Finite Group Databasebeta

Subgroups — #22625813

Order 9344.

260 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 72 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17989597	C₂	1	2	yes	maximal, complemented
2	#17989566	C₂	1	2	yes	maximal
3	#17989455	C₂	1	2	yes	maximal, complemented
4	C₂×#14556597	C₂	1	1	yes	maximal, characteristic
5	#14556670	C₂²	1	4	yes	—
6	#14556660	—	2	2	no	—
7	#14556659	—	2	2	no	—
8	#14556650	—	2	2	no	—
9	#14556649	—	2	2	no	—
10	C₂×#11924028	C₂²	1	1	yes	characteristic
11	#14556597	—	2	2	no	—
12	#14556597	—	2	2	no	—
13	C₂×#11924012	C₄	1	1	yes	characteristic, complemented
14	C₂×#11924011	C₄	1	1	yes	characteristic, complemented
15	C₂×#11924011	C₂²	1	1	yes	characteristic
16	C₂×C₈×D₇₃	C₄	1	1	yes	characteristic, complemented
17	C₂×C₈×D₇₃	C₂²	1	1	yes	characteristic
18	C₂²×C₈×C₇₃	C₄	1	1	yes	characteristic, complemented
19	C₄⋊#64408	—	2	4	no	—
20	C₂²⋊#64408	—	2	4	no	—
21	C₄×#64408	—	2	4	no	—
22	#11924028	—	2	2	no	—
23	#11924028	—	2	2	no	—
24	C₂×Dic₁₄₆	Q₈	1	1	yes	characteristic
25	C₂×D₂₉₂	D₄	1	1	yes	characteristic
26	C₂×C₂²⋊D₇₃	D₄	1	1	yes	characteristic
27	C₂×C₂²⋊D₇₃	Q₈	1	1	yes	characteristic
28	C₂×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×D₇₃	C₂³	1	1	yes	characteristic
30	#11924012	—	2	2	no	—
31	#11924011	—	2	2	no	—
32	#11924011	—	2	2	no	—
33	C₂×#10596831	C₂×C₄	1	1	yes	characteristic
34	C₂×#10596831	C₂×C₄	1	1	yes	characteristic
35	C₈×D₇₃	—	2	1	no	—
36	C₈×D₇₃	—	2	1	no	—
37	C₈×D₇₃	D₄	1	2	yes	—
38	C₈×D₇₃	D₄	1	2	yes	—
39	C₂²×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₈×C₇₃	—	2	1	no	—
41	C₂×C₈×C₇₃	—	2	1	no	—
42	C₂×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
44	C₂³×C₇₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂²×D₇₃	C₄⋊C₄	1	1	yes	characteristic
46	C₂²×D₇₃	C₂×D₄	1	1	yes	characteristic
47	C₂×#64408	—	4	4	no	—
48	C₂×C₄×C₇₃	—	2	1	no	—
49	C₂×C₄×C₇₃	—	2	1	no	—
50	C₂×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
52	C₂²⋊D₇₃	—	2	2	no	—
53	C₂²⋊D₇₃	—	2	2	no	—
54	C₂×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
55	C₂×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
56	D₂₉₂	—	2	2	no	—
57	C₄×D₇₃	—	2	1	no	—
58	C₄×D₇₃	—	2	1	no	—
59	C₄×D₇₃	C₂×D₄	1	2	yes	—
60	C₄×D₇₃	#41	1	2	yes	—
61	Dic₁₄₆	—	2	2	no	—
62	C₈×C₇₃	—	2	1	no	—
63	C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
65	#10596831	—	2	1	no	—
66	#10596831	C₂²⋊C₄	1	2	yes	—
67	C₂²×C₇₃	—	2	1	no	—
68	C₂²×C₇₃	—	4	1	no	—
69	C₂²×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×D₇₃	—	2	1	no	—
71	C₂×D₇₃	—	4	1	no	—
72	C₂×D₇₃	—	4	1	no	—
73	C₂×D₇₃	#121	1	2	yes	—
74	#64408	—	8	4	no	—
75	C₄×C₇₃	—	2	1	no	—
76	C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₄×C₇₃	#117	1	1	yes	characteristic
78	Dic₇₃	—	2	1	no	—
79	Dic₇₃	C₂²⋊Q₈	1	2	yes	—
80	C₂×C₇₃	—	2	1	no	—
81	C₂×C₇₃	—	4	1	no	—
82	C₂×C₇₃	#380	1	1	yes	characteristic
83	D₇₃	—	2	2	no	—
84	D₇₃	—	4	1	no	—
85	#1812	—	73	1	no	maximal, Sylow
86	C₇₃	#1812	1	1	yes	characteristic, complemented, Sylow
87	C₂×#131	—	73	1	no	—
88	#427	—	73	2	no	—
89	#410	—	73	2	no	—
90	H(3,ℤ/4)	—	73	2	no	—
91	C₂×#41	—	73	1	no	—
92	#131	—	146	2	no	—
93	#131	—	146	2	no	—
94	C₂×M₁₆	—	73	1	no	—
95	C₂×M₁₆	—	73	1	no	—
96	C₂×M₁₆	—	73	1	no	—
97	C₂²×C₈	—	73	1	no	—
98	C₂²×C₈	—	73	1	no	—
99	C₂²×C₈	#64408	1	1	yes	characteristic, complemented
100	#117	—	73	4	no	—
101	#107	—	146	2	no	—
102	#106	—	146	2	no	—
103	#99	—	146	2	no	—
104	#99	—	146	2	no	—
105	#41	—	146	2	no	—
106	#41	—	146	2	no	—
107	C₂×Q₈	—	73	1	no	—
108	C₂×D₄	—	73	1	no	—
109	C₂×D₄	—	73	1	no	—
110	C₂×D₄	—	73	1	no	—
111	C₂²×C₄	—	73	1	no	—
112	C₂²×C₄	—	73	1	no	—
113	C₂²×C₄	C₂×#64408	1	1	yes	characteristic
114	M₁₆	—	146	2	no	—
115	M₁₆	—	146	2	no	—
116	M₁₆	—	146	2	no	—
117	C₂×C₈	—	2	1	no	—
118	C₂×C₈	—	2	1	no	—
119	C₂×C₈	—	73	1	no	—
120	C₂×C₈	—	73	2	no	—
121	C₂×C₈	—	73	2	no	—
122	C₂×C₈	—	73	1	no	—
123	C₂×C₈	—	146	1	no	—
124	C₂×C₈	—	146	1	no	—
125	C₂×C₈	C₂×#64408	1	1	yes	characteristic
126	C₂×C₈	C₂×#64408	1	1	yes	characteristic
127	C₄⋊C₄	—	146	4	no	—
128	C₂²⋊C₄	—	146	4	no	—
129	C₄²	—	146	4	no	—
130	C₂³	—	73	1	no	—
131	C₂³	—	73	1	no	—
132	C₂³	C₄⋊#64408	1	1	yes	characteristic
133	Q₈	—	146	2	no	—
134	D₄	—	146	2	no	—
135	D₄	—	146	2	no	—
136	D₄	—	146	2	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	292	4	no	—
140	C₂×C₄	—	73	1	no	—
141	C₂×C₄	—	73	2	no	—
142	C₂×C₄	—	73	2	no	—
143	C₂×C₄	—	73	1	no	—
144	C₂×C₄	—	146	1	no	—
145	C₂×C₄	—	146	1	no	—
146	C₂×C₄	C₄⋊#64408	1	1	yes	characteristic
147	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
148	C₈	—	2	1	no	—
149	C₈	—	73	2	no	—
150	C₈	—	146	1	no	—
151	C₈	C₂²⋊#64408	1	1	yes	characteristic
152	C₈	C₂²⋊#64408	1	1	yes	characteristic
153	C₂²	—	2	1	no	—
154	C₂²	—	4	1	no	—
155	C₂²	—	292	1	no	—
156	C₂²	—	292	1	no	—
157	C₂²	—	73	2	no	—
158	C₂²	—	146	1	no	—
159	C₂²	C₂×C₄⋊#64408	1	1	yes	characteristic
160	C₄	—	2	1	no	—
161	C₄	—	584	4	no	—
162	C₄	—	73	2	no	—
163	C₄	—	146	1	no	—
164	C₄	C₂×C₂²⋊#64408	1	1	yes	characteristic
165	C₄	#14556670	1	1	yes	characteristic
166	C₂	—	2	1	no	—
167	C₂	—	4	1	no	—
168	C₂	—	292	1	no	—
169	C₂	—	146	2	no	—
170	C₂	#17989529	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#14556670
commutator	C₂×C₄×C₇₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#64408
Fitting	C₂²×C₈×C₇₃	C₄
socle	C₂×C₇₃	#380
hypercenter	C₂²×C₈	#64408
hypocenter	C₇₃	#1812
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
1. #17989597	C₂	semidirect
3. #17989455	C₂	semidirect
13. C₂×#11924012	C₄	semidirect
14. C₂×#11924011	C₄	semidirect
16. C₂×C₈×D₇₃	C₄	semidirect
18. C₂²×C₈×C₇₃	C₄	semidirect
86. C₇₃	#1812	semidirect
99. C₂²×C₈	#64408	semidirect