Subgroups — C₁₁₃×#15178 #43283198

Order 28928.

2446 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊Q₈×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₂²⋊Q₈×C₁₁₃	C₂	1	2	yes	maximal
3	C₂×#391×C₁₁₃	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×C₂²⋊C₄×C₁₁₃	C₂	1	1	yes	maximal, characteristic
5	C₁₁₃×#1444	C₂	1	4	yes	maximal, complemented
6	C₁₁₃×#1439	C₂	1	4	yes	maximal
7	C₁₁₃×C₄⋊#95	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×Q₈×C₁₁₃	C₂²	1	2	yes	—
9	C₂⁴×C₄×C₁₁₃	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Q₈×C₁₁₃	—	2	8	no	—
11	C₂×C₄⋊Q₈×C₁₁₃	—	2	4	no	—
12	C₂×C₂²⋊Q₈×C₁₁₃	—	2	8	no	—
13	C₂×C₂²⋊Q₈×C₁₁₃	—	2	8	no	—
14	C₂×C₂²⋊Q₈×C₁₁₃	—	2	8	no	—
15	C₂²×C₄⋊C₄×C₁₁₃	C₂²	1	4	yes	—
16	C₂²×C₄⋊C₄×C₁₁₃	C₂²	1	4	yes	—
17	C₂²×C₄⋊C₄×C₁₁₃	C₂²	1	2	yes	—
18	C₂²×C₂²⋊C₄×C₁₁₃	C₂²	1	4	yes	—
19	C₂²×C₂²⋊C₄×C₁₁₃	C₂²	1	2	yes	—
20	C₂²×C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
21	C₂²×C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
22	#391×C₁₁₃	—	2	8	no	—
23	C₄×C₂²⋊C₄×C₁₁₃	—	2	4	no	—
24	C₂×#95×C₁₁₃	C₂²	1	8	yes	—
25	C₂×#95×C₁₁₃	C₂²	1	4	yes	—
26	C₂×#95×C₁₁₃	C₂²	1	2	yes	—
27	C₂⁵×C₁₁₃	C₂³	1	1	yes	characteristic
28	C₂²×Q₈×C₁₁₃	—	2	4	no	—
29	C₂²×Q₈×C₁₁₃	—	2	4	no	—
30	C₂²×Q₈×C₁₁₃	—	2	4	no	—
31	C₂²×Q₈×C₁₁₃	—	2	4	no	—
32	C₂²×Q₈×C₁₁₃	—	2	4	no	—
33	C₂²×Q₈×C₁₁₃	—	2	4	no	—
34	C₂²×Q₈×C₁₁₃	D₄	1	8	yes	—
35	C₂³×C₄×C₁₁₃	—	2	4	no	—
36	C₂³×C₄×C₁₁₃	—	2	4	no	—
37	C₂³×C₄×C₁₁₃	—	2	2	no	—
38	C₂³×C₄×C₁₁₃	—	2	2	no	—
39	C₂³×C₄×C₁₁₃	D₄	1	4	yes	—
40	C₂³×C₄×C₁₁₃	C₂³	1	4	yes	—
41	C₂³×C₄×C₁₁₃	C₂³	1	4	yes	—
42	C₂³×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
43	C₂³×C₄×C₁₁₃	C₂³	1	2	yes	—
44	C₂³×C₄×C₁₁₃	C₂³	1	2	yes	—
45	C₂³×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
46	C₄⋊Q₈×C₁₁₃	—	2	32	no	—
47	C₂²⋊Q₈×C₁₁₃	—	2	64	no	—
48	C₂×C₄⋊C₄×C₁₁₃	—	2	8	no	—
49	C₂×C₄⋊C₄×C₁₁₃	—	2	8	no	—
50	C₂×C₄⋊C₄×C₁₁₃	—	2	8	no	—
51	C₂×C₄⋊C₄×C₁₁₃	—	2	4	no	—
52	C₂×C₄⋊C₄×C₁₁₃	—	2	4	no	—
53	C₂×C₄⋊C₄×C₁₁₃	—	2	16	no	—
54	C₂×C₄⋊C₄×C₁₁₃	—	2	8	no	—
55	C₂×C₄⋊C₄×C₁₁₃	D₄	1	8	yes	—
56	C₂×C₂²⋊C₄×C₁₁₃	—	2	8	no	—
57	C₂×C₂²⋊C₄×C₁₁₃	—	2	8	no	—
58	C₂×C₂²⋊C₄×C₁₁₃	—	2	8	no	—
59	C₂×C₂²⋊C₄×C₁₁₃	—	2	4	no	—
60	C₂×C₂²⋊C₄×C₁₁₃	—	2	4	no	—
61	C₂×C₂²⋊C₄×C₁₁₃	Q₈	1	8	yes	complemented
62	C₂×C₄²×C₁₁₃	—	2	4	no	—
63	C₂×C₄²×C₁₁₃	—	2	2	no	—
64	C₂×C₄²×C₁₁₃	—	2	4	no	—
65	C₂×C₄²×C₁₁₃	D₄	1	4	yes	—
66	#95×C₁₁₃	—	2	4	no	—
67	#95×C₁₁₃	—	2	16	no	—
68	#95×C₁₁₃	—	2	8	no	—
69	C₂⁴×C₁₁₃	—	2	2	no	—
70	C₂⁴×C₁₁₃	—	2	2	no	—
71	C₂⁴×C₁₁₃	—	2	1	no	—
72	C₂⁴×C₁₁₃	—	2	2	no	—
73	C₂⁴×C₁₁₃	—	2	1	no	—
74	C₂⁴×C₁₁₃	—	2	4	no	—
75	C₂⁴×C₁₁₃	C₂×D₄	1	2	yes	—
76	C₂⁴×C₁₁₃	C₂×Q₈	1	4	yes	—
77	C₂⁴×C₁₁₃	C₂⁴	1	1	yes	characteristic
78	C₂×Q₈×C₁₁₃	—	2	16	no	—
79	C₂×Q₈×C₁₁₃	—	2	16	no	—
80	C₂×Q₈×C₁₁₃	—	2	16	no	—
81	C₂×Q₈×C₁₁₃	—	4	8	no	—
82	C₂×Q₈×C₁₁₃	—	4	8	no	—
83	C₂×Q₈×C₁₁₃	—	4	8	no	—
84	C₂×Q₈×C₁₁₃	—	4	8	no	—
85	C₂²×C₄×C₁₁₃	—	2	4	no	—
86	C₂²×C₄×C₁₁₃	—	2	4	no	—
87	C₂²×C₄×C₁₁₃	—	2	4	no	—
88	C₂²×C₄×C₁₁₃	—	2	1	no	—
89	C₂²×C₄×C₁₁₃	—	2	4	no	—
90	C₂²×C₄×C₁₁₃	—	2	2	no	—
91	C₂²×C₄×C₁₁₃	—	2	4	no	—
92	C₂²×C₄×C₁₁₃	—	2	4	no	—
93	C₂²×C₄×C₁₁₃	—	2	4	no	—
94	C₂²×C₄×C₁₁₃	—	2	4	no	—
95	C₂²×C₄×C₁₁₃	—	2	2	no	—
96	C₂²×C₄×C₁₁₃	—	2	2	no	—
97	C₂²×C₄×C₁₁₃	—	2	4	no	—
98	C₂²×C₄×C₁₁₃	—	2	4	no	—
99	C₂²×C₄×C₁₁₃	—	2	8	no	—
100	C₂²×C₄×C₁₁₃	—	2	2	no	—
101	C₂²×C₄×C₁₁₃	—	2	1	no	—
102	C₂²×C₄×C₁₁₃	—	2	4	no	—
103	C₂²×C₄×C₁₁₃	—	2	8	no	—
104	C₂²×C₄×C₁₁₃	—	2	4	no	—
105	C₂²×C₄×C₁₁₃	—	2	16	no	—
106	C₂²×C₄×C₁₁₃	—	2	8	no	—
107	C₂²×C₄×C₁₁₃	—	4	4	no	—
108	C₂²×C₄×C₁₁₃	—	4	8	no	—
109	C₂²×C₄×C₁₁₃	—	4	4	no	—
110	C₂²×C₄×C₁₁₃	C₂×D₄	1	8	yes	—
111	C₂²×C₄×C₁₁₃	C₂×D₄	1	8	yes	—
112	C₂²×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
113	C₂²×C₄×C₁₁₃	C₂×D₄	1	4	yes	—
114	C₂²×C₄×C₁₁₃	C₂×D₄	1	4	yes	—
115	C₂²×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
116	C₂²×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
117	C₂²×C₄×C₁₁₃	C₂×D₄	1	4	yes	—
118	C₂²×C₄×C₁₁₃	C₂×Q₈	1	4	yes	—
119	C₂²×C₄×C₁₁₃	C₂×Q₈	1	4	yes	—
120	C₂²×C₄×C₁₁₃	#41	1	4	yes	—
121	C₂²×C₄×C₁₁₃	#41	1	2	yes	—
122	C₄⋊C₄×C₁₁₃	—	2	32	no	—
123	C₄⋊C₄×C₁₁₃	—	2	32	no	—
124	C₄⋊C₄×C₁₁₃	—	2	16	no	—
125	C₂²⋊C₄×C₁₁₃	—	2	32	no	—
126	C₂²⋊C₄×C₁₁₃	—	2	16	no	—
127	C₄²×C₁₁₃	—	2	8	no	—
128	C₄²×C₁₁₃	—	4	4	no	—
129	C₂³×C₁₁₃	—	2	4	no	—
130	C₂³×C₁₁₃	—	2	2	no	—
131	C₂³×C₁₁₃	—	2	4	no	—
132	C₂³×C₁₁₃	—	2	2	no	—
133	C₂³×C₁₁₃	—	2	4	no	—
134	C₂³×C₁₁₃	—	2	4	no	—
135	C₂³×C₁₁₃	—	2	8	no	—
136	C₂³×C₁₁₃	—	2	8	no	—
137	C₂³×C₁₁₃	—	4	1	no	—
138	C₂³×C₁₁₃	—	4	2	no	—
139	C₂³×C₁₁₃	—	4	1	no	—
140	C₂³×C₁₁₃	—	4	2	no	—
141	C₂³×C₁₁₃	—	4	2	no	—
142	C₂³×C₁₁₃	—	4	4	no	—
143	C₂³×C₁₁₃	—	4	4	no	—
144	C₂³×C₁₁₃	C₄⋊Q₈	1	4	yes	complemented
145	C₂³×C₁₁₃	C₂²×D₄	1	1	yes	characteristic
146	C₂³×C₁₁₃	C₂²×D₄	1	2	yes	—
147	C₂³×C₁₁₃	C₂²×D₄	1	1	yes	characteristic
148	C₂³×C₁₁₃	C₂²×D₄	1	2	yes	—
149	C₂³×C₁₁₃	C₂²×Q₈	1	2	yes	—
150	C₂³×C₁₁₃	C₂×#41	1	1	yes	characteristic
151	C₂³×C₁₁₃	C₂×#41	1	2	yes	—
152	C₂³×C₁₁₃	2₋¹⁺⁴	1	4	yes	—
153	Q₈×C₁₁₃	—	4	32	no	—
154	C₂×C₄×C₁₁₃	—	2	4	no	—
155	C₂×C₄×C₁₁₃	—	2	4	no	—
156	C₂×C₄×C₁₁₃	—	2	8	no	—
157	C₂×C₄×C₁₁₃	—	2	8	no	—
158	C₂×C₄×C₁₁₃	—	2	8	no	—
159	C₂×C₄×C₁₁₃	—	2	8	no	—
160	C₂×C₄×C₁₁₃	—	2	4	no	—
161	C₂×C₄×C₁₁₃	—	2	8	no	—
162	C₂×C₄×C₁₁₃	—	2	2	no	—
163	C₂×C₄×C₁₁₃	—	2	8	no	—
164	C₂×C₄×C₁₁₃	—	2	4	no	—
165	C₂×C₄×C₁₁₃	—	2	16	no	—
166	C₂×C₄×C₁₁₃	—	2	8	no	—
167	C₂×C₄×C₁₁₃	—	4	1	no	—
168	C₂×C₄×C₁₁₃	—	4	4	no	—
169	C₂×C₄×C₁₁₃	—	4	2	no	—
170	C₂×C₄×C₁₁₃	—	4	4	no	—
171	C₂×C₄×C₁₁₃	—	4	4	no	—
172	C₂×C₄×C₁₁₃	—	4	2	no	—
173	C₂×C₄×C₁₁₃	—	4	1	no	—
174	C₂×C₄×C₁₁₃	—	4	4	no	—
175	C₂×C₄×C₁₁₃	—	4	4	no	—
176	C₂×C₄×C₁₁₃	—	4	4	no	—
177	C₂×C₄×C₁₁₃	—	4	2	no	—
178	C₂×C₄×C₁₁₃	—	4	4	no	—
179	C₂×C₄×C₁₁₃	—	4	8	no	—
180	C₂×C₄×C₁₁₃	—	4	16	no	—
181	C₂×C₄×C₁₁₃	#120	1	4	yes	—
182	C₂×C₄×C₁₁₃	#120	1	8	yes	—
183	C₂×C₄×C₁₁₃	#121	1	4	yes	—
184	C₂×C₄×C₁₁₃	#121	1	8	yes	—
185	C₂×C₄×C₁₁₃	C₂²⋊Q₈	1	8	yes	complemented
186	C₂×C₄×C₁₁₃	#127	1	4	yes	—
187	C₂²×C₁₁₃	—	2	4	no	—
188	C₂²×C₁₁₃	—	2	8	no	—
189	C₂²×C₁₁₃	—	4	2	no	—
190	C₂²×C₁₁₃	—	4	2	no	—
191	C₂²×C₁₁₃	—	4	4	no	—
192	C₂²×C₁₁₃	—	4	8	no	—
193	C₂²×C₁₁₃	—	4	4	no	—
194	C₂²×C₁₁₃	—	4	4	no	—
195	C₂²×C₁₁₃	C₂×#120	1	1	yes	characteristic
196	C₂²×C₁₁₃	C₂×#120	1	2	yes	—
197	C₂²×C₁₁₃	C₂×#121	1	1	yes	characteristic
198	C₂²×C₁₁₃	C₂×#121	1	2	yes	—
199	C₂²×C₁₁₃	C₂×C₂²⋊Q₈	1	2	yes	—
200	C₂²×C₁₁₃	C₂×#127	1	1	yes	characteristic
201	C₂²×C₁₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
202	C₂²×C₁₁₃	#532	1	1	yes	characteristic
203	C₂²×C₁₁₃	#533	1	2	yes	—
204	C₂²×C₁₁₃	#536	1	2	yes	—
205	C₂²×C₁₁₃	#536	1	2	yes	—
206	C₂²×C₁₁₃	#541	1	2	yes	—
207	C₂²×C₁₁₃	#547	1	4	yes	—
208	C₂²×C₁₁₃	#548	1	4	yes	—
209	C₂²×C₁₁₃	D₄×Q₈	1	4	yes	—
210	C₂²×C₁₁₃	D₄×Q₈	1	4	yes	—
211	C₄×C₁₁₃	—	2	8	no	—
212	C₄×C₁₁₃	—	4	2	no	—
213	C₄×C₁₁₃	—	4	8	no	—
214	C₄×C₁₁₃	—	4	8	no	—
215	C₄×C₁₁₃	—	4	4	no	—
216	C₄×C₁₁₃	#392	1	8	yes	—
217	#15178	C₁₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
218	C₂×C₁₁₃	—	4	4	no	—
219	C₂×C₁₁₃	C₂×#392	1	1	yes	characteristic
220	C₂×C₁₁₃	#2409	1	2	yes	—
221	C₂×C₁₁₃	#2429	1	4	yes	—
222	C₂×C₁₁₃	#2432	1	2	yes	—
223	C₂×C₁₁₃	#2434	1	2	yes	—
224	C₂×C₁₁₃	#2542	1	2	yes	—
225	C₂×C₁₁₃	#2621	1	1	yes	characteristic
226	C₂×C₁₁₃	#2857	1	1	yes	characteristic
227	C₂²×C₄⋊Q₈	C₂×C₁₁₃	1	1	yes	characteristic, complemented
228	C₂²×C₂²⋊Q₈	C₂×C₁₁₃	1	2	yes	—
229	C₂×#391	C₂×C₁₁₃	1	2	yes	complemented
230	C₂×C₄×C₂²⋊C₄	C₂×C₁₁₃	1	1	yes	characteristic
231	#1444	C₂×C₁₁₃	1	4	yes	complemented
232	#1439	C₂×C₁₁₃	1	4	yes	—
233	C₄⋊#95	C₂×C₁₁₃	1	1	yes	characteristic, complemented
234	C₁₁₃	#15178	1	1	yes	characteristic, direct product factor, Sylow
235	C₂³×Q₈	C₂²×C₁₁₃	1	2	yes	—
236	C₂⁴×C₄	C₂²×C₁₁₃	1	1	yes	characteristic
237	C₂×C₄⋊Q₈	—	2	8	no	—
238	C₂×C₄⋊Q₈	—	2	4	no	—
239	C₂×C₂²⋊Q₈	—	2	8	no	—
240	C₂×C₂²⋊Q₈	—	2	8	no	—
241	C₂×C₂²⋊Q₈	—	2	8	no	—
242	C₂²×C₄⋊C₄	C₂²×C₁₁₃	1	4	yes	—
243	C₂²×C₄⋊C₄	C₂²×C₁₁₃	1	4	yes	—
244	C₂²×C₄⋊C₄	C₂²×C₁₁₃	1	2	yes	—
245	C₂²×C₂²⋊C₄	C₂²×C₁₁₃	1	4	yes	—
246	C₂²×C₂²⋊C₄	C₂²×C₁₁₃	1	2	yes	—
247	C₂²×C₄²	C₂²×C₁₁₃	1	1	yes	characteristic
248	C₂²×C₄²	C₂²×C₁₁₃	1	1	yes	characteristic
249	#391	—	2	8	no	—
250	C₄×C₂²⋊C₄	—	2	4	no	—
251	C₂×#95	C₂²×C₁₁₃	1	4	yes	—
252	C₂×#95	C₂²×C₁₁₃	1	8	yes	—
253	C₂×#95	C₂²×C₁₁₃	1	2	yes	—
254	C₂⁵	C₂³×C₁₁₃	1	1	yes	characteristic
255	C₂²×Q₈	—	2	4	no	—
256	C₂²×Q₈	—	2	4	no	—
257	C₂²×Q₈	—	2	4	no	—
258	C₂²×Q₈	—	2	4	no	—
259	C₂²×Q₈	—	2	4	no	—
260	C₂²×Q₈	—	2	4	no	—
261	C₂²×Q₈	D₄×C₁₁₃	1	8	yes	—
262	C₂³×C₄	—	2	4	no	—
263	C₂³×C₄	—	2	4	no	—
264	C₂³×C₄	—	2	2	no	—
265	C₂³×C₄	—	2	2	no	—
266	C₂³×C₄	D₄×C₁₁₃	1	4	yes	—
267	C₂³×C₄	C₂³×C₁₁₃	1	4	yes	—
268	C₂³×C₄	C₂³×C₁₁₃	1	4	yes	—
269	C₂³×C₄	C₂³×C₁₁₃	1	1	yes	characteristic
270	C₂³×C₄	C₂³×C₁₁₃	1	2	yes	—
271	C₂³×C₄	C₂³×C₁₁₃	1	2	yes	—
272	C₂³×C₄	C₂³×C₁₁₃	1	1	yes	characteristic
273	C₄⋊Q₈	—	2	32	no	—
274	C₂²⋊Q₈	—	2	64	no	—
275	C₂×C₄⋊C₄	—	2	8	no	—
276	C₂×C₄⋊C₄	—	2	8	no	—
277	C₂×C₄⋊C₄	—	2	8	no	—
278	C₂×C₄⋊C₄	—	2	4	no	—
279	C₂×C₄⋊C₄	—	2	4	no	—
280	C₂×C₄⋊C₄	—	2	16	no	—
281	C₂×C₄⋊C₄	—	2	8	no	—
282	C₂×C₄⋊C₄	D₄×C₁₁₃	1	8	yes	—
283	C₂×C₂²⋊C₄	—	2	8	no	—
284	C₂×C₂²⋊C₄	—	2	8	no	—
285	C₂×C₂²⋊C₄	—	2	8	no	—
286	C₂×C₂²⋊C₄	—	2	4	no	—
287	C₂×C₂²⋊C₄	—	2	4	no	—
288	C₂×C₂²⋊C₄	Q₈×C₁₁₃	1	8	yes	complemented
289	C₂×C₄²	—	2	4	no	—
290	C₂×C₄²	—	2	2	no	—
291	C₂×C₄²	—	2	4	no	—
292	C₂×C₄²	D₄×C₁₁₃	1	4	yes	—
293	#95	—	2	4	no	—
294	#95	—	2	16	no	—
295	#95	—	2	8	no	—
296	C₂⁴	—	2	2	no	—
297	C₂⁴	—	2	2	no	—
298	C₂⁴	—	2	1	no	—
299	C₂⁴	—	2	2	no	—
300	C₂⁴	—	2	1	no	—
301	C₂⁴	—	2	4	no	—
302	C₂⁴	C₂×D₄×C₁₁₃	1	2	yes	—
303	C₂⁴	C₂×Q₈×C₁₁₃	1	4	yes	—
304	C₂⁴	C₂⁴×C₁₁₃	1	1	yes	characteristic
305	C₂×Q₈	—	2	16	no	—
306	C₂×Q₈	—	2	16	no	—
307	C₂×Q₈	—	2	16	no	—
308	C₂×Q₈	—	4	8	no	—
309	C₂×Q₈	—	4	8	no	—
310	C₂×Q₈	—	4	8	no	—
311	C₂×Q₈	—	4	8	no	—
312	C₂²×C₄	—	2	4	no	—
313	C₂²×C₄	—	2	4	no	—
314	C₂²×C₄	—	2	4	no	—
315	C₂²×C₄	—	2	1	no	—
316	C₂²×C₄	—	2	4	no	—
317	C₂²×C₄	—	2	2	no	—
318	C₂²×C₄	—	2	4	no	—
319	C₂²×C₄	—	2	4	no	—
320	C₂²×C₄	—	2	4	no	—
321	C₂²×C₄	—	2	4	no	—
322	C₂²×C₄	—	2	2	no	—
323	C₂²×C₄	—	2	2	no	—
324	C₂²×C₄	—	2	4	no	—
325	C₂²×C₄	—	2	4	no	—
326	C₂²×C₄	—	2	8	no	—
327	C₂²×C₄	—	2	2	no	—
328	C₂²×C₄	—	2	1	no	—
329	C₂²×C₄	—	2	4	no	—
330	C₂²×C₄	—	2	8	no	—
331	C₂²×C₄	—	2	4	no	—
332	C₂²×C₄	—	2	16	no	—
333	C₂²×C₄	—	2	8	no	—
334	C₂²×C₄	—	4	4	no	—
335	C₂²×C₄	—	4	8	no	—
336	C₂²×C₄	—	4	4	no	—
337	C₂²×C₄	C₂×D₄×C₁₁₃	1	8	yes	—
338	C₂²×C₄	C₂×D₄×C₁₁₃	1	8	yes	—
339	C₂²×C₄	C₂×D₄×C₁₁₃	1	2	yes	—
340	C₂²×C₄	C₂×D₄×C₁₁₃	1	4	yes	—
341	C₂²×C₄	C₂×D₄×C₁₁₃	1	4	yes	—
342	C₂²×C₄	C₂×D₄×C₁₁₃	1	2	yes	—
343	C₂²×C₄	C₂×D₄×C₁₁₃	1	2	yes	—
344	C₂²×C₄	C₂×D₄×C₁₁₃	1	4	yes	—
345	C₂²×C₄	C₂×Q₈×C₁₁₃	1	4	yes	—
346	C₂²×C₄	C₂×Q₈×C₁₁₃	1	4	yes	—
347	C₂²×C₄	#41×C₁₁₃	1	4	yes	—
348	C₂²×C₄	#41×C₁₁₃	1	2	yes	—
349	C₄⋊C₄	—	2	32	no	—
350	C₄⋊C₄	—	2	32	no	—
351	C₄⋊C₄	—	2	16	no	—
352	C₂²⋊C₄	—	2	32	no	—
353	C₂²⋊C₄	—	2	16	no	—
354	C₄²	—	2	8	no	—
355	C₄²	—	4	4	no	—
356	C₂³	—	2	4	no	—
357	C₂³	—	2	2	no	—
358	C₂³	—	2	4	no	—
359	C₂³	—	2	2	no	—
360	C₂³	—	2	4	no	—
361	C₂³	—	2	4	no	—
362	C₂³	—	2	8	no	—
363	C₂³	—	2	8	no	—
364	C₂³	—	4	1	no	—
365	C₂³	—	4	2	no	—
366	C₂³	—	4	1	no	—
367	C₂³	—	4	2	no	—
368	C₂³	—	4	2	no	—
369	C₂³	—	4	4	no	—
370	C₂³	—	4	4	no	—
371	C₂³	C₄⋊Q₈×C₁₁₃	1	4	yes	complemented
372	C₂³	C₂²×D₄×C₁₁₃	1	1	yes	characteristic
373	C₂³	C₂²×D₄×C₁₁₃	1	2	yes	—
374	C₂³	C₂²×D₄×C₁₁₃	1	1	yes	characteristic
375	C₂³	C₂²×D₄×C₁₁₃	1	2	yes	—
376	C₂³	C₂²×Q₈×C₁₁₃	1	2	yes	—
377	C₂³	C₂×#41×C₁₁₃	1	1	yes	characteristic
378	C₂³	C₂×#41×C₁₁₃	1	2	yes	—
379	C₂³	2₋¹⁺⁴×C₁₁₃	1	4	yes	—
380	Q₈	—	4	32	no	—
381	C₂×C₄	—	2	4	no	—
382	C₂×C₄	—	2	4	no	—
383	C₂×C₄	—	2	8	no	—
384	C₂×C₄	—	2	8	no	—
385	C₂×C₄	—	2	8	no	—
386	C₂×C₄	—	2	8	no	—
387	C₂×C₄	—	2	4	no	—
388	C₂×C₄	—	2	8	no	—
389	C₂×C₄	—	2	2	no	—
390	C₂×C₄	—	2	8	no	—
391	C₂×C₄	—	2	4	no	—
392	C₂×C₄	—	2	16	no	—
393	C₂×C₄	—	2	8	no	—
394	C₂×C₄	—	4	1	no	—
395	C₂×C₄	—	4	4	no	—
396	C₂×C₄	—	4	2	no	—
397	C₂×C₄	—	4	4	no	—
398	C₂×C₄	—	4	4	no	—
399	C₂×C₄	—	4	2	no	—
400	C₂×C₄	—	4	1	no	—
401	C₂×C₄	—	4	4	no	—
402	C₂×C₄	—	4	4	no	—
403	C₂×C₄	—	4	2	no	—
404	C₂×C₄	—	4	4	no	—
405	C₂×C₄	—	4	4	no	—
406	C₂×C₄	—	4	8	no	—
407	C₂×C₄	—	4	16	no	—
408	C₂×C₄	#120×C₁₁₃	1	4	yes	—
409	C₂×C₄	#120×C₁₁₃	1	8	yes	—
410	C₂×C₄	#121×C₁₁₃	1	4	yes	—
411	C₂×C₄	#121×C₁₁₃	1	8	yes	—
412	C₂×C₄	C₂²⋊Q₈×C₁₁₃	1	8	yes	complemented
413	C₂×C₄	#127×C₁₁₃	1	4	yes	—
414	C₂²	—	2	4	no	—
415	C₂²	—	2	8	no	—
416	C₂²	—	4	4	no	—
417	C₂²	—	4	2	no	—
418	C₂²	—	4	2	no	—
419	C₂²	—	4	4	no	—
420	C₂²	—	4	8	no	—
421	C₂²	—	4	4	no	—
422	C₂²	C₂×#120×C₁₁₃	1	1	yes	characteristic
423	C₂²	C₂×#120×C₁₁₃	1	2	yes	—
424	C₂²	C₂×#121×C₁₁₃	1	1	yes	characteristic
425	C₂²	C₂×#121×C₁₁₃	1	2	yes	—
426	C₂²	C₂×C₂²⋊Q₈×C₁₁₃	1	2	yes	—
427	C₂²	C₂×#127×C₁₁₃	1	1	yes	characteristic
428	C₂²	C₂×C₄⋊Q₈×C₁₁₃	1	1	yes	characteristic
429	C₂²	#532×C₁₁₃	1	1	yes	characteristic
430	C₂²	#533×C₁₁₃	1	2	yes	—
431	C₂²	#536×C₁₁₃	1	2	yes	—
432	C₂²	#536×C₁₁₃	1	2	yes	—
433	C₂²	#541×C₁₁₃	1	2	yes	—
434	C₂²	#547×C₁₁₃	1	4	yes	—
435	C₂²	#548×C₁₁₃	1	4	yes	—
436	C₂²	D₄×Q₈×C₁₁₃	1	4	yes	—
437	C₂²	D₄×Q₈×C₁₁₃	1	4	yes	—
438	C₄	—	2	8	no	—
439	C₄	—	4	4	no	—
440	C₄	—	4	2	no	—
441	C₄	—	4	8	no	—
442	C₄	—	4	8	no	—
443	C₄	#392×C₁₁₃	1	8	yes	—
444	C₂	—	4	4	no	—
445	C₂	C₂×#392×C₁₁₃	1	1	yes	characteristic
446	C₂	C₁₁₃×#2409	1	2	yes	—
447	C₂	C₁₁₃×#2429	1	4	yes	—
448	C₂	C₁₁₃×#2432	1	2	yes	—
449	C₂	C₁₁₃×#2434	1	2	yes	—
450	C₂	C₁₁₃×#2542	1	2	yes	—
451	C₂	C₁₁₃×#2621	1	1	yes	characteristic
452	C₂	C₁₁₃×#2857	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₁₁₃	C₂⁴
commutator, Frattini, cosocle	C₂⁴	C₂⁴×C₁₁₃
prime radical	C₂⁵×C₁₁₃	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₄⋊Q₈×C₁₁₃	C₂	semidirect
3. C₂×#391×C₁₁₃	C₂	semidirect
5. C₁₁₃×#1444	C₂	semidirect
7. C₁₁₃×C₄⋊#95	C₂	semidirect
61. C₂×C₂²⋊C₄×C₁₁₃	Q₈	semidirect
144. C₂³×C₁₁₃	C₄⋊Q₈	semidirect
185. C₂×C₄×C₁₁₃	C₂²⋊Q₈	semidirect
217. #15178	C₁₁₃	direct (N × H)
227. C₂²×C₄⋊Q₈	C₂×C₁₁₃	semidirect
229. C₂×#391	C₂×C₁₁₃	semidirect
231. #1444	C₂×C₁₁₃	semidirect
233. C₄⋊#95	C₂×C₁₁₃	semidirect
234. C₁₁₃	#15178	direct (N × H)
288. C₂×C₂²⋊C₄	Q₈×C₁₁₃	semidirect
371. C₂³	C₄⋊Q₈×C₁₁₃	semidirect
412. C₂×C₄	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
C₂²×C₄⋊Q₈×C₁₁₃	2
C₂²×C₂²⋊Q₈×C₁₁₃	2
C₂×#391×C₁₁₃	2
C₂×C₄×C₂²⋊C₄×C₁₁₃	2
C₁₁₃×#1444	2
C₁₁₃×#1439	2
C₁₁₃×C₄⋊#95	2
#15178	113

Subgroups — C113×#15178 #43283198

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁₃×#15178 #43283198