Subgroups — #22618934

Order 9344.

370 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17989124	C₂	1	1	yes	maximal, characteristic
2	#17988718	C₂	1	1	yes	maximal, characteristic, complemented
3	#17988666	C₂	1	1	yes	maximal, characteristic, complemented
4	#17988666	C₂	1	2	yes	maximal, complemented
5	#17988657	C₂	1	2	yes	maximal, complemented
6	#17988652	C₂	1	1	yes	maximal, characteristic, complemented
7	#17988647	C₂	1	2	yes	maximal, complemented
8	#17988639	C₂	1	1	yes	maximal, characteristic, complemented
9	#17988541	C₂	1	2	yes	maximal, complemented
10	C₄×#11924011	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×Q₈×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#11924011	C₂²	1	1	yes	characteristic
13	C₂×#11924011	C₂²	1	1	yes	characteristic
14	C₂×#11924011	C₂²	1	2	yes	—
15	C₂×C₈×D₇₃	C₂²	1	2	yes	—
16	#14556586	C₄	1	2	yes	—
17	#14556582	C₄	1	1	yes	characteristic
18	#14556580	C₄	1	1	yes	characteristic
19	#14556576	C₄	1	2	yes	—
20	#14556567	C₄	1	1	yes	characteristic
21	#14556566	C₄	1	1	yes	characteristic
22	C₄×D₂₉₂	C₂²	1	1	yes	characteristic
23	C₄×D₂₉₂	C₂²	1	2	yes	—
24	#14556548	C₂²	1	2	yes	—
25	C₄²×D₇₃	C₂²	1	1	yes	characteristic
26	C₄⋊#10596831	C₂²	1	1	yes	characteristic
27	C₄⋊#10596831	C₂²	1	2	yes	—
28	#14556503	C₂²	1	2	yes	—
29	#14556503	C₂²	1	2	yes	—
30	#14556503	C₂²	1	2	yes	—
31	#14556502	C₂²	1	2	yes	—
32	#14556501	C₂²	1	1	yes	characteristic
33	#14556501	C₂²	1	1	yes	characteristic
34	C₄×#10596831	C₂²	1	1	yes	characteristic
35	C₈×Dic₇₃	C₂²	1	2	yes	—
36	C₄×Q₈×C₇₃	C₂²	1	1	yes	characteristic
37	C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
38	C₄⋊C₈×C₇₃	C₂²	1	2	yes	—
39	C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
40	C₄×C₈×C₇₃	C₂²	1	2	yes	—
41	C₂×D₂₉₂	C₂×C₄	1	1	yes	characteristic
42	C₂×D₂₉₂	C₂×C₄	1	2	yes	—
43	C₂×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×D₇₃	C₂³	1	1	yes	characteristic
46	C₂×C₄×D₇₃	C₂³	1	2	yes	—
47	#11924011	—	2	2	no	—
48	#11924011	—	2	4	no	—
49	#11924011	—	2	2	no	—
50	C₂×#10596831	C₂³	1	2	yes	—
51	C₂×#10596831	C₂³	1	1	yes	characteristic
52	C₂×#10596831	C₂³	1	1	yes	characteristic
53	C₈×D₇₃	—	2	4	no	—
54	C₄⋊Dic₇₃	C₂×C₄	1	1	yes	characteristic
55	C₄⋊Dic₇₃	C₂×C₄	1	2	yes	—
56	#11924007	C₂×C₄	1	2	yes	—
57	#11924007	C₂×C₄	1	2	yes	—
58	#11924005	C₂×C₄	1	2	yes	—
59	#11924005	C₂×C₄	1	2	yes	—
60	#11924005	C₂×C₄	1	2	yes	—
61	#11924005	C₂×C₄	1	2	yes	—
62	#11924005	C₂×C₄	1	2	yes	—
63	C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
64	C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
65	C₄×Dic₇₃	C₂³	1	1	yes	characteristic
66	C₂×Q₈×C₇₃	C₂×C₄	1	1	yes	characteristic
67	C₂×C₈×C₇₃	C₂³	1	2	yes	—
68	C₂×C₈×C₇₃	C₂³	1	1	yes	characteristic
69	C₂×C₈×C₇₃	C₂³	1	1	yes	characteristic
70	C₄⋊C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
71	C₄⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
72	C₄²×C₇₃	C₂³	1	1	yes	characteristic
73	C₄²×C₇₃	C₂³	1	2	yes	—
74	C₂²×D₇₃	C₂²×C₄	1	2	yes	—
75	C₂²×D₇₃	C₂²×C₄	1	1	yes	characteristic
76	Q₈×C₇₃	—	2	2	no	—
77	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×C₇₃	C₂²×C₄	1	2	yes	—
79	C₂×C₄×C₇₃	C₂²×C₄	1	2	yes	—
80	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
82	C₂×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
83	C₂×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
84	C₂×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×Dic₇₃	C₂²×C₄	1	2	yes	—
86	D₂₉₂	—	2	2	no	—
87	D₂₉₂	—	4	2	no	—
88	C₄×D₇₃	—	2	2	no	—
89	C₄×D₇₃	—	2	2	no	—
90	C₄×D₇₃	—	2	1	no	—
91	C₄×D₇₃	—	2	1	no	—
92	C₄×D₇₃	—	2	2	no	—
93	C₄×D₇₃	—	4	1	no	—
94	C₈×C₇₃	—	2	2	no	—
95	C₈×C₇₃	—	2	1	no	—
96	C₈×C₇₃	#41	1	2	yes	—
97	#10596831	—	2	2	no	—
98	#10596831	—	2	1	no	—
99	#10596831	#41	1	2	yes	—
100	C₂²×C₇₃	C₂³×C₄	1	1	yes	characteristic
101	C₂×D₇₃	—	2	2	no	—
102	C₂×D₇₃	—	2	1	no	—
103	C₂×D₇₃	—	2	1	no	—
104	C₂×D₇₃	—	2	2	no	—
105	C₂×D₇₃	—	2	1	no	—
106	C₂×D₇₃	—	2	2	no	—
107	C₄×C₇₃	—	2	2	no	—
108	C₄×C₇₃	—	2	1	no	—
109	C₄×C₇₃	—	2	2	no	—
110	C₄×C₇₃	#131	1	2	yes	—
111	C₄×C₇₃	C₂×#41	1	1	yes	characteristic
112	C₄×C₇₃	C₂×#41	1	1	yes	characteristic
113	Dic₇₃	—	2	2	no	—
114	Dic₇₃	—	2	1	no	—
115	Dic₇₃	—	2	1	no	—
116	Dic₇₃	—	2	1	no	—
117	C₂×C₇₃	C₄×#41	1	1	yes	characteristic
118	C₂×C₇₃	C₂×#131	1	1	yes	characteristic
119	C₂×C₇₃	#568	1	1	yes	characteristic
120	D₇₃	—	4	1	no	—
121	D₇₃	—	4	2	no	—
122	#2969	—	73	1	no	maximal, Sylow
123	C₇₃	#2969	1	1	yes	characteristic, complemented, Sylow
124	#529	—	73	1	no	—
125	#446	—	73	1	no	—
126	C₈×Q₈	D₇₃	1	1	yes	characteristic, complemented
127	#436	—	73	1	no	—
128	#436	—	73	2	no	—
129	#435	—	73	2	no	—
130	C₈×D₄	—	73	1	no	—
131	#433	—	73	2	no	—
132	#431	—	73	1	no	—
133	#405	—	73	2	no	—
134	C₄×M₁₆	—	73	1	no	—
135	C₂×M₁₆	—	73	1	no	—
136	C₂×M₁₆	—	73	1	no	—
137	C₂×M₁₆	—	73	2	no	—
138	C₂²×C₈	—	73	2	no	—
139	#126	—	73	2	no	—
140	#125	—	73	1	no	—
141	#124	—	73	1	no	—
142	#123	—	73	2	no	—
143	C₂²⋊Q₈	—	73	1	no	—
144	#121	—	73	1	no	—
145	C₄×Q₈	C₂×D₇₃	1	1	yes	characteristic
146	C₄×D₄	—	73	1	no	—
147	C₄×D₄	—	73	2	no	—
148	#117	—	73	2	no	—
149	C₂×C₄²	—	73	1	no	—
150	C₄⋊C₈	—	73	1	no	—
151	C₄⋊C₈	—	73	2	no	—
152	C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
153	C₄⋊C₈	C₂×D₇₃	1	2	yes	—
154	C₂²⋊C₈	—	73	2	no	—
155	C₂²⋊C₈	—	73	2	no	—
156	C₂²⋊C₈	—	73	2	no	—
157	#97	—	73	1	no	—
158	#97	—	73	1	no	—
159	#97	—	73	2	no	—
160	C₄×C₈	—	73	2	no	—
161	C₄×C₈	—	73	1	no	—
162	C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
163	C₄×C₈	C₂×D₇₃	1	2	yes	—
164	C₂×Q₈	C₄×D₇₃	1	1	yes	characteristic
165	C₂×D₄	—	73	2	no	—
166	C₂×D₄	—	73	1	no	—
167	C₂²×C₄	—	73	1	no	—
168	C₂²×C₄	—	73	1	no	—
169	C₂²×C₄	—	73	1	no	—
170	C₂²×C₄	—	73	2	no	—
171	M₁₆	—	146	2	no	—
172	M₁₆	—	146	2	no	—
173	M₁₆	—	146	4	no	—
174	C₂×C₈	—	73	1	no	—
175	C₂×C₈	—	73	2	no	—
176	C₂×C₈	—	73	1	no	—
177	C₂×C₈	—	146	4	no	—
178	C₂×C₈	C₂²×D₇₃	1	2	yes	—
179	C₂×C₈	C₂²×D₇₃	1	1	yes	characteristic
180	C₂×C₈	C₂²×D₇₃	1	1	yes	characteristic
181	C₄⋊C₄	—	73	2	no	—
182	C₄⋊C₄	—	73	2	no	—
183	C₄⋊C₄	—	73	1	no	—
184	C₄⋊C₄	—	73	2	no	—
185	C₄⋊C₄	C₄×D₇₃	1	1	yes	characteristic
186	C₄⋊C₄	C₄×D₇₃	1	2	yes	—
187	C₂²⋊C₄	—	73	2	no	—
188	C₂²⋊C₄	—	73	2	no	—
189	C₂²⋊C₄	—	73	2	no	—
190	C₂²⋊C₄	—	73	2	no	—
191	C₂²⋊C₄	—	73	2	no	—
192	C₄²	—	73	1	no	—
193	C₄²	—	73	1	no	—
194	C₄²	—	73	1	no	—
195	C₄²	C₂²×D₇₃	1	1	yes	characteristic
196	C₄²	C₂²×D₇₃	1	2	yes	—
197	C₂³	—	73	2	no	—
198	C₂³	—	73	1	no	—
199	Q₈	—	2	2	no	—
200	D₄	—	292	2	no	—
201	D₄	—	146	2	no	—
202	C₂×C₄	—	292	1	no	—
203	C₂×C₄	—	73	1	no	—
204	C₂×C₄	—	73	1	no	—
205	C₂×C₄	—	73	1	no	—
206	C₂×C₄	—	73	2	no	—
207	C₂×C₄	—	146	2	no	—
208	C₂×C₄	—	146	2	no	—
209	C₂×C₄	—	146	1	no	—
210	C₂×C₄	—	146	1	no	—
211	C₂×C₄	—	146	2	no	—
212	C₂×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
213	C₂×C₄	C₂×C₄×D₇₃	1	2	yes	—
214	C₂×C₄	C₂×C₄×D₇₃	1	2	yes	—
215	C₂×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
216	C₂×C₄	C₂³×D₇₃	1	1	yes	characteristic
217	C₈	—	2	2	no	—
218	C₈	—	2	1	no	—
219	C₈	—	73	2	no	—
220	C₈	—	146	2	no	—
221	C₈	—	146	1	no	—
222	C₈	#11924029	1	2	yes	—
223	C₂²	—	146	2	no	—
224	C₂²	—	146	1	no	—
225	C₂²	—	146	1	no	—
226	C₂²	—	146	2	no	—
227	C₂²	—	146	1	no	—
228	C₂²	—	146	2	no	—
229	C₂²	C₂²×C₄×D₇₃	1	1	yes	characteristic
230	C₄	—	2	2	no	—
231	C₄	—	2	1	no	—
232	C₄	—	2	2	no	—
233	C₄	—	146	1	no	—
234	C₄	—	146	2	no	—
235	C₄	—	146	1	no	—
236	C₄	—	146	1	no	—
237	C₄	#14556597	1	2	yes	—
238	C₄	#41×D₇₃	1	1	yes	characteristic
239	C₄	C₂×#11924029	1	1	yes	characteristic
240	C₂	—	292	1	no	—
241	C₂	—	292	2	no	—
242	C₂	C₄×#11924029	1	1	yes	characteristic
243	C₂	C₂×#14556597	1	1	yes	characteristic
244	C₂	#17989352	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₇₃
commutator, socle	C₂²×C₇₃	C₂³×C₄
Fitting	C₈×Q₈×C₇₃	C₂
hypercenter	C₈×Q₈	D₇₃
hypocenter	C₇₃	#2969
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₈×Q₈

Complements

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

N	H ≅ G/N	type
2. #17988718	C₂	semidirect
3. #17988666	C₂	semidirect
4. #17988666	C₂	semidirect
5. #17988657	C₂	semidirect
6. #17988652	C₂	semidirect
7. #17988647	C₂	semidirect
8. #17988639	C₂	semidirect
9. #17988541	C₂	semidirect
10. C₄×#11924011	C₂	semidirect
11. C₈×Q₈×C₇₃	C₂	semidirect
123. C₇₃	#2969	semidirect
126. C₈×Q₈	D₇₃	semidirect