Subgroups — C₁₁₃×#2699 #28387495

Order 14464.

612 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#121×C₁₁₃	C₂	1	2	yes	maximal, complemented
2	#398×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#394×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#394×C₁₁₃	C₂	1	2	yes	maximal, complemented
5	#392×C₁₁₃	C₂	1	2	yes	maximal, complemented
6	#390×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
7	#388×C₁₁₃	C₂	1	2	yes	maximal, complemented
8	#386×C₁₁₃	C₂	1	2	yes	maximal, complemented
9	#381×C₁₁₃	C₂	1	2	yes	maximal, complemented
10	C₂²×D₄×C₁₁₃	C₂²	1	2	yes	—
11	C₂²×D₄×C₁₁₃	C₂²	1	2	yes	—
12	C₂²×D₄×C₁₁₃	C₂²	1	1	yes	characteristic
13	C₂²×D₄×C₁₁₃	C₂²	1	1	yes	characteristic
14	C₂³×C₄×C₁₁₃	C₂²	1	2	yes	—
15	#121×C₁₁₃	—	2	8	no	—
16	C₂×C₄⋊C₄×C₁₁₃	C₂²	1	2	yes	—
17	C₂×C₄⋊C₄×C₁₁₃	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₁₁₃	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊C₄×C₁₁₃	C₂²	1	2	yes	—
20	C₂×C₂²⋊C₄×C₁₁₃	C₂²	1	2	yes	—
21	C₂×C₂²⋊C₄×C₁₁₃	C₂²	1	2	yes	—
22	C₂×C₂²⋊C₄×C₁₁₃	C₂²	1	2	yes	—
23	C₂×C₂²⋊C₄×C₁₁₃	C₂²	1	2	yes	—
24	C₂×C₂²⋊C₄×C₁₁₃	C₂²	1	2	yes	—
25	C₂×C₂²⋊C₄×C₁₁₃	C₂²	1	2	yes	—
26	C₂×C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
27	#95×C₁₁₃	C₂²	1	1	yes	characteristic
28	#95×C₁₁₃	C₂²	1	2	yes	—
29	#95×C₁₁₃	C₂²	1	2	yes	—
30	#95×C₁₁₃	C₂²	1	2	yes	—
31	#95×C₁₁₃	C₂²	1	1	yes	characteristic
32	C₂⁴×C₁₁₃	C₂³	1	2	yes	—
33	C₂⁴×C₁₁₃	C₂³	1	2	yes	—
34	C₂×D₄×C₁₁₃	—	2	4	no	—
35	C₂×D₄×C₁₁₃	—	2	4	no	—
36	C₂×D₄×C₁₁₃	—	2	4	no	—
37	C₂×D₄×C₁₁₃	—	4	1	no	—
38	C₂×D₄×C₁₁₃	—	4	1	no	—
39	C₂×D₄×C₁₁₃	—	4	2	no	—
40	C₂×D₄×C₁₁₃	—	4	2	no	—
41	C₂×D₄×C₁₁₃	—	4	2	no	—
42	C₂×D₄×C₁₁₃	—	4	2	no	—
43	C₂×D₄×C₁₁₃	—	4	1	no	—
44	C₂×D₄×C₁₁₃	D₄	1	4	yes	complemented
45	C₂²×C₄×C₁₁₃	—	2	4	no	—
46	C₂²×C₄×C₁₁₃	—	2	4	no	—
47	C₂²×C₄×C₁₁₃	—	4	2	no	—
48	C₂²×C₄×C₁₁₃	C₂³	1	2	yes	—
49	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₁₁₃	C₂³	1	2	yes	—
51	C₂²×C₄×C₁₁₃	C₂³	1	2	yes	—
52	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₁₁₃	C₂³	1	2	yes	—
55	C₄⋊C₄×C₁₁₃	—	2	4	no	—
56	C₄⋊C₄×C₁₁₃	—	4	1	no	—
57	C₄⋊C₄×C₁₁₃	—	4	1	no	—
58	C₂²⋊C₄×C₁₁₃	—	2	4	no	—
59	C₂²⋊C₄×C₁₁₃	—	2	4	no	—
60	C₂²⋊C₄×C₁₁₃	—	2	4	no	—
61	C₂²⋊C₄×C₁₁₃	—	4	2	no	—
62	C₂²⋊C₄×C₁₁₃	—	4	2	no	—
63	C₂²⋊C₄×C₁₁₃	—	4	2	no	—
64	C₂²⋊C₄×C₁₁₃	—	4	2	no	—
65	C₄²×C₁₁₃	—	4	1	no	—
66	C₂³×C₁₁₃	—	2	2	no	—
67	C₂³×C₁₁₃	—	2	2	no	—
68	C₂³×C₁₁₃	—	2	2	no	—
69	C₂³×C₁₁₃	—	2	2	no	—
70	C₂³×C₁₁₃	—	2	2	no	—
71	C₂³×C₁₁₃	—	2	2	no	—
72	C₂³×C₁₁₃	—	2	2	no	—
73	C₂³×C₁₁₃	—	2	2	no	—
74	C₂³×C₁₁₃	—	2	2	no	—
75	C₂³×C₁₁₃	—	2	2	no	—
76	C₂³×C₁₁₃	—	2	2	no	—
77	C₂³×C₁₁₃	—	2	2	no	—
78	C₂³×C₁₁₃	—	2	2	no	—
79	C₂³×C₁₁₃	C₂×D₄	1	4	yes	—
80	C₂³×C₁₁₃	C₂⁴	1	1	yes	characteristic
81	D₄×C₁₁₃	—	4	4	no	—
82	D₄×C₁₁₃	—	4	8	no	—
83	D₄×C₁₁₃	—	4	8	no	—
84	D₄×C₁₁₃	—	4	4	no	—
85	C₂×C₄×C₁₁₃	—	2	2	no	—
86	C₂×C₄×C₁₁₃	—	2	1	no	—
87	C₂×C₄×C₁₁₃	—	2	1	no	—
88	C₂×C₄×C₁₁₃	—	2	2	no	—
89	C₂×C₄×C₁₁₃	—	2	2	no	—
90	C₂×C₄×C₁₁₃	—	2	2	no	—
91	C₂×C₄×C₁₁₃	—	2	2	no	—
92	C₂×C₄×C₁₁₃	—	2	2	no	—
93	C₂×C₄×C₁₁₃	—	2	2	no	—
94	C₂×C₄×C₁₁₃	—	2	2	no	—
95	C₂×C₄×C₁₁₃	—	2	2	no	—
96	C₂×C₄×C₁₁₃	—	2	2	no	—
97	C₂×C₄×C₁₁₃	—	2	2	no	—
98	C₂×C₄×C₁₁₃	—	2	2	no	—
99	C₂×C₄×C₁₁₃	—	2	1	no	—
100	C₂×C₄×C₁₁₃	—	2	2	no	—
101	C₂×C₄×C₁₁₃	—	2	1	no	—
102	C₂×C₄×C₁₁₃	—	4	8	no	—
103	C₂×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
104	C₂×C₄×C₁₁₃	#41	1	4	yes	—
105	C₂²×C₁₁₃	—	2	4	no	—
106	C₂²×C₁₁₃	—	2	4	no	—
107	C₂²×C₁₁₃	—	2	4	no	—
108	C₂²×C₁₁₃	—	4	2	no	—
109	C₂²×C₁₁₃	—	4	2	no	—
110	C₂²×C₁₁₃	—	4	2	no	—
111	C₂²×C₁₁₃	—	4	2	no	—
112	C₂²×C₁₁₃	—	4	2	no	—
113	C₂²×C₁₁₃	—	4	2	no	—
114	C₂²×C₁₁₃	—	4	2	no	—
115	C₂²×C₁₁₃	—	4	2	no	—
116	C₂²×C₁₁₃	—	4	2	no	—
117	C₂²×C₁₁₃	—	4	2	no	—
118	C₂²×C₁₁₃	—	4	2	no	—
119	C₂²×C₁₁₃	C₂²×D₄	1	1	yes	characteristic
120	C₂²×C₁₁₃	C₂×#41	1	2	yes	—
121	C₂²×C₁₁₃	H(5,𝔽₂)	1	1	yes	characteristic
122	C₂²×C₁₁₃	H(5,𝔽₂)	1	1	yes	characteristic
123	C₂²×C₁₁₃	H(5,𝔽₂)	1	2	yes	—
124	C₄×C₁₁₃	—	2	2	no	—
125	C₄×C₁₁₃	—	2	4	no	—
126	C₄×C₁₁₃	—	4	2	no	—
127	C₄×C₁₁₃	—	4	2	no	—
128	C₄×C₁₁₃	—	4	2	no	—
129	C₄×C₁₁₃	—	4	1	no	—
130	C₄×C₁₁₃	—	4	1	no	—
131	C₂×C₁₁₃	—	4	4	no	—
132	C₂×C₁₁₃	—	8	2	no	—
133	C₂×C₁₁₃	#534	1	1	yes	characteristic
134	C₂×C₁₁₃	#540	1	2	yes	—
135	C₂×C₁₁₃	#546	1	2	yes	—
136	C₂×C₁₁₃	#555	1	1	yes	characteristic
137	C₂×C₁₁₃	#560	1	1	yes	characteristic
138	#2699	C₁₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
139	C₁₁₃	#2699	1	1	yes	characteristic, direct product factor, Sylow
140	C₂×#121	C₂×C₁₁₃	1	2	yes	complemented
141	#398	C₂×C₁₁₃	1	1	yes	characteristic, complemented
142	#394	C₂×C₁₁₃	1	1	yes	characteristic, complemented
143	#394	C₂×C₁₁₃	1	2	yes	complemented
144	#392	C₂×C₁₁₃	1	2	yes	complemented
145	#390	C₂×C₁₁₃	1	1	yes	characteristic, complemented
146	#388	C₂×C₁₁₃	1	2	yes	complemented
147	#386	C₂×C₁₁₃	1	2	yes	complemented
148	#381	C₂×C₁₁₃	1	2	yes	complemented
149	C₂²×D₄	C₂²×C₁₁₃	1	2	yes	—
150	C₂²×D₄	C₂²×C₁₁₃	1	2	yes	—
151	C₂²×D₄	C₂²×C₁₁₃	1	1	yes	characteristic
152	C₂²×D₄	C₂²×C₁₁₃	1	1	yes	characteristic
153	C₂³×C₄	C₂²×C₁₁₃	1	2	yes	—
154	#121	—	2	8	no	—
155	C₂×C₄⋊C₄	C₂²×C₁₁₃	1	2	yes	—
156	C₂×C₄⋊C₄	C₂²×C₁₁₃	1	1	yes	characteristic
157	C₂×C₄⋊C₄	C₂²×C₁₁₃	1	1	yes	characteristic
158	C₂×C₂²⋊C₄	C₂²×C₁₁₃	1	2	yes	—
159	C₂×C₂²⋊C₄	C₂²×C₁₁₃	1	2	yes	—
160	C₂×C₂²⋊C₄	C₂²×C₁₁₃	1	2	yes	—
161	C₂×C₂²⋊C₄	C₂²×C₁₁₃	1	2	yes	—
162	C₂×C₂²⋊C₄	C₂²×C₁₁₃	1	2	yes	—
163	C₂×C₂²⋊C₄	C₂²×C₁₁₃	1	2	yes	—
164	C₂×C₂²⋊C₄	C₂²×C₁₁₃	1	2	yes	—
165	C₂×C₄²	C₂²×C₁₁₃	1	1	yes	characteristic
166	#95	C₂²×C₁₁₃	1	1	yes	characteristic
167	#95	C₂²×C₁₁₃	1	2	yes	—
168	#95	C₂²×C₁₁₃	1	2	yes	—
169	#95	C₂²×C₁₁₃	1	2	yes	—
170	#95	C₂²×C₁₁₃	1	1	yes	characteristic
171	C₂⁴	C₂³×C₁₁₃	1	2	yes	—
172	C₂⁴	C₂³×C₁₁₃	1	2	yes	—
173	C₂×D₄	—	2	4	no	—
174	C₂×D₄	—	2	4	no	—
175	C₂×D₄	—	2	4	no	—
176	C₂×D₄	—	4	1	no	—
177	C₂×D₄	—	4	1	no	—
178	C₂×D₄	—	4	2	no	—
179	C₂×D₄	—	4	2	no	—
180	C₂×D₄	—	4	2	no	—
181	C₂×D₄	—	4	2	no	—
182	C₂×D₄	—	4	1	no	—
183	C₂×D₄	D₄×C₁₁₃	1	4	yes	complemented
184	C₂²×C₄	—	2	4	no	—
185	C₂²×C₄	—	2	4	no	—
186	C₂²×C₄	—	4	2	no	—
187	C₂²×C₄	C₂³×C₁₁₃	1	2	yes	—
188	C₂²×C₄	C₂³×C₁₁₃	1	1	yes	characteristic
189	C₂²×C₄	C₂³×C₁₁₃	1	2	yes	—
190	C₂²×C₄	C₂³×C₁₁₃	1	2	yes	—
191	C₂²×C₄	C₂³×C₁₁₃	1	1	yes	characteristic
192	C₂²×C₄	C₂³×C₁₁₃	1	1	yes	characteristic
193	C₂²×C₄	C₂³×C₁₁₃	1	2	yes	—
194	C₄⋊C₄	—	2	4	no	—
195	C₄⋊C₄	—	4	1	no	—
196	C₄⋊C₄	—	4	1	no	—
197	C₂²⋊C₄	—	2	4	no	—
198	C₂²⋊C₄	—	2	4	no	—
199	C₂²⋊C₄	—	2	4	no	—
200	C₂²⋊C₄	—	4	2	no	—
201	C₂²⋊C₄	—	4	2	no	—
202	C₂²⋊C₄	—	4	2	no	—
203	C₂²⋊C₄	—	4	2	no	—
204	C₄²	—	4	1	no	—
205	C₂³	—	2	2	no	—
206	C₂³	—	2	2	no	—
207	C₂³	—	2	2	no	—
208	C₂³	—	2	2	no	—
209	C₂³	—	2	2	no	—
210	C₂³	—	2	2	no	—
211	C₂³	—	2	2	no	—
212	C₂³	—	2	2	no	—
213	C₂³	—	2	2	no	—
214	C₂³	—	2	2	no	—
215	C₂³	—	2	2	no	—
216	C₂³	—	2	2	no	—
217	C₂³	—	2	2	no	—
218	C₂³	C₂×D₄×C₁₁₃	1	4	yes	—
219	C₂³	C₂⁴×C₁₁₃	1	1	yes	characteristic
220	D₄	—	4	4	no	—
221	D₄	—	4	8	no	—
222	D₄	—	4	8	no	—
223	D₄	—	4	4	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	4	8	no	—
242	C₂×C₄	C₂×D₄×C₁₁₃	1	2	yes	—
243	C₂×C₄	#41×C₁₁₃	1	4	yes	—
244	C₂²	—	2	4	no	—
245	C₂²	—	2	4	no	—
246	C₂²	—	2	4	no	—
247	C₂²	—	4	2	no	—
248	C₂²	—	4	2	no	—
249	C₂²	—	4	2	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	4	2	no	—
252	C₂²	—	4	2	no	—
253	C₂²	—	4	2	no	—
254	C₂²	—	4	2	no	—
255	C₂²	—	4	2	no	—
256	C₂²	—	4	2	no	—
257	C₂²	—	4	2	no	—
258	C₂²	C₂²×D₄×C₁₁₃	1	1	yes	characteristic
259	C₂²	C₂×#41×C₁₁₃	1	2	yes	—
260	C₂²	H(5,𝔽₂)×C₁₁₃	1	1	yes	characteristic
261	C₂²	H(5,𝔽₂)×C₁₁₃	1	1	yes	characteristic
262	C₂²	H(5,𝔽₂)×C₁₁₃	1	2	yes	—
263	C₄	—	2	2	no	—
264	C₄	—	2	4	no	—
265	C₄	—	4	2	no	—
266	C₄	—	4	2	no	—
267	C₄	—	4	2	no	—
268	C₄	—	4	1	no	—
269	C₄	—	4	1	no	—
270	C₂	—	4	4	no	—
271	C₂	—	8	2	no	—
272	C₂	#534×C₁₁₃	1	1	yes	characteristic
273	C₂	#540×C₁₁₃	1	2	yes	—
274	C₂	#546×C₁₁₃	1	2	yes	—
275	C₂	#555×C₁₁₃	1	1	yes	characteristic
276	C₂	#560×C₁₁₃	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂×#121×C₁₁₃	C₂	semidirect
2. #398×C₁₁₃	C₂	semidirect
3. #394×C₁₁₃	C₂	semidirect
4. #394×C₁₁₃	C₂	semidirect
5. #392×C₁₁₃	C₂	semidirect
6. #390×C₁₁₃	C₂	semidirect
7. #388×C₁₁₃	C₂	semidirect
8. #386×C₁₁₃	C₂	semidirect
9. #381×C₁₁₃	C₂	semidirect
44. C₂×D₄×C₁₁₃	D₄	semidirect
138. #2699	C₁₁₃	direct (N × H)
139. C₁₁₃	#2699	direct (N × H)
140. C₂×#121	C₂×C₁₁₃	semidirect
141. #398	C₂×C₁₁₃	semidirect
142. #394	C₂×C₁₁₃	semidirect
143. #394	C₂×C₁₁₃	semidirect
144. #392	C₂×C₁₁₃	semidirect
145. #390	C₂×C₁₁₃	semidirect
146. #388	C₂×C₁₁₃	semidirect
147. #386	C₂×C₁₁₃	semidirect
148. #381	C₂×C₁₁₃	semidirect
183. C₂×D₄	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×#121×C₁₁₃	2
#398×C₁₁₃	2
#394×C₁₁₃	2
#392×C₁₁₃	2
#390×C₁₁₃	2
#388×C₁₁₃	2
#386×C₁₁₃	2
#381×C₁₁₃	2
#2699	113

Subgroups — C113×#2699 #28387495

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁₃×#2699 #28387495