Finite Group Databasebeta

Subgroups — #22626211

Order 9344.

270 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 52 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	#17989568	C₂	1	2	yes	maximal, complemented
2	#17989492	C₂	1	2	yes	maximal, complemented
3	#17989491	C₂	1	2	yes	maximal, complemented
4	#17989155	C₂	1	1	yes	maximal, characteristic
5	#14556670	—	2	2	no	—
6	#14556650	—	2	2	no	—
7	#14556650	C₂²	1	4	yes	complemented
8	C₂×#11924029	C₂²	1	1	yes	characteristic
9	C₂×D₄×D₇₃	C₄	1	1	yes	characteristic, complemented
10	#14556588	C₂²	1	1	yes	characteristic
11	#14556587	C₂²	1	1	yes	characteristic
12	#14556580	C₄	1	1	yes	characteristic, complemented
13	#14556564	—	2	2	no	—
14	#14556561	—	2	2	no	—
15	#14556548	C₄	1	1	yes	characteristic, complemented
16	#124×C₇₃	C₄	1	1	yes	characteristic, complemented
17	C₄⋊#64408	—	4	2	no	—
18	C₂²⋊#64408	—	2	4	no	—
19	C₄×#64408	—	4	2	no	—
20	C₂³×D₇₃	—	2	1	no	—
21	#11924029	—	4	2	no	—
22	C₂×D₂₉₂	—	2	2	no	—
23	C₂×D₂₉₂	C₂×C₄	1	1	yes	characteristic
24	C₂×D₂₉₂	D₄	1	2	yes	—
25	C₂×D₂₉₂	C₂³	1	1	yes	characteristic
26	C₂×C₂²⋊D₇₃	—	2	2	no	—
27	C₂×C₂²⋊D₇₃	—	2	1	no	—
28	D₄×D₇₃	—	4	2	no	—
29	C₂×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×D₇₃	D₄	1	2	yes	—
31	#11924007	—	2	1	no	—
32	#11924005	—	2	1	no	—
33	#11924005	—	2	2	no	—
34	C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
35	C₂×Q₈×C₇₃	C₂×C₄	1	1	yes	characteristic
36	C₂×D₄×C₇₃	C₂×C₄	1	1	yes	characteristic
37	C₂²⋊C₄×C₇₃	—	2	2	no	—
38	C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
39	C₂³×C₇₃	—	2	1	no	—
40	C₂²×D₇₃	—	2	1	no	—
41	C₂²×D₇₃	—	4	2	no	—
42	C₂²×D₇₃	—	8	2	no	—
43	C₂²×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
44	C₂²×D₇₃	C₂×D₄	1	2	yes	—
45	C₂×#64408	—	4	4	no	—
46	Q₈×C₇₃	—	4	1	no	—
47	D₄×C₇₃	—	4	1	no	—
48	C₂×C₄×C₇₃	—	2	1	no	—
49	C₂×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
50	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
51	C₂²⋊D₇₃	—	4	2	no	—
52	C₂²⋊D₇₃	—	8	2	no	—
53	C₂×Dic₇₃	—	2	1	no	—
54	C₂×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
55	D₂₉₂	—	4	2	no	—
56	D₂₉₂	—	4	1	no	—
57	D₂₉₂	—	4	1	no	—
58	D₂₉₂	—	8	2	no	—
59	C₄×D₇₃	—	2	2	no	—
60	C₄×D₇₃	—	2	2	no	—
61	C₄×D₇₃	—	2	2	no	—
62	C₂²×C₇₃	—	4	2	no	—
63	C₂²×C₇₃	—	4	1	no	—
64	C₂²×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₂×D₇₃	—	2	2	no	—
66	C₂×D₇₃	—	2	1	no	—
67	C₂×D₇₃	—	4	2	no	—
68	C₂×D₇₃	—	4	1	no	—
69	C₂×D₇₃	—	4	2	no	—
70	C₂×D₇₃	—	4	1	no	—
71	C₂×D₇₃	—	8	4	no	—
72	#64408	—	8	4	no	—
73	C₄×C₇₃	—	2	1	no	—
74	C₄×C₇₃	—	2	2	no	—
75	C₄×C₇₃	—	4	1	no	—
76	Dic₇₃	—	4	1	no	—
77	Dic₇₃	#99	1	2	yes	—
78	C₂×C₇₃	—	2	1	no	—
79	C₂×C₇₃	—	8	1	no	—
80	C₂×C₇₃	C₂×#99	1	1	yes	characteristic
81	D₇₃	—	4	2	no	—
82	D₇₃	—	4	1	no	—
83	D₇₃	—	8	1	no	—
84	#2128	—	73	1	no	maximal, Sylow
85	C₇₃	#2128	1	1	yes	characteristic, complemented, Sylow
86	#535	—	73	1	no	—
87	#410	—	73	2	no	—
88	#353	—	73	2	no	—
89	#353	—	73	2	no	—
90	C₂×#41	—	73	1	no	—
91	C₂²×D₄	—	73	1	no	—
92	#127	—	73	1	no	—
93	#127	—	73	1	no	—
94	#124	—	73	1	no	—
95	#124	#64408	1	1	yes	characteristic, complemented
96	#121	—	146	2	no	—
97	#120	—	146	2	no	—
98	#117	—	73	1	no	—
99	#117	—	146	2	no	—
100	#99	—	73	4	no	—
101	#99	—	146	2	no	—
102	C₂⁴	—	146	1	no	—
103	#41	—	292	2	no	—
104	C₂×Q₈	C₂×#64408	1	1	yes	characteristic
105	C₂×D₄	—	292	2	no	—
106	C₂×D₄	—	73	2	no	—
107	C₂×D₄	—	73	1	no	—
108	C₂×D₄	—	73	1	no	—
109	C₂×D₄	—	146	2	no	—
110	C₂×D₄	—	146	1	no	—
111	C₂×D₄	—	146	2	no	—
112	C₂×D₄	C₂×#64408	1	1	yes	characteristic
113	C₂²×C₄	—	73	2	no	—
114	C₂²×C₄	—	73	1	no	—
115	C₄⋊C₄	—	292	2	no	—
116	C₄⋊C₄	—	146	1	no	—
117	C₂²⋊C₄	—	2	2	no	—
118	C₂²⋊C₄	—	146	1	no	—
119	C₂²⋊C₄	—	146	4	no	—
120	C₂²⋊C₄	—	146	2	no	—
121	C₄²	—	292	2	no	—
122	C₄²	—	73	1	no	—
123	C₄²	C₂×#64408	1	1	yes	characteristic
124	C₂³	—	2	1	no	—
125	C₂³	—	292	2	no	—
126	C₂³	—	584	2	no	—
127	C₂³	—	73	2	no	—
128	C₂³	—	73	1	no	—
129	C₂³	—	146	1	no	—
130	Q₈	—	4	1	no	—
131	D₄	—	4	1	no	—
132	D₄	—	292	1	no	—
133	D₄	—	292	2	no	—
134	D₄	—	292	1	no	—
135	D₄	—	292	2	no	—
136	D₄	—	584	2	no	—
137	D₄	—	584	2	no	—
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	292	4	no	—
140	C₂×C₄	—	73	1	no	—
141	C₂×C₄	—	146	2	no	—
142	C₂×C₄	—	146	2	no	—
143	C₂×C₄	—	146	2	no	—
144	C₂×C₄	—	146	1	no	—
145	C₂×C₄	C₂²⋊#64408	1	2	yes	—
146	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
147	C₂²	—	4	2	no	—
148	C₂²	—	4	1	no	—
149	C₂²	—	292	2	no	—
150	C₂²	—	292	1	no	—
151	C₂²	—	292	2	no	—
152	C₂²	—	292	1	no	—
153	C₂²	—	584	4	no	—
154	C₂²	—	146	2	no	—
155	C₂²	—	146	1	no	—
156	C₂²	C₂×C₂²⋊#64408	1	1	yes	characteristic
157	C₄	—	2	2	no	—
158	C₄	—	2	1	no	—
159	C₄	—	4	1	no	—
160	C₄	—	292	1	no	—
161	C₄	—	584	4	no	—
162	C₄	—	73	2	no	—
163	C₂	—	2	1	no	—
164	C₂	—	8	1	no	—
165	C₂	—	292	2	no	—
166	C₂	—	292	1	no	—
167	C₂	—	584	1	no	—
168	C₂	D₇₃⋊#99	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₇₃⋊#99
commutator	C₂×C₄×C₇₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#64408
Fitting	#124×C₇₃	C₄
socle	C₂×C₇₃	C₂×#99
hypercenter	#124	#64408
hypocenter	C₇₃	#2128
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×Q₈ ◁ #124

Complements

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

N	H ≅ G/N	type
1. #17989568	C₂	semidirect
2. #17989492	C₂	semidirect
3. #17989491	C₂	semidirect
7. #14556650	C₂²	semidirect
9. C₂×D₄×D₇₃	C₄	semidirect
12. #14556580	C₄	semidirect
15. #14556548	C₄	semidirect
16. #124×C₇₃	C₄	semidirect
85. C₇₃	#2128	semidirect
95. #124	#64408	semidirect