Subgroups — #106×#10596831 #32977210

Order 18688.

446 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊C₄×#10596831	C₂	1	2	yes	maximal
2	C₂×#17988338	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×#10596831	C₂	1	1	yes	maximal, characteristic
4	#22607772	C₂	1	2	yes	maximal
5	C₄×#106×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#10596831	C₂²	1	2	yes	—
7	C₂×C₄×#10596831	—	2	2	no	—
8	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
9	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
10	#17988342	—	2	2	no	—
11	#17988338	C₄	1	4	yes	complemented
12	C₈×#10596831	C₄	1	4	yes	complemented
13	C₂×#106×C₇₃	C₄	1	2	yes	—
14	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
15	C₄×C₄⋊C₄×C₇₃	C₂²	1	2	yes	—
16	#340×C₇₃	C₄	1	2	yes	—
17	C₂²×#10596831	—	2	2	no	—
18	C₂²×#10596831	—	2	1	no	—
19	C₂²×#10596831	D₄	1	1	yes	characteristic
20	C₂²×#10596831	D₄	1	1	yes	characteristic
21	C₄⋊#10596831	—	2	2	no	—
22	C₄⋊#10596831	C₂×C₄	1	4	yes	—
23	C₄×#10596831	—	2	1	no	—
24	C₄×#10596831	—	4	4	no	—
25	C₄×#10596831	C₂×C₄	1	2	yes	—
26	C₄×#10596831	C₂×C₄	1	2	yes	—
27	C₄×#10596831	Q₈	1	2	yes	—
28	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄⋊C₄×C₇₃	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₇₃	—	2	2	no	—
32	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
33	#106×C₇₃	C₈	1	8	yes	complemented
34	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
36	#95×C₇₃	—	2	2	no	—
37	C₂×#10596831	—	2	4	no	—
38	C₂×#10596831	—	2	2	no	—
39	C₂×#10596831	—	2	2	no	—
40	C₂×#10596831	—	2	4	no	—
41	C₂×#10596831	—	4	2	no	—
42	C₂×#10596831	—	4	2	no	—
43	C₂×#10596831	C₄⋊C₄	1	2	yes	—
44	C₂×#10596831	C₄⋊C₄	1	2	yes	—
45	C₂×#10596831	SD₁₆	1	4	yes	—
46	C₂²×C₄×C₇₃	—	2	4	no	—
47	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
50	C₂×C₈×C₇₃	—	2	2	no	—
51	C₂×C₈×C₇₃	C₄²	1	2	yes	—
52	C₂×C₈×C₇₃	C₄²	1	2	yes	—
53	C₂×C₈×C₇₃	C₂×C₈	1	4	yes	—
54	C₄⋊C₄×C₇₃	—	2	4	no	—
55	C₄⋊C₄×C₇₃	C₂×C₈	1	8	yes	—
56	C₄²×C₇₃	—	4	2	no	—
57	C₄²×C₇₃	—	4	2	no	—
58	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
59	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
60	C₄²×C₇₃	C₂×Q₈	1	1	yes	characteristic
61	C₄²×C₇₃	#41	1	1	yes	characteristic
62	C₂³×C₇₃	C₄×D₄	1	1	yes	characteristic
63	C₂×C₄×C₇₃	—	2	2	no	—
64	C₂×C₄×C₇₃	—	2	8	no	—
65	C₂×C₄×C₇₃	—	4	4	no	—
66	C₂×C₄×C₇₃	—	4	4	no	—
67	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
68	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
70	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
71	C₂×C₄×C₇₃	C₄×Q₈	1	1	yes	characteristic
72	C₂×C₄×C₇₃	C₄×Q₈	1	1	yes	characteristic
73	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
74	C₂×C₄×C₇₃	#131	1	2	yes	—
75	C₂×C₄×C₇₃	C₂×SD₁₆	1	2	yes	—
76	C₂×C₄×C₇₃	#135	1	2	yes	—
77	C₈×C₇₃	—	2	2	no	—
78	C₈×C₇₃	C₄×C₈	1	4	yes	complemented
79	#10596831	—	2	4	no	—
80	#10596831	—	2	2	no	—
81	#10596831	—	4	4	no	—
82	#10596831	#106	1	4	yes	direct product factor
83	C₂²×C₇₃	C₄×C₄⋊C₄	1	1	yes	characteristic
84	C₂²×C₇₃	C₈×D₄	1	2	yes	—
85	C₂²×C₇₃	C₄×SD₁₆	1	2	yes	—
86	C₂²×C₇₃	#443	1	2	yes	—
87	C₄×C₇₃	—	2	1	no	—
88	C₄×C₇₃	—	2	1	no	—
89	C₄×C₇₃	—	4	8	no	—
90	C₄×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
91	C₄×C₇₃	#405	1	1	yes	characteristic
92	C₄×C₇₃	C₂×#106	1	1	yes	characteristic
93	C₄×C₇₃	#427	1	1	yes	characteristic
94	C₄×C₇₃	C₄×SD₁₆	1	2	yes	—
95	C₄×C₇₃	C₈×Q₈	1	2	yes	—
96	C₈×#106	—	73	1	no	maximal, Sylow
97	C₂×C₇₃	C₈×SD₁₆	1	4	yes	—
98	C₂×C₇₃	C₈×C₄⋊C₄	1	1	yes	characteristic
99	C₂×C₇₃	#1772	1	1	yes	characteristic
100	C₂×C₇₃	C₄×#106	1	1	yes	characteristic
101	C₄×#106	D₇₃	1	1	yes	characteristic
102	C₈×C₄⋊C₄	—	73	2	no	—
103	C₂×#334	—	73	1	no	—
104	C₂×C₈²	—	73	1	no	—
105	#1279	—	73	2	no	—
106	C₇₃	C₈×#106	1	1	yes	characteristic, complemented, Sylow
107	C₂×#106	Dic₇₃	1	2	yes	—
108	C₂×C₄⋊C₈	—	73	2	no	—
109	C₂×C₄×C₈	—	73	1	no	—
110	C₂×C₄×C₈	—	73	1	no	—
111	C₂×C₄×C₈	—	146	2	no	—
112	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
113	C₄×C₄⋊C₄	C₂×D₇₃	1	2	yes	—
114	#340	Dic₇₃	1	2	yes	—
115	#336	—	146	2	no	—
116	#334	—	73	4	no	—
117	C₈²	—	73	4	no	—
118	C₂²×C₈	—	73	1	no	—
119	C₂²×C₈	—	73	1	no	—
120	C₂²×C₈	—	146	2	no	—
121	C₂²×C₈	—	146	1	no	—
122	C₂²×C₈	C₂×Dic₇₃	1	1	yes	characteristic
123	C₂²×C₈	C₂×Dic₇₃	1	1	yes	characteristic
124	C₂×C₄⋊C₄	C₂×Dic₇₃	1	4	yes	—
125	C₂×C₄²	—	2	2	no	—
126	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
127	#106	#10596831	1	8	yes	direct product factor
128	C₄⋊C₈	—	73	4	no	—
129	C₄⋊C₈	—	146	2	no	—
130	C₄×C₈	—	292	4	no	—
131	C₄×C₈	—	73	2	no	—
132	C₄×C₈	—	73	2	no	—
133	C₄×C₈	—	73	2	no	—
134	C₄×C₈	—	146	1	no	—
135	C₄×C₈	C₄×D₇₃	1	2	yes	—
136	C₄×C₈	C₄×D₇₃	1	2	yes	—
137	#95	—	2	2	no	—
138	C₂²×C₄	—	2	4	no	—
139	C₂²×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
140	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
141	C₂²×C₄	#11924027	1	1	yes	characteristic
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	292	2	no	—
144	C₂×C₈	—	292	2	no	—
145	C₂×C₈	—	73	2	no	—
146	C₂×C₈	—	73	4	no	—
147	C₂×C₈	—	73	2	no	—
148	C₂×C₈	—	146	4	no	—
149	C₂×C₈	—	146	4	no	—
150	C₂×C₈	—	146	2	no	—
151	C₂×C₈	—	146	2	no	—
152	C₂×C₈	C₄×Dic₇₃	1	2	yes	—
153	C₂×C₈	C₄×Dic₇₃	1	2	yes	—
154	C₂×C₈	C₂×#10596831	1	4	yes	—
155	C₄⋊C₄	—	2	4	no	—
156	C₄⋊C₄	C₂×#10596831	1	8	yes	—
157	C₄²	—	4	2	no	—
158	C₄²	—	4	2	no	—
159	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
160	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
161	C₄²	Q₈×D₇₃	1	1	yes	characteristic
162	C₄²	#11924029	1	1	yes	characteristic
163	C₂³	D₄×Dic₇₃	1	1	yes	characteristic
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	8	no	—
166	C₂×C₄	—	4	4	no	—
167	C₂×C₄	—	4	4	no	—
168	C₂×C₄	C₂×C₄×Dic₇₃	1	1	yes	characteristic
169	C₂×C₄	C₄⋊C₄×D₇₃	1	1	yes	characteristic
170	C₂×C₄	#14556549	1	1	yes	characteristic
171	C₂×C₄	#14556553	1	1	yes	characteristic
172	C₂×C₄	#14556556	1	1	yes	characteristic
173	C₂×C₄	Q₈×Dic₇₃	1	1	yes	characteristic
174	C₂×C₄	C₂²×#10596831	1	2	yes	—
175	C₂×C₄	#14556598	1	2	yes	—
176	C₂×C₄	SD₁₆×D₇₃	1	2	yes	—
177	C₂×C₄	#14556612	1	2	yes	—
178	C₈	—	2	2	no	—
179	C₈	—	292	4	no	—
180	C₈	—	73	4	no	—
181	C₈	—	146	4	no	—
182	C₈	—	146	2	no	—
183	C₈	C₄×#10596831	1	4	yes	complemented
184	C₂²	C₄⋊C₄×Dic₇₃	1	1	yes	characteristic
185	C₂²	D₄×#10596831	1	2	yes	—
186	C₂²	SD₁₆×Dic₇₃	1	2	yes	—
187	C₂²	#17988704	1	2	yes	—
188	C₄	—	2	1	no	—
189	C₄	—	2	1	no	—
190	C₄	—	4	8	no	—
191	C₄	C₂×C₄×#10596831	1	1	yes	characteristic
192	C₄	#17988543	1	1	yes	characteristic
193	C₄	#106×D₇₃	1	1	yes	characteristic
194	C₄	#17988622	1	1	yes	characteristic
195	C₄	#17988674	1	2	yes	—
196	C₄	Q₈×#10596831	1	2	yes	—
197	C₂	SD₁₆×#10596831	1	4	yes	—
198	C₂	C₄⋊C₄×#10596831	1	1	yes	characteristic
199	C₂	#22609812	1	1	yes	characteristic
200	C₂	#106×Dic₇₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₇₃
commutator	C₄×C₇₃	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₄×#106×C₇₃	C₂
socle, prime radical	C₂³×C₇₃	C₄×D₄
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	C₄×#106	D₇₃
hypocenter	C₇₃	C₈×#106
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×#106

Complements

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

N	H ≅ G/N	type
11. #17988338	C₄	semidirect
12. C₈×#10596831	C₄	semidirect
33. #106×C₇₃	C₈	semidirect
78. C₈×C₇₃	C₄×C₈	semidirect
82. #10596831	#106	direct (N × H)
106. C₇₃	C₈×#106	semidirect
127. #106	#10596831	direct (N × H)
183. C₈	C₄×#10596831	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊C₄×#10596831	2
C₂×#17988338	2
C₂×C₈×#10596831	2
#22607772	2
C₄×#106×C₇₃	2
C₈×#106	73