Subgroups — #22626685

Order 9344.

492 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41×#64408	C₂	1	2	yes	maximal, complemented
2	#17989628	C₂	1	2	yes	maximal, complemented
3	#17989627	C₂	1	2	yes	maximal, complemented
4	#17989626	C₂	1	4	yes	maximal, complemented
5	#17989597	C₂	1	1	yes	maximal, characteristic, complemented
6	#17989577	C₂	1	2	yes	maximal, complemented
7	C₂×#14556645	C₂	1	1	yes	maximal, characteristic, complemented
8	#135×D₇₃	C₂	1	1	yes	maximal, characteristic
9	C₂×#11924036	C₂²	1	1	yes	characteristic
10	Q₈×#64408	C₂²	1	4	yes	—
11	D₄×#64408	—	2	4	no	—
12	D₄×#64408	C₂²	1	4	yes	—
13	#14556670	—	2	2	no	—
14	#14556670	C₂²	1	2	yes	—
15	C₂×C₄×#64408	—	2	2	no	—
16	C₂×C₄×#64408	C₂²	1	1	yes	characteristic
17	#14556660	C₂²	1	2	yes	—
18	#14556659	C₂²	1	2	yes	—
19	#14556654	C₂²	1	4	yes	—
20	#14556653	C₂²	1	4	yes	—
21	#14556645	C₂²	1	2	yes	—
22	#14556645	C₂²	1	2	yes	—
23	#41×D₇₃	C₂²	1	2	yes	—
24	#14556614	C₄	1	1	yes	characteristic, complemented
25	#14556613	C₄	1	1	yes	characteristic, complemented
26	#14556612	C₄	1	2	yes	complemented
27	#14556611	C₄	1	2	yes	complemented
28	#14556610	C₄	1	1	yes	characteristic, complemented
29	Q₁₆×D₇₃	C₂²	1	1	yes	characteristic
30	SD₁₆×D₇₃	C₂²	1	2	yes	—
31	D₈×D₇₃	C₂²	1	1	yes	characteristic
32	C₂×C₈×D₇₃	C₂²	1	1	yes	characteristic
33	#135×C₇₃	C₄	1	1	yes	characteristic, complemented
34	C₂²×#64408	—	2	4	no	—
35	C₂²×#64408	D₄	1	2	yes	—
36	#11924036	—	2	1	no	—
37	#11924036	C₂³	1	2	yes	—
38	C₂×#10596833	—	2	1	no	—
39	C₄⋊#64408	—	2	4	no	—
40	C₄⋊#64408	C₂³	1	4	yes	—
41	C₂²⋊#64408	—	2	2	no	—
42	C₂²⋊#64408	—	2	4	no	—
43	C₄×#64408	—	2	4	no	—
44	C₄×#64408	—	2	2	no	—
45	C₄×#64408	D₄	1	2	yes	—
46	C₄×#64408	C₂³	1	2	yes	—
47	#11924029	C₂×C₄	1	2	yes	complemented
48	#11924028	—	2	2	no	—
49	#11924028	C₂×C₄	1	2	yes	complemented
50	#11924027	—	2	2	no	—
51	#11924027	C₂×C₄	1	2	yes	complemented
52	Q₈×D₇₃	C₂³	1	2	yes	—
53	D₄×D₇₃	—	2	2	no	—
54	D₄×D₇₃	C₂³	1	2	yes	—
55	C₂×C₄×D₇₃	—	2	2	no	—
56	C₂×C₄×D₇₃	C₂³	1	1	yes	characteristic
57	Dic₂₉₂	C₂×C₄	1	1	yes	characteristic, complemented
58	#11924018	C₂×C₄	1	2	yes	complemented
59	#11924017	C₂×C₄	1	2	yes	complemented
60	#11924016	C₂×C₄	1	2	yes	complemented
61	#11924015	C₂×C₄	1	2	yes	complemented
62	D₅₈₄	C₂×C₄	1	1	yes	characteristic, complemented
63	#11924013	C₂×C₄	1	2	yes	complemented
64	C₂×#10596831	C₂×C₄	1	1	yes	characteristic, complemented
65	C₈×D₇₃	C₂×C₄	1	1	yes	characteristic, complemented
66	C₈×D₇₃	C₂×C₄	1	1	yes	characteristic, complemented
67	C₈×D₇₃	C₂³	1	1	yes	characteristic
68	C₈×D₇₃	C₂³	1	1	yes	characteristic
69	#41×C₇₃	C₂×C₄	1	2	yes	complemented
70	Q₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic, complemented
71	SD₁₆×C₇₃	C₂×C₄	1	2	yes	complemented
72	D₈×C₇₃	C₂×C₄	1	1	yes	characteristic, complemented
73	C₂×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic, complemented
74	C₂²×D₇₃	—	2	2	no	—
75	C₂²×D₇₃	C₂×D₄	1	1	yes	characteristic
76	C₂×#64408	—	2	4	no	—
77	C₂×#64408	—	4	2	no	—
78	C₂×#64408	—	4	8	no	—
79	C₂×#64408	C₂×D₄	1	4	yes	—
80	Q₈×C₇₃	C₂²×C₄	1	2	yes	—
81	D₄×C₇₃	—	2	2	no	—
82	D₄×C₇₃	C₂²×C₄	1	2	yes	—
83	C₂×C₄×C₇₃	—	2	2	no	—
84	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
85	C₂²⋊D₇₃	—	2	2	no	—
86	C₂²⋊D₇₃	—	2	2	no	—
87	C₂²⋊D₇₃	—	2	2	no	—
88	C₂×Dic₇₃	—	2	2	no	—
89	C₂×Dic₇₃	#41	1	1	yes	characteristic
90	D₂₉₂	—	2	2	no	—
91	D₂₉₂	C₂²×C₄	1	2	yes	—
92	C₄×D₇₃	—	2	2	no	—
93	C₄×D₇₃	—	2	2	no	—
94	C₄×D₇₃	—	2	2	no	—
95	C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
96	C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
97	C₄×D₇₃	#41	1	1	yes	characteristic
98	C₄×D₇₃	C₂⁴	1	1	yes	characteristic
99	Dic₁₄₆	—	2	2	no	—
100	Dic₁₄₆	C₂²×C₄	1	2	yes	—
101	#10596833	—	2	2	no	—
102	C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
103	C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
104	#10596831	C₂²×C₄	1	1	yes	characteristic
105	#10596831	C₂²×C₄	1	1	yes	characteristic
106	C₂²×C₇₃	—	2	2	no	—
107	C₂²×C₇₃	C₄×D₄	1	1	yes	characteristic
108	C₂×D₇₃	—	2	2	no	—
109	C₂×D₇₃	—	2	2	no	—
110	C₂×D₇₃	—	4	4	no	—
111	C₂×D₇₃	C₄×D₄	1	1	yes	characteristic
112	C₂×D₇₃	C₂²×D₄	1	1	yes	characteristic
113	#64408	—	2	4	no	—
114	#64408	—	4	4	no	—
115	C₄×C₇₃	—	2	2	no	—
116	C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
117	C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
118	Dic₇₃	—	2	2	no	—
119	Dic₇₃	C₄×D₄	1	1	yes	characteristic
120	Dic₇₃	C₂×#41	1	1	yes	characteristic
121	C₂×C₇₃	—	2	1	no	—
122	C₂×C₇₃	—	4	2	no	—
123	C₂×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
124	D₇₃	—	2	1	no	—
125	D₇₃	—	4	2	no	—
126	D₇₃	#575	1	2	yes	—
127	#2943	—	73	1	no	maximal, Sylow
128	C₇₃	#2943	1	1	yes	characteristic, complemented, Sylow
129	C₂×#135	—	73	1	no	—
130	C₄×#41	—	73	2	no	—
131	#442	—	73	2	no	—
132	#441	—	73	2	no	—
133	#440	—	73	4	no	—
134	#427	—	73	1	no	—
135	#416	—	73	2	no	—
136	C₂×#97	—	73	1	no	—
137	C₂×#41	—	73	2	no	—
138	#135	—	73	2	no	—
139	#135	—	73	1	no	—
140	#135	—	73	2	no	—
141	#135	—	73	1	no	—
142	#135	—	73	1	no	—
143	#135	#64408	1	1	yes	characteristic, complemented
144	C₂×Q₁₆	—	73	1	no	—
145	C₂×SD₁₆	—	73	2	no	—
146	C₂×D₈	—	73	1	no	—
147	C₂²×C₈	—	73	1	no	—
148	C₂²×C₈	—	73	1	no	—
149	C₄×Q₈	—	73	4	no	—
150	C₄×D₄	—	73	4	no	—
151	C₄×D₄	—	146	4	no	—
152	#117	—	73	2	no	—
153	#117	—	146	2	no	—
154	C₂×C₄²	—	73	1	no	—
155	C₂×C₄²	—	146	2	no	—
156	#107	—	73	2	no	—
157	#106	—	73	2	no	—
158	#103	—	73	4	no	—
159	#102	—	73	4	no	—
160	#97	—	73	2	no	—
161	#97	—	73	2	no	—
162	#41	—	73	2	no	—
163	#41	—	73	2	no	—
164	#41	—	73	2	no	—
165	#41	—	146	2	no	—
166	#41	—	146	2	no	—
167	#41	C₂×#64408	1	2	yes	complemented
168	C₂×Q₈	—	73	2	no	—
169	C₂×D₄	—	73	2	no	—
170	C₂×D₄	—	146	2	no	—
171	C₂²×C₄	—	73	2	no	—
172	C₂²×C₄	—	73	1	no	—
173	C₂²×C₄	—	146	2	no	—
174	C₂²×C₄	—	146	4	no	—
175	Q₁₆	—	73	2	no	—
176	Q₁₆	—	73	1	no	—
177	Q₁₆	C₂×#64408	1	1	yes	characteristic, complemented
178	SD₁₆	—	73	2	no	—
179	SD₁₆	—	73	2	no	—
180	SD₁₆	—	73	2	no	—
181	SD₁₆	C₂×#64408	1	2	yes	complemented
182	D₈	—	73	2	no	—
183	D₈	—	73	1	no	—
184	D₈	C₂×#64408	1	1	yes	characteristic, complemented
185	C₂×C₈	—	73	1	no	—
186	C₂×C₈	—	73	1	no	—
187	C₂×C₈	—	73	1	no	—
188	C₂×C₈	—	73	1	no	—
189	C₂×C₈	—	73	1	no	—
190	C₂×C₈	—	73	2	no	—
191	C₂×C₈	—	146	1	no	—
192	C₂×C₈	—	146	1	no	—
193	C₂×C₈	C₂×#64408	1	1	yes	characteristic, complemented
194	C₄⋊C₄	—	73	4	no	—
195	C₄⋊C₄	—	146	4	no	—
196	C₂²⋊C₄	—	146	4	no	—
197	C₂²⋊C₄	—	146	2	no	—
198	C₄²	—	73	2	no	—
199	C₄²	—	73	2	no	—
200	C₄²	—	146	2	no	—
201	C₄²	—	146	4	no	—
202	C₂³	—	73	1	no	—
203	C₂³	—	146	2	no	—
204	Q₈	—	73	2	no	—
205	Q₈	—	146	2	no	—
206	Q₈	C₂²×#64408	1	2	yes	—
207	D₄	—	2	2	no	—
208	D₄	—	73	2	no	—
209	D₄	—	146	2	no	—
210	D₄	—	146	2	no	—
211	D₄	—	146	2	no	—
212	D₄	—	146	2	no	—
213	D₄	C₂²×#64408	1	2	yes	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	292	8	no	—
216	C₂×C₄	—	292	2	no	—
217	C₂×C₄	—	73	1	no	—
218	C₂×C₄	—	73	1	no	—
219	C₂×C₄	—	73	1	no	—
220	C₂×C₄	—	73	1	no	—
221	C₂×C₄	—	73	1	no	—
222	C₂×C₄	—	73	4	no	—
223	C₂×C₄	—	146	2	no	—
224	C₂×C₄	—	146	2	no	—
225	C₂×C₄	—	146	4	no	—
226	C₂×C₄	—	146	2	no	—
227	C₂×C₄	—	146	2	no	—
228	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
229	C₈	—	73	1	no	—
230	C₈	—	73	1	no	—
231	C₈	—	146	2	no	—
232	C₈	C₂²×#64408	1	1	yes	characteristic
233	C₈	C₂²×#64408	1	1	yes	characteristic
234	C₂²	—	2	2	no	—
235	C₂²	—	292	4	no	—
236	C₂²	—	73	1	no	—
237	C₂²	—	73	1	no	—
238	C₂²	—	146	2	no	—
239	C₂²	—	146	2	no	—
240	C₂²	D₄×#64408	1	1	yes	characteristic
241	C₄	—	2	2	no	—
242	C₄	—	292	4	no	—
243	C₄	—	73	1	no	—
244	C₄	—	73	1	no	—
245	C₄	—	146	4	no	—
246	C₄	—	146	2	no	—
247	C₄	D₄×#64408	1	1	yes	characteristic
248	C₄	C₂³×#64408	1	1	yes	characteristic
249	C₂	—	2	1	no	—
250	C₂	—	4	2	no	—
251	C₂	—	292	2	no	—
252	C₂	—	73	2	no	—
253	C₂	—	146	1	no	—
254	C₂	C₂×D₄×#64408	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄×#64408
commutator	C₄×C₇₃	C₂³×C₄
Frattini	C₄	C₂³×#64408
Fitting	#135×C₇₃	C₄
socle	C₂×C₇₃	C₂×C₄×D₄
hypercenter	#135	#64408
hypocenter	C₇₃	#2943
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #135

Complements

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

N	H ≅ G/N	type
1. #41×#64408	C₂	semidirect
2. #17989628	C₂	semidirect
3. #17989627	C₂	semidirect
4. #17989626	C₂	semidirect
5. #17989597	C₂	semidirect
6. #17989577	C₂	semidirect
7. C₂×#14556645	C₂	semidirect
24. #14556614	C₄	semidirect
25. #14556613	C₄	semidirect
26. #14556612	C₄	semidirect
27. #14556611	C₄	semidirect
28. #14556610	C₄	semidirect
33. #135×C₇₃	C₄	semidirect
47. #11924029	C₂×C₄	semidirect
49. #11924028	C₂×C₄	semidirect
51. #11924027	C₂×C₄	semidirect
57. Dic₂₉₂	C₂×C₄	semidirect
58. #11924018	C₂×C₄	semidirect
59. #11924017	C₂×C₄	semidirect
60. #11924016	C₂×C₄	semidirect
61. #11924015	C₂×C₄	semidirect
62. D₅₈₄	C₂×C₄	semidirect
63. #11924013	C₂×C₄	semidirect
64. C₂×#10596831	C₂×C₄	semidirect
65. C₈×D₇₃	C₂×C₄	semidirect
66. C₈×D₇₃	C₂×C₄	semidirect
69. #41×C₇₃	C₂×C₄	semidirect
70. Q₁₆×C₇₃	C₂×C₄	semidirect
71. SD₁₆×C₇₃	C₂×C₄	semidirect
72. D₈×C₇₃	C₂×C₄	semidirect
73. C₂×C₈×C₇₃	C₂×C₄	semidirect
128. C₇₃	#2943	semidirect
143. #135	#64408	semidirect
167. #41	C₂×#64408	semidirect
177. Q₁₆	C₂×#64408	semidirect
181. SD₁₆	C₂×#64408	semidirect
184. D₈	C₂×#64408	semidirect
193. C₂×C₈	C₂×#64408	semidirect