	semidirect
3. C₂×#452×C₁₁₃	C₂	semidirect
4. C₂×#441×C₁₁₃	C₂	semidirect
5. C₁₁₃×#2536	C₂	semidirect
6. C₁₁₃×#2034	C₂	semidirect
7. C₁₁₃×#2000	C₂	semidirect
8. C₁₁₃×#1961	C₂	semidirect
9. C₁₁₃×#1955	C₂	semidirect
10. C₁₁₃×#1805	C₂	semidirect
11. C₁₁₃×#1788	C₂	semidirect
64. C₂×Q₁₆×C₁₁₃	D₄	semidirect
294. #24254	C₁₁₃	direct (N × H)
303. C₂×#460	C₂×C₁₁₃	semidirect
304. C₂×#452	C₂×C₁₁₃	semidirect
305. C₂×#441	C₂×C₁₁₃	semidirect
306. #2536	C₂×C₁₁₃	semidirect
307. #2034	C₂×C₁₁₃	semidirect
308. #2000	C₂×C₁₁₃	semidirect
309. #1961	C₂×C₁₁₃	semidirect
310. #1955	C₂×C₁₁₃	semidirect
311. #1805	C₂×C₁₁₃	semidirect
312. #1788	C₂×C₁₁₃	semidirect
313. C₁₁₃	#24254	direct (N × H)
366. C₂×Q₁₆	D₄×C₁₁₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#550×C₁₁₃	2
C₂×#460×C₁₁₃	2
C₂×#452×C₁₁₃	2
C₂×#441×C₁₁₃	2
C₁₁₃×#2536	2
C₁₁₃×#2034	2
C₁₁₃×#2000	2
C₁₁₃×#1961	2
C₁₁₃×#1955	2
C₁₁₃×#1805	2
C₁₁₃×#1788	2
#24254	113

Subgroups — C113×#24254 #43292274

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁₃×#24254 #43292274