Subgroups — #135×#6923 #21462569

Order 7808.

500 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41×#6923	C₂	1	2	yes	maximal, complemented
2	Q₁₆×#6923	C₂	1	2	yes	maximal, complemented
3	SD₁₆×#6923	C₂	1	4	yes	maximal, complemented
4	D₈×#6923	C₂	1	2	yes	maximal, complemented
5	#16840772	C₂	1	1	yes	maximal, characteristic, complemented
6	#16840752	C₂	1	2	yes	maximal, complemented
7	C₂×C₈×#6923	C₂	1	1	yes	maximal, characteristic, complemented
8	#135×D₆₁	C₂	1	1	yes	maximal, characteristic
9	C₂×D₆₁⋊C₈	C₂²	1	1	yes	characteristic
10	Q₈×#6923	C₂²	1	4	yes	—
11	D₄×#6923	—	2	4	no	—
12	D₄×#6923	C₂²	1	4	yes	—
13	#14519856	—	2	2	no	—
14	#14519856	C₂²	1	2	yes	—
15	C₂×C₄×#6923	—	2	2	no	—
16	C₂×C₄×#6923	C₂²	1	1	yes	characteristic
17	#14519846	C₂²	1	2	yes	—
18	#14519845	C₂²	1	2	yes	—
19	#14519840	C₂²	1	4	yes	—
20	#14519839	C₂²	1	4	yes	—
21	C₈×#6923	C₂²	1	2	yes	—
22	C₈×#6923	C₂²	1	2	yes	—
23	#41×D₆₁	C₂²	1	2	yes	—
24	#14519800	C₄	1	1	yes	characteristic, complemented
25	#14519799	C₄	1	1	yes	characteristic, complemented
26	#14519798	C₄	1	2	yes	complemented
27	#14519797	C₄	1	2	yes	complemented
28	#14519796	C₄	1	1	yes	characteristic, complemented
29	Q₁₆×D₆₁	C₂²	1	1	yes	characteristic
30	SD₁₆×D₆₁	C₂²	1	2	yes	—
31	D₈×D₆₁	C₂²	1	1	yes	characteristic
32	C₂×C₈×D₆₁	C₂²	1	1	yes	characteristic
33	#135×C₆₁	C₄	1	1	yes	characteristic, complemented
34	C₂²×#6923	—	2	4	no	—
35	C₂²×#6923	D₄	1	2	yes	—
36	#41×C₆₁	C₂×C₄	1	2	yes	complemented
37	#11758226	C₂×C₄	1	2	yes	complemented
38	Q₈×D₆₁	C₂³	1	2	yes	—
39	#11758224	—	2	2	no	—
40	#11758224	C₂×C₄	1	2	yes	complemented
41	D₄×D₆₁	—	2	2	no	—
42	D₄×D₆₁	C₂³	1	2	yes	—
43	#11758222	—	2	2	no	—
44	#11758222	C₂×C₄	1	2	yes	complemented
45	C₂×C₄×D₆₁	—	2	2	no	—
46	C₂×C₄×D₆₁	C₂³	1	1	yes	characteristic
47	C₂²⋊#6923	—	2	2	no	—
48	C₂²⋊#6923	—	2	4	no	—
49	C₂×#92420	—	2	1	no	—
50	C₄⋊#6923	—	2	4	no	—
51	C₄⋊#6923	C₂³	1	4	yes	—
52	C₄×#6923	—	2	4	no	—
53	C₄×#6923	—	2	2	no	—
54	C₄×#6923	D₄	1	1	yes	characteristic
55	C₄×#6923	D₄	1	1	yes	characteristic
56	C₄×#6923	C₂³	1	2	yes	—
57	D₆₁⋊C₈	—	2	1	no	—
58	D₆₁⋊C₈	C₂³	1	2	yes	—
59	Q₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic, complemented
60	SD₁₆×C₆₁	C₂×C₄	1	2	yes	complemented
61	D₈×C₆₁	C₂×C₄	1	1	yes	characteristic, complemented
62	C₂×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic, complemented
63	#11758202	C₂×C₄	1	2	yes	complemented
64	#11758201	C₂×C₄	1	2	yes	complemented
65	#11758200	C₂×C₄	1	2	yes	complemented
66	#11758199	C₂×C₄	1	2	yes	complemented
67	C₂×#92418	C₂×C₄	1	1	yes	characteristic, complemented
68	Dic₂₄₄	C₂×C₄	1	1	yes	characteristic, complemented
69	D₄₈₈	C₂×C₄	1	1	yes	characteristic, complemented
70	#11758190	C₂×C₄	1	2	yes	complemented
71	C₈×D₆₁	C₂×C₄	1	1	yes	characteristic, complemented
72	C₈×D₆₁	C₂×C₄	1	1	yes	characteristic, complemented
73	C₈×D₆₁	C₂³	1	1	yes	characteristic
74	C₈×D₆₁	C₂³	1	1	yes	characteristic
75	C₂²×D₆₁	—	2	2	no	—
76	C₂²×D₆₁	C₂×D₄	1	1	yes	characteristic
77	C₂×#6923	—	2	4	no	—
78	C₂×#6923	—	2	4	no	—
79	C₂×#6923	—	4	8	no	—
80	C₂×#6923	C₂×D₄	1	2	yes	—
81	C₂×#6923	C₂×D₄	1	2	yes	—
82	Q₈×C₆₁	C₂²×C₄	1	2	yes	—
83	D₄×C₆₁	—	2	2	no	—
84	D₄×C₆₁	C₂²×C₄	1	2	yes	—
85	C₂×C₄×C₆₁	—	2	2	no	—
86	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
87	C₂²⋊D₆₁	—	2	2	no	—
88	C₂²⋊D₆₁	—	2	2	no	—
89	C₂²⋊D₆₁	—	2	2	no	—
90	C₂×Dic₆₁	—	2	2	no	—
91	C₂×Dic₆₁	#41	1	1	yes	characteristic
92	D₂₄₄	—	2	2	no	—
93	D₂₄₄	C₂²×C₄	1	2	yes	—
94	C₄×D₆₁	—	2	2	no	—
95	C₄×D₆₁	—	2	2	no	—
96	C₄×D₆₁	—	2	2	no	—
97	C₄×D₆₁	C₂²×C₄	1	1	yes	characteristic
98	C₄×D₆₁	C₂×D₄	1	1	yes	characteristic
99	C₄×D₆₁	#41	1	1	yes	characteristic
100	C₄×D₆₁	C₂⁴	1	1	yes	characteristic
101	Dic₁₂₂	—	2	2	no	—
102	Dic₁₂₂	C₂²×C₄	1	2	yes	—
103	#92420	—	2	2	no	—
104	C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
105	C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
106	#92418	C₂²×C₄	1	1	yes	characteristic
107	#92418	C₂²×C₄	1	1	yes	characteristic
108	C₂²×C₆₁	—	2	2	no	—
109	C₂²×C₆₁	C₄×D₄	1	1	yes	characteristic
110	C₂×D₆₁	—	2	2	no	—
111	C₂×D₆₁	—	2	2	no	—
112	C₂×D₆₁	—	4	4	no	—
113	C₂×D₆₁	C₄×D₄	1	1	yes	characteristic
114	C₂×D₆₁	C₂²×D₄	1	1	yes	characteristic
115	#6923	—	2	2	no	—
116	#6923	—	4	4	no	—
117	#6923	#135	1	4	yes	direct product factor
118	C₄×C₆₁	—	2	2	no	—
119	C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
120	C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
121	Dic₆₁	—	2	2	no	—
122	Dic₆₁	C₄×D₄	1	1	yes	characteristic
123	Dic₆₁	C₂×#41	1	1	yes	characteristic
124	C₄×#135	—	61	1	no	maximal, Sylow
125	C₂×C₆₁	—	2	1	no	—
126	C₂×C₆₁	—	4	2	no	—
127	C₂×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
128	D₆₁	—	2	1	no	—
129	D₆₁	—	4	2	no	—
130	D₆₁	C₂×#135	1	2	yes	—
131	C₂×#135	—	61	1	no	—
132	C₄×#41	—	61	2	no	—
133	C₄×Q₁₆	—	61	2	no	—
134	C₄×SD₁₆	—	61	4	no	—
135	C₄×D₈	—	61	2	no	—
136	#427	—	61	1	no	—
137	#416	—	61	2	no	—
138	C₂×C₄×C₈	—	61	1	no	—
139	C₆₁	C₄×#135	1	1	yes	characteristic, complemented, Sylow
140	C₂×#41	—	61	2	no	—
141	#135	—	61	2	no	—
142	#135	—	61	1	no	—
143	#135	—	61	2	no	—
144	#135	—	61	1	no	—
145	#135	—	61	1	no	—
146	#135	#6923	1	1	yes	characteristic, direct product factor
147	C₂×Q₁₆	—	61	1	no	—
148	C₂×SD₁₆	—	61	2	no	—
149	C₂×D₈	—	61	1	no	—
150	C₂²×C₈	—	61	1	no	—
151	C₂²×C₈	—	61	1	no	—
152	C₄×Q₈	—	61	4	no	—
153	C₄×D₄	—	61	4	no	—
154	C₄×D₄	—	122	4	no	—
155	#117	—	61	2	no	—
156	#117	—	122	2	no	—
157	C₂×C₄²	—	61	1	no	—
158	C₂×C₄²	—	122	2	no	—
159	#107	—	61	2	no	—
160	#106	—	61	2	no	—
161	#103	—	61	4	no	—
162	#102	—	61	4	no	—
163	C₄×C₈	—	61	2	no	—
164	C₄×C₈	—	61	2	no	—
165	#41	—	61	2	no	—
166	#41	—	61	2	no	—
167	#41	—	61	2	no	—
168	#41	—	122	2	no	—
169	#41	—	122	2	no	—
170	#41	C₂×#6923	1	2	yes	complemented
171	C₂×Q₈	—	61	2	no	—
172	C₂×D₄	—	61	2	no	—
173	C₂×D₄	—	122	2	no	—
174	C₂²×C₄	—	61	1	no	—
175	C₂²×C₄	—	61	2	no	—
176	C₂²×C₄	—	122	2	no	—
177	C₂²×C₄	—	122	4	no	—
178	Q₁₆	—	61	2	no	—
179	Q₁₆	—	61	1	no	—
180	Q₁₆	C₂×#6923	1	1	yes	characteristic, complemented
181	SD₁₆	—	61	2	no	—
182	SD₁₆	—	61	2	no	—
183	SD₁₆	—	61	2	no	—
184	SD₁₆	C₂×#6923	1	2	yes	complemented
185	D₈	—	61	2	no	—
186	D₈	—	61	1	no	—
187	D₈	C₂×#6923	1	1	yes	characteristic, complemented
188	C₂×C₈	—	61	1	no	—
189	C₂×C₈	—	61	1	no	—
190	C₂×C₈	—	61	1	no	—
191	C₂×C₈	—	61	1	no	—
192	C₂×C₈	—	61	1	no	—
193	C₂×C₈	—	61	2	no	—
194	C₂×C₈	—	122	1	no	—
195	C₂×C₈	—	122	1	no	—
196	C₂×C₈	C₂×#6923	1	1	yes	characteristic, complemented
197	C₄⋊C₄	—	61	4	no	—
198	C₄⋊C₄	—	122	4	no	—
199	C₂²⋊C₄	—	122	4	no	—
200	C₂²⋊C₄	—	122	2	no	—
201	C₄²	—	61	2	no	—
202	C₄²	—	61	1	no	—
203	C₄²	—	61	1	no	—
204	C₄²	—	122	2	no	—
205	C₄²	—	122	4	no	—
206	C₂³	—	61	1	no	—
207	C₂³	—	122	2	no	—
208	Q₈	—	61	2	no	—
209	Q₈	—	122	2	no	—
210	Q₈	C₂²×#6923	1	2	yes	—
211	D₄	—	2	2	no	—
212	D₄	—	61	2	no	—
213	D₄	—	122	2	no	—
214	D₄	—	122	2	no	—
215	D₄	—	122	2	no	—
216	D₄	—	122	2	no	—
217	D₄	C₂²×#6923	1	2	yes	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	61	2	no	—
220	C₂×C₄	—	61	1	no	—
221	C₂×C₄	—	61	1	no	—
222	C₂×C₄	—	61	1	no	—
223	C₂×C₄	—	61	1	no	—
224	C₂×C₄	—	61	1	no	—
225	C₂×C₄	—	61	2	no	—
226	C₂×C₄	—	122	2	no	—
227	C₂×C₄	—	122	2	no	—
228	C₂×C₄	—	122	2	no	—
229	C₂×C₄	—	122	4	no	—
230	C₂×C₄	—	122	2	no	—
231	C₂×C₄	—	122	4	no	—
232	C₂×C₄	—	244	8	no	—
233	C₂×C₄	C₂²×#6923	1	1	yes	characteristic
234	C₈	—	61	1	no	—
235	C₈	—	61	1	no	—
236	C₈	—	122	2	no	—
237	C₈	C₂²×#6923	1	1	yes	characteristic
238	C₈	C₂²×#6923	1	1	yes	characteristic
239	C₂²	—	2	2	no	—
240	C₂²	—	61	1	no	—
241	C₂²	—	61	1	no	—
242	C₂²	—	122	2	no	—
243	C₂²	—	122	2	no	—
244	C₂²	—	244	4	no	—
245	C₂²	D₄×#6923	1	1	yes	characteristic
246	C₄	—	2	2	no	—
247	C₄	—	61	4	no	—
248	C₄	—	61	1	no	—
249	C₄	—	61	1	no	—
250	C₄	—	122	2	no	—
251	C₄	—	122	2	no	—
252	C₄	—	244	4	no	—
253	C₄	D₄×#6923	1	1	yes	characteristic
254	C₄	C₂³×#6923	1	1	yes	characteristic
255	C₂	—	2	1	no	—
256	C₂	—	4	2	no	—
257	C₂	—	61	2	no	—
258	C₂	—	122	1	no	—
259	C₂	—	244	2	no	—
260	C₂	C₂×D₄×#6923	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄×#6923
commutator	C₄×C₆₁	C₂³×C₄
Frattini	C₄	C₂³×#6923
Fitting	#135×C₆₁	C₄
socle	C₂×C₆₁	C₂×C₄×D₄
hypercenter	#135	#6923
hypocenter	C₆₁	C₄×#135
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #135

Complements

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

N	H ≅ G/N	type
1. #41×#6923	C₂	semidirect
2. Q₁₆×#6923	C₂	semidirect
3. SD₁₆×#6923	C₂	semidirect
4. D₈×#6923	C₂	semidirect
5. #16840772	C₂	semidirect
6. #16840752	C₂	semidirect
7. C₂×C₈×#6923	C₂	semidirect
24. #14519800	C₄	semidirect
25. #14519799	C₄	semidirect
26. #14519798	C₄	semidirect
27. #14519797	C₄	semidirect
28. #14519796	C₄	semidirect
33. #135×C₆₁	C₄	semidirect
36. #41×C₆₁	C₂×C₄	semidirect
37. #11758226	C₂×C₄	semidirect
40. #11758224	C₂×C₄	semidirect
44. #11758222	C₂×C₄	semidirect
59. Q₁₆×C₆₁	C₂×C₄	semidirect
60. SD₁₆×C₆₁	C₂×C₄	semidirect
61. D₈×C₆₁	C₂×C₄	semidirect
62. C₂×C₈×C₆₁	C₂×C₄	semidirect
63. #11758202	C₂×C₄	semidirect
64. #11758201	C₂×C₄	semidirect
65. #11758200	C₂×C₄	semidirect
66. #11758199	C₂×C₄	semidirect
67. C₂×#92418	C₂×C₄	semidirect
68. Dic₂₄₄	C₂×C₄	semidirect
69. D₄₈₈	C₂×C₄	semidirect
70. #11758190	C₂×C₄	semidirect
71. C₈×D₆₁	C₂×C₄	semidirect
72. C₈×D₆₁	C₂×C₄	semidirect
117. #6923	#135	direct (N × H)
139. C₆₁	C₄×#135	semidirect
146. #135	#6923	direct (N × H)
170. #41	C₂×#6923	semidirect
180. Q₁₆	C₂×#6923	semidirect
184. SD₁₆	C₂×#6923	semidirect
187. D₈	C₂×#6923	semidirect
196. C₂×C₈	C₂×#6923	semidirect