Subgroups — #22626708

Order 9344.

452 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17989666	C₂	1	2	yes	maximal, complemented
2	#17989632	C₂	1	4	yes	maximal, complemented
3	#17989631	C₂	1	2	yes	maximal, complemented
4	#17989629	C₂	1	2	yes	maximal, complemented
5	C₂×#14556662	C₂	1	1	yes	maximal, characteristic, complemented
6	#17989584	C₂	1	2	yes	maximal, complemented
7	#17989550	C₂	1	1	yes	maximal, characteristic, complemented
8	#135×D₇₃	C₂	1	1	yes	maximal, characteristic
9	#14556679	C₂²	1	4	yes	complemented
10	#14556678	—	2	4	no	—
11	#14556678	C₂²	1	4	yes	complemented
12	C₂×#11924038	—	2	2	no	—
13	C₂×#11924038	C₂²	1	2	yes	—
14	D₇₃⋊M₁₆	—	2	2	no	—
15	D₇₃⋊M₁₆	C₂²	1	1	yes	characteristic
16	#14556670	C₂²	1	1	yes	characteristic
17	#14556662	C₂²	1	2	yes	complemented
18	#14556662	C₂²	1	2	yes	complemented
19	#14556656	C₂²	1	4	yes	complemented
20	#14556655	C₂²	1	4	yes	complemented
21	#14556645	C₂²	1	2	yes	complemented
22	C₈×#64408	C₂²	1	2	yes	complemented
23	#41×D₇₃	C₂²	1	2	yes	—
24	#14556614	C₂²	1	1	yes	characteristic
25	#14556613	C₄	1	1	yes	characteristic, complemented
26	#14556612	C₂²	1	2	yes	—
27	#14556611	C₄	1	2	yes	complemented
28	#14556610	C₂²	1	1	yes	characteristic
29	Q₁₆×D₇₃	C₄	1	1	yes	characteristic, complemented
30	SD₁₆×D₇₃	C₄	1	2	yes	complemented
31	D₈×D₇₃	C₄	1	1	yes	characteristic, complemented
32	C₂×C₈×D₇₃	C₂²	1	1	yes	characteristic
33	#135×C₇₃	C₄	1	1	yes	characteristic, complemented
34	#11924038	—	2	2	no	—
35	#11924038	—	2	4	no	—
36	#11924038	D₄	1	1	yes	characteristic
37	#11924038	D₄	1	1	yes	characteristic
38	#11924038	C₂³	1	4	yes	—
39	#11924037	—	2	4	no	—
40	#11924037	D₄	1	2	yes	—
41	#11924036	—	2	4	no	—
42	#11924036	C₂³	1	2	yes	—
43	C₂×#10596833	—	2	2	no	—
44	C₂×#10596833	—	2	4	no	—
45	C₄⋊#64408	—	2	1	no	—
46	C₂²⋊#64408	—	2	1	no	—
47	C₄×#64408	C₂³	1	2	yes	—
48	#11924029	C₂³	1	2	yes	—
49	#11924028	—	2	2	no	—
50	#11924028	C₂×C₄	1	2	yes	—
51	#11924027	—	2	2	no	—
52	#11924027	C₂³	1	2	yes	—
53	Q₈×D₇₃	C₂×C₄	1	2	yes	—
54	D₄×D₇₃	—	2	2	no	—
55	D₄×D₇₃	C₂×C₄	1	2	yes	—
56	C₂×C₄×D₇₃	—	2	2	no	—
57	C₂×C₄×D₇₃	C₂³	1	1	yes	characteristic
58	Dic₂₉₂	C₂×C₄	1	1	yes	characteristic
59	#11924018	C₂×C₄	1	2	yes	—
60	#11924017	C₂×C₄	1	2	yes	—
61	#11924016	C₂×C₄	1	2	yes	—
62	#11924015	C₂×C₄	1	2	yes	—
63	D₅₈₄	C₂×C₄	1	1	yes	characteristic
64	#11924013	C₂×C₄	1	2	yes	—
65	C₂×#10596831	C₂×C₄	1	1	yes	characteristic
66	C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
67	C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
68	C₈×D₇₃	C₂³	1	1	yes	characteristic
69	C₈×D₇₃	C₂³	1	1	yes	characteristic
70	#41×C₇₃	C₂×C₄	1	2	yes	—
71	Q₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
72	SD₁₆×C₇₃	C₂×C₄	1	2	yes	—
73	D₈×C₇₃	C₂×C₄	1	1	yes	characteristic
74	C₂×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
75	C₂²×D₇₃	—	2	2	no	—
76	C₂²×D₇₃	#41	1	1	yes	characteristic
77	C₂×#64408	—	2	2	no	—
78	Q₈×C₇₃	C₂²×C₄	1	2	yes	—
79	D₄×C₇₃	—	2	2	no	—
80	D₄×C₇₃	C₂²×C₄	1	2	yes	—
81	C₂×C₄×C₇₃	—	2	2	no	—
82	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
83	C₂²⋊D₇₃	—	2	2	no	—
84	C₂²⋊D₇₃	—	2	2	no	—
85	C₂²⋊D₇₃	—	2	2	no	—
86	C₂×Dic₇₃	—	2	2	no	—
87	C₂×Dic₇₃	C₂×D₄	1	1	yes	characteristic
88	D₂₉₂	—	2	2	no	—
89	D₂₉₂	C₂²×C₄	1	2	yes	—
90	C₄×D₇₃	—	2	2	no	—
91	C₄×D₇₃	—	2	2	no	—
92	C₄×D₇₃	—	2	2	no	—
93	C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
94	C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
95	C₄×D₇₃	#41	1	1	yes	characteristic
96	C₄×D₇₃	C₂⁴	1	1	yes	characteristic
97	Dic₁₄₆	—	2	2	no	—
98	Dic₁₄₆	C₂²×C₄	1	2	yes	—
99	#10596833	—	2	4	no	—
100	#10596833	C₂×D₄	1	2	yes	—
101	#10596833	C₂×D₄	1	2	yes	—
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	—	4	1	no	—
110	C₂×D₇₃	—	8	2	no	—
111	C₂×D₇₃	C₄×D₄	1	1	yes	characteristic
112	C₂×D₇₃	C₂×#41	1	1	yes	characteristic
113	#64408	—	4	2	no	—
114	C₄×C₇₃	—	2	2	no	—
115	C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
116	C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
117	Dic₇₃	—	2	2	no	—
118	Dic₇₃	C₄×D₄	1	1	yes	characteristic
119	Dic₇₃	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₇₃	—	2	1	no	—
121	C₂×C₇₃	—	4	2	no	—
122	C₂×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
123	D₇₃	—	2	1	no	—
124	D₇₃	—	2	1	no	—
125	D₇₃	—	4	2	no	—
126	#2957	—	73	1	no	maximal, Sylow
127	C₇₃	#2957	1	1	yes	characteristic, complemented, Sylow
128	C₂×#135	—	73	1	no	—
129	#568	—	73	2	no	—
130	#444	—	73	4	no	—
131	#443	—	73	2	no	—
132	#443	—	73	2	no	—
133	C₂×#108	—	73	1	no	—
134	#421	—	73	2	no	—
135	#405	—	73	1	no	—
136	C₂×#41	—	73	2	no	—
137	#135	—	73	2	no	—
138	#135	—	73	1	no	—
139	#135	—	73	2	no	—
140	#135	—	73	1	no	—
141	#135	—	73	1	no	—
142	#135	#64408	1	1	yes	characteristic, complemented
143	C₂×Q₁₆	—	73	1	no	—
144	C₂×SD₁₆	—	73	2	no	—
145	C₂×D₈	—	73	1	no	—
146	#131	—	73	4	no	—
147	#131	—	73	4	no	—
148	#131	—	146	4	no	—
149	C₂×M₁₆	—	73	1	no	—
150	C₂×M₁₆	—	73	2	no	—
151	C₂×M₁₆	—	146	2	no	—
152	C₂×M₁₆	—	146	2	no	—
153	C₂²×C₈	—	73	1	no	—
154	#117	—	73	1	no	—
155	#108	—	73	2	no	—
156	#108	—	73	2	no	—
157	#104	—	73	4	no	—
158	#104	—	73	4	no	—
159	#97	—	73	2	no	—
160	C₄×C₈	—	73	2	no	—
161	#41	—	73	2	no	—
162	#41	—	73	2	no	—
163	#41	—	73	2	no	—
164	#41	—	146	2	no	—
165	#41	—	146	2	no	—
166	#41	C₂×#64408	1	2	yes	—
167	C₂×Q₈	—	73	2	no	—
168	C₂×D₄	—	73	2	no	—
169	C₂×D₄	—	146	2	no	—
170	C₂²×C₄	—	73	1	no	—
171	C₂²×C₄	—	146	2	no	—
172	Q₁₆	—	73	2	no	—
173	Q₁₆	—	73	1	no	—
174	Q₁₆	C₂×#64408	1	1	yes	characteristic
175	SD₁₆	—	73	2	no	—
176	SD₁₆	—	73	2	no	—
177	SD₁₆	—	73	2	no	—
178	SD₁₆	C₂×#64408	1	2	yes	—
179	D₈	—	73	2	no	—
180	D₈	—	73	1	no	—
181	D₈	C₂×#64408	1	1	yes	characteristic
182	M₁₆	—	73	2	no	—
183	M₁₆	—	73	1	no	—
184	M₁₆	—	73	1	no	—
185	M₁₆	—	73	4	no	—
186	M₁₆	—	146	4	no	—
187	M₁₆	—	146	4	no	—
188	M₁₆	—	146	2	no	—
189	C₂×C₈	—	73	2	no	—
190	C₂×C₈	—	73	1	no	—
191	C₂×C₈	—	73	1	no	—
192	C₂×C₈	—	73	1	no	—
193	C₂×C₈	—	73	1	no	—
194	C₂×C₈	—	73	1	no	—
195	C₂×C₈	—	146	4	no	—
196	C₂×C₈	—	146	4	no	—
197	C₂×C₈	—	146	2	no	—
198	C₂×C₈	C₂×#64408	1	1	yes	characteristic
199	C₄⋊C₄	—	146	1	no	—
200	C₂²⋊C₄	—	146	1	no	—
201	C₄²	—	73	2	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	—	73	1	no	—
216	C₂×C₄	—	73	1	no	—
217	C₂×C₄	—	73	1	no	—
218	C₂×C₄	—	73	1	no	—
219	C₂×C₄	—	73	1	no	—
220	C₂×C₄	—	146	2	no	—
221	C₂×C₄	—	146	2	no	—
222	C₂×C₄	—	146	2	no	—
223	C₂×C₄	—	146	2	no	—
224	C₂×C₄	—	146	2	no	—
225	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
226	C₈	—	73	2	no	—
227	C₈	—	73	2	no	—
228	C₈	—	73	1	no	—
229	C₈	—	73	1	no	—
230	C₈	—	146	4	no	—
231	C₈	C₂²×#64408	1	1	yes	characteristic
232	C₈	C₂²×#64408	1	1	yes	characteristic
233	C₂²	—	2	2	no	—
234	C₂²	—	292	1	no	—
235	C₂²	—	584	2	no	—
236	C₂²	—	73	1	no	—
237	C₂²	—	73	1	no	—
238	C₂²	—	146	2	no	—
239	C₂²	D₄×#64408	1	1	yes	characteristic
240	C₄	—	2	2	no	—
241	C₄	—	292	2	no	—
242	C₄	—	73	1	no	—
243	C₄	—	73	1	no	—
244	C₄	—	146	2	no	—
245	C₄	D₄×#64408	1	1	yes	characteristic
246	C₄	C₂³×#64408	1	1	yes	characteristic
247	C₂	—	2	1	no	—
248	C₂	—	4	2	no	—
249	C₂	—	292	2	no	—
250	C₂	—	146	1	no	—
251	C₂	—	146	1	no	—
252	C₂	C₂×D₄×#64408	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×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	#2957
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #135

Complements

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

N	H ≅ G/N	type
1. #17989666	C₂	semidirect
2. #17989632	C₂	semidirect
3. #17989631	C₂	semidirect
4. #17989629	C₂	semidirect
5. C₂×#14556662	C₂	semidirect
6. #17989584	C₂	semidirect
7. #17989550	C₂	semidirect
9. #14556679	C₂²	semidirect
11. #14556678	C₂²	semidirect
17. #14556662	C₂²	semidirect
18. #14556662	C₂²	semidirect
19. #14556656	C₂²	semidirect
20. #14556655	C₂²	semidirect
21. #14556645	C₂²	semidirect
22. C₈×#64408	C₂²	semidirect
25. #14556613	C₄	semidirect
27. #14556611	C₄	semidirect
29. Q₁₆×D₇₃	C₄	semidirect
30. SD₁₆×D₇₃	C₄	semidirect
31. D₈×D₇₃	C₄	semidirect
33. #135×C₇₃	C₄	semidirect
127. C₇₃	#2957	semidirect
142. #135	#64408	semidirect