Subgroups — #22618596

Order 9344.

392 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#11924028	C₂	1	1	yes	maximal, characteristic
2	#17988647	C₂	1	4	yes	maximal, complemented
3	#17988643	C₂	1	2	yes	maximal, complemented
4	#17988639	C₂	1	2	yes	maximal, complemented
5	#17988611	C₂	1	1	yes	maximal, characteristic, complemented
6	#17988553	C₂	1	2	yes	maximal
7	#17988542	C₂	1	2	yes	maximal, complemented
8	C₂×C₄⋊C₈×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#11924028	C₂²	1	1	yes	characteristic
10	C₂×#11924012	C₂²	1	2	yes	—
11	C₄×Dic₁₄₆	C₄	1	1	yes	characteristic
12	C₄×Dic₁₄₆	C₄	1	1	yes	characteristic
13	C₄×D₂₉₂	C₄	1	1	yes	characteristic
14	C₄×D₂₉₂	C₄	1	1	yes	characteristic
15	C₄×C₂²⋊D₇₃	C₄	1	2	yes	—
16	C₄×C₂²⋊D₇₃	C₄	1	2	yes	—
17	#14556550	C₂²	1	1	yes	characteristic
18	#14556548	C₂²	1	2	yes	—
19	C₄²×D₇₃	C₂²	1	2	yes	—
20	C₄⋊#10596831	C₂²	1	2	yes	—
21	C₄⋊#10596831	C₂²	1	2	yes	—
22	#14556503	C₂²	1	4	yes	—
23	#14556503	C₂²	1	4	yes	—
24	#14556501	C₂²	1	4	yes	—
25	C₈×Dic₇₃	C₂²	1	4	yes	—
26	C₂²×C₈×C₇₃	C₂²	1	2	yes	—
27	C₂×C₄²×C₇₃	C₂²	1	1	yes	characteristic
28	C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
29	C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
30	C₄⋊C₈×C₇₃	C₂²	1	2	yes	—
31	#11924028	—	2	2	no	—
32	#11924028	—	2	2	no	—
33	C₂×Dic₁₄₆	C₂×C₄	1	1	yes	characteristic
34	C₂×D₂₉₂	C₂×C₄	1	1	yes	characteristic
35	C₂×C₂²⋊D₇₃	C₂×C₄	1	2	yes	—
36	C₂×C₄×D₇₃	C₂×C₄	1	2	yes	—
37	C₂×C₄×D₇₃	C₂×C₄	1	2	yes	—
38	C₂×C₄×D₇₃	C₂³	1	2	yes	—
39	#11924012	—	2	4	no	—
40	C₂×#10596831	C₂³	1	4	yes	—
41	C₄⋊Dic₇₃	C₂×C₄	1	1	yes	characteristic
42	C₄⋊Dic₇₃	C₂×C₄	1	1	yes	characteristic
43	#11924007	C₂×C₄	1	2	yes	—
44	#11924007	C₂×C₄	1	2	yes	—
45	C₂²⋊Dic₇₃	C₂×C₄	1	2	yes	—
46	#11924005	C₂×C₄	1	2	yes	—
47	#11924005	C₂×C₄	1	2	yes	—
48	C₄×Dic₇₃	C₂×C₄	1	2	yes	—
49	C₄×Dic₇₃	C₂×C₄	1	2	yes	—
50	C₄×Dic₇₃	C₂³	1	2	yes	—
51	C₂²×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
54	C₂×C₈×C₇₃	—	2	4	no	—
55	C₂×C₈×C₇₃	C₂³	1	2	yes	—
56	C₂×C₈×C₇₃	C₂³	1	2	yes	—
57	C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
58	C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
59	C₄²×C₇₃	C₂³	1	1	yes	characteristic
60	C₄²×C₇₃	C₂³	1	1	yes	characteristic
61	C₂³×C₇₃	C₂²×C₄	1	1	yes	characteristic
62	C₂²×D₇₃	C₂²×C₄	1	2	yes	—
63	C₂×C₄×C₇₃	—	2	2	no	—
64	C₂×C₄×C₇₃	—	4	1	no	—
65	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₄×C₇₃	#41	1	2	yes	—
71	C₂×C₄×C₇₃	#41	1	2	yes	—
72	C₂×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
73	C₂²⋊D₇₃	—	2	4	no	—
74	C₂×Dic₇₃	C₂²×C₄	1	2	yes	—
75	C₂×Dic₇₃	C₂²×C₄	1	2	yes	—
76	C₂×Dic₇₃	C₂²×C₄	1	2	yes	—
77	D₂₉₂	—	2	2	no	—
78	C₄×D₇₃	—	2	4	no	—
79	C₄×D₇₃	—	2	2	no	—
80	C₄×D₇₃	—	2	2	no	—
81	C₄×D₇₃	—	4	2	no	—
82	Dic₁₄₆	—	2	2	no	—
83	C₈×C₇₃	—	2	2	no	—
84	C₈×C₇₃	—	2	2	no	—
85	#10596831	—	2	4	no	—
86	C₂²×C₇₃	—	2	1	no	—
87	C₂²×C₇₃	—	2	1	no	—
88	C₂²×C₇₃	#117	1	2	yes	—
89	C₂²×C₇₃	C₂³×C₄	1	1	yes	characteristic
90	C₂×D₇₃	—	2	2	no	—
91	C₂×D₇₃	—	2	2	no	—
92	C₂×D₇₃	—	2	2	no	—
93	C₄×C₇₃	—	2	1	no	—
94	C₄×C₇₃	—	2	1	no	—
95	C₄×C₇₃	—	2	1	no	—
96	C₄×C₇₃	#117	1	2	yes	—
97	C₄×C₇₃	#131	1	2	yes	—
98	C₄×C₇₃	C₂×#41	1	1	yes	characteristic
99	C₄×C₇₃	C₂×#41	1	1	yes	characteristic
100	Dic₇₃	—	2	2	no	—
101	Dic₇₃	—	2	2	no	—
102	Dic₇₃	—	2	2	no	—
103	C₂×C₇₃	—	2	2	no	—
104	C₂×C₇₃	C₂×#117	1	1	yes	characteristic
105	C₂×C₇₃	C₂×#131	1	1	yes	characteristic
106	C₂×C₇₃	#568	1	1	yes	characteristic
107	D₇₃	—	4	2	no	—
108	#2925	—	73	1	no	maximal, Sylow
109	C₇₃	#2925	1	1	yes	characteristic, complemented, Sylow
110	C₄×#41	—	73	1	no	—
111	#433	—	73	4	no	—
112	#432	—	73	2	no	—
113	#431	—	73	2	no	—
114	#423	—	73	1	no	—
115	C₂×C₄⋊C₈	D₇₃	1	1	yes	characteristic, complemented
116	#408	—	73	2	no	—
117	#405	—	73	2	no	—
118	C₂×#41	—	73	1	no	—
119	C₂×M₁₆	—	73	2	no	—
120	C₂²×C₈	C₂×D₇₃	1	2	yes	—
121	C₄×Q₈	—	73	1	no	—
122	C₄×Q₈	—	73	1	no	—
123	C₄×D₄	—	73	2	no	—
124	C₄×D₄	—	73	2	no	—
125	C₄×D₄	—	73	1	no	—
126	C₄×D₄	—	73	1	no	—
127	#117	—	73	1	no	—
128	#117	—	73	2	no	—
129	C₂×C₄²	—	73	2	no	—
130	C₂×C₄²	C₂×D₇₃	1	1	yes	characteristic
131	C₄⋊C₈	—	73	2	no	—
132	C₄⋊C₈	—	73	2	no	—
133	C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
134	C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
135	C₄⋊C₈	C₂×D₇₃	1	2	yes	—
136	C₂²⋊C₈	—	73	4	no	—
137	C₂²⋊C₈	—	73	4	no	—
138	#97	—	73	4	no	—
139	C₄×C₈	—	73	4	no	—
140	#41	—	146	2	no	—
141	#41	—	146	2	no	—
142	C₂×Q₈	—	73	1	no	—
143	C₂×D₄	—	73	2	no	—
144	C₂×D₄	—	73	1	no	—
145	C₂²×C₄	—	73	2	no	—
146	C₂²×C₄	—	73	2	no	—
147	C₂²×C₄	—	73	2	no	—
148	C₂²×C₄	C₄×D₇₃	1	1	yes	characteristic
149	C₂²×C₄	C₄×D₇₃	1	1	yes	characteristic
150	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
151	M₁₆	—	146	4	no	—
152	C₂×C₈	—	2	4	no	—
153	C₂×C₈	—	73	4	no	—
154	C₂×C₈	C₂²×D₇₃	1	2	yes	—
155	C₂×C₈	C₂²×D₇₃	1	2	yes	—
156	C₄⋊C₄	—	73	2	no	—
157	C₄⋊C₄	—	73	2	no	—
158	C₄⋊C₄	—	73	1	no	—
159	C₄⋊C₄	—	73	1	no	—
160	C₂²⋊C₄	—	73	2	no	—
161	C₂²⋊C₄	—	73	2	no	—
162	C₂²⋊C₄	—	73	2	no	—
163	C₄²	—	73	2	no	—
164	C₄²	—	73	2	no	—
165	C₄²	—	73	2	no	—
166	C₄²	C₄×D₇₃	1	1	yes	characteristic
167	C₄²	C₄×D₇₃	1	1	yes	characteristic
168	C₄²	C₂²×D₇₃	1	1	yes	characteristic
169	C₄²	C₂²×D₇₃	1	1	yes	characteristic
170	C₂³	—	73	2	no	—
171	C₂³	C₂×C₄×D₇₃	1	1	yes	characteristic
172	Q₈	—	146	2	no	—
173	D₄	—	146	4	no	—
174	D₄	—	146	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	4	1	no	—
177	C₂×C₄	—	292	2	no	—
178	C₂×C₄	—	73	2	no	—
179	C₂×C₄	—	73	2	no	—
180	C₂×C₄	—	73	2	no	—
181	C₂×C₄	—	146	4	no	—
182	C₂×C₄	—	146	2	no	—
183	C₂×C₄	—	146	2	no	—
184	C₂×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
185	C₂×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
186	C₂×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
187	C₂×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
188	C₂×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
189	C₂×C₄	#11924027	1	2	yes	—
190	C₂×C₄	#11924029	1	2	yes	—
191	C₂×C₄	C₂³×D₇₃	1	1	yes	characteristic
192	C₈	—	2	2	no	—
193	C₈	—	2	2	no	—
194	C₈	—	146	4	no	—
195	C₂²	—	2	1	no	—
196	C₂²	—	2	1	no	—
197	C₂²	—	146	2	no	—
198	C₂²	—	146	2	no	—
199	C₂²	—	146	2	no	—
200	C₂²	#14556549	1	2	yes	—
201	C₂²	C₂²×C₄×D₇₃	1	1	yes	characteristic
202	C₄	—	2	1	no	—
203	C₄	—	2	1	no	—
204	C₄	—	2	1	no	—
205	C₄	—	146	2	no	—
206	C₄	—	146	2	no	—
207	C₄	—	146	2	no	—
208	C₄	#14556549	1	2	yes	—
209	C₄	#14556597	1	2	yes	—
210	C₄	C₂×#11924027	1	1	yes	characteristic
211	C₄	C₂×#11924029	1	1	yes	characteristic
212	C₂	—	2	2	no	—
213	C₂	—	292	2	no	—
214	C₂	C₂×#14556549	1	1	yes	characteristic
215	C₂	C₂×#14556597	1	1	yes	characteristic
216	C₂	#17989353	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₇₃
commutator, socle	C₂²×C₇₃	C₂³×C₄
Fitting	C₂×C₄⋊C₈×C₇₃	C₂
hypercenter	C₂×C₄⋊C₈	D₇₃
hypocenter	C₇₃	#2925
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. #17988647	C₂	semidirect
3. #17988643	C₂	semidirect
4. #17988639	C₂	semidirect
5. #17988611	C₂	semidirect
7. #17988542	C₂	semidirect
8. C₂×C₄⋊C₈×C₇₃	C₂	semidirect
109. C₇₃	#2925	semidirect
115. C₂×C₄⋊C₈	D₇₃	semidirect