Subgroups — C₁₁₃×#10769 #43278789

Order 28928.

1378 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#518×C₁₁₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×#100×C₁₁₃	C₂	1	2	yes	maximal
3	C₂×#344×C₁₁₃	C₂	1	4	yes	maximal, complemented
4	C₂³×D₄×C₁₁₃	C₂²	1	1	yes	characteristic
5	C₂²×M₁₆×C₁₁₃	C₂²	1	4	yes	—
6	#518×C₁₁₃	—	2	8	no	—
7	C₂×C₄×D₄×C₁₁₃	—	2	4	no	—
8	C₂×#117×C₁₁₃	C₂²	1	2	yes	—
9	C₂²×C₂²⋊C₄×C₁₁₃	C₄	1	4	yes	—
10	C₂×#100×C₁₁₃	—	2	4	no	—
11	C₂×#100×C₁₁₃	—	2	4	no	—
12	C₂×#100×C₁₁₃	C₄	1	8	yes	complemented
13	#344×C₁₁₃	—	2	8	no	—
14	C₂⁵×C₁₁₃	C₂×C₄	1	2	yes	—
15	C₂²×D₄×C₁₁₃	—	2	4	no	—
16	C₂²×D₄×C₁₁₃	—	2	4	no	—
17	C₂²×D₄×C₁₁₃	C₂×C₄	1	4	yes	—
18	C₂²×D₄×C₁₁₃	D₄	1	2	yes	—
19	C₂²×D₄×C₁₁₃	D₄	1	4	yes	—
20	C₂²×D₄×C₁₁₃	Q₈	1	2	yes	—
21	C₂³×C₄×C₁₁₃	—	2	4	no	—
22	C₂³×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
23	C₂×M₁₆×C₁₁₃	—	2	4	no	—
24	C₂×M₁₆×C₁₁₃	—	2	4	no	—
25	C₂×M₁₆×C₁₁₃	—	2	4	no	—
26	C₂×M₁₆×C₁₁₃	—	2	4	no	—
27	C₂×M₁₆×C₁₁₃	—	2	4	no	—
28	C₂×M₁₆×C₁₁₃	C₂×C₄	1	8	yes	complemented
29	C₂²×C₈×C₁₁₃	—	2	4	no	—
30	C₄×D₄×C₁₁₃	—	2	32	no	—
31	#117×C₁₁₃	—	2	4	no	—
32	#117×C₁₁₃	D₄	1	8	yes	—
33	C₂×C₄⋊C₄×C₁₁₃	—	2	2	no	—
34	C₂×C₂²⋊C₄×C₁₁₃	—	2	4	no	—
35	C₂×C₂²⋊C₄×C₁₁₃	—	2	16	no	—
36	C₂×C₂²⋊C₄×C₁₁₃	C₂×C₄	1	4	yes	—
37	C₂×C₄²×C₁₁₃	—	2	2	no	—
38	#100×C₁₁₃	—	2	16	no	—
39	C₂⁴×C₁₁₃	—	2	4	no	—
40	C₂⁴×C₁₁₃	—	2	4	no	—
41	C₂⁴×C₁₁₃	—	8	4	no	—
42	C₂⁴×C₁₁₃	C₄²	1	4	yes	—
43	C₂⁴×C₁₁₃	C₂²⋊C₄	1	4	yes	—
44	C₂⁴×C₁₁₃	C₄⋊C₄	1	4	yes	—
45	C₂⁴×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
46	C₂×D₄×C₁₁₃	—	2	8	no	—
47	C₂×D₄×C₁₁₃	—	2	8	no	—
48	C₂×D₄×C₁₁₃	—	2	8	no	—
49	C₂×D₄×C₁₁₃	—	2	8	no	—
50	C₂×D₄×C₁₁₃	—	2	8	no	—
51	C₂×D₄×C₁₁₃	—	2	8	no	—
52	C₂×D₄×C₁₁₃	C₂²⋊C₄	1	8	yes	—
53	C₂×D₄×C₁₁₃	C₄⋊C₄	1	8	yes	—
54	C₂²×C₄×C₁₁₃	—	2	4	no	—
55	C₂²×C₄×C₁₁₃	—	2	16	no	—
56	C₂²×C₄×C₁₁₃	—	8	8	no	—
57	C₂²×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
58	C₂²×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
60	C₂²×C₄×C₁₁₃	C₂×D₄	1	4	yes	—
61	C₂²×C₄×C₁₁₃	C₂×Q₈	1	1	yes	characteristic
62	C₂²×C₄×C₁₁₃	#41	1	4	yes	—
63	M₁₆×C₁₁₃	—	2	8	no	—
64	M₁₆×C₁₁₃	—	2	8	no	—
65	M₁₆×C₁₁₃	—	4	4	no	—
66	M₁₆×C₁₁₃	—	4	4	no	—
67	C₂×C₈×C₁₁₃	—	2	8	no	—
68	C₂×C₈×C₁₁₃	—	4	4	no	—
69	C₂×C₈×C₁₁₃	—	4	4	no	—
70	C₄⋊C₄×C₁₁₃	—	2	8	no	—
71	C₂²⋊C₄×C₁₁₃	—	2	16	no	—
72	C₂²⋊C₄×C₁₁₃	—	2	4	no	—
73	C₂²⋊C₄×C₁₁₃	C₂²⋊C₄	1	8	yes	—
74	C₄²×C₁₁₃	—	2	8	no	—
75	C₂³×C₁₁₃	—	2	4	no	—
76	C₂³×C₁₁₃	—	2	4	no	—
77	C₂³×C₁₁₃	—	2	8	no	—
78	C₂³×C₁₁₃	—	2	8	no	—
79	C₂³×C₁₁₃	—	4	2	no	—
80	C₂³×C₁₁₃	—	4	8	no	—
81	C₂³×C₁₁₃	—	8	8	no	—
82	C₂³×C₁₁₃	—	8	4	no	—
83	C₂³×C₁₁₃	—	8	4	no	—
84	C₂³×C₁₁₃	—	8	8	no	—
85	C₂³×C₁₁₃	#95	1	8	yes	—
86	C₂³×C₁₁₃	C₂×C₄²	1	1	yes	characteristic
87	C₂³×C₁₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₂³×C₁₁₃	C₂×C₂²⋊C₄	1	2	yes	—
89	C₂³×C₁₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₂³×C₁₁₃	#117	1	2	yes	—
91	D₄×C₁₁₃	—	2	16	no	—
92	D₄×C₁₁₃	—	2	16	no	—
93	C₂×C₄×C₁₁₃	—	2	8	no	—
94	C₂×C₄×C₁₁₃	—	4	8	no	—
95	C₂×C₄×C₁₁₃	—	8	16	no	—
96	C₂×C₄×C₁₁₃	C₂×C₂²⋊C₄	1	2	yes	—
97	C₂×C₄×C₁₁₃	C₂×C₄⋊C₄	1	2	yes	—
98	C₂×C₄×C₁₁₃	C₄×D₄	1	4	yes	—
99	C₂×C₄×C₁₁₃	C₄×D₄	1	4	yes	—
100	C₂×C₄×C₁₁₃	#120	1	4	yes	—
101	C₂×C₄×C₁₁₃	#121	1	4	yes	—
102	C₂×C₄×C₁₁₃	C₂²⋊Q₈	1	4	yes	—
103	C₂×C₄×C₁₁₃	#123	1	4	yes	—
104	C₈×C₁₁₃	—	4	8	no	—
105	C₂²×C₁₁₃	—	2	2	no	—
106	C₂²×C₁₁₃	—	2	4	no	—
107	C₂²×C₁₁₃	—	2	8	no	—
108	C₂²×C₁₁₃	—	4	1	no	—
109	C₂²×C₁₁₃	—	4	2	no	—
110	C₂²×C₁₁₃	—	4	1	no	—
111	C₂²×C₁₁₃	—	4	8	no	—
112	C₂²×C₁₁₃	—	4	8	no	—
113	C₂²×C₁₁₃	—	4	8	no	—
114	C₂²×C₁₁₃	—	8	8	no	—
115	C₂²×C₁₁₃	—	8	8	no	—
116	C₂²×C₁₁₃	C₂×#95	1	1	yes	characteristic
117	C₂²×C₁₁₃	C₄×C₂²⋊C₄	1	2	yes	—
118	C₂²×C₁₁₃	#379	1	1	yes	characteristic
119	C₂²×C₁₁₃	#380	1	2	yes	—
120	C₂²×C₁₁₃	#381	1	1	yes	characteristic
121	C₄×C₁₁₃	—	4	8	no	—
122	C₄×C₁₁₃	#385	1	4	yes	—
123	C₄×C₁₁₃	#386	1	4	yes	—
124	#10769	C₁₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
125	C₂×C₁₁₃	—	2	1	no	—
126	C₂×C₁₁₃	—	2	1	no	—
127	C₂×C₁₁₃	—	2	2	no	—
128	C₂×C₁₁₃	—	2	2	no	—
129	C₂×C₁₁₃	—	4	8	no	—
130	C₂×C₁₁₃	C₂²⋊#95	1	1	yes	characteristic
131	C₂×C₁₁₃	#1829	1	2	yes	—
132	C₂×#518	C₂×C₁₁₃	1	1	yes	characteristic
133	C₂²×#100	C₂×C₁₁₃	1	2	yes	—
134	C₂×#344	C₂×C₁₁₃	1	4	yes	complemented
135	C₁₁₃	#10769	1	1	yes	characteristic, direct product factor, Sylow
136	C₂³×D₄	C₂²×C₁₁₃	1	1	yes	characteristic
137	C₂²×M₁₆	C₂²×C₁₁₃	1	4	yes	—
138	#518	—	2	8	no	—
139	C₂×C₄×D₄	—	2	4	no	—
140	C₂×#117	C₂²×C₁₁₃	1	2	yes	—
141	C₂²×C₂²⋊C₄	C₄×C₁₁₃	1	4	yes	—
142	C₂×#100	—	2	4	no	—
143	C₂×#100	—	2	4	no	—
144	C₂×#100	C₄×C₁₁₃	1	8	yes	complemented
145	#344	—	2	8	no	—
146	C₂⁵	C₂×C₄×C₁₁₃	1	2	yes	—
147	C₂²×D₄	—	2	4	no	—
148	C₂²×D₄	—	2	4	no	—
149	C₂²×D₄	C₂×C₄×C₁₁₃	1	4	yes	—
150	C₂²×D₄	D₄×C₁₁₃	1	2	yes	—
151	C₂²×D₄	D₄×C₁₁₃	1	4	yes	—
152	C₂²×D₄	Q₈×C₁₁₃	1	2	yes	—
153	C₂³×C₄	—	2	4	no	—
154	C₂³×C₄	C₂³×C₁₁₃	1	1	yes	characteristic
155	C₂×M₁₆	—	2	4	no	—
156	C₂×M₁₆	—	2	4	no	—
157	C₂×M₁₆	—	2	4	no	—
158	C₂×M₁₆	—	2	4	no	—
159	C₂×M₁₆	—	2	4	no	—
160	C₂×M₁₆	C₂×C₄×C₁₁₃	1	8	yes	complemented
161	C₂²×C₈	—	2	4	no	—
162	C₄×D₄	—	2	32	no	—
163	#117	—	2	4	no	—
164	#117	D₄×C₁₁₃	1	8	yes	—
165	C₂×C₄⋊C₄	—	2	2	no	—
166	C₂×C₂²⋊C₄	—	2	4	no	—
167	C₂×C₂²⋊C₄	—	2	16	no	—
168	C₂×C₂²⋊C₄	C₂×C₄×C₁₁₃	1	4	yes	—
169	C₂×C₄²	—	2	2	no	—
170	#100	—	2	16	no	—
171	C₂⁴	—	2	4	no	—
172	C₂⁴	—	2	4	no	—
173	C₂⁴	—	8	4	no	—
174	C₂⁴	C₄²×C₁₁₃	1	4	yes	—
175	C₂⁴	C₂²⋊C₄×C₁₁₃	1	4	yes	—
176	C₂⁴	C₄⋊C₄×C₁₁₃	1	4	yes	—
177	C₂⁴	C₂²×C₄×C₁₁₃	1	1	yes	characteristic
178	C₂×D₄	—	2	8	no	—
179	C₂×D₄	—	2	8	no	—
180	C₂×D₄	—	2	8	no	—
181	C₂×D₄	—	2	8	no	—
182	C₂×D₄	—	2	8	no	—
183	C₂×D₄	—	2	8	no	—
184	C₂×D₄	C₂²⋊C₄×C₁₁₃	1	8	yes	—
185	C₂×D₄	C₄⋊C₄×C₁₁₃	1	8	yes	—
186	C₂²×C₄	—	2	4	no	—
187	C₂²×C₄	—	2	16	no	—
188	C₂²×C₄	—	8	8	no	—
189	C₂²×C₄	C₂²×C₄×C₁₁₃	1	2	yes	—
190	C₂²×C₄	C₂×D₄×C₁₁₃	1	1	yes	characteristic
191	C₂²×C₄	C₂×D₄×C₁₁₃	1	2	yes	—
192	C₂²×C₄	C₂×D₄×C₁₁₃	1	4	yes	—
193	C₂²×C₄	C₂×Q₈×C₁₁₃	1	1	yes	characteristic
194	C₂²×C₄	#41×C₁₁₃	1	4	yes	—
195	M₁₆	—	2	8	no	—
196	M₁₆	—	2	8	no	—
197	M₁₆	—	4	4	no	—
198	M₁₆	—	4	4	no	—
199	C₂×C₈	—	2	8	no	—
200	C₂×C₈	—	4	4	no	—
201	C₂×C₈	—	4	4	no	—
202	C₄⋊C₄	—	2	8	no	—
203	C₂²⋊C₄	—	2	16	no	—
204	C₂²⋊C₄	—	2	4	no	—
205	C₂²⋊C₄	C₂²⋊C₄×C₁₁₃	1	8	yes	—
206	C₄²	—	2	8	no	—
207	C₂³	—	2	4	no	—
208	C₂³	—	2	4	no	—
209	C₂³	—	2	8	no	—
210	C₂³	—	2	8	no	—
211	C₂³	—	4	2	no	—
212	C₂³	—	4	8	no	—
213	C₂³	—	8	8	no	—
214	C₂³	—	8	4	no	—
215	C₂³	—	8	4	no	—
216	C₂³	—	8	8	no	—
217	C₂³	#95×C₁₁₃	1	8	yes	—
218	C₂³	C₂×C₄²×C₁₁₃	1	1	yes	characteristic
219	C₂³	C₂×C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
220	C₂³	C₂×C₂²⋊C₄×C₁₁₃	1	2	yes	—
221	C₂³	C₂×C₄⋊C₄×C₁₁₃	1	1	yes	characteristic
222	C₂³	#117×C₁₁₃	1	2	yes	—
223	D₄	—	2	16	no	—
224	D₄	—	2	16	no	—
225	C₂×C₄	—	2	8	no	—
226	C₂×C₄	—	4	8	no	—
227	C₂×C₄	—	8	16	no	—
228	C₂×C₄	C₂×C₂²⋊C₄×C₁₁₃	1	2	yes	—
229	C₂×C₄	C₂×C₄⋊C₄×C₁₁₃	1	2	yes	—
230	C₂×C₄	C₄×D₄×C₁₁₃	1	4	yes	—
231	C₂×C₄	C₄×D₄×C₁₁₃	1	4	yes	—
232	C₂×C₄	#120×C₁₁₃	1	4	yes	—
233	C₂×C₄	#121×C₁₁₃	1	4	yes	—
234	C₂×C₄	C₂²⋊Q₈×C₁₁₃	1	4	yes	—
235	C₂×C₄	#123×C₁₁₃	1	4	yes	—
236	C₈	—	4	8	no	—
237	C₂²	—	2	2	no	—
238	C₂²	—	2	4	no	—
239	C₂²	—	2	8	no	—
240	C₂²	—	4	1	no	—
241	C₂²	—	4	2	no	—
242	C₂²	—	4	1	no	—
243	C₂²	—	4	8	no	—
244	C₂²	—	4	8	no	—
245	C₂²	—	4	8	no	—
246	C₂²	—	8	8	no	—
247	C₂²	—	8	8	no	—
248	C₂²	C₂×#95×C₁₁₃	1	1	yes	characteristic
249	C₂²	C₄×C₂²⋊C₄×C₁₁₃	1	2	yes	—
250	C₂²	#379×C₁₁₃	1	1	yes	characteristic
251	C₂²	#380×C₁₁₃	1	2	yes	—
252	C₂²	#381×C₁₁₃	1	1	yes	characteristic
253	C₄	—	4	8	no	—
254	C₄	#385×C₁₁₃	1	4	yes	—
255	C₄	#386×C₁₁₃	1	4	yes	—
256	C₂	—	2	1	no	—
257	C₂	—	2	1	no	—
258	C₂	—	2	2	no	—
259	C₂	—	2	2	no	—
260	C₂	—	4	8	no	—
261	C₂	C₁₁₃×C₂²⋊#95	1	1	yes	characteristic
262	C₂	C₁₁₃×#1829	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₁₃	#381
commutator	C₂³	C₂×C₄²×C₁₁₃
Frattini, cosocle	C₂³×C₄	C₂³×C₁₁₃
prime radical	C₂³×D₄×C₁₁₃	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₁₃ ◁ C₂³×C₄×C₁₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×#344×C₁₁₃	C₂	semidirect
12. C₂×#100×C₁₁₃	C₄	semidirect
28. C₂×M₁₆×C₁₁₃	C₂×C₄	semidirect
124. #10769	C₁₁₃	direct (N × H)
134. C₂×#344	C₂×C₁₁₃	semidirect
135. C₁₁₃	#10769	direct (N × H)
144. C₂×#100	C₄×C₁₁₃	semidirect
160. C₂×M₁₆	C₂×C₄×C₁₁₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#518×C₁₁₃	2
C₂²×#100×C₁₁₃	2
C₂×#344×C₁₁₃	2
#10769	113

Subgroups — C113×#10769 #43278789

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁₃×#10769 #43278789