Subgroups — #41×#11924003 #33239503

Order 18688.

602 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#11924003	C₂	1	2	yes	maximal, complemented
2	D₄×#11924003	C₂	1	6	yes	maximal, complemented
3	#22611730	C₂	1	3	yes	maximal, complemented
4	C₂×C₄×#11924003	C₂	1	3	yes	maximal, complemented
5	C₈×#41×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11924003	C₂²	1	6	yes	—
7	C₄⋊#11924003	C₂²	1	6	yes	—
8	C₂²⋊#11924003	C₂²	1	6	yes	—
9	C₄×#11924003	C₂²	1	3	yes	—
10	C₄×#11924003	C₂²	1	6	yes	—
11	C₄×#11924003	C₂²	1	1	yes	characteristic
12	C₄×#41×C₇₃	C₄	1	1	yes	characteristic
13	C₈×Q₈×C₇₃	C₄	1	1	yes	characteristic
14	C₈×Q₈×C₇₃	C₂²	1	1	yes	characteristic
15	C₈×D₄×C₇₃	C₄	1	3	yes	—
16	C₈×D₄×C₇₃	C₂²	1	3	yes	—
17	#431×C₇₃	C₄	1	3	yes	—
18	C₂×C₄×C₈×C₇₃	C₂²	1	3	yes	—
19	C₂×#11924003	—	2	12	no	—
20	C₂×#11924003	C₂³	1	6	yes	—
21	C₂×#11924003	C₂³	1	2	yes	—
22	C₂×#41×C₇₃	C₈	1	1	yes	characteristic
23	C₂²×C₈×C₇₃	C₂×C₄	1	3	yes	—
24	C₂²×C₈×C₇₃	C₂³	1	3	yes	—
25	C₄×Q₈×C₇₃	C₈	1	1	yes	characteristic
26	C₄×Q₈×C₇₃	C₂×C₄	1	1	yes	characteristic
27	C₄×D₄×C₇₃	C₈	1	3	yes	—
28	C₄×D₄×C₇₃	C₂×C₄	1	3	yes	—
29	#117×C₇₃	C₈	1	3	yes	—
30	C₂×C₄²×C₇₃	C₂×C₄	1	3	yes	—
31	C₄⋊C₈×C₇₃	C₂×C₄	1	3	yes	—
32	C₄⋊C₈×C₇₃	C₂×C₄	1	3	yes	—
33	C₂²⋊C₈×C₇₃	C₂×C₄	1	6	yes	—
34	C₄×C₈×C₇₃	C₂×C₄	1	3	yes	—
35	C₄×C₈×C₇₃	C₂×C₄	1	3	yes	—
36	C₄×C₈×C₇₃	C₂³	1	3	yes	—
37	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
38	#11924003	—	2	6	no	—
39	#11924003	#41	1	4	yes	direct product factor
40	#41×C₇₃	C₁₆	1	8	yes	complemented
41	C₂×Q₈×C₇₃	C₂×C₈	1	1	yes	characteristic
42	C₂×D₄×C₇₃	C₂×C₈	1	3	yes	—
43	C₂²×C₄×C₇₃	C₂×C₈	1	3	yes	—
44	C₂²×C₄×C₇₃	C₂×C₈	1	3	yes	—
45	C₂²×C₄×C₇₃	C₂²×C₄	1	3	yes	—
46	C₂×C₈×C₇₃	—	2	6	no	—
47	C₂×C₈×C₇₃	—	2	6	no	—
48	C₂×C₈×C₇₃	C₂²×C₄	1	3	yes	—
49	C₂×C₈×C₇₃	C₂²×C₄	1	3	yes	—
50	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
52	C₄⋊C₄×C₇₃	C₂×C₈	1	3	yes	—
53	C₄⋊C₄×C₇₃	C₂×C₈	1	3	yes	—
54	C₂²⋊C₄×C₇₃	C₂×C₈	1	6	yes	—
55	C₄²×C₇₃	C₂×C₈	1	3	yes	—
56	C₄²×C₇₃	C₂×C₈	1	3	yes	—
57	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
58	C₄²×C₇₃	C₂²×C₄	1	3	yes	—
59	C₂³×C₇₃	C₂²×C₈	1	3	yes	—
60	Q₈×C₇₃	C₂×C₁₆	1	4	yes	complemented
61	D₄×C₇₃	C₂×C₁₆	1	12	yes	complemented
62	C₂×C₄×C₇₃	—	2	3	no	—
63	C₂×C₄×C₇₃	—	2	3	no	—
64	C₂×C₄×C₇₃	—	2	6	no	—
65	C₂×C₄×C₇₃	C₂×C₁₆	1	12	yes	complemented
66	C₂×C₄×C₇₃	C₂²×C₈	1	3	yes	—
67	C₂×C₄×C₇₃	C₂²×C₈	1	3	yes	—
68	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
69	C₂×C₄×C₇₃	C₂²×C₈	1	3	yes	—
70	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
71	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
72	C₈×C₇₃	—	2	3	no	—
73	C₈×C₇₃	—	2	3	no	—
74	C₈×C₇₃	#131	1	2	yes	—
75	C₈×C₇₃	C₂×#41	1	2	yes	—
76	C₂²×C₇₃	—	2	3	no	—
77	C₂²×C₇₃	—	2	3	no	—
78	C₂²×C₇₃	C₂²×C₁₆	1	6	yes	—
79	C₂²×C₇₃	C₂³×C₈	1	1	yes	characteristic
80	C₄×C₇₃	—	2	3	no	—
81	C₄×C₇₃	—	2	3	no	—
82	C₄×C₇₃	C₂²×C₁₆	1	2	yes	—
83	C₄×C₇₃	C₂²×C₁₆	1	6	yes	—
84	C₄×C₇₃	#504	1	2	yes	—
85	C₄×C₇₃	C₄×#41	1	1	yes	characteristic
86	C₄×C₇₃	C₂×#131	1	1	yes	characteristic
87	C₁₆×#41	—	73	1	no	maximal, Sylow
88	C₂×C₇₃	—	2	6	no	—
89	C₂×C₇₃	C₈×#41	1	1	yes	characteristic
90	C₂×C₇₃	C₂³×C₁₆	1	1	yes	characteristic
91	C₂×C₇₃	C₂×#504	1	1	yes	characteristic
92	C₈×#41	D₇₃	1	1	yes	characteristic
93	Q₈×C₁₆	—	73	2	no	—
94	D₄×C₁₆	—	73	6	no	—
95	#2162	—	73	3	no	—
96	C₂×C₄×C₁₆	—	73	3	no	—
97	C₇₃	C₁₆×#41	1	1	yes	characteristic, complemented, Sylow
98	C₄×#41	Dic₇₃	1	1	yes	characteristic
99	C₂²×C₁₆	—	73	6	no	—
100	C₈×Q₈	Dic₇₃	1	1	yes	characteristic
101	C₈×Q₈	C₂×D₇₃	1	1	yes	characteristic
102	C₈×D₄	Dic₇₃	1	3	yes	—
103	C₈×D₄	C₂×D₇₃	1	3	yes	—
104	#431	Dic₇₃	1	3	yes	—
105	C₂×C₄×C₈	C₂×D₇₃	1	3	yes	—
106	C₄⋊C₁₆	—	73	6	no	—
107	C₂²⋊C₁₆	—	73	6	no	—
108	C₄×C₁₆	—	73	6	no	—
109	C₄×C₁₆	—	73	1	no	—
110	C₄×C₁₆	—	73	3	no	—
111	C₂×#41	#10596831	1	1	yes	characteristic
112	C₂²×C₈	C₂×Dic₇₃	1	3	yes	—
113	C₂²×C₈	C₂²×D₇₃	1	3	yes	—
114	C₄×Q₈	#10596831	1	1	yes	characteristic
115	C₄×Q₈	C₂×Dic₇₃	1	1	yes	characteristic
116	C₄×D₄	#10596831	1	3	yes	—
117	C₄×D₄	C₂×Dic₇₃	1	3	yes	—
118	#117	#10596831	1	3	yes	—
119	C₂×C₄²	C₂×Dic₇₃	1	3	yes	—
120	C₂×C₁₆	—	73	2	no	—
121	C₂×C₁₆	—	73	6	no	—
122	C₂×C₁₆	—	146	12	no	—
123	C₄⋊C₈	C₂×Dic₇₃	1	3	yes	—
124	C₄⋊C₈	C₂×Dic₇₃	1	3	yes	—
125	C₂²⋊C₈	C₂×Dic₇₃	1	6	yes	—
126	C₄×C₈	C₂×Dic₇₃	1	3	yes	—
127	C₄×C₈	C₂×Dic₇₃	1	3	yes	—
128	C₄×C₈	C₂²×D₇₃	1	3	yes	—
129	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
130	#41	#11924003	1	8	yes	direct product factor
131	C₂×Q₈	C₂×#10596831	1	1	yes	characteristic
132	C₂×D₄	C₂×#10596831	1	3	yes	—
133	C₂²×C₄	C₂×#10596831	1	3	yes	—
134	C₂²×C₄	C₂×#10596831	1	3	yes	—
135	C₂²×C₄	C₂²×Dic₇₃	1	3	yes	—
136	C₂×C₈	—	2	6	no	—
137	C₂×C₈	—	2	6	no	—
138	C₂×C₈	C₂²×Dic₇₃	1	3	yes	—
139	C₂×C₈	C₂²×Dic₇₃	1	3	yes	—
140	C₂×C₈	C₂²×Dic₇₃	1	1	yes	characteristic
141	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
142	C₄⋊C₄	C₂×#10596831	1	3	yes	—
143	C₄⋊C₄	C₂×#10596831	1	3	yes	—
144	C₂²⋊C₄	C₂×#10596831	1	6	yes	—
145	C₄²	C₂×#10596831	1	3	yes	—
146	C₄²	C₂×#10596831	1	3	yes	—
147	C₄²	C₂²×Dic₇₃	1	1	yes	characteristic
148	C₄²	C₂²×Dic₇₃	1	3	yes	—
149	C₁₆	—	73	4	no	—
150	C₁₆	—	146	6	no	—
151	C₂³	C₂²×#10596831	1	3	yes	—
152	Q₈	C₂×#11924003	1	4	yes	complemented
153	D₄	C₂×#11924003	1	12	yes	complemented
154	C₂×C₄	—	2	6	no	—
155	C₂×C₄	—	2	3	no	—
156	C₂×C₄	—	2	3	no	—
157	C₂×C₄	C₂×#11924003	1	12	yes	complemented
158	C₂×C₄	C₂²×#10596831	1	3	yes	—
159	C₂×C₄	C₂²×#10596831	1	3	yes	—
160	C₂×C₄	C₂²×#10596831	1	1	yes	characteristic
161	C₂×C₄	C₂²×#10596831	1	3	yes	—
162	C₂×C₄	C₂²×#10596831	1	1	yes	characteristic
163	C₂×C₄	C₂³×Dic₇₃	1	1	yes	characteristic
164	C₈	—	2	3	no	—
165	C₈	—	2	3	no	—
166	C₈	#14556598	1	2	yes	—
167	C₈	#41×D₇₃	1	2	yes	—
168	C₂²	—	2	3	no	—
169	C₂²	—	2	3	no	—
170	C₂²	C₂²×#11924003	1	6	yes	—
171	C₂²	C₂³×#10596831	1	1	yes	characteristic
172	C₄	—	2	3	no	—
173	C₄	—	2	3	no	—
174	C₄	C₂²×#11924003	1	2	yes	—
175	C₄	C₂²×#11924003	1	6	yes	—
176	C₄	#17989004	1	2	yes	—
177	C₄	#41×Dic₇₃	1	1	yes	characteristic
178	C₄	C₂×#14556598	1	1	yes	characteristic
179	C₂	—	2	6	no	—
180	C₂	#41×#10596831	1	1	yes	characteristic
181	C₂	C₂³×#11924003	1	1	yes	characteristic
182	C₂	C₂×#17989004	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₈	C₂²×D₇₃
commutator	C₂×C₇₃	C₂³×C₁₆
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₈×#41×C₇₃	C₂
socle	C₂²×C₇₃	C₂³×C₈
prime radical	C₂×#41×C₇₃	C₈
cosocle	C₂×C₈×C₇₃	C₂⁴
hypercenter	C₈×#41	D₇₃
hypocenter	C₇₃	C₁₆×#41
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄×C₈ ◁ C₈×#41

Complements

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

N	H ≅ G/N	type
1. Q₈×#11924003	C₂	semidirect
2. D₄×#11924003	C₂	semidirect
3. #22611730	C₂	semidirect
4. C₂×C₄×#11924003	C₂	semidirect
39. #11924003	#41	direct (N × H)
40. #41×C₇₃	C₁₆	semidirect
60. Q₈×C₇₃	C₂×C₁₆	semidirect
61. D₄×C₇₃	C₂×C₁₆	semidirect
65. C₂×C₄×C₇₃	C₂×C₁₆	semidirect
97. C₇₃	C₁₆×#41	semidirect
130. #41	#11924003	direct (N × H)
152. Q₈	C₂×#11924003	semidirect
153. D₄	C₂×#11924003	semidirect
157. C₂×C₄	C₂×#11924003	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#11924003	2
D₄×#11924003	2
#22611730	2
C₂×C₄×#11924003	2
C₈×#41×C₇₃	2
C₁₆×#41	73