Subgroups — #108×#10596831 #32977278

Order 18688.

374 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22608679	C₂	1	1	yes	maximal, characteristic
2	M₁₆×#10596831	C₂	1	2	yes	maximal
3	C₂×C₈×#10596831	C₂	1	1	yes	maximal, characteristic
4	#22607769	C₂	1	2	yes	maximal
5	C₄×#108×C₇₃	C₂	1	1	yes	maximal, characteristic
6	#17988640	C₂²	1	2	yes	—
7	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
8	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
9	#17988340	C₄	1	2	yes	—
10	#17988338	C₄	1	2	yes	—
11	#17988308	—	2	2	no	—
12	C₈×#10596831	—	2	2	no	—
13	C₈×#10596831	C₄	1	2	yes	—
14	C₈×#10596831	C₄	1	2	yes	—
15	C₂×#108×C₇₃	C₄	1	2	yes	—
16	C₄×M₁₆×C₇₃	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
18	#341×C₇₃	C₄	1	2	yes	—
19	C₂²×#10596831	—	2	1	no	—
20	C₂²×#10596831	Q₈	1	2	yes	—
21	C₄⋊#10596831	C₂×C₄	1	4	yes	—
22	C₂²⋊#10596831	—	2	2	no	—
23	C₄×#10596831	—	2	2	no	—
24	C₄×#10596831	—	2	1	no	—
25	C₄×#10596831	C₂×C₄	1	2	yes	—
26	C₄×#10596831	C₂×C₄	1	2	yes	—
27	C₄×#10596831	D₄	1	1	yes	characteristic
28	C₄×#10596831	D₄	1	1	yes	characteristic
29	C₂×M₁₆×C₇₃	C₂×C₄	1	4	yes	—
30	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
33	#108×C₇₃	C₈	1	8	yes	complemented
34	#97×C₇₃	—	2	2	no	—
35	C₄×C₈×C₇₃	—	2	2	no	—
36	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
38	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
40	C₂×#10596831	—	2	4	no	—
41	C₂×#10596831	—	2	2	no	—
42	C₂×#10596831	—	2	4	no	—
43	C₂×#10596831	—	2	4	no	—
44	C₂×#10596831	C₄⋊C₄	1	2	yes	—
45	C₂×#10596831	C₄⋊C₄	1	2	yes	—
46	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
48	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
49	M₁₆×C₇₃	—	2	4	no	—
50	M₁₆×C₇₃	C₂×C₈	1	8	yes	—
51	C₂×C₈×C₇₃	—	2	4	no	—
52	C₂×C₈×C₇₃	—	2	2	no	—
53	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
54	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
55	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
56	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
57	C₂×C₈×C₇₃	C₂×C₈	1	4	yes	—
58	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
59	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
60	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
61	C₄²×C₇₃	#41	1	1	yes	characteristic
62	C₂³×C₇₃	C₄×Q₈	1	1	yes	characteristic
63	C₂×C₄×C₇₃	—	2	2	no	—
64	C₂×C₄×C₇₃	—	2	2	no	—
65	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
66	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
68	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
69	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₇₃	C₄×Q₈	1	1	yes	characteristic
71	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
72	C₂×C₄×C₇₃	#131	1	2	yes	—
73	C₈×C₇₃	—	2	1	no	—
74	C₈×C₇₃	—	2	1	no	—
75	C₈×C₇₃	—	2	8	no	—
76	C₈×C₇₃	C₄×C₈	1	2	yes	—
77	C₈×C₇₃	C₄×C₈	1	2	yes	—
78	#10596831	—	2	4	no	—
79	#10596831	—	2	2	no	—
80	#10596831	—	4	4	no	—
81	#10596831	#108	1	4	yes	direct product factor
82	C₂²×C₇₃	—	2	1	no	—
83	C₂²×C₇₃	—	2	1	no	—
84	C₂²×C₇₃	C₄×C₄⋊C₄	1	1	yes	characteristic
85	C₂²×C₇₃	C₈×Q₈	1	2	yes	—
86	C₄×C₇₃	—	2	1	no	—
87	C₄×C₇₃	—	2	1	no	—
88	C₄×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
89	C₄×C₇₃	#405	1	1	yes	characteristic
90	C₄×C₇₃	C₂×#108	1	2	yes	—
91	C₄×C₇₃	C₈×D₄	1	2	yes	—
92	C₄×C₇₃	#443	1	2	yes	—
93	C₈×#108	—	73	1	no	maximal, Sylow
94	C₂×C₇₃	—	2	2	no	—
95	C₂×C₇₃	C₈×C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₇₃	#1772	1	1	yes	characteristic
97	C₂×C₇₃	C₄×#108	1	1	yes	characteristic
98	C₄×#108	D₇₃	1	1	yes	characteristic
99	#1564	—	73	1	no	—
100	C₈×M₁₆	—	73	2	no	—
101	C₂×C₈²	—	73	1	no	—
102	#1278	—	73	2	no	—
103	C₇₃	C₈×#108	1	1	yes	characteristic, complemented, Sylow
104	#431	—	73	2	no	—
105	C₂×#108	Dic₇₃	1	2	yes	—
106	C₄×M₁₆	C₂×D₇₃	1	2	yes	—
107	C₂×C₄×C₈	—	73	1	no	—
108	C₂×C₄×C₈	—	73	1	no	—
109	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
110	#341	Dic₇₃	1	2	yes	—
111	#335	—	73	2	no	—
112	#334	—	73	2	no	—
113	#322	—	146	2	no	—
114	C₈²	—	73	2	no	—
115	C₈²	—	73	2	no	—
116	C₈²	—	146	2	no	—
117	C₂×M₁₆	C₂×Dic₇₃	1	4	yes	—
118	C₂²×C₈	—	73	2	no	—
119	C₂²×C₈	—	146	1	no	—
120	C₂²×C₈	C₂×Dic₇₃	1	1	yes	characteristic
121	C₂²×C₈	C₂×Dic₇₃	1	1	yes	characteristic
122	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
123	#108	#10596831	1	8	yes	direct product factor
124	C₄⋊C₈	—	73	4	no	—
125	C₂²⋊C₈	—	146	2	no	—
126	#97	—	2	2	no	—
127	C₄×C₈	—	2	2	no	—
128	C₄×C₈	—	73	2	no	—
129	C₄×C₈	—	73	2	no	—
130	C₄×C₈	—	73	1	no	—
131	C₄×C₈	—	73	1	no	—
132	C₄×C₈	—	146	2	no	—
133	C₄×C₈	—	146	1	no	—
134	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
135	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
136	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
137	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
138	C₂²×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
139	C₂²×C₄	Q₈×D₇₃	1	1	yes	characteristic
140	C₂²×C₄	#11924029	1	1	yes	characteristic
141	M₁₆	—	2	4	no	—
142	M₁₆	C₂×#10596831	1	8	yes	—
143	C₂×C₈	—	2	4	no	—
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	—	73	2	no	—
146	C₂×C₈	—	73	2	no	—
147	C₂×C₈	—	146	4	no	—
148	C₂×C₈	—	146	4	no	—
149	C₂×C₈	—	146	4	no	—
150	C₂×C₈	—	146	2	no	—
151	C₂×C₈	C₄×Dic₇₃	1	1	yes	characteristic
152	C₂×C₈	C₄×Dic₇₃	1	1	yes	characteristic
153	C₂×C₈	C₄×Dic₇₃	1	1	yes	characteristic
154	C₂×C₈	C₄×Dic₇₃	1	1	yes	characteristic
155	C₂×C₈	C₂×#10596831	1	4	yes	—
156	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
157	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
158	C₄²	D₄×D₇₃	1	1	yes	characteristic
159	C₄²	#11924027	1	1	yes	characteristic
160	C₂³	Q₈×Dic₇₃	1	1	yes	characteristic
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	C₂×C₄×Dic₇₃	1	1	yes	characteristic
164	C₂×C₄	C₄⋊C₄×D₇₃	1	1	yes	characteristic
165	C₂×C₄	#14556549	1	1	yes	characteristic
166	C₂×C₄	#14556553	1	1	yes	characteristic
167	C₂×C₄	D₄×Dic₇₃	1	1	yes	characteristic
168	C₂×C₄	#14556556	1	1	yes	characteristic
169	C₂×C₄	C₂²×#10596831	1	2	yes	—
170	C₂×C₄	#14556598	1	2	yes	—
171	C₈	—	2	8	no	—
172	C₈	—	2	1	no	—
173	C₈	—	2	1	no	—
174	C₈	—	292	4	no	—
175	C₈	—	73	4	no	—
176	C₈	—	146	4	no	—
177	C₈	—	146	2	no	—
178	C₈	C₄×#10596831	1	2	yes	—
179	C₈	C₄×#10596831	1	2	yes	—
180	C₂²	—	2	1	no	—
181	C₂²	—	2	1	no	—
182	C₂²	C₄⋊C₄×Dic₇₃	1	1	yes	characteristic
183	C₂²	Q₈×#10596831	1	2	yes	—
184	C₄	—	2	1	no	—
185	C₄	—	2	1	no	—
186	C₄	C₂×C₄×#10596831	1	1	yes	characteristic
187	C₄	#17988543	1	1	yes	characteristic
188	C₄	#108×D₇₃	1	2	yes	—
189	C₄	D₄×#10596831	1	2	yes	—
190	C₄	#17988703	1	2	yes	—
191	C₂	—	2	2	no	—
192	C₂	C₄⋊C₄×#10596831	1	1	yes	characteristic
193	C₂	#22609813	1	1	yes	characteristic
194	C₂	#108×Dic₇₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₇₃
commutator	C₄×C₇₃	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₄×#108×C₇₃	C₂
socle	C₂²×C₇₃	C₄×C₄⋊C₄
prime radical	C₂³×C₇₃	C₄×Q₈
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	C₄×#108	D₇₃
hypocenter	C₇₃	C₈×#108
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄×#108

Complements

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

N	H ≅ G/N	type
33. #108×C₇₃	C₈	semidirect
81. #10596831	#108	direct (N × H)
103. C₇₃	C₈×#108	semidirect
123. #108	#10596831	direct (N × H)

Maximal subgroups

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

subgroup	index
#22608679	2
M₁₆×#10596831	2
C₂×C₈×#10596831	2
#22607769	2
C₄×#108×C₇₃	2
C₈×#108	73