Subgroups — #22614849

Order 9344.

620 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14556567	C₂	1	2	yes	maximal, complemented
2	C₂×C₄×D₂₉₂	C₂	1	1	yes	maximal, characteristic
3	#17988509	C₂	1	4	yes	maximal, complemented
4	#17988489	C₂	1	2	yes	maximal
5	#17988485	C₂	1	2	yes	maximal, complemented
6	#17988457	C₂	1	1	yes	maximal, characteristic, complemented
7	#17988444	C₂	1	2	yes	maximal, complemented
8	C₂×C₄×Q₈×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₂₉₂	C₂²	1	1	yes	characteristic
10	C₂²×C₄×D₇₃	C₂²	1	2	yes	—
11	#14556567	—	2	8	no	—
12	C₄×D₂₉₂	—	2	4	no	—
13	C₂×C₄⋊Dic₇₃	C₂²	1	2	yes	—
14	C₂×C₄⋊Dic₇₃	C₂²	1	2	yes	—
15	C₂×C₄⋊Dic₇₃	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₇₃	C₂²	1	2	yes	—
17	C₂×#11924007	C₂²	1	4	yes	—
18	C₂×#11924005	C₂²	1	4	yes	—
19	C₂×#11924005	C₂²	1	2	yes	—
20	C₂×#11924005	C₂²	1	4	yes	—
21	#14556498	C₂²	1	4	yes	—
22	C₂²×Q₈×C₇₃	C₂²	1	1	yes	characteristic
23	C₄×Q₈×C₇₃	—	2	4	no	—
24	C₂×C₄⋊C₄×C₇₃	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊C₄×C₇₃	C₂²	1	2	yes	—
26	C₂×C₄²×C₇₃	C₂²	1	2	yes	—
27	C₂×C₄²×C₇₃	C₂²	1	1	yes	characteristic
28	C₂³×D₇₃	C₂³	1	2	yes	—
29	C₂×D₂₉₂	—	2	2	no	—
30	C₂×D₂₉₂	—	2	2	no	—
31	C₂×D₂₉₂	Q₈	1	4	yes	complemented
32	C₂²×Dic₇₃	C₂³	1	2	yes	—
33	C₂²×Dic₇₃	C₂³	1	4	yes	—
34	C₂×C₄×D₇₃	—	2	4	no	—
35	C₂×C₄×D₇₃	—	2	4	no	—
36	C₂×C₄×D₇₃	—	2	4	no	—
37	C₄⋊Dic₇₃	—	2	2	no	—
38	C₄⋊Dic₇₃	—	2	4	no	—
39	C₄⋊Dic₇₃	—	2	4	no	—
40	C₄⋊Dic₇₃	—	4	2	no	—
41	#11924007	—	2	8	no	—
42	#11924005	—	2	4	no	—
43	#11924005	—	2	8	no	—
44	#11924005	—	4	4	no	—
45	C₂×Q₈×C₇₃	—	2	2	no	—
46	C₂×Q₈×C₇₃	—	2	2	no	—
47	C₂×Q₈×C₇₃	D₄	1	4	yes	complemented
48	C₂²×C₄×C₇₃	C₂³	1	2	yes	—
49	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₇₃	C₂³	1	2	yes	—
51	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
53	C₄⋊C₄×C₇₃	—	2	4	no	—
54	C₄⋊C₄×C₇₃	—	2	2	no	—
55	C₄²×C₇₃	—	2	4	no	—
56	C₄²×C₇₃	D₄	1	4	yes	—
57	C₂³×C₇₃	C₂⁴	1	1	yes	characteristic
58	C₂²×D₇₃	—	2	2	no	—
59	C₂²×D₇₃	—	2	2	no	—
60	C₂²×D₇₃	—	2	2	no	—
61	C₂²×D₇₃	—	2	2	no	—
62	C₂²×D₇₃	—	2	2	no	—
63	C₂²×D₇₃	—	2	2	no	—
64	C₂²×D₇₃	C₂×Q₈	1	4	yes	—
65	Q₈×C₇₃	—	2	8	no	—
66	C₂×C₄×C₇₃	—	2	2	no	—
67	C₂×C₄×C₇₃	—	2	2	no	—
68	C₂×C₄×C₇₃	—	2	1	no	—
69	C₂×C₄×C₇₃	—	2	1	no	—
70	C₂×C₄×C₇₃	—	2	2	no	—
71	C₂×C₄×C₇₃	—	2	2	no	—
72	C₂×C₄×C₇₃	—	2	2	no	—
73	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
74	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
75	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
76	C₂×C₄×C₇₃	C₂×D₄	1	4	yes	—
77	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
78	C₂×C₄×C₇₃	C₂×Q₈	1	2	yes	—
79	C₂×C₄×C₇₃	#41	1	2	yes	—
80	C₂×C₄×C₇₃	#41	1	2	yes	—
81	C₂×Dic₇₃	—	2	4	no	—
82	C₂×Dic₇₃	—	2	2	no	—
83	C₂×Dic₇₃	—	2	2	no	—
84	C₂×Dic₇₃	—	2	4	no	—
85	C₂×Dic₇₃	—	2	2	no	—
86	C₂×Dic₇₃	—	2	4	no	—
87	D₂₉₂	—	2	8	no	—
88	C₄×D₇₃	—	4	8	no	—
89	C₂²×C₇₃	C₂²×D₄	1	1	yes	characteristic
90	C₂²×C₇₃	C₂²×D₄	1	1	yes	characteristic
91	C₂²×C₇₃	C₂²×Q₈	1	1	yes	characteristic
92	C₂²×C₇₃	C₂×#41	1	1	yes	characteristic
93	C₂²×C₇₃	C₂×#41	1	1	yes	characteristic
94	C₂²×C₇₃	H(5,𝔽₂)	1	1	yes	characteristic
95	C₂²×C₇₃	2₋¹⁺⁴	1	1	yes	characteristic
96	C₂×D₇₃	—	2	4	no	—
97	C₂×D₇₃	—	2	4	no	—
98	C₂×D₇₃	—	2	4	no	—
99	C₂×D₇₃	—	4	2	no	—
100	C₂×D₇₃	—	4	2	no	—
101	C₂×D₇₃	—	4	2	no	—
102	C₂×D₇₃	—	4	2	no	—
103	C₄×C₇₃	—	2	4	no	—
104	C₄×C₇₃	—	2	2	no	—
105	C₄×C₇₃	—	4	2	no	—
106	C₄×C₇₃	#121	1	4	yes	—
107	C₄×C₇₃	C₂²⋊Q₈	1	4	yes	complemented
108	Dic₇₃	—	4	2	no	—
109	Dic₇₃	—	4	4	no	—
110	C₂×C₇₃	C₂×#121	1	1	yes	characteristic
111	C₂×C₇₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
112	C₂×C₇₃	#537	1	1	yes	characteristic
113	C₂×C₇₃	D₄×Q₈	1	1	yes	characteristic
114	C₂×C₇₃	#550	1	1	yes	characteristic
115	C₂×C₇₃	#554	1	1	yes	characteristic
116	C₂×C₇₃	#556	1	1	yes	characteristic
117	D₇₃	—	4	4	no	—
118	#2538	—	73	1	no	maximal, Sylow
119	C₇₃	#2538	1	1	yes	characteristic, complemented, Sylow
120	C₂×C₂²⋊Q₈	—	73	2	no	—
121	C₂×C₄×Q₈	D₇₃	1	1	yes	characteristic, complemented
122	C₂×C₄×D₄	—	73	1	no	—
123	#396	—	73	4	no	—
124	#390	—	73	2	no	—
125	#389	—	73	2	no	—
126	#384	—	73	1	no	—
127	#380	—	73	2	no	—
128	C₂²×Q₈	C₂×D₇₃	1	1	yes	characteristic
129	C₂²×D₄	—	73	1	no	—
130	C₂³×C₄	—	73	2	no	—
131	C₂²⋊Q₈	—	146	8	no	—
132	C₄×Q₈	—	2	4	no	—
133	C₄×D₄	—	146	4	no	—
134	C₂×C₄⋊C₄	—	73	2	no	—
135	C₂×C₄⋊C₄	—	73	1	no	—
136	C₂×C₄⋊C₄	—	73	2	no	—
137	C₂×C₄⋊C₄	—	73	4	no	—
138	C₂×C₄⋊C₄	—	73	2	no	—
139	C₂×C₄⋊C₄	C₂×D₇₃	1	1	yes	characteristic
140	C₂×C₄⋊C₄	C₂×D₇₃	1	2	yes	—
141	C₂×C₂²⋊C₄	—	73	4	no	—
142	C₂×C₂²⋊C₄	—	73	2	no	—
143	C₂×C₂²⋊C₄	—	73	4	no	—
144	C₂×C₄²	C₂×D₇₃	1	2	yes	—
145	C₂×C₄²	C₂×D₇₃	1	1	yes	characteristic
146	#95	—	73	4	no	—
147	C₂⁴	—	73	2	no	—
148	C₂×Q₈	—	2	2	no	—
149	C₂×Q₈	—	2	2	no	—
150	C₂×Q₈	D₂₉₂	1	4	yes	complemented
151	C₂×D₄	—	73	4	no	—
152	C₂×D₄	—	146	2	no	—
153	C₂×D₄	—	146	2	no	—
154	C₂²×C₄	—	73	2	no	—
155	C₂²×C₄	—	73	4	no	—
156	C₂²×C₄	—	146	4	no	—
157	C₂²×C₄	—	146	4	no	—
158	C₂²×C₄	—	146	4	no	—
159	C₂²×C₄	C₂²×D₇₃	1	2	yes	—
160	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
161	C₂²×C₄	C₂²×D₇₃	1	2	yes	—
162	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
163	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
164	C₄⋊C₄	—	2	4	no	—
165	C₄⋊C₄	—	2	2	no	—
166	C₄⋊C₄	—	292	2	no	—
167	C₄⋊C₄	—	146	2	no	—
168	C₄⋊C₄	—	146	8	no	—
169	C₄⋊C₄	—	146	4	no	—
170	C₄⋊C₄	—	146	4	no	—
171	C₂²⋊C₄	—	292	4	no	—
172	C₂²⋊C₄	—	146	4	no	—
173	C₂²⋊C₄	—	146	8	no	—
174	C₄²	—	2	4	no	—
175	C₄²	C₂²⋊D₇₃	1	4	yes	—
176	C₂³	—	73	4	no	—
177	C₂³	—	146	2	no	—
178	C₂³	—	146	2	no	—
179	C₂³	—	146	2	no	—
180	C₂³	—	146	2	no	—
181	C₂³	—	146	2	no	—
182	C₂³	—	146	2	no	—
183	C₂³	C₂³×D₇₃	1	1	yes	characteristic
184	Q₈	—	2	8	no	—
185	D₄	—	146	8	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	292	8	no	—
194	C₂×C₄	—	146	2	no	—
195	C₂×C₄	—	146	4	no	—
196	C₂×C₄	—	146	2	no	—
197	C₂×C₄	—	146	2	no	—
198	C₂×C₄	—	146	4	no	—
199	C₂×C₄	—	146	4	no	—
200	C₂×C₄	C₂×C₂²⋊D₇₃	1	2	yes	—
201	C₂×C₄	C₂×C₂²⋊D₇₃	1	2	yes	—
202	C₂×C₄	C₂×C₂²⋊D₇₃	1	2	yes	—
203	C₂×C₄	C₂×D₂₉₂	1	4	yes	—
204	C₂×C₄	C₂×D₂₉₂	1	2	yes	—
205	C₂×C₄	Q₈×D₇₃	1	2	yes	—
206	C₂×C₄	#11924028	1	2	yes	—
207	C₂×C₄	#11924029	1	2	yes	—
208	C₂²	—	292	2	no	—
209	C₂²	—	292	2	no	—
210	C₂²	—	292	2	no	—
211	C₂²	—	292	2	no	—
212	C₂²	—	146	4	no	—
213	C₂²	—	146	4	no	—
214	C₂²	—	146	4	no	—
215	C₂²	C₂²×C₂²⋊D₇₃	1	1	yes	characteristic
216	C₂²	C₂²×D₂₉₂	1	1	yes	characteristic
217	C₂²	C₂×Q₈×D₇₃	1	1	yes	characteristic
218	C₂²	C₂×#11924028	1	1	yes	characteristic
219	C₂²	C₂×#11924029	1	1	yes	characteristic
220	C₂²	#14556637	1	1	yes	characteristic
221	C₂²	#14556638	1	1	yes	characteristic
222	C₄	—	2	2	no	—
223	C₄	—	2	4	no	—
224	C₄	—	4	2	no	—
225	C₄	—	292	4	no	—
226	C₄	—	292	2	no	—
227	C₄	#14556565	1	4	yes	—
228	C₄	#14556567	1	4	yes	complemented
229	C₂	—	292	4	no	—
230	C₂	C₂×#14556565	1	1	yes	characteristic
231	C₂	C₂×#14556567	1	1	yes	characteristic
232	C₂	#17989165	1	1	yes	characteristic
233	C₂	Q₈×D₂₉₂	1	1	yes	characteristic
234	C₂	#17989259	1	1	yes	characteristic
235	C₂	#17989297	1	1	yes	characteristic
236	C₂	#17989310	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₇₃
commutator, socle	C₂³×C₇₃	C₂⁴
Fitting	C₂×C₄×Q₈×C₇₃	C₂
hypercenter	C₂×C₄×Q₈	D₇₃
hypocenter	C₇₃	#2538
LCS	G ▷ C₂³×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×#14556567	C₂	semidirect
3. #17988509	C₂	semidirect
5. #17988485	C₂	semidirect
6. #17988457	C₂	semidirect
7. #17988444	C₂	semidirect
8. C₂×C₄×Q₈×C₇₃	C₂	semidirect
31. C₂×D₂₉₂	Q₈	semidirect
47. C₂×Q₈×C₇₃	D₄	semidirect
107. C₄×C₇₃	C₂²⋊Q₈	semidirect
119. C₇₃	#2538	semidirect
121. C₂×C₄×Q₈	D₇₃	semidirect
150. C₂×Q₈	D₂₉₂	semidirect
228. C₄	#14556567	semidirect