Subgroups — C₁₁₃×#26484 #43294504

Order 28928.

918 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#500×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×D₈×C₁₁₃	C₂	1	1	yes	maximal, characteristic
3	C₁₁₃×#2452	C₂	1	2	yes	maximal, complemented
4	C₁₁₃×#2067	C₂	1	2	yes	maximal, complemented
5	C₁₁₃×#2023	C₂	1	2	yes	maximal, complemented
6	C₁₁₃×#1970	C₂	1	1	yes	maximal, characteristic
7	C₁₁₃×#1970	C₂	1	1	yes	maximal, characteristic
8	C₁₁₃×#1942	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₁₃×#1942	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₁₃×#1862	C₂	1	2	yes	maximal, complemented
11	C₈×C₄⋊C₄×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₈×C₁₁₃	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Q₈×C₁₁₃	C₂²	1	2	yes	—
14	C₂×C₂²⋊Q₈×C₁₁₃	—	2	2	no	—
15	C₂×C₄×D₄×C₁₁₃	C₂²	1	2	yes	—
16	#500×C₁₁₃	—	2	4	no	—
17	C₄×D₈×C₁₁₃	—	2	4	no	—
18	C₂×#107×C₁₁₃	C₂²	1	2	yes	—
19	C₂×#107×C₁₁₃	C₂²	1	1	yes	characteristic
20	C₂×#107×C₁₁₃	C₂²	1	1	yes	characteristic
21	C₂×#107×C₁₁₃	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₈×C₁₁₃	C₂²	1	1	yes	characteristic
23	C₂×#102×C₁₁₃	C₂²	1	2	yes	—
24	C₂×#102×C₁₁₃	C₂²	1	2	yes	—
25	C₂×#102×C₁₁₃	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₁₁₃	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₁₁₃	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₁₁₃	C₂²	1	1	yes	characteristic
29	#396×C₁₁₃	—	2	2	no	—
30	#393×C₁₁₃	—	2	2	no	—
31	#390×C₁₁₃	C₂²	1	2	yes	—
32	#390×C₁₁₃	C₂²	1	2	yes	—
33	#389×C₁₁₃	C₂²	1	2	yes	—
34	#389×C₁₁₃	C₂²	1	2	yes	—
35	#385×C₁₁₃	—	2	2	no	—
36	C₄×C₄⋊C₄×C₁₁₃	C₂²	1	1	yes	characteristic
37	#340×C₁₁₃	C₂²	1	2	yes	—
38	#340×C₁₁₃	C₂²	1	2	yes	—
39	#336×C₁₁₃	C₂²	1	1	yes	characteristic
40	#336×C₁₁₃	C₂²	1	1	yes	characteristic
41	C₂²×Q₈×C₁₁₃	—	2	2	no	—
42	C₂²×D₄×C₁₁₃	C₂³	1	2	yes	—
43	C₂³×C₄×C₁₁₃	—	2	2	no	—
44	C₂×D₈×C₁₁₃	—	2	2	no	—
45	C₂×D₈×C₁₁₃	—	2	2	no	—
46	C₂×D₈×C₁₁₃	Q₈	1	4	yes	complemented
47	C₂²×C₈×C₁₁₃	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₁₁₃	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₁₁₃	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₁₁₃	C₂³	1	1	yes	characteristic
51	C₄⋊Q₈×C₁₁₃	—	2	4	no	—
52	C₄⋊Q₈×C₁₁₃	—	4	2	no	—
53	C₂²⋊Q₈×C₁₁₃	—	4	8	no	—
54	C₄×D₄×C₁₁₃	—	2	4	no	—
55	C₄×D₄×C₁₁₃	—	4	2	no	—
56	C₂×C₄⋊C₄×C₁₁₃	—	2	2	no	—
57	C₂×C₄⋊C₄×C₁₁₃	—	2	2	no	—
58	C₂×C₄⋊C₄×C₁₁₃	—	2	2	no	—
59	C₂×C₄⋊C₄×C₁₁₃	D₄	1	1	yes	characteristic
60	C₂×C₄⋊C₄×C₁₁₃	D₄	1	1	yes	characteristic
61	C₂×C₄⋊C₄×C₁₁₃	C₂³	1	2	yes	—
62	C₂×C₄⋊C₄×C₁₁₃	C₂³	1	2	yes	—
63	C₂×C₄⋊C₄×C₁₁₃	C₂³	1	2	yes	—
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×C₂²⋊C₄×C₁₁₃	—	2	2	no	—
69	C₂×C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
70	C₂×C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
71	C₂×C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
72	#107×C₁₁₃	—	2	2	no	—
73	#107×C₁₁₃	—	2	2	no	—
74	#107×C₁₁₃	—	2	4	no	—
75	#107×C₁₁₃	D₄	1	4	yes	—
76	C₄⋊C₈×C₁₁₃	—	2	2	no	—
77	#102×C₁₁₃	—	2	4	no	—
78	#102×C₁₁₃	—	2	4	no	—
79	#102×C₁₁₃	—	4	2	no	—
80	C₄×C₈×C₁₁₃	—	2	2	no	—
81	C₄×C₈×C₁₁₃	—	2	2	no	—
82	C₄×C₈×C₁₁₃	—	2	2	no	—
83	#95×C₁₁₃	—	2	2	no	—
84	#95×C₁₁₃	—	2	2	no	—
85	#95×C₁₁₃	D₄	1	1	yes	characteristic
86	#95×C₁₁₃	D₄	1	1	yes	characteristic
87	C₂⁴×C₁₁₃	—	2	2	no	—
88	C₂×Q₈×C₁₁₃	—	4	4	no	—
89	C₂×Q₈×C₁₁₃	—	4	4	no	—
90	C₂×Q₈×C₁₁₃	—	8	2	no	—
91	C₂×D₄×C₁₁₃	—	2	2	no	—
92	C₂×D₄×C₁₁₃	—	2	2	no	—
93	C₂×D₄×C₁₁₃	—	2	2	no	—
94	C₂×D₄×C₁₁₃	—	2	2	no	—
95	C₂×D₄×C₁₁₃	—	2	2	no	—
96	C₂×D₄×C₁₁₃	C₂×Q₈	1	4	yes	—
97	C₂²×C₄×C₁₁₃	—	2	2	no	—
98	C₂²×C₄×C₁₁₃	—	2	2	no	—
99	C₂²×C₄×C₁₁₃	—	2	2	no	—
100	C₂²×C₄×C₁₁₃	—	4	4	no	—
101	C₂²×C₄×C₁₁₃	—	4	4	no	—
102	C₂²×C₄×C₁₁₃	—	8	2	no	—
103	C₂²×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
108	C₂²×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
110	D₈×C₁₁₃	—	2	8	no	—
111	C₂×C₈×C₁₁₃	—	2	1	no	—
112	C₂×C₈×C₁₁₃	—	2	1	no	—
113	C₂×C₈×C₁₁₃	—	2	1	no	—
114	C₂×C₈×C₁₁₃	—	2	1	no	—
115	C₂×C₈×C₁₁₃	—	2	1	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	C₂×D₄	1	2	yes	—
120	C₂×C₈×C₁₁₃	C₂×Q₈	1	2	yes	—
121	C₂×C₈×C₁₁₃	#41	1	2	yes	—
122	C₂×C₈×C₁₁₃	#41	1	2	yes	—
123	C₄⋊C₄×C₁₁₃	—	2	2	no	—
124	C₄⋊C₄×C₁₁₃	—	2	2	no	—
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	—	4	4	no	—
130	C₄⋊C₄×C₁₁₃	—	4	4	no	—
131	C₄⋊C₄×C₁₁₃	—	4	4	no	—
132	C₄⋊C₄×C₁₁₃	D₈	1	4	yes	complemented
133	C₄⋊C₄×C₁₁₃	C₂×D₄	1	4	yes	—
134	C₂²⋊C₄×C₁₁₃	—	4	4	no	—
135	C₂²⋊C₄×C₁₁₃	—	4	4	no	—
136	C₂²⋊C₄×C₁₁₃	—	4	4	no	—
137	C₂²⋊C₄×C₁₁₃	—	4	4	no	—
138	C₂²⋊C₄×C₁₁₃	—	8	2	no	—
139	C₄²×C₁₁₃	—	2	1	no	—
140	C₄²×C₁₁₃	—	2	1	no	—
141	C₄²×C₁₁₃	—	2	1	no	—
142	C₄²×C₁₁₃	—	2	1	no	—
143	C₄²×C₁₁₃	—	2	1	no	—
144	C₄²×C₁₁₃	—	2	1	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	C₂²×D₄	1	1	yes	characteristic
153	Q₈×C₁₁₃	—	8	8	no	—
154	D₄×C₁₁₃	—	2	4	no	—
155	D₄×C₁₁₃	—	2	4	no	—
156	D₄×C₁₁₃	—	4	2	no	—
157	D₄×C₁₁₃	—	4	2	no	—
158	C₂×C₄×C₁₁₃	—	2	1	no	—
159	C₂×C₄×C₁₁₃	—	2	1	no	—
160	C₂×C₄×C₁₁₃	—	2	1	no	—
161	C₂×C₄×C₁₁₃	—	2	1	no	—
162	C₂×C₄×C₁₁₃	—	2	1	no	—
163	C₂×C₄×C₁₁₃	—	2	1	no	—
164	C₂×C₄×C₁₁₃	—	2	1	no	—
165	C₂×C₄×C₁₁₃	—	2	1	no	—
166	C₂×C₄×C₁₁₃	—	2	1	no	—
167	C₂×C₄×C₁₁₃	—	2	1	no	—
168	C₂×C₄×C₁₁₃	—	2	4	no	—
169	C₂×C₄×C₁₁₃	—	2	1	no	—
170	C₂×C₄×C₁₁₃	—	2	1	no	—
171	C₂×C₄×C₁₁₃	—	2	1	no	—
172	C₂×C₄×C₁₁₃	—	2	1	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	C₂×D₈	1	2	yes	—
183	C₂×C₄×C₁₁₃	C₂×D₈	1	2	yes	—
184	C₂×C₄×C₁₁₃	C₂×D₈	1	2	yes	—
185	C₂×C₄×C₁₁₃	#135	1	2	yes	—
186	C₂×C₄×C₁₁₃	C₂²×D₄	1	1	yes	characteristic
187	C₂×C₄×C₁₁₃	C₂²×Q₈	1	1	yes	characteristic
188	C₂×C₄×C₁₁₃	C₂×#41	1	1	yes	characteristic
189	C₂×C₄×C₁₁₃	C₂×#41	1	1	yes	characteristic
190	C₂×C₄×C₁₁₃	H(5,𝔽₂)	1	1	yes	characteristic
191	C₂×C₄×C₁₁₃	2₋¹⁺⁴	1	1	yes	characteristic
192	C₈×C₁₁₃	—	2	2	no	—
193	C₈×C₁₁₃	—	4	1	no	—
194	C₈×C₁₁₃	—	4	1	no	—
195	C₈×C₁₁₃	C₂²⋊Q₈	1	4	yes	complemented
196	C₂²×C₁₁₃	—	4	4	no	—
197	C₂²×C₁₁₃	—	4	4	no	—
198	C₂²×C₁₁₃	—	4	4	no	—
199	C₂²×C₁₁₃	—	8	2	no	—
200	C₂²×C₁₁₃	—	8	2	no	—
201	C₂²×C₁₁₃	—	8	2	no	—
202	C₂²×C₁₁₃	—	8	2	no	—
203	C₂²×C₁₁₃	#532	1	1	yes	characteristic
204	C₂²×C₁₁₃	D₄²	1	1	yes	characteristic
205	C₂²×C₁₁₃	D₄×Q₈	1	1	yes	characteristic
206	C₂²×C₁₁₃	C₂²×D₈	1	1	yes	characteristic
207	C₂²×C₁₁₃	C₂×#135	1	1	yes	characteristic
208	C₂²×C₁₁₃	#576	1	1	yes	characteristic
209	C₂²×C₁₁₃	#578	1	1	yes	characteristic
210	C₄×C₁₁₃	—	2	2	no	—
211	C₄×C₁₁₃	—	2	2	no	—
212	C₄×C₁₁₃	—	2	2	no	—
213	C₄×C₁₁₃	—	2	2	no	—
214	C₄×C₁₁₃	—	4	1	no	—
215	C₄×C₁₁₃	—	4	4	no	—
216	C₄×C₁₁₃	—	4	1	no	—
217	C₄×C₁₁₃	—	8	2	no	—
218	C₄×C₁₁₃	—	8	2	no	—
219	C₄×C₁₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
220	C₄×C₁₁₃	#542	1	1	yes	characteristic
221	C₄×C₁₁₃	#548	1	1	yes	characteristic
222	C₄×C₁₁₃	#554	1	1	yes	characteristic
223	#26484	C₁₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
224	C₂×C₁₁₃	—	8	4	no	—
225	C₂×C₁₁₃	#2452	1	1	yes	characteristic
226	C₂×C₁₁₃	#3169	1	1	yes	characteristic
227	C₂×C₁₁₃	#3175	1	1	yes	characteristic
228	C₂×C₁₁₃	D₄×D₈	1	1	yes	characteristic
229	C₂×C₁₁₃	#3285	1	1	yes	characteristic
230	C₂×C₁₁₃	Q₈×D₈	1	1	yes	characteristic
231	C₂×C₁₁₃	#3380	1	1	yes	characteristic
232	C₂×#500	C₂×C₁₁₃	1	1	yes	characteristic, complemented
233	C₂×C₄×D₈	C₂×C₁₁₃	1	1	yes	characteristic
234	#2452	C₂×C₁₁₃	1	2	yes	complemented
235	#2067	C₂×C₁₁₃	1	2	yes	complemented
236	#2023	C₂×C₁₁₃	1	2	yes	complemented
237	#1970	C₂×C₁₁₃	1	1	yes	characteristic
238	#1970	C₂×C₁₁₃	1	1	yes	characteristic
239	#1942	C₂×C₁₁₃	1	1	yes	characteristic, complemented
240	#1942	C₂×C₁₁₃	1	1	yes	characteristic, complemented
241	#1862	C₂×C₁₁₃	1	2	yes	complemented
242	C₈×C₄⋊C₄	C₂×C₁₁₃	1	1	yes	characteristic, complemented
243	C₁₁₃	#26484	1	1	yes	characteristic, direct product factor, Sylow
244	C₂²×D₈	C₂²×C₁₁₃	1	1	yes	characteristic
245	C₂×C₄⋊Q₈	C₂²×C₁₁₃	1	2	yes	—
246	C₂×C₂²⋊Q₈	—	2	2	no	—
247	C₂×C₄×D₄	C₂²×C₁₁₃	1	2	yes	—
248	#500	—	2	4	no	—
249	C₄×D₈	—	2	4	no	—
250	C₂×#107	C₂²×C₁₁₃	1	2	yes	—
251	C₂×#107	C₂²×C₁₁₃	1	1	yes	characteristic
252	C₂×#107	C₂²×C₁₁₃	1	1	yes	characteristic
253	C₂×#107	C₂²×C₁₁₃	1	1	yes	characteristic
254	C₂×C₄⋊C₈	C₂²×C₁₁₃	1	1	yes	characteristic
255	C₂×#102	C₂²×C₁₁₃	1	2	yes	—
256	C₂×#102	C₂²×C₁₁₃	1	2	yes	—
257	C₂×#102	C₂²×C₁₁₃	1	2	yes	—
258	C₂×C₄×C₈	C₂²×C₁₁₃	1	1	yes	characteristic
259	C₂×C₄×C₈	C₂²×C₁₁₃	1	1	yes	characteristic
260	C₂×C₄×C₈	C₂²×C₁₁₃	1	1	yes	characteristic
261	#396	—	2	2	no	—
262	#393	—	2	2	no	—
263	#390	C₂²×C₁₁₃	1	2	yes	—
264	#390	C₂²×C₁₁₃	1	2	yes	—
265	#389	C₂²×C₁₁₃	1	2	yes	—
266	#389	C₂²×C₁₁₃	1	2	yes	—
267	#385	—	2	2	no	—
268	C₄×C₄⋊C₄	C₂²×C₁₁₃	1	1	yes	characteristic
269	#340	C₂²×C₁₁₃	1	2	yes	—
270	#340	C₂²×C₁₁₃	1	2	yes	—
271	#336	C₂²×C₁₁₃	1	1	yes	characteristic
272	#336	C₂²×C₁₁₃	1	1	yes	characteristic
273	C₂²×Q₈	—	2	2	no	—
274	C₂²×D₄	C₂³×C₁₁₃	1	2	yes	—
275	C₂³×C₄	—	2	2	no	—
276	C₂×D₈	—	2	2	no	—
277	C₂×D₈	—	2	2	no	—
278	C₂×D₈	Q₈×C₁₁₃	1	4	yes	complemented
279	C₂²×C₈	C₂³×C₁₁₃	1	1	yes	characteristic
280	C₂²×C₈	C₂³×C₁₁₃	1	1	yes	characteristic
281	C₂²×C₈	C₂³×C₁₁₃	1	1	yes	characteristic
282	C₂²×C₈	C₂³×C₁₁₃	1	1	yes	characteristic
283	C₄⋊Q₈	—	2	4	no	—
284	C₄⋊Q₈	—	4	2	no	—
285	C₂²⋊Q₈	—	4	8	no	—
286	C₄×D₄	—	2	4	no	—
287	C₄×D₄	—	4	2	no	—
288	C₂×C₄⋊C₄	—	2	2	no	—
289	C₂×C₄⋊C₄	—	2	2	no	—
290	C₂×C₄⋊C₄	—	2	2	no	—
291	C₂×C₄⋊C₄	D₄×C₁₁₃	1	1	yes	characteristic
292	C₂×C₄⋊C₄	D₄×C₁₁₃	1	1	yes	characteristic
293	C₂×C₄⋊C₄	C₂³×C₁₁₃	1	2	yes	—
294	C₂×C₄⋊C₄	C₂³×C₁₁₃	1	2	yes	—
295	C₂×C₄⋊C₄	C₂³×C₁₁₃	1	2	yes	—
296	C₂×C₂²⋊C₄	—	2	2	no	—
297	C₂×C₂²⋊C₄	—	2	2	no	—
298	C₂×C₂²⋊C₄	—	2	2	no	—
299	C₂×C₂²⋊C₄	—	2	2	no	—
300	C₂×C₂²⋊C₄	—	2	2	no	—
301	C₂×C₄²	C₂³×C₁₁₃	1	1	yes	characteristic
302	C₂×C₄²	C₂³×C₁₁₃	1	1	yes	characteristic
303	C₂×C₄²	C₂³×C₁₁₃	1	1	yes	characteristic
304	#107	—	2	2	no	—
305	#107	—	2	2	no	—
306	#107	—	2	4	no	—
307	#107	D₄×C₁₁₃	1	4	yes	—
308	C₄⋊C₈	—	2	2	no	—
309	#102	—	2	4	no	—
310	#102	—	2	4	no	—
311	#102	—	4	2	no	—
312	C₄×C₈	—	2	2	no	—
313	C₄×C₈	—	2	2	no	—
314	C₄×C₈	—	2	2	no	—
315	#95	—	2	2	no	—
316	#95	—	2	2	no	—
317	#95	D₄×C₁₁₃	1	1	yes	characteristic
318	#95	D₄×C₁₁₃	1	1	yes	characteristic
319	C₂⁴	—	2	2	no	—
320	C₂×Q₈	—	4	4	no	—
321	C₂×Q₈	—	4	4	no	—
322	C₂×Q₈	—	8	2	no	—
323	C₂×D₄	—	2	2	no	—
324	C₂×D₄	—	2	2	no	—
325	C₂×D₄	—	2	2	no	—
326	C₂×D₄	—	2	2	no	—
327	C₂×D₄	—	2	2	no	—
328	C₂×D₄	C₂×Q₈×C₁₁₃	1	4	yes	—
329	C₂²×C₄	—	2	2	no	—
330	C₂²×C₄	—	2	2	no	—
331	C₂²×C₄	—	2	2	no	—
332	C₂²×C₄	—	4	4	no	—
333	C₂²×C₄	—	4	4	no	—
334	C₂²×C₄	—	8	2	no	—
335	C₂²×C₄	C₂×D₄×C₁₁₃	1	1	yes	characteristic
336	C₂²×C₄	C₂×D₄×C₁₁₃	1	1	yes	characteristic
337	C₂²×C₄	C₂×D₄×C₁₁₃	1	1	yes	characteristic
338	C₂²×C₄	C₂×D₄×C₁₁₃	1	1	yes	characteristic
339	C₂²×C₄	C₂×D₄×C₁₁₃	1	1	yes	characteristic
340	C₂²×C₄	C₂×D₄×C₁₁₃	1	1	yes	characteristic
341	C₂²×C₄	C₂⁴×C₁₁₃	1	1	yes	characteristic
342	D₈	—	2	8	no	—
343	C₂×C₈	—	2	1	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	1	no	—
349	C₂×C₈	—	2	1	no	—
350	C₂×C₈	—	2	1	no	—
351	C₂×C₈	C₂×D₄×C₁₁₃	1	2	yes	—
352	C₂×C₈	C₂×Q₈×C₁₁₃	1	2	yes	—
353	C₂×C₈	#41×C₁₁₃	1	2	yes	—
354	C₂×C₈	#41×C₁₁₃	1	2	yes	—
355	C₄⋊C₄	—	2	2	no	—
356	C₄⋊C₄	—	2	2	no	—
357	C₄⋊C₄	—	2	2	no	—
358	C₄⋊C₄	—	2	2	no	—
359	C₄⋊C₄	—	2	2	no	—
360	C₄⋊C₄	—	2	2	no	—
361	C₄⋊C₄	—	4	4	no	—
362	C₄⋊C₄	—	4	4	no	—
363	C₄⋊C₄	—	4	4	no	—
364	C₄⋊C₄	D₈×C₁₁₃	1	4	yes	complemented
365	C₄⋊C₄	C₂×D₄×C₁₁₃	1	4	yes	—
366	C₂²⋊C₄	—	4	4	no	—
367	C₂²⋊C₄	—	4	4	no	—
368	C₂²⋊C₄	—	4	4	no	—
369	C₂²⋊C₄	—	4	4	no	—
370	C₂²⋊C₄	—	8	2	no	—
371	C₄²	—	2	1	no	—
372	C₄²	—	2	1	no	—
373	C₄²	—	2	1	no	—
374	C₄²	—	2	1	no	—
375	C₄²	—	2	1	no	—
376	C₄²	—	2	1	no	—
377	C₂³	—	2	4	no	—
378	C₂³	—	4	2	no	—
379	C₂³	—	4	2	no	—
380	C₂³	—	4	2	no	—
381	C₂³	—	4	2	no	—
382	C₂³	—	4	2	no	—
383	C₂³	—	4	2	no	—
384	C₂³	C₂²×D₄×C₁₁₃	1	1	yes	characteristic
385	Q₈	—	8	8	no	—
386	D₄	—	2	4	no	—
387	D₄	—	2	4	no	—
388	D₄	—	4	2	no	—
389	D₄	—	4	2	no	—
390	C₂×C₄	—	2	1	no	—
391	C₂×C₄	—	2	1	no	—
392	C₂×C₄	—	2	1	no	—
393	C₂×C₄	—	2	1	no	—
394	C₂×C₄	—	2	1	no	—
395	C₂×C₄	—	2	1	no	—
396	C₂×C₄	—	2	1	no	—
397	C₂×C₄	—	2	1	no	—
398	C₂×C₄	—	2	1	no	—
399	C₂×C₄	—	2	1	no	—
400	C₂×C₄	—	2	4	no	—
401	C₂×C₄	—	2	1	no	—
402	C₂×C₄	—	2	1	no	—
403	C₂×C₄	—	2	1	no	—
404	C₂×C₄	—	2	1	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	—	4	2	no	—
410	C₂×C₄	—	4	2	no	—
411	C₂×C₄	—	4	2	no	—
412	C₂×C₄	—	4	2	no	—
413	C₂×C₄	—	8	8	no	—
414	C₂×C₄	C₂×D₈×C₁₁₃	1	2	yes	—
415	C₂×C₄	C₂×D₈×C₁₁₃	1	2	yes	—
416	C₂×C₄	C₂×D₈×C₁₁₃	1	2	yes	—
417	C₂×C₄	#135×C₁₁₃	1	2	yes	—
418	C₂×C₄	C₂²×D₄×C₁₁₃	1	1	yes	characteristic
419	C₂×C₄	C₂²×Q₈×C₁₁₃	1	1	yes	characteristic
420	C₂×C₄	C₂×#41×C₁₁₃	1	1	yes	characteristic
421	C₂×C₄	C₂×#41×C₁₁₃	1	1	yes	characteristic
422	C₂×C₄	H(5,𝔽₂)×C₁₁₃	1	1	yes	characteristic
423	C₂×C₄	2₋¹⁺⁴×C₁₁₃	1	1	yes	characteristic
424	C₈	—	2	2	no	—
425	C₈	—	4	1	no	—
426	C₈	—	4	1	no	—
427	C₈	C₂²⋊Q₈×C₁₁₃	1	4	yes	complemented
428	C₂²	—	4	4	no	—
429	C₂²	—	4	4	no	—
430	C₂²	—	4	4	no	—
431	C₂²	—	8	2	no	—
432	C₂²	—	8	2	no	—
433	C₂²	—	8	2	no	—
434	C₂²	—	8	2	no	—
435	C₂²	#532×C₁₁₃	1	1	yes	characteristic
436	C₂²	D₄²×C₁₁₃	1	1	yes	characteristic
437	C₂²	D₄×Q₈×C₁₁₃	1	1	yes	characteristic
438	C₂²	C₂²×D₈×C₁₁₃	1	1	yes	characteristic
439	C₂²	C₂×#135×C₁₁₃	1	1	yes	characteristic
440	C₂²	#576×C₁₁₃	1	1	yes	characteristic
441	C₂²	#578×C₁₁₃	1	1	yes	characteristic
442	C₄	—	2	2	no	—
443	C₄	—	2	2	no	—
444	C₄	—	2	2	no	—
445	C₄	—	2	2	no	—
446	C₄	—	4	1	no	—
447	C₄	—	4	4	no	—
448	C₄	—	4	1	no	—
449	C₄	—	8	2	no	—
450	C₄	—	8	2	no	—
451	C₄	C₂×C₂²⋊Q₈×C₁₁₃	1	1	yes	characteristic
452	C₄	#542×C₁₁₃	1	1	yes	characteristic
453	C₄	#548×C₁₁₃	1	1	yes	characteristic
454	C₄	#554×C₁₁₃	1	1	yes	characteristic
455	C₂	—	8	4	no	—
456	C₂	C₁₁₃×#2452	1	1	yes	characteristic
457	C₂	C₁₁₃×#3169	1	1	yes	characteristic
458	C₂	C₁₁₃×#3175	1	1	yes	characteristic
459	C₂	D₄×D₈×C₁₁₃	1	1	yes	characteristic
460	C₂	C₁₁₃×#3285	1	1	yes	characteristic
461	C₂	Q₈×D₈×C₁₁₃	1	1	yes	characteristic
462	C₂	C₁₁₃×#3380	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₁₃
prime radical	C₂²×D₈×C₁₁₃	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₁₃ ◁ C₄×C₄⋊C₄×C₁₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#500×C₁₁₃	C₂	semidirect
3. C₁₁₃×#2452	C₂	semidirect
4. C₁₁₃×#2067	C₂	semidirect
5. C₁₁₃×#2023	C₂	semidirect
8. C₁₁₃×#1942	C₂	semidirect
9. C₁₁₃×#1942	C₂	semidirect
10. C₁₁₃×#1862	C₂	semidirect
11. C₈×C₄⋊C₄×C₁₁₃	C₂	semidirect
46. C₂×D₈×C₁₁₃	Q₈	semidirect
132. C₄⋊C₄×C₁₁₃	D₈	semidirect
195. C₈×C₁₁₃	C₂²⋊Q₈	semidirect
223. #26484	C₁₁₃	direct (N × H)
232. C₂×#500	C₂×C₁₁₃	semidirect
234. #2452	C₂×C₁₁₃	semidirect
235. #2067	C₂×C₁₁₃	semidirect
236. #2023	C₂×C₁₁₃	semidirect
239. #1942	C₂×C₁₁₃	semidirect
240. #1942	C₂×C₁₁₃	semidirect
241. #1862	C₂×C₁₁₃	semidirect
242. C₈×C₄⋊C₄	C₂×C₁₁₃	semidirect
243. C₁₁₃	#26484	direct (N × H)
278. C₂×D₈	Q₈×C₁₁₃	semidirect
364. C₄⋊C₄	D₈×C₁₁₃	semidirect
427. C₈	C₂²⋊Q₈×C₁₁₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#500×C₁₁₃	2
C₂×C₄×D₈×C₁₁₃	2
C₁₁₃×#2452	2
C₁₁₃×#2067	2
C₁₁₃×#2023	2
C₁₁₃×#1970	2
C₁₁₃×#1942	2
C₁₁₃×#1862	2
C₈×C₄⋊C₄×C₁₁₃	2
#26484	113

Subgroups — C113×#26484 #43294504

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁₃×#26484 #43294504