Finite Group Databasebeta

Subgroups — #22625978

Order 9344.

314 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 66 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	#17989655	C₂	1	1	yes	maximal, characteristic, complemented
2	#17989586	C₂	1	2	yes	maximal
3	D₇₃⋊#101	C₂	1	1	yes	maximal, characteristic, complemented
4	#17989454	C₂	1	2	yes	maximal, complemented
5	#17989162	C₂	1	1	yes	maximal, characteristic
6	D₇₃⋊M₁₆	C₂²	1	2	yes	—
7	Q₈×#64408	—	2	4	no	—
8	#14556670	—	2	2	no	—
9	#14556670	C₂²	1	2	yes	—
10	#14556656	—	2	4	no	—
11	#14556652	—	2	2	no	—
12	C₂×#11924029	C₂²	1	1	yes	characteristic
13	C₂×Q₈×D₇₃	C₂²	1	1	yes	characteristic
14	#14556589	C₄	1	1	yes	characteristic, complemented
15	#14556580	C₄	1	1	yes	characteristic, complemented
16	#14556579	C₄	1	1	yes	characteristic, complemented
17	#14556576	—	2	2	no	—
18	#14556569	—	2	2	no	—
19	#14556548	C₂²	1	1	yes	characteristic
20	C₄⋊Q₈×C₇₃	C₄	1	1	yes	characteristic, complemented
21	#11924038	—	2	2	no	—
22	#11924037	—	2	2	no	—
23	#11924036	—	2	2	no	—
24	C₄⋊#64408	—	2	4	no	—
25	C₄⋊#64408	—	2	2	no	—
26	C₂²⋊#64408	—	2	2	no	—
27	C₄×#64408	—	2	4	no	—
28	C₄×#64408	D₄	1	4	yes	—
29	#11924029	—	2	2	no	—
30	#11924029	—	2	2	no	—
31	C₂×Dic₁₄₆	C₂×C₄	1	1	yes	characteristic
32	C₂×Dic₁₄₆	D₄	1	2	yes	—
33	Q₈×D₇₃	—	2	2	no	—
34	Q₈×D₇₃	—	2	2	no	—
35	C₂×D₂₉₂	—	2	1	no	—
36	C₂×D₂₉₂	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×D₇₃	—	2	1	no	—
38	C₂×C₄×D₇₃	D₄	1	2	yes	—
39	C₂×C₄×D₇₃	C₂³	1	1	yes	characteristic
40	#11924007	—	2	2	no	—
41	#11924007	—	2	1	no	—
42	#11924005	—	2	1	no	—
43	#11924005	—	2	2	no	—
44	#11924005	—	2	2	no	—
45	C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
46	C₂×Q₈×C₇₃	C₂×C₄	1	1	yes	characteristic
47	C₂×Q₈×C₇₃	C₂×C₄	1	1	yes	characteristic
48	C₄⋊C₄×C₇₃	—	2	2	no	—
49	C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
50	C₂²×D₇₃	—	2	1	no	—
51	C₂²×D₇₃	C₂×D₄	1	1	yes	characteristic
52	C₂×#64408	—	2	4	no	—
53	Q₈×C₇₃	—	2	2	no	—
54	Q₈×C₇₃	—	2	2	no	—
55	C₂×C₄×C₇₃	—	2	1	no	—
56	C₂×C₄×C₇₃	—	2	1	no	—
57	C₂×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
58	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
59	C₂×Dic₇₃	—	2	1	no	—
60	C₂×Dic₇₃	C₂²⋊C₄	1	2	yes	—
61	C₂×Dic₇₃	C₂×D₄	1	1	yes	characteristic
62	D₂₉₂	—	2	2	no	—
63	D₂₉₂	—	4	1	no	—
64	D₂₉₂	—	4	1	no	—
65	C₄×D₇₃	—	2	2	no	—
66	C₄×D₇₃	—	2	2	no	—
67	C₄×D₇₃	—	4	2	no	—
68	C₄×D₇₃	C₂×D₄	1	2	yes	—
69	C₄×D₇₃	#41	1	2	yes	—
70	Dic₁₄₆	—	2	2	no	—
71	Dic₁₄₆	—	2	4	no	—
72	#10596833	—	4	2	no	—
73	C₂²×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₂×D₇₃	—	4	1	no	—
75	C₂×D₇₃	—	4	2	no	—
76	C₂×D₇₃	—	4	1	no	—
77	C₂×D₇₃	#120	1	1	yes	characteristic
78	C₂×D₇₃	#123	1	1	yes	characteristic
79	#64408	—	4	4	no	—
80	C₄×C₇₃	—	2	2	no	—
81	C₄×C₇₃	—	4	1	no	—
82	C₄×C₇₃	—	4	1	no	—
83	C₄×C₇₃	C₄×D₄	1	2	yes	—
84	Dic₇₃	—	2	2	no	—
85	Dic₇₃	—	4	1	no	—
86	Dic₇₃	#121	1	2	yes	—
87	C₂×C₇₃	—	2	1	no	—
88	C₂×C₇₃	#386	1	1	yes	characteristic
89	D₇₃	—	2	1	no	—
90	D₇₃	—	2	1	no	—
91	D₇₃	—	8	1	no	—
92	#1912	—	73	1	no	maximal, Sylow
93	C₇₃	#1912	1	1	yes	characteristic, complemented, Sylow
94	#536	—	73	1	no	—
95	#519	—	73	1	no	—
96	#421	—	73	2	no	—
97	C₂×#101	—	73	1	no	—
98	H(3,ℤ/4)	—	73	2	no	—
99	C₂×#41	—	73	1	no	—
100	C₂²×Q₈	—	73	1	no	—
101	C₂×M₁₆	—	73	2	no	—
102	C₄⋊Q₈	—	73	1	no	—
103	C₄⋊Q₈	#64408	1	1	yes	characteristic, complemented
104	#124	—	73	1	no	—
105	#124	—	73	1	no	—
106	#123	—	146	2	no	—
107	C₂²⋊Q₈	—	146	2	no	—
108	C₄×Q₈	—	146	4	no	—
109	#117	—	73	2	no	—
110	#117	—	73	1	no	—
111	#117	—	146	2	no	—
112	#104	—	146	4	no	—
113	#101	—	146	2	no	—
114	#41	—	146	2	no	—
115	#41	—	146	2	no	—
116	C₂×Q₈	—	73	2	no	—
117	C₂×Q₈	—	73	1	no	—
118	C₂×Q₈	—	146	2	no	—
119	C₂×Q₈	—	146	2	no	—
120	C₂×Q₈	C₂×#64408	1	1	yes	characteristic
121	C₂×Q₈	C₂×#64408	1	1	yes	characteristic
122	C₂×D₄	—	73	1	no	—
123	C₂×D₄	—	146	1	no	—
124	C₂²×C₄	—	73	1	no	—
125	C₂²×C₄	—	73	2	no	—
126	C₂²×C₄	—	146	1	no	—
127	M₁₆	—	146	2	no	—
128	M₁₆	—	146	2	no	—
129	C₂×C₈	—	146	2	no	—
130	C₄⋊C₄	—	2	2	no	—
131	C₄⋊C₄	—	146	1	no	—
132	C₄⋊C₄	—	146	2	no	—
133	C₄⋊C₄	—	146	2	no	—
134	C₄⋊C₄	—	146	4	no	—
135	C₂²⋊C₄	—	146	2	no	—
136	C₂²⋊C₄	—	146	1	no	—
137	C₂²⋊C₄	—	146	2	no	—
138	C₂²⋊C₄	—	146	2	no	—
139	C₄²	—	73	4	no	—
140	C₄²	—	73	1	no	—
141	C₄²	—	146	4	no	—
142	C₄²	C₂×#64408	1	1	yes	characteristic
143	C₂³	—	73	1	no	—
144	C₂³	—	146	1	no	—
145	Q₈	—	2	2	no	—
146	Q₈	—	2	2	no	—
147	Q₈	—	146	4	no	—
148	Q₈	—	146	2	no	—
149	D₄	—	292	1	no	—
150	D₄	—	292	1	no	—
151	D₄	—	146	2	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	292	2	no	—
155	C₂×C₄	—	73	1	no	—
156	C₂×C₄	—	73	2	no	—
157	C₂×C₄	—	73	2	no	—
158	C₂×C₄	—	73	2	no	—
159	C₂×C₄	—	146	2	no	—
160	C₂×C₄	—	146	2	no	—
161	C₂×C₄	—	146	4	no	—
162	C₂×C₄	—	146	1	no	—
163	C₂×C₄	C₂²⋊#64408	1	2	yes	—
164	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
165	C₈	—	292	2	no	—
166	C₂²	—	292	2	no	—
167	C₂²	—	292	1	no	—
168	C₂²	—	292	1	no	—
169	C₂²	—	73	1	no	—
170	C₂²	—	73	1	no	—
171	C₂²	C₂×C₂²⋊#64408	1	1	yes	characteristic
172	C₄	—	2	2	no	—
173	C₄	—	4	1	no	—
174	C₄	—	4	1	no	—
175	C₄	—	292	1	no	—
176	C₄	—	292	4	no	—
177	C₄	—	73	2	no	—
178	C₄	—	146	2	no	—
179	C₄	D₄×#64408	1	2	yes	—
180	C₂	—	2	1	no	—
181	C₂	—	584	1	no	—
182	C₂	—	146	1	no	—
183	C₂	—	146	1	no	—
184	C₂	#17989535	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#17989535
commutator	C₂×C₄×C₇₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#64408
Fitting	C₄⋊Q₈×C₇₃	C₄
socle	C₂×C₇₃	#386
hypercenter	C₄⋊Q₈	#64408
hypocenter	C₇₃	#1912
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. #17989655	C₂	semidirect
3. D₇₃⋊#101	C₂	semidirect
4. #17989454	C₂	semidirect
14. #14556589	C₄	semidirect
15. #14556580	C₄	semidirect
16. #14556579	C₄	semidirect
20. C₄⋊Q₈×C₇₃	C₄	semidirect
93. C₇₃	#1912	semidirect
103. C₄⋊Q₈	#64408	semidirect