Subgroups — #22618874

Order 9344.

356 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17989144	C₂	1	1	yes	maximal, characteristic
2	#17988716	C₂	1	4	yes	maximal, complemented
3	#17988714	C₂	1	2	yes	maximal, complemented
4	#17988711	C₂	1	2	yes	maximal, complemented
5	#17988613	C₂	1	2	yes	maximal
6	#17988542	C₂	1	2	yes	maximal
7	C₄×#11924012	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊C₈×C₇₃	C₂	1	1	yes	maximal, characteristic
9	C₂×#11924012	C₂²	1	1	yes	characteristic
10	C₂×#11924012	C₂²	1	1	yes	characteristic
11	#14556591	C₄	1	1	yes	characteristic
12	#14556591	C₄	1	1	yes	characteristic
13	#14556583	C₄	1	2	yes	—
14	#14556570	C₄	1	2	yes	—
15	#14556570	C₄	1	2	yes	—
16	C₄×Dic₁₄₆	C₂²	1	2	yes	—
17	C₄×Dic₁₄₆	C₂²	1	2	yes	—
18	#14556550	C₂²	1	2	yes	—
19	#14556520	C₂²	1	4	yes	—
20	#14556520	C₂²	1	4	yes	—
21	#14556502	C₂²	1	2	yes	—
22	#14556501	C₂²	1	4	yes	—
23	C₄×#10596831	C₂²	1	2	yes	—
24	C₈×Dic₇₃	C₂²	1	4	yes	—
25	C₂²×C₈×C₇₃	C₂²	1	2	yes	—
26	C₂×C₄²×C₇₃	C₂²	1	1	yes	characteristic
27	C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
28	C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
29	C₄⋊C₈×C₇₃	C₂²	1	2	yes	—
30	C₂×Dic₁₄₆	C₂×C₄	1	2	yes	—
31	C₂×Dic₁₄₆	C₂×C₄	1	2	yes	—
32	#11924012	—	2	2	no	—
33	#11924012	Q₈	1	4	yes	complemented
34	C₂×#10596831	C₂³	1	2	yes	—
35	C₂×#10596831	C₂³	1	2	yes	—
36	C₄⋊Dic₇₃	C₂×C₄	1	2	yes	—
37	C₄⋊Dic₇₃	C₂×C₄	1	2	yes	—
38	C₄⋊Dic₇₃	C₂×C₄	1	2	yes	—
39	C₄⋊Dic₇₃	C₂×C₄	1	2	yes	—
40	#11924007	C₂×C₄	1	4	yes	—
41	#11924007	C₂×C₄	1	4	yes	—
42	C₂²⋊Dic₇₃	C₂×C₄	1	4	yes	—
43	C₄×Dic₇₃	C₂³	1	4	yes	—
44	C₂²×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
47	C₂×C₈×C₇₃	—	2	4	no	—
48	C₂×C₈×C₇₃	C₂³	1	2	yes	—
49	C₂×C₈×C₇₃	C₂³	1	2	yes	—
50	C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
51	C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
52	C₄²×C₇₃	C₂³	1	1	yes	characteristic
53	C₄²×C₇₃	C₂³	1	1	yes	characteristic
54	C₂³×C₇₃	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₇₃	—	2	2	no	—
56	C₂×C₄×C₇₃	—	4	1	no	—
57	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄×C₇₃	C₂×Q₈	1	2	yes	—
63	C₂×C₄×C₇₃	#41	1	2	yes	—
64	C₂×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
65	C₂×Dic₇₃	C₂²×C₄	1	4	yes	—
66	C₂×Dic₇₃	C₂²×C₄	1	4	yes	—
67	Dic₁₄₆	—	2	4	no	—
68	Dic₁₄₆	—	4	2	no	—
69	C₈×C₇₃	—	2	2	no	—
70	C₈×C₇₃	—	2	2	no	—
71	#10596831	—	2	2	no	—
72	#10596831	C₂×Q₈	1	4	yes	—
73	C₂²×C₇₃	—	2	1	no	—
74	C₂²×C₇₃	—	2	1	no	—
75	C₂²×C₇₃	C₄×Q₈	1	2	yes	—
76	C₂²×C₇₃	C₂³×C₄	1	1	yes	characteristic
77	C₄×C₇₃	—	2	1	no	—
78	C₄×C₇₃	—	2	1	no	—
79	C₄×C₇₃	—	2	1	no	—
80	C₄×C₇₃	C₄×Q₈	1	2	yes	—
81	C₄×C₇₃	#131	1	2	yes	—
82	C₄×C₇₃	C₂²×Q₈	1	1	yes	characteristic
83	C₄×C₇₃	C₂×#41	1	1	yes	characteristic
84	Dic₇₃	—	2	4	no	—
85	Dic₇₃	—	2	4	no	—
86	C₂×C₇₃	—	2	2	no	—
87	C₂×C₇₃	C₂×C₄×Q₈	1	1	yes	characteristic
88	C₂×C₇₃	C₂×#131	1	1	yes	characteristic
89	C₂×C₇₃	#568	1	1	yes	characteristic
90	#2962	—	73	1	no	maximal, Sylow
91	C₇₃	#2962	1	1	yes	characteristic, complemented, Sylow
92	#533	—	73	1	no	—
93	#446	—	73	2	no	—
94	#446	—	73	4	no	—
95	C₈×Q₈	—	73	2	no	—
96	#424	—	73	2	no	—
97	C₂×C₄⋊C₈	D₇₃	1	1	yes	characteristic
98	#405	—	73	2	no	—
99	C₄×M₁₆	—	73	1	no	—
100	C₂×M₁₆	—	73	1	no	—
101	C₂×M₁₆	—	73	1	no	—
102	C₂²×C₈	C₂×D₇₃	1	2	yes	—
103	C₄⋊Q₈	—	73	1	no	—
104	C₄⋊Q₈	—	73	1	no	—
105	#125	—	73	2	no	—
106	C₂²⋊Q₈	—	73	2	no	—
107	C₂²⋊Q₈	—	73	2	no	—
108	C₄×Q₈	—	73	2	no	—
109	C₄×Q₈	—	73	2	no	—
110	#117	—	73	2	no	—
111	C₂×C₄²	C₂×D₇₃	1	1	yes	characteristic
112	C₄⋊C₈	—	73	4	no	—
113	C₄⋊C₈	—	73	4	no	—
114	C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
115	C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
116	C₄⋊C₈	C₂×D₇₃	1	2	yes	—
117	#97	—	73	4	no	—
118	#97	—	73	2	no	—
119	C₄×C₈	—	73	4	no	—
120	C₄×C₈	—	73	2	no	—
121	C₂×Q₈	—	73	2	no	—
122	C₂×Q₈	—	73	2	no	—
123	C₂²×C₄	C₄×D₇₃	1	1	yes	characteristic
124	C₂²×C₄	C₄×D₇₃	1	1	yes	characteristic
125	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
126	M₁₆	—	73	4	no	—
127	M₁₆	—	146	2	no	—
128	C₂×C₈	—	2	4	no	—
129	C₂×C₈	—	73	2	no	—
130	C₂×C₈	—	73	2	no	—
131	C₂×C₈	C₂²×D₇₃	1	2	yes	—
132	C₂×C₈	C₂²×D₇₃	1	2	yes	—
133	C₄⋊C₄	—	73	4	no	—
134	C₄⋊C₄	—	73	4	no	—
135	C₄⋊C₄	—	73	2	no	—
136	C₄⋊C₄	—	73	2	no	—
137	C₄⋊C₄	—	73	2	no	—
138	C₄⋊C₄	—	73	2	no	—
139	C₂²⋊C₄	—	73	4	no	—
140	C₄²	—	73	4	no	—
141	C₄²	C₄×D₇₃	1	1	yes	characteristic
142	C₄²	C₄×D₇₃	1	1	yes	characteristic
143	C₄²	C₂²×D₇₃	1	1	yes	characteristic
144	C₄²	C₂²×D₇₃	1	1	yes	characteristic
145	C₂³	C₂×C₄×D₇₃	1	1	yes	characteristic
146	Q₈	—	292	2	no	—
147	Q₈	—	146	4	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	4	1	no	—
150	C₂×C₄	—	73	4	no	—
151	C₂×C₄	—	73	4	no	—
152	C₂×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
153	C₂×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
154	C₂×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
155	C₂×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
156	C₂×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
157	C₂×C₄	Q₈×D₇₃	1	2	yes	—
158	C₂×C₄	#11924027	1	2	yes	—
159	C₂×C₄	C₂³×D₇₃	1	1	yes	characteristic
160	C₈	—	2	2	no	—
161	C₈	—	2	2	no	—
162	C₈	—	73	4	no	—
163	C₈	—	146	2	no	—
164	C₂²	—	2	1	no	—
165	C₂²	—	2	1	no	—
166	C₂²	#14556556	1	2	yes	—
167	C₂²	C₂²×C₄×D₇₃	1	1	yes	characteristic
168	C₄	—	2	1	no	—
169	C₄	—	2	1	no	—
170	C₄	—	2	1	no	—
171	C₄	—	146	4	no	—
172	C₄	—	146	4	no	—
173	C₄	#14556556	1	2	yes	—
174	C₄	#14556597	1	2	yes	—
175	C₄	C₂×Q₈×D₇₃	1	1	yes	characteristic
176	C₄	C₂×#11924027	1	1	yes	characteristic
177	C₂	—	2	2	no	—
178	C₂	C₂×#14556556	1	1	yes	characteristic
179	C₂	C₂×#14556597	1	1	yes	characteristic
180	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	#2962
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. #17988716	C₂	semidirect
3. #17988714	C₂	semidirect
4. #17988711	C₂	semidirect
33. #11924012	Q₈	semidirect
91. C₇₃	#2962	semidirect