Finite Group Databasebeta

Subgroups — #34034471

Order 18688.

184 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 71 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	#22627165	C₂	1	1	yes	maximal, characteristic, complemented
2	#22627164	C₂	1	1	yes	maximal, characteristic, complemented
3	#22627032	C₂	1	1	yes	maximal, characteristic, complemented
4	#22627028	C₂	1	1	yes	maximal, characteristic, complemented
5	#22627027	C₂	1	1	yes	maximal, characteristic, complemented
6	#22627016	C₂	1	1	yes	maximal, characteristic, complemented
7	#22626319	C₂	1	1	yes	maximal, characteristic
8	#17989696	—	2	1	no	—
9	#17989695	—	2	1	no	—
10	#17989695	C₂²	1	1	yes	characteristic
11	#17989695	C₂²	1	1	yes	characteristic
12	#17989694	—	2	1	no	—
13	C₂×#14556681	—	2	1	no	—
14	C₂×#14556681	C₂²	1	1	yes	characteristic
15	C₂×#14556681	C₂²	1	1	yes	characteristic
16	#17989646	C₂²	1	1	yes	characteristic
17	#17989645	C₂²	1	1	yes	characteristic
18	#17989510	C₄	1	1	yes	characteristic
19	#17989486	C₄	1	1	yes	characteristic
20	#17989485	C₄	1	1	yes	characteristic
21	C₄×#11924031	C₂²	1	1	yes	characteristic
22	#17988701	C₄	1	1	yes	characteristic
23	#14556681	—	2	1	no	—
24	#14556681	—	2	1	no	—
25	#14556681	—	2	1	no	—
26	#14556681	D₄	1	1	yes	characteristic
27	#14556681	D₄	1	1	yes	characteristic
28	#14556666	—	2	1	no	—
29	#14556665	—	2	1	no	—
30	#14556665	C₂×C₄	1	1	yes	characteristic
31	#14556665	C₂×C₄	1	1	yes	characteristic
32	#14556664	—	2	1	no	—
33	C₂×#11924031	—	2	1	no	—
34	C₂×#11924031	C₂×C₄	1	1	yes	characteristic
35	C₂×#11924031	C₂³	1	1	yes	characteristic
36	#14556611	C₈	1	1	yes	characteristic
37	#14556599	C₂×C₄	1	1	yes	characteristic
38	#14556598	C₂×C₄	1	1	yes	characteristic
39	#14556526	C₈	1	1	yes	characteristic
40	#14556519	C₈	1	1	yes	characteristic
41	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
42	#104×C₇₃	C₈	1	1	yes	characteristic
43	#11924031	—	2	1	no	—
44	#11924031	—	2	1	no	—
45	#11924031	—	2	1	no	—
46	#11924031	C₂×D₄	1	1	yes	characteristic
47	#11924031	#41	1	1	yes	characteristic
48	#11924028	C₂×C₈	1	1	yes	characteristic
49	#11924018	C₁₆	1	1	yes	characteristic
50	#11924016	C₁₆	1	1	yes	characteristic
51	#11924015	C₁₆	1	1	yes	characteristic
52	#11924013	C₁₆	1	1	yes	characteristic
53	#11924012	—	2	1	no	—
54	#11924012	C₂×C₈	1	1	yes	characteristic
55	#11924011	—	2	1	no	—
56	C₂×#10596831	—	2	1	no	—
57	C₂×#10596831	C₂×C₈	1	1	yes	characteristic
58	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
59	C₈×D₇₃	—	2	1	no	—
60	#41×C₇₃	C₂×C₈	1	1	yes	characteristic
61	M₁₆×C₇₃	C₂×C₈	1	1	yes	characteristic
62	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
63	Q₈×C₇₃	C₂×C₁₆	1	1	yes	characteristic
64	D₄×C₇₃	—	2	1	no	—
65	D₄×C₇₃	C₂×C₁₆	1	1	yes	characteristic
66	C₂×C₄×C₇₃	—	2	1	no	—
67	C₂×C₄×C₇₃	—	2	1	no	—
68	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
69	C₂²⋊D₇₃	—	2	1	no	—
70	D₂₉₂	C₂×C₁₆	1	1	yes	characteristic
71	C₄×D₇₃	—	2	1	no	—
72	Dic₁₄₆	C₂×C₁₆	1	1	yes	characteristic
73	C₈×C₇₃	—	2	1	no	—
74	#10596831	—	2	1	no	—
75	#10596831	C₂×C₁₆	1	1	yes	characteristic
76	#10596831	C₂×C₁₆	1	1	yes	characteristic
77	#10596831	C₄×D₄	1	1	yes	characteristic
78	#10596831	#131	1	1	yes	characteristic
79	C₂²×C₇₃	—	2	1	no	—
80	C₂²×C₇₃	#504	1	1	yes	characteristic
81	C₂×D₇₃	—	2	1	no	—
82	C₄×C₇₃	—	2	1	no	—
83	C₄×C₇₃	—	4	1	no	—
84	C₄×C₇₃	C₈×D₄	1	1	yes	characteristic
85	C₄×C₇₃	C₂²×C₁₆	1	1	yes	characteristic
86	Dic₇₃	—	2	1	no	—
87	#13648	—	73	1	no	maximal, Sylow
88	C₂×C₇₃	—	2	1	no	—
89	C₂×C₇₃	—	4	1	no	—
90	C₂×C₇₃	D₄×C₁₆	1	1	yes	characteristic
91	D₇₃	—	4	1	no	—
92	#2258	—	73	1	no	—
93	#2258	—	73	1	no	—
94	#2170	—	73	1	no	—
95	#1422	—	73	1	no	—
96	#1401	—	73	1	no	—
97	#1401	—	73	1	no	—
98	#1397	—	73	1	no	—
99	C₇₃	#13648	1	1	yes	characteristic, complemented, Sylow
100	#504	—	73	1	no	—
101	#504	—	73	1	no	—
102	#443	—	73	1	no	—
103	M₆₄	—	73	1	no	—
104	M₆₄	—	73	1	no	—
105	M₆₄	—	146	1	no	—
106	M₆₄	—	146	1	no	—
107	C₂×C₃₂	—	73	1	no	—
108	C₂×C₃₂	—	73	1	no	—
109	C₂×C₃₂	—	146	1	no	—
110	C₂×C₃₂	—	146	1	no	—
111	#364	—	73	1	no	—
112	#350	—	73	1	no	—
113	#350	—	73	1	no	—
114	C₄×C₁₆	—	73	1	no	—
115	#135	—	73	1	no	—
116	#131	—	73	1	no	—
117	#131	—	73	1	no	—
118	M₃₂	—	73	1	no	—
119	M₃₂	—	73	1	no	—
120	M₃₂	—	146	1	no	—
121	M₃₂	—	146	1	no	—
122	C₂×C₁₆	—	73	1	no	—
123	C₂×C₁₆	—	73	1	no	—
124	C₂×C₁₆	—	146	1	no	—
125	C₂×C₁₆	—	146	1	no	—
126	#108	—	73	1	no	—
127	#104	—	73	1	no	—
128	#104	#10596842	1	1	yes	characteristic
129	C₄×C₈	—	73	1	no	—
130	C₃₂	—	73	1	no	—
131	C₃₂	—	73	1	no	—
132	C₃₂	—	146	1	no	—
133	C₃₂	—	146	1	no	—
134	C₃₂	—	146	1	no	—
135	#41	—	73	1	no	—
136	#41	C₂×#10596842	1	1	yes	characteristic
137	Q₁₆	—	73	1	no	—
138	SD₁₆	—	73	1	no	—
139	SD₁₆	—	73	1	no	—
140	D₈	—	73	1	no	—
141	M₁₆	—	73	1	no	—
142	M₁₆	—	146	1	no	—
143	M₁₆	—	146	1	no	—
144	M₁₆	C₂×#10596842	1	1	yes	characteristic
145	C₂×C₈	—	73	1	no	—
146	C₂×C₈	—	73	1	no	—
147	C₂×C₈	—	146	1	no	—
148	C₂×C₈	—	146	1	no	—
149	C₄²	C₂×#10596842	1	1	yes	characteristic
150	C₁₆	—	73	1	no	—
151	C₁₆	—	73	1	no	—
152	C₁₆	—	146	1	no	—
153	C₁₆	—	146	1	no	—
154	C₁₆	—	146	1	no	—
155	Q₈	—	73	1	no	—
156	Q₈	#14556686	1	1	yes	characteristic
157	D₄	—	2	1	no	—
158	D₄	—	73	1	no	—
159	D₄	—	146	1	no	—
160	D₄	#14556686	1	1	yes	characteristic
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	146	1	no	—
164	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
165	C₈	—	2	1	no	—
166	C₈	—	73	1	no	—
167	C₈	—	73	1	no	—
168	C₈	—	73	1	no	—
169	C₈	—	73	1	no	—
170	C₈	—	146	1	no	—
171	C₂²	—	2	1	no	—
172	C₂²	—	146	1	no	—
173	C₂²	#17989708	1	1	yes	characteristic
174	C₄	—	2	1	no	—
175	C₄	—	4	1	no	—
176	C₄	—	146	1	no	—
177	C₄	D₄×#10596842	1	1	yes	characteristic
178	C₄	C₂×#14556686	1	1	yes	characteristic
179	C₂	—	2	1	no	—
180	C₂	—	4	1	no	—
181	C₂	—	292	1	no	—
182	C₂	#22627139	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄×#10596842
commutator	C₄×C₇₃	C₂²×C₁₆
Frattini	C₂×C₄	C₂²×#10596842
Fitting	#104×C₇₃	C₈
socle	C₂×C₇₃	D₄×C₁₆
prime radical	#14556611	C₈
cosocle	C₂×#11924031	C₂³
hypercenter	#104	#10596842
hypocenter	C₇₃	#13648
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #104

Complements

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

N	H ≅ G/N	type
1. #22627165	C₂	semidirect
2. #22627164	C₂	semidirect
3. #22627032	C₂	semidirect
4. #22627028	C₂	semidirect
5. #22627027	C₂	semidirect
6. #22627016	C₂	semidirect
99. C₇₃	#13648	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#22627165	2
#22627164	2
#22627032	2
#22627028	2
#22627027	2
#22627016	2
#22626319	2
#13648	73