Subgroups — #22613708

Order 9344.

460 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	C₂×#14556589	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₁₄₆	C₂	1	1	yes	maximal, characteristic
3	#17988528	C₂	1	2	yes	maximal
4	#17988521	C₂	1	2	yes	maximal
5	#17988494	C₂	1	1	yes	maximal, characteristic
6	#17988492	C₂	1	1	yes	maximal, characteristic
7	#17988486	C₂	1	1	yes	maximal, characteristic
8	#17988470	C₂	1	2	yes	maximal
9	#17988469	C₂	1	2	yes	maximal
10	C₄×#11924007	C₂	1	1	yes	maximal, characteristic
11	#391×C₇₃	C₂	1	1	yes	maximal, characteristic
12	C₂²×Dic₁₄₆	C₂²	1	1	yes	characteristic
13	#14556589	—	2	4	no	—
14	C₄×Dic₁₄₆	—	2	4	no	—
15	C₂×C₄⋊Dic₇₃	C₂²	1	2	yes	—
16	C₂×C₄⋊Dic₇₃	C₂²	1	1	yes	characteristic
17	C₂×#11924007	C₂²	1	2	yes	—
18	C₂×#11924007	C₂²	1	2	yes	—
19	C₂×#11924007	C₂²	1	1	yes	characteristic
20	C₂×#11924007	C₂²	1	2	yes	—
21	C₂×#11924007	C₂²	1	1	yes	characteristic
22	C₂×C₄×Dic₇₃	C₂²	1	1	yes	characteristic
23	C₂×C₄×Dic₇₃	C₂²	1	1	yes	characteristic
24	C₂×C₄×Dic₇₃	C₂²	1	2	yes	—
25	#14556498	C₂²	1	1	yes	characteristic
26	#14556498	C₂²	1	1	yes	characteristic
27	#14556498	C₂²	1	2	yes	—
28	#14556498	C₂²	1	2	yes	—
29	#14556498	C₂²	1	2	yes	—
30	#14556498	C₂²	1	2	yes	—
31	#14556498	C₂²	1	2	yes	—
32	C₂²×Q₈×C₇₃	C₂²	1	1	yes	characteristic
33	C₂×C₄⋊C₄×C₇₃	C₂²	1	1	yes	characteristic
34	C₂×C₄²×C₇₃	C₂²	1	1	yes	characteristic
35	#95×C₇₃	C₂²	1	2	yes	—
36	#95×C₇₃	C₂²	1	2	yes	—
37	C₂×Dic₁₄₆	—	2	2	no	—
38	C₂×Dic₁₄₆	—	2	2	no	—
39	C₂×Dic₁₄₆	D₄	1	4	yes	—
40	C₂²×Dic₇₃	C₂³	1	1	yes	characteristic
41	C₂²×Dic₇₃	C₂³	1	2	yes	—
42	C₂²×Dic₇₃	C₂³	1	2	yes	—
43	C₂²×Dic₇₃	C₂³	1	1	yes	characteristic
44	C₂²×Dic₇₃	C₂³	1	1	yes	characteristic
45	C₂²×Dic₇₃	C₂³	1	1	yes	characteristic
46	C₄⋊Dic₇₃	—	2	2	no	—
47	C₄⋊Dic₇₃	—	2	4	no	—
48	#11924007	—	2	4	no	—
49	#11924007	—	2	4	no	—
50	#11924007	—	2	4	no	—
51	#11924007	—	2	2	no	—
52	#11924007	Q₈	1	4	yes	complemented
53	C₄×Dic₇₃	—	2	2	no	—
54	C₄×Dic₇₃	—	2	2	no	—
55	C₄×Dic₇₃	—	2	4	no	—
56	C₂×Q₈×C₇₃	—	2	2	no	—
57	C₂×Q₈×C₇₃	—	2	2	no	—
58	C₂×Q₈×C₇₃	—	2	2	no	—
59	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₇₃	C₂³	1	2	yes	—
61	C₂²×C₄×C₇₃	C₂³	1	2	yes	—
62	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
63	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
64	C₄⋊C₄×C₇₃	—	2	2	no	—
65	C₄²×C₇₃	—	2	2	no	—
66	C₂³×C₇₃	C₂⁴	1	1	yes	characteristic
67	Q₈×C₇₃	—	4	4	no	—
68	C₂×C₄×C₇₃	—	2	1	no	—
69	C₂×C₄×C₇₃	—	2	1	no	—
70	C₂×C₄×C₇₃	—	2	1	no	—
71	C₂×C₄×C₇₃	—	2	2	no	—
72	C₂×C₄×C₇₃	—	2	2	no	—
73	C₂×C₄×C₇₃	—	2	2	no	—
74	C₂×C₄×C₇₃	—	2	2	no	—
75	C₂×C₄×C₇₃	—	2	1	no	—
76	C₂×C₄×C₇₃	—	2	2	no	—
77	C₂×C₄×C₇₃	—	2	2	no	—
78	C₂×C₄×C₇₃	—	2	1	no	—
79	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
80	C₂×C₄×C₇₃	C₂×Q₈	1	2	yes	—
81	C₂×C₄×C₇₃	#41	1	2	yes	—
82	C₂×C₄×C₇₃	#41	1	2	yes	—
83	C₂×Dic₇₃	—	2	1	no	—
84	C₂×Dic₇₃	—	2	1	no	—
85	C₂×Dic₇₃	—	2	1	no	—
86	C₂×Dic₇₃	—	2	1	no	—
87	C₂×Dic₇₃	—	2	2	no	—
88	C₂×Dic₇₃	—	2	2	no	—
89	C₂×Dic₇₃	—	2	1	no	—
90	C₂×Dic₇₃	—	2	1	no	—
91	C₂×Dic₇₃	—	2	2	no	—
92	C₂×Dic₇₃	—	2	1	no	—
93	C₂×Dic₇₃	—	2	2	no	—
94	C₂×Dic₇₃	—	2	1	no	—
95	C₂×Dic₇₃	—	2	1	no	—
96	C₂×Dic₇₃	—	2	2	no	—
97	C₂×Dic₇₃	C₂×D₄	1	4	yes	—
98	C₂×Dic₇₃	C₂×Q₈	1	2	yes	—
99	C₂×Dic₇₃	C₂×Q₈	1	2	yes	—
100	C₂×Dic₇₃	#41	1	2	yes	—
101	Dic₁₄₆	—	2	8	no	—
102	C₂²×C₇₃	C₂²×D₄	1	1	yes	characteristic
103	C₂²×C₇₃	C₂²×Q₈	1	1	yes	characteristic
104	C₂²×C₇₃	C₂×#41	1	1	yes	characteristic
105	C₂²×C₇₃	C₂×#41	1	1	yes	characteristic
106	C₂²×C₇₃	C₂×#41	1	1	yes	characteristic
107	C₂²×C₇₃	2₋¹⁺⁴	1	1	yes	characteristic
108	C₂²×C₇₃	2₋¹⁺⁴	1	1	yes	characteristic
109	C₄×C₇₃	—	2	2	no	—
110	C₄×C₇₃	—	4	2	no	—
111	C₄×C₇₃	—	4	1	no	—
112	C₄×C₇₃	—	4	2	no	—
113	C₄×C₇₃	#123	1	4	yes	—
114	Dic₇₃	—	2	4	no	—
115	Dic₇₃	—	2	2	no	—
116	Dic₇₃	—	2	2	no	—
117	Dic₇₃	—	2	2	no	—
118	Dic₇₃	—	4	1	no	—
119	Dic₇₃	—	4	2	no	—
120	C₂×C₇₃	C₂×#123	1	1	yes	characteristic
121	C₂×C₇₃	#533	1	1	yes	characteristic
122	C₂×C₇₃	#541	1	1	yes	characteristic
123	C₂×C₇₃	#548	1	1	yes	characteristic
124	C₂×C₇₃	D₄×Q₈	1	1	yes	characteristic
125	C₂×C₇₃	#556	1	1	yes	characteristic
126	C₂×C₇₃	#557	1	1	yes	characteristic
127	#2446	—	73	1	no	maximal, Sylow
128	C₇₃	#2446	1	1	yes	characteristic, complemented, Sylow
129	C₂×C₄⋊Q₈	—	73	1	no	—
130	C₂×C₄×Q₈	—	73	1	no	—
131	#400	—	73	2	no	—
132	#398	—	73	2	no	—
133	#391	—	73	1	no	—
134	#391	—	73	1	no	—
135	#391	D₇₃	1	1	yes	characteristic
136	#389	—	73	1	no	—
137	#387	—	73	2	no	—
138	#387	—	73	2	no	—
139	C₄×C₄⋊C₄	—	73	1	no	—
140	C₂²×Q₈	—	73	1	no	—
141	C₂²×Q₈	C₂×D₇₃	1	1	yes	characteristic
142	C₄⋊Q₈	—	146	4	no	—
143	C₄×Q₈	—	146	4	no	—
144	C₂×C₄⋊C₄	—	73	1	no	—
145	C₂×C₄⋊C₄	—	73	2	no	—
146	C₂×C₄⋊C₄	—	73	1	no	—
147	C₂×C₄⋊C₄	—	73	2	no	—
148	C₂×C₄⋊C₄	—	73	2	no	—
149	C₂×C₄⋊C₄	—	73	1	no	—
150	C₂×C₄⋊C₄	—	73	2	no	—
151	C₂×C₄⋊C₄	C₂×D₇₃	1	1	yes	characteristic
152	C₂×C₄²	—	73	2	no	—
153	C₂×C₄²	—	73	1	no	—
154	C₂×C₄²	—	73	1	no	—
155	C₂×C₄²	C₂×D₇₃	1	1	yes	characteristic
156	#95	—	73	1	no	—
157	#95	—	73	1	no	—
158	#95	—	73	2	no	—
159	#95	—	73	2	no	—
160	#95	—	73	2	no	—
161	#95	—	73	2	no	—
162	#95	—	73	2	no	—
163	#95	C₂×D₇₃	1	2	yes	—
164	#95	C₂×D₇₃	1	2	yes	—
165	C₂×Q₈	—	2	2	no	—
166	C₂×Q₈	—	2	2	no	—
167	C₂×Q₈	—	2	2	no	—
168	C₂×Q₈	—	73	4	no	—
169	C₂×Q₈	—	146	2	no	—
170	C₂×Q₈	—	146	2	no	—
171	C₂²×C₄	—	73	1	no	—
172	C₂²×C₄	—	73	1	no	—
173	C₂²×C₄	—	73	2	no	—
174	C₂²×C₄	—	73	1	no	—
175	C₂²×C₄	—	73	1	no	—
176	C₂²×C₄	—	73	2	no	—
177	C₂²×C₄	C₂²×D₇₃	1	2	yes	—
178	C₂²×C₄	C₂²×D₇₃	1	2	yes	—
179	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
180	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
181	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
182	C₄⋊C₄	—	2	2	no	—
183	C₄⋊C₄	—	73	4	no	—
184	C₄⋊C₄	—	146	2	no	—
185	C₄⋊C₄	—	146	4	no	—
186	C₄⋊C₄	—	146	4	no	—
187	C₄⋊C₄	—	146	4	no	—
188	C₄⋊C₄	—	146	4	no	—
189	C₄⋊C₄	—	146	2	no	—
190	C₄²	—	2	2	no	—
191	C₄²	—	146	4	no	—
192	C₄²	—	146	2	no	—
193	C₄²	—	146	2	no	—
194	C₂³	C₂³×D₇₃	1	1	yes	characteristic
195	Q₈	—	4	4	no	—
196	Q₈	—	146	8	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	73	4	no	—
209	C₂×C₄	—	73	2	no	—
210	C₂×C₄	—	73	2	no	—
211	C₂×C₄	—	73	2	no	—
212	C₂×C₄	—	146	2	no	—
213	C₂×C₄	—	146	2	no	—
214	C₂×C₄	—	146	1	no	—
215	C₂×C₄	—	146	1	no	—
216	C₂×C₄	—	146	1	no	—
217	C₂×C₄	—	146	2	no	—
218	C₂×C₄	—	146	1	no	—
219	C₂×C₄	—	146	1	no	—
220	C₂×C₄	—	146	1	no	—
221	C₂×C₄	—	146	2	no	—
222	C₂×C₄	—	146	1	no	—
223	C₂×C₄	—	146	2	no	—
224	C₂×C₄	—	146	1	no	—
225	C₂×C₄	—	146	1	no	—
226	C₂×C₄	D₄×D₇₃	1	2	yes	—
227	C₂×C₄	Q₈×D₇₃	1	2	yes	—
228	C₂×C₄	#11924027	1	2	yes	—
229	C₂×C₄	#11924028	1	2	yes	—
230	C₂²	C₂×D₄×D₇₃	1	1	yes	characteristic
231	C₂²	C₂×Q₈×D₇₃	1	1	yes	characteristic
232	C₂²	#41×D₇₃	1	1	yes	characteristic
233	C₂²	C₂×#11924027	1	1	yes	characteristic
234	C₂²	C₂×#11924028	1	1	yes	characteristic
235	C₂²	#14556638	1	1	yes	characteristic
236	C₂²	#14556639	1	1	yes	characteristic
237	C₄	—	2	2	no	—
238	C₄	—	4	1	no	—
239	C₄	—	4	2	no	—
240	C₄	—	4	2	no	—
241	C₄	—	292	1	no	—
242	C₄	—	292	2	no	—
243	C₄	—	146	2	no	—
244	C₄	—	146	2	no	—
245	C₄	—	146	2	no	—
246	C₄	—	146	4	no	—
247	C₄	#14556573	1	4	yes	—
248	C₂	C₂×#14556573	1	1	yes	characteristic
249	C₂	#17989141	1	1	yes	characteristic
250	C₂	#17989195	1	1	yes	characteristic
251	C₂	#17989247	1	1	yes	characteristic
252	C₂	#17989250	1	1	yes	characteristic
253	C₂	#17989306	1	1	yes	characteristic
254	C₂	#17989312	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₇₃
commutator, socle	C₂³×C₇₃	C₂⁴
Fitting	#391×C₇₃	C₂
hypercenter	#391	D₇₃
hypocenter	C₇₃	#2446
LCS	G ▷ C₂³×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ #391

Complements

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

N	H ≅ G/N	type
52. #11924007	Q₈	semidirect
128. C₇₃	#2446	semidirect