Subgroups — C₁₁₃×#21212 #43289232

Order 28928.

1658 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#532×C₁₁₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#410×C₁₁₃	C₂	1	4	yes	maximal, complemented
3	C₁₁₃×#2285	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×#117×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₁₃×#1964	C₂	1	8	yes	maximal, complemented
6	C₂²×#41×C₁₁₃	C₂²	1	2	yes	—
7	#532×C₁₁₃	—	2	1	no	—
8	#532×C₁₁₃	—	2	1	no	—
9	#532×C₁₁₃	—	2	2	no	—
10	#532×C₁₁₃	—	2	2	no	—
11	#532×C₁₁₃	—	2	1	no	—
12	#532×C₁₁₃	—	2	1	no	—
13	C₂×C₄⋊Q₈×C₁₁₃	C₄	1	1	yes	characteristic, complemented
14	C₂×#127×C₁₁₃	C₄	1	1	yes	characteristic, complemented
15	C₂×#124×C₁₁₃	C₄	1	2	yes	complemented
16	C₂×#121×C₁₁₃	C₂²	1	4	yes	—
17	C₂×C₄×D₄×C₁₁₃	C₄	1	4	yes	complemented
18	C₂×#117×C₁₁₃	C₂²	1	2	yes	—
19	C₂×#117×C₁₁₃	C₂²	1	2	yes	—
20	C₂²×C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
21	#410×C₁₁₃	—	2	16	no	—
22	C₂×#99×C₁₁₃	C₂²	1	16	yes	—
23	#384×C₁₁₃	—	2	1	no	—
24	#382×C₁₁₃	—	2	1	no	—
25	#380×C₁₁₃	C₂²	1	4	yes	—
26	C₄×C₄⋊C₄×C₁₁₃	—	2	2	no	—
27	C₄×C₄⋊C₄×C₁₁₃	—	2	2	no	—
28	C₄×C₂²⋊C₄×C₁₁₃	C₂²	1	4	yes	—
29	#376×C₁₁₃	—	2	1	no	—
30	C₄³×C₁₁₃	—	2	1	no	—
31	C₂×#41×C₁₁₃	—	2	2	no	—
32	C₂×#41×C₁₁₃	—	2	2	no	—
33	C₂×#41×C₁₁₃	—	2	2	no	—
34	C₂×#41×C₁₁₃	—	2	2	no	—
35	C₂×#41×C₁₁₃	—	2	8	no	—
36	C₂×#41×C₁₁₃	—	2	8	no	—
37	C₂²×Q₈×C₁₁₃	C₂×C₄	1	2	yes	—
38	C₂²×D₄×C₁₁₃	C₂×C₄	1	2	yes	—
39	C₂²×D₄×C₁₁₃	C₂×C₄	1	4	yes	—
40	C₂²×D₄×C₁₁₃	C₂³	1	4	yes	—
41	C₂³×C₄×C₁₁₃	C₂×C₄	1	4	yes	—
42	C₂³×C₄×C₁₁₃	C₂³	1	2	yes	—
43	C₂³×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
44	C₄⋊Q₈×C₁₁₃	—	2	1	no	—
45	C₄⋊Q₈×C₁₁₃	—	2	2	no	—
46	C₄⋊Q₈×C₁₁₃	—	2	1	no	—
47	#127×C₁₁₃	—	2	1	no	—
48	#127×C₁₁₃	—	2	2	no	—
49	#127×C₁₁₃	—	2	1	no	—
50	#124×C₁₁₃	—	2	2	no	—
51	#124×C₁₁₃	—	2	2	no	—
52	#124×C₁₁₃	—	2	4	no	—
53	#121×C₁₁₃	—	2	4	no	—
54	#121×C₁₁₃	—	2	4	no	—
55	#121×C₁₁₃	—	2	4	no	—
56	#121×C₁₁₃	—	2	4	no	—
57	C₄×D₄×C₁₁₃	—	2	4	no	—
58	C₄×D₄×C₁₁₃	—	2	4	no	—
59	C₄×D₄×C₁₁₃	—	2	4	no	—
60	C₄×D₄×C₁₁₃	—	2	4	no	—
61	#117×C₁₁₃	—	2	4	no	—
62	#117×C₁₁₃	—	2	8	no	—
63	#117×C₁₁₃	D₄	1	8	yes	complemented
64	C₂×C₄⋊C₄×C₁₁₃	—	2	2	no	—
65	C₂×C₄⋊C₄×C₁₁₃	—	2	2	no	—
66	C₂×C₄⋊C₄×C₁₁₃	—	2	4	no	—
67	C₂×C₄⋊C₄×C₁₁₃	C₂×C₄	1	4	yes	—
68	C₂×C₂²⋊C₄×C₁₁₃	C₂×C₄	1	8	yes	—
69	C₂×C₂²⋊C₄×C₁₁₃	C₂³	1	4	yes	—
70	C₂×C₂²⋊C₄×C₁₁₃	C₂³	1	4	yes	—
71	C₂×C₄²×C₁₁₃	—	2	2	no	—
72	C₂×C₄²×C₁₁₃	—	2	2	no	—
73	C₂×C₄²×C₁₁₃	—	2	4	no	—
74	C₂×C₄²×C₁₁₃	C₂×C₄	1	2	yes	—
75	C₂×C₄²×C₁₁₃	C₂×C₄	1	1	yes	characteristic
76	C₂×C₄²×C₁₁₃	C₂×C₄	1	1	yes	characteristic
77	C₂×C₄²×C₁₁₃	D₄	1	1	yes	characteristic
78	C₂×C₄²×C₁₁₃	D₄	1	1	yes	characteristic
79	C₂×C₄²×C₁₁₃	D₄	1	2	yes	—
80	C₂×C₄²×C₁₁₃	D₄	1	2	yes	—
81	C₂×C₄²×C₁₁₃	D₄	1	1	yes	characteristic
82	C₂×C₄²×C₁₁₃	D₄	1	1	yes	characteristic
83	#99×C₁₁₃	—	2	32	no	—
84	#95×C₁₁₃	—	2	4	no	—
85	#95×C₁₁₃	—	2	4	no	—
86	C₂⁴×C₁₁₃	C₂²×C₄	1	4	yes	—
87	C₂⁴×C₁₁₃	C₂⁴	1	1	yes	characteristic
88	#41×C₁₁₃	—	4	8	no	—
89	#41×C₁₁₃	—	4	8	no	—
90	#41×C₁₁₃	—	4	4	no	—
91	#41×C₁₁₃	—	4	4	no	—
92	#41×C₁₁₃	—	4	4	no	—
93	#41×C₁₁₃	—	4	4	no	—
94	C₂×Q₈×C₁₁₃	—	2	2	no	—
95	C₂×Q₈×C₁₁₃	—	2	2	no	—
96	C₂×Q₈×C₁₁₃	—	2	8	no	—
97	C₂×D₄×C₁₁₃	—	2	8	no	—
98	C₂×D₄×C₁₁₃	—	2	4	no	—
99	C₂×D₄×C₁₁₃	—	2	4	no	—
100	C₂×D₄×C₁₁₃	—	2	8	no	—
101	C₂×D₄×C₁₁₃	—	2	4	no	—
102	C₂×D₄×C₁₁₃	—	2	4	no	—
103	C₂×D₄×C₁₁₃	—	2	2	no	—
104	C₂×D₄×C₁₁₃	—	2	2	no	—
105	C₂×D₄×C₁₁₃	—	2	8	no	—
106	C₂×D₄×C₁₁₃	—	4	8	no	—
107	C₂²×C₄×C₁₁₃	—	2	4	no	—
108	C₂²×C₄×C₁₁₃	—	2	4	no	—
109	C₂²×C₄×C₁₁₃	—	2	4	no	—
110	C₂²×C₄×C₁₁₃	—	2	16	no	—
111	C₂²×C₄×C₁₁₃	—	2	4	no	—
112	C₂²×C₄×C₁₁₃	—	4	4	no	—
113	C₂²×C₄×C₁₁₃	C₂²×C₄	1	4	yes	—
114	C₂²×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
115	C₂²×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
116	C₂²×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
117	C₂²×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
118	C₂²×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
119	C₂²×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
120	C₂²×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
121	C₂²×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
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	1	yes	characteristic
125	C₂²×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
126	C₂²×C₄×C₁₁₃	C₂×D₄	1	4	yes	—
127	C₂²×C₄×C₁₁₃	#41	1	4	yes	—
128	C₄⋊C₄×C₁₁₃	—	2	4	no	—
129	C₄⋊C₄×C₁₁₃	—	2	4	no	—
130	C₄⋊C₄×C₁₁₃	—	2	8	no	—
131	C₄⋊C₄×C₁₁₃	—	4	8	no	—
132	C₄⋊C₄×C₁₁₃	—	4	4	no	—
133	C₂²⋊C₄×C₁₁₃	—	2	4	no	—
134	C₂²⋊C₄×C₁₁₃	—	2	8	no	—
135	C₂²⋊C₄×C₁₁₃	—	2	8	no	—
136	C₂²⋊C₄×C₁₁₃	—	2	8	no	—
137	C₂²⋊C₄×C₁₁₃	C₂×D₄	1	8	yes	—
138	C₄²×C₁₁₃	—	2	8	no	—
139	C₄²×C₁₁₃	—	4	8	no	—
140	C₄²×C₁₁₃	—	4	4	no	—
141	C₄²×C₁₁₃	C₂²⋊C₄	1	2	yes	—
142	C₄²×C₁₁₃	C₂²⋊C₄	1	2	yes	—
143	C₄²×C₁₁₃	C₂²⋊C₄	1	1	yes	characteristic
144	C₄²×C₁₁₃	C₂²⋊C₄	1	1	yes	characteristic
145	C₄²×C₁₁₃	C₂²⋊C₄	1	2	yes	—
146	C₄²×C₁₁₃	C₂²⋊C₄	1	1	yes	characteristic
147	C₄²×C₁₁₃	C₂²⋊C₄	1	2	yes	—
148	C₄²×C₁₁₃	C₂²⋊C₄	1	1	yes	characteristic
149	C₄²×C₁₁₃	C₂²⋊C₄	1	2	yes	—
150	C₄²×C₁₁₃	C₂²⋊C₄	1	2	yes	—
151	C₂³×C₁₁₃	—	2	4	no	—
152	C₂³×C₁₁₃	—	2	8	no	—
153	C₂³×C₁₁₃	—	4	2	no	—
154	C₂³×C₁₁₃	—	4	8	no	—
155	C₂³×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
156	C₂³×C₁₁₃	C₂²×D₄	1	1	yes	characteristic
157	C₂³×C₁₁₃	C₂²×D₄	1	1	yes	characteristic
158	C₂³×C₁₁₃	C₂²×D₄	1	2	yes	—
159	C₂³×C₁₁₃	C₂×#41	1	2	yes	—
160	Q₈×C₁₁₃	—	4	4	no	—
161	Q₈×C₁₁₃	—	4	4	no	—
162	D₄×C₁₁₃	—	4	8	no	—
163	D₄×C₁₁₃	—	4	8	no	—
164	D₄×C₁₁₃	—	4	4	no	—
165	D₄×C₁₁₃	—	4	4	no	—
166	D₄×C₁₁₃	—	4	4	no	—
167	D₄×C₁₁₃	—	4	4	no	—
168	D₄×C₁₁₃	—	4	8	no	—
169	C₂×C₄×C₁₁₃	—	2	4	no	—
170	C₂×C₄×C₁₁₃	—	2	4	no	—
171	C₂×C₄×C₁₁₃	—	2	4	no	—
172	C₂×C₄×C₁₁₃	—	2	4	no	—
173	C₂×C₄×C₁₁₃	—	2	8	no	—
174	C₂×C₄×C₁₁₃	—	2	8	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	8	no	—
179	C₂×C₄×C₁₁₃	—	4	8	no	—
180	C₂×C₄×C₁₁₃	—	4	8	no	—
181	C₂×C₄×C₁₁₃	—	4	8	no	—
182	C₂×C₄×C₁₁₃	—	4	4	no	—
183	C₂×C₄×C₁₁₃	C₂×C₂²⋊C₄	1	2	yes	—
184	C₂×C₄×C₁₁₃	C₂×C₂²⋊C₄	1	2	yes	—
185	C₂×C₄×C₁₁₃	C₂×C₂²⋊C₄	1	2	yes	—
186	C₂×C₄×C₁₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
187	C₂×C₄×C₁₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
188	C₂×C₄×C₁₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
189	C₂×C₄×C₁₁₃	C₂×C₂²⋊C₄	1	2	yes	—
190	C₂×C₄×C₁₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
191	C₂×C₄×C₁₁₃	C₄×D₄	1	4	yes	—
192	C₂×C₄×C₁₁₃	C₄×D₄	1	4	yes	—
193	C₂×C₄×C₁₁₃	#121	1	4	yes	—
194	C₂×C₄×C₁₁₃	#121	1	4	yes	—
195	C₂×C₄×C₁₁₃	#124	1	2	yes	—
196	C₂×C₄×C₁₁₃	#124	1	2	yes	—
197	C₂×C₄×C₁₁₃	#127	1	2	yes	—
198	C₂×C₄×C₁₁₃	#127	1	2	yes	—
199	C₂²×C₁₁₃	—	2	2	no	—
200	C₂²×C₁₁₃	—	2	2	no	—
201	C₂²×C₁₁₃	—	2	2	no	—
202	C₂²×C₁₁₃	—	4	2	no	—
203	C₂²×C₁₁₃	—	4	2	no	—
204	C₂²×C₁₁₃	—	4	8	no	—
205	C₂²×C₁₁₃	—	4	4	no	—
206	C₂²×C₁₁₃	—	8	8	no	—
207	C₂²×C₁₁₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
208	C₂²×C₁₁₃	C₂×C₄×D₄	1	2	yes	—
209	C₂²×C₁₁₃	C₂×#121	1	2	yes	—
210	C₂²×C₁₁₃	C₂×#124	1	1	yes	characteristic
211	C₂²×C₁₁₃	C₂×#127	1	1	yes	characteristic
212	C₄×C₁₁₃	—	2	2	no	—
213	C₄×C₁₁₃	—	2	2	no	—
214	C₄×C₁₁₃	—	2	2	no	—
215	C₄×C₁₁₃	—	4	8	no	—
216	C₄×C₁₁₃	—	4	4	no	—
217	C₄×C₁₁₃	—	8	4	no	—
218	C₄×C₁₁₃	#390	1	2	yes	—
219	C₄×C₁₁₃	#390	1	2	yes	—
220	C₄×C₁₁₃	#390	1	2	yes	—
221	C₄×C₁₁₃	#390	1	2	yes	—
222	C₄×C₁₁₃	#410	1	4	yes	complemented
223	#21212	C₁₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
224	C₂×C₁₁₃	—	2	2	no	—
225	C₂×C₁₁₃	—	2	1	no	—
226	C₂×C₁₁₃	—	2	2	no	—
227	C₂×C₁₁₃	—	2	1	no	—
228	C₂×C₁₁₃	—	8	4	no	—
229	C₂×C₁₁₃	C₂×#390	1	1	yes	characteristic
230	C₂×C₁₁₃	C₂×#410	1	2	yes	—
231	C₂×#532	C₂×C₁₁₃	1	1	yes	characteristic
232	C₂×#410	C₂×C₁₁₃	1	4	yes	complemented
233	#2285	C₂×C₁₁₃	1	1	yes	characteristic, complemented
234	C₄×#117	C₂×C₁₁₃	1	1	yes	characteristic, complemented
235	#1964	C₂×C₁₁₃	1	8	yes	complemented
236	C₁₁₃	#21212	1	1	yes	characteristic, direct product factor, Sylow
237	C₂²×#41	C₂²×C₁₁₃	1	2	yes	—
238	#532	—	2	1	no	—
239	#532	—	2	1	no	—
240	#532	—	2	2	no	—
241	#532	—	2	2	no	—
242	#532	—	2	1	no	—
243	#532	—	2	1	no	—
244	C₂×C₄⋊Q₈	C₄×C₁₁₃	1	1	yes	characteristic, complemented
245	C₂×#127	C₄×C₁₁₃	1	1	yes	characteristic, complemented
246	C₂×#124	C₄×C₁₁₃	1	2	yes	complemented
247	C₂×#121	C₂²×C₁₁₃	1	4	yes	—
248	C₂×C₄×D₄	C₄×C₁₁₃	1	4	yes	complemented
249	C₂×#117	C₂²×C₁₁₃	1	2	yes	—
250	C₂×#117	C₂²×C₁₁₃	1	2	yes	—
251	C₂²×C₄²	C₂²×C₁₁₃	1	1	yes	characteristic
252	#410	—	2	16	no	—
253	C₂×#99	C₂²×C₁₁₃	1	16	yes	—
254	#384	—	2	1	no	—
255	#382	—	2	1	no	—
256	#380	C₂²×C₁₁₃	1	4	yes	—
257	C₄×C₄⋊C₄	—	2	2	no	—
258	C₄×C₄⋊C₄	—	2	2	no	—
259	C₄×C₂²⋊C₄	C₂²×C₁₁₃	1	4	yes	—
260	#376	—	2	1	no	—
261	C₄³	—	2	1	no	—
262	C₂×#41	—	2	2	no	—
263	C₂×#41	—	2	2	no	—
264	C₂×#41	—	2	2	no	—
265	C₂×#41	—	2	2	no	—
266	C₂×#41	—	2	8	no	—
267	C₂×#41	—	2	8	no	—
268	C₂²×Q₈	C₂×C₄×C₁₁₃	1	2	yes	—
269	C₂²×D₄	C₂×C₄×C₁₁₃	1	2	yes	—
270	C₂²×D₄	C₂×C₄×C₁₁₃	1	4	yes	—
271	C₂²×D₄	C₂³×C₁₁₃	1	4	yes	—
272	C₂³×C₄	C₂×C₄×C₁₁₃	1	4	yes	—
273	C₂³×C₄	C₂³×C₁₁₃	1	2	yes	—
274	C₂³×C₄	C₂³×C₁₁₃	1	1	yes	characteristic
275	C₄⋊Q₈	—	2	1	no	—
276	C₄⋊Q₈	—	2	2	no	—
277	C₄⋊Q₈	—	2	1	no	—
278	#127	—	2	1	no	—
279	#127	—	2	2	no	—
280	#127	—	2	1	no	—
281	#124	—	2	2	no	—
282	#124	—	2	2	no	—
283	#124	—	2	4	no	—
284	#121	—	2	4	no	—
285	#121	—	2	4	no	—
286	#121	—	2	4	no	—
287	#121	—	2	4	no	—
288	C₄×D₄	—	2	4	no	—
289	C₄×D₄	—	2	4	no	—
290	C₄×D₄	—	2	4	no	—
291	C₄×D₄	—	2	4	no	—
292	#117	—	2	4	no	—
293	#117	—	2	8	no	—
294	#117	D₄×C₁₁₃	1	8	yes	complemented
295	C₂×C₄⋊C₄	—	2	2	no	—
296	C₂×C₄⋊C₄	—	2	2	no	—
297	C₂×C₄⋊C₄	—	2	4	no	—
298	C₂×C₄⋊C₄	C₂×C₄×C₁₁₃	1	4	yes	—
299	C₂×C₂²⋊C₄	C₂×C₄×C₁₁₃	1	8	yes	—
300	C₂×C₂²⋊C₄	C₂³×C₁₁₃	1	4	yes	—
301	C₂×C₂²⋊C₄	C₂³×C₁₁₃	1	4	yes	—
302	C₂×C₄²	—	2	2	no	—
303	C₂×C₄²	—	2	2	no	—
304	C₂×C₄²	—	2	4	no	—
305	C₂×C₄²	C₂×C₄×C₁₁₃	1	2	yes	—
306	C₂×C₄²	C₂×C₄×C₁₁₃	1	1	yes	characteristic
307	C₂×C₄²	C₂×C₄×C₁₁₃	1	1	yes	characteristic
308	C₂×C₄²	D₄×C₁₁₃	1	1	yes	characteristic
309	C₂×C₄²	D₄×C₁₁₃	1	1	yes	characteristic
310	C₂×C₄²	D₄×C₁₁₃	1	2	yes	—
311	C₂×C₄²	D₄×C₁₁₃	1	2	yes	—
312	C₂×C₄²	D₄×C₁₁₃	1	1	yes	characteristic
313	C₂×C₄²	D₄×C₁₁₃	1	1	yes	characteristic
314	#99	—	2	32	no	—
315	#95	—	2	4	no	—
316	#95	—	2	4	no	—
317	C₂⁴	C₂²×C₄×C₁₁₃	1	4	yes	—
318	C₂⁴	C₂⁴×C₁₁₃	1	1	yes	characteristic
319	#41	—	4	8	no	—
320	#41	—	4	8	no	—
321	#41	—	4	4	no	—
322	#41	—	4	4	no	—
323	#41	—	4	4	no	—
324	#41	—	4	4	no	—
325	C₂×Q₈	—	2	2	no	—
326	C₂×Q₈	—	2	2	no	—
327	C₂×Q₈	—	2	8	no	—
328	C₂×D₄	—	2	8	no	—
329	C₂×D₄	—	2	4	no	—
330	C₂×D₄	—	2	4	no	—
331	C₂×D₄	—	2	8	no	—
332	C₂×D₄	—	2	4	no	—
333	C₂×D₄	—	2	4	no	—
334	C₂×D₄	—	2	2	no	—
335	C₂×D₄	—	2	2	no	—
336	C₂×D₄	—	2	8	no	—
337	C₂×D₄	—	4	8	no	—
338	C₂²×C₄	—	2	4	no	—
339	C₂²×C₄	—	2	4	no	—
340	C₂²×C₄	—	2	4	no	—
341	C₂²×C₄	—	2	16	no	—
342	C₂²×C₄	—	2	4	no	—
343	C₂²×C₄	—	4	4	no	—
344	C₂²×C₄	C₂²×C₄×C₁₁₃	1	4	yes	—
345	C₂²×C₄	C₂²×C₄×C₁₁₃	1	2	yes	—
346	C₂²×C₄	C₂²×C₄×C₁₁₃	1	2	yes	—
347	C₂²×C₄	C₂²×C₄×C₁₁₃	1	1	yes	characteristic
348	C₂²×C₄	C₂²×C₄×C₁₁₃	1	1	yes	characteristic
349	C₂²×C₄	C₂×D₄×C₁₁₃	1	2	yes	—
350	C₂²×C₄	C₂×D₄×C₁₁₃	1	2	yes	—
351	C₂²×C₄	C₂×D₄×C₁₁₃	1	2	yes	—
352	C₂²×C₄	C₂×D₄×C₁₁₃	1	2	yes	—
353	C₂²×C₄	C₂×D₄×C₁₁₃	1	1	yes	characteristic
354	C₂²×C₄	C₂×D₄×C₁₁₃	1	1	yes	characteristic
355	C₂²×C₄	C₂×D₄×C₁₁₃	1	1	yes	characteristic
356	C₂²×C₄	C₂×D₄×C₁₁₃	1	1	yes	characteristic
357	C₂²×C₄	C₂×D₄×C₁₁₃	1	4	yes	—
358	C₂²×C₄	#41×C₁₁₃	1	4	yes	—
359	C₄⋊C₄	—	2	4	no	—
360	C₄⋊C₄	—	2	4	no	—
361	C₄⋊C₄	—	2	8	no	—
362	C₄⋊C₄	—	4	8	no	—
363	C₄⋊C₄	—	4	4	no	—
364	C₂²⋊C₄	—	2	4	no	—
365	C₂²⋊C₄	—	2	8	no	—
366	C₂²⋊C₄	—	2	8	no	—
367	C₂²⋊C₄	—	2	8	no	—
368	C₂²⋊C₄	C₂×D₄×C₁₁₃	1	8	yes	—
369	C₄²	—	2	8	no	—
370	C₄²	—	4	8	no	—
371	C₄²	—	4	4	no	—
372	C₄²	C₂²⋊C₄×C₁₁₃	1	2	yes	—
373	C₄²	C₂²⋊C₄×C₁₁₃	1	2	yes	—
374	C₄²	C₂²⋊C₄×C₁₁₃	1	2	yes	—
375	C₄²	C₂²⋊C₄×C₁₁₃	1	2	yes	—
376	C₄²	C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
377	C₄²	C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
378	C₄²	C₂²⋊C₄×C₁₁₃	1	2	yes	—
379	C₄²	C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
380	C₄²	C₂²⋊C₄×C₁₁₃	1	2	yes	—
381	C₄²	C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
382	C₂³	—	2	4	no	—
383	C₂³	—	2	8	no	—
384	C₂³	—	4	2	no	—
385	C₂³	—	4	8	no	—
386	C₂³	C₂³×C₄×C₁₁₃	1	1	yes	characteristic
387	C₂³	C₂²×D₄×C₁₁₃	1	1	yes	characteristic
388	C₂³	C₂²×D₄×C₁₁₃	1	1	yes	characteristic
389	C₂³	C₂²×D₄×C₁₁₃	1	2	yes	—
390	C₂³	C₂×#41×C₁₁₃	1	2	yes	—
391	Q₈	—	4	4	no	—
392	Q₈	—	4	4	no	—
393	D₄	—	4	8	no	—
394	D₄	—	4	8	no	—
395	D₄	—	4	4	no	—
396	D₄	—	4	4	no	—
397	D₄	—	4	4	no	—
398	D₄	—	4	4	no	—
399	D₄	—	4	8	no	—
400	C₂×C₄	—	2	4	no	—
401	C₂×C₄	—	2	4	no	—
402	C₂×C₄	—	2	4	no	—
403	C₂×C₄	—	2	4	no	—
404	C₂×C₄	—	2	8	no	—
405	C₂×C₄	—	2	8	no	—
406	C₂×C₄	—	4	2	no	—
407	C₂×C₄	—	4	2	no	—
408	C₂×C₄	—	4	2	no	—
409	C₂×C₄	—	4	8	no	—
410	C₂×C₄	—	4	8	no	—
411	C₂×C₄	—	4	8	no	—
412	C₂×C₄	—	4	8	no	—
413	C₂×C₄	—	4	4	no	—
414	C₂×C₄	C₂×C₂²⋊C₄×C₁₁₃	1	2	yes	—
415	C₂×C₄	C₂×C₂²⋊C₄×C₁₁₃	1	2	yes	—
416	C₂×C₄	C₂×C₂²⋊C₄×C₁₁₃	1	2	yes	—
417	C₂×C₄	C₂×C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
418	C₂×C₄	C₂×C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
419	C₂×C₄	C₂×C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
420	C₂×C₄	C₂×C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
421	C₂×C₄	C₂×C₂²⋊C₄×C₁₁₃	1	2	yes	—
422	C₂×C₄	C₄×D₄×C₁₁₃	1	4	yes	—
423	C₂×C₄	C₄×D₄×C₁₁₃	1	4	yes	—
424	C₂×C₄	#121×C₁₁₃	1	4	yes	—
425	C₂×C₄	#121×C₁₁₃	1	4	yes	—
426	C₂×C₄	#124×C₁₁₃	1	2	yes	—
427	C₂×C₄	#124×C₁₁₃	1	2	yes	—
428	C₂×C₄	#127×C₁₁₃	1	2	yes	—
429	C₂×C₄	#127×C₁₁₃	1	2	yes	—
430	C₂²	—	2	2	no	—
431	C₂²	—	2	2	no	—
432	C₂²	—	2	2	no	—
433	C₂²	—	4	2	no	—
434	C₂²	—	4	2	no	—
435	C₂²	—	4	4	no	—
436	C₂²	—	4	8	no	—
437	C₂²	—	8	8	no	—
438	C₂²	C₂²×C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
439	C₂²	C₂×C₄×D₄×C₁₁₃	1	2	yes	—
440	C₂²	C₂×#121×C₁₁₃	1	2	yes	—
441	C₂²	C₂×#124×C₁₁₃	1	1	yes	characteristic
442	C₂²	C₂×#127×C₁₁₃	1	1	yes	characteristic
443	C₄	—	2	2	no	—
444	C₄	—	2	2	no	—
445	C₄	—	2	2	no	—
446	C₄	—	4	8	no	—
447	C₄	—	4	4	no	—
448	C₄	—	8	4	no	—
449	C₄	#390×C₁₁₃	1	2	yes	—
450	C₄	#390×C₁₁₃	1	2	yes	—
451	C₄	#390×C₁₁₃	1	2	yes	—
452	C₄	#390×C₁₁₃	1	2	yes	—
453	C₄	#410×C₁₁₃	1	4	yes	complemented
454	C₂	—	2	2	no	—
455	C₂	—	2	1	no	—
456	C₂	—	2	2	no	—
457	C₂	—	2	1	no	—
458	C₂	—	8	4	no	—
459	C₂	C₂×#390×C₁₁₃	1	1	yes	characteristic
460	C₂	C₂×#410×C₁₁₃	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₁₃	C₂×C₂²⋊C₄
commutator	C₂³	C₂³×C₄×C₁₁₃
Frattini, cosocle	C₂⁴	C₂⁴×C₁₁₃
socle	C₂²×C₁₁₃	C₂²×C₂²⋊C₄
prime radical	C₂×#532×C₁₁₃	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₁₃ ◁ C₂²×C₄²×C₁₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#410×C₁₁₃	C₂	semidirect
3. C₁₁₃×#2285	C₂	semidirect
4. C₄×#117×C₁₁₃	C₂	semidirect
5. C₁₁₃×#1964	C₂	semidirect
13. C₂×C₄⋊Q₈×C₁₁₃	C₄	semidirect
14. C₂×#127×C₁₁₃	C₄	semidirect
15. C₂×#124×C₁₁₃	C₄	semidirect
17. C₂×C₄×D₄×C₁₁₃	C₄	semidirect
63. #117×C₁₁₃	D₄	semidirect
222. C₄×C₁₁₃	#410	semidirect
223. #21212	C₁₁₃	direct (N × H)
232. C₂×#410	C₂×C₁₁₃	semidirect
233. #2285	C₂×C₁₁₃	semidirect
234. C₄×#117	C₂×C₁₁₃	semidirect
235. #1964	C₂×C₁₁₃	semidirect
236. C₁₁₃	#21212	direct (N × H)
244. C₂×C₄⋊Q₈	C₄×C₁₁₃	semidirect
245. C₂×#127	C₄×C₁₁₃	semidirect
246. C₂×#124	C₄×C₁₁₃	semidirect
248. C₂×C₄×D₄	C₄×C₁₁₃	semidirect
294. #117	D₄×C₁₁₃	semidirect
453. C₄	#410×C₁₁₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#532×C₁₁₃	2
C₂×#410×C₁₁₃	2
C₁₁₃×#2285	2
C₄×#117×C₁₁₃	2
C₁₁₃×#1964	2
#21212	113

Subgroups — C113×#21212 #43289232

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁₃×#21212 #43289232