Subgroups — #108×#10596833 #34004051

Order 18688.

374 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22625478	C₂	1	1	yes	maximal, characteristic
2	M₁₆×#10596833	C₂	1	2	yes	maximal
3	C₂×C₈×#10596833	C₂	1	1	yes	maximal, characteristic
4	#22624764	C₂	1	2	yes	maximal
5	#108×Dic₇₃	C₂	1	1	yes	maximal, characteristic
6	#17989603	C₂²	1	2	yes	—
7	C₂×C₄×#10596833	C₂²	1	1	yes	characteristic
8	C₂×C₄×#10596833	C₂²	1	1	yes	characteristic
9	#17989450	C₄	1	2	yes	—
10	#17989448	C₄	1	2	yes	—
11	#17989428	—	2	2	no	—
12	C₈×#10596833	—	2	2	no	—
13	C₈×#10596833	C₄	1	2	yes	—
14	C₈×#10596833	C₄	1	2	yes	—
15	C₂×#14556527	C₄	1	1	yes	characteristic
16	M₁₆×Dic₇₃	C₂²	1	2	yes	—
17	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
18	#17988351	C₄	1	2	yes	—
19	C₂×#108×C₇₃	C₄	1	1	yes	characteristic
20	C₂²×#10596833	—	2	1	no	—
21	C₂²×#10596833	Q₈	1	2	yes	—
22	C₄⋊#10596833	C₂×C₄	1	4	yes	—
23	C₂²⋊#10596833	—	2	2	no	—
24	C₄×#10596833	—	2	2	no	—
25	C₄×#10596833	—	2	1	no	—
26	C₄×#10596833	C₂×C₄	1	2	yes	—
27	C₄×#10596833	C₂×C₄	1	2	yes	—
28	C₄×#10596833	D₄	1	1	yes	characteristic
29	C₄×#10596833	D₄	1	1	yes	characteristic
30	C₂×#11924012	C₂×C₄	1	2	yes	—
31	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×Dic₇₃	C₂³	1	1	yes	characteristic
33	#14556527	C₈	1	4	yes	complemented
34	#14556501	—	2	2	no	—
35	C₈×Dic₇₃	—	2	2	no	—
36	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
37	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
38	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
39	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
40	C₂×M₁₆×C₇₃	C₂×C₄	1	2	yes	—
41	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
42	#108×C₇₃	C₈	1	4	yes	complemented
43	C₂×#10596833	—	2	4	no	—
44	C₂×#10596833	—	2	2	no	—
45	C₂×#10596833	—	2	4	no	—
46	C₂×#10596833	—	2	4	no	—
47	C₂×#10596833	C₄⋊C₄	1	2	yes	—
48	C₂×#10596833	C₄⋊C₄	1	2	yes	—
49	C₂²×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
50	C₂²×Dic₇₃	#41	1	1	yes	characteristic
51	#11924012	—	2	2	no	—
52	#11924012	C₂×C₈	1	4	yes	—
53	C₂×#10596831	—	2	2	no	—
54	C₂×#10596831	—	2	1	no	—
55	C₂×#10596831	—	2	1	no	—
56	C₂×#10596831	C₄²	1	1	yes	characteristic
57	C₂×#10596831	C₄²	1	1	yes	characteristic
58	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
59	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
60	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
61	C₄×Dic₇₃	#41	1	1	yes	characteristic
62	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
63	M₁₆×C₇₃	—	2	2	no	—
64	M₁₆×C₇₃	C₂×C₈	1	4	yes	—
65	C₂×C₈×C₇₃	—	2	2	no	—
66	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
67	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
68	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
69	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
70	C₂³×C₇₃	C₄×Q₈	1	1	yes	characteristic
71	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
72	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
73	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
74	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
75	C₂×C₄×C₇₃	#131	1	1	yes	characteristic
76	C₂×C₄×C₇₃	#131	1	1	yes	characteristic
77	C₂×Dic₇₃	—	2	2	no	—
78	C₂×Dic₇₃	—	2	2	no	—
79	C₂×Dic₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₂×Dic₇₃	#117	1	1	yes	characteristic
81	C₂×Dic₇₃	C₄×D₄	1	1	yes	characteristic
82	C₂×Dic₇₃	C₄×Q₈	1	1	yes	characteristic
83	#10596833	—	2	4	no	—
84	#10596833	—	2	2	no	—
85	#10596833	—	4	4	no	—
86	#10596833	#108	1	4	yes	direct product factor
87	C₈×C₇₃	—	2	4	no	—
88	C₈×C₇₃	C₄×C₈	1	2	yes	—
89	C₈×C₇₃	C₄×C₈	1	2	yes	—
90	#10596831	—	2	1	no	—
91	#10596831	—	2	1	no	—
92	#10596831	—	2	4	no	—
93	C₂²×C₇₃	—	2	1	no	—
94	C₂²×C₇₃	—	2	1	no	—
95	C₂²×C₇₃	C₄×C₄⋊C₄	1	1	yes	characteristic
96	C₂²×C₇₃	C₈×Q₈	1	1	yes	characteristic
97	C₂²×C₇₃	C₈×Q₈	1	1	yes	characteristic
98	C₄×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
99	C₄×C₇₃	#405	1	1	yes	characteristic
100	C₄×C₇₃	C₈×D₄	1	1	yes	characteristic
101	C₄×C₇₃	C₈×D₄	1	1	yes	characteristic
102	Dic₇₃	—	2	1	no	—
103	Dic₇₃	—	2	1	no	—
104	Dic₇₃	C₂×#108	1	2	yes	—
105	Dic₇₃	#443	1	2	yes	—
106	C₈×#108	—	73	1	no	maximal, Sylow
107	C₂×C₇₃	—	2	1	no	—
108	C₂×C₇₃	—	2	1	no	—
109	C₂×C₇₃	C₈×C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₇₃	#1772	1	1	yes	characteristic
111	C₂×C₇₃	C₄×#108	1	1	yes	characteristic
112	C₄×#108	—	73	1	no	—
113	#1564	—	73	1	no	—
114	C₈×M₁₆	—	73	2	no	—
115	C₂×C₈²	—	73	1	no	—
116	#1278	—	73	2	no	—
117	C₇₃	C₈×#108	1	1	yes	characteristic, complemented, Sylow
118	#431	—	73	2	no	—
119	C₂×#108	—	73	1	no	—
120	C₂×#108	#64408	1	1	yes	characteristic
121	C₄×M₁₆	—	73	2	no	—
122	C₂×C₄×C₈	—	73	1	no	—
123	C₂×C₄×C₈	—	73	1	no	—
124	C₂×C₄×C₈	—	73	1	no	—
125	#341	—	73	2	no	—
126	#335	—	73	2	no	—
127	#334	—	73	2	no	—
128	#322	—	146	2	no	—
129	C₈²	—	73	2	no	—
130	C₈²	—	73	2	no	—
131	C₈²	—	146	2	no	—
132	C₂×M₁₆	—	73	2	no	—
133	C₂×M₁₆	C₂×#64408	1	2	yes	—
134	C₂²×C₈	—	73	2	no	—
135	C₂²×C₈	—	73	1	no	—
136	C₂²×C₈	—	146	1	no	—
137	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
138	C₂×C₄²	—	73	1	no	—
139	#108	—	73	4	no	—
140	#108	#10596833	1	4	yes	direct product factor
141	C₄⋊C₈	—	73	4	no	—
142	C₂²⋊C₈	—	146	2	no	—
143	#97	—	146	2	no	—
144	C₄×C₈	—	73	2	no	—
145	C₄×C₈	—	73	1	no	—
146	C₄×C₈	—	73	1	no	—
147	C₄×C₈	—	73	1	no	—
148	C₄×C₈	—	73	1	no	—
149	C₄×C₈	—	73	1	no	—
150	C₄×C₈	—	73	1	no	—
151	C₄×C₈	—	73	2	no	—
152	C₄×C₈	—	146	2	no	—
153	C₄×C₈	—	146	2	no	—
154	C₄×C₈	—	146	1	no	—
155	C₂²×C₄	—	73	1	no	—
156	C₂²×C₄	—	73	1	no	—
157	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
158	M₁₆	—	2	2	no	—
159	M₁₆	—	73	4	no	—
160	M₁₆	—	146	2	no	—
161	M₁₆	C₂×#10596833	1	4	yes	—
162	C₂×C₈	—	2	2	no	—
163	C₂×C₈	—	73	2	no	—
164	C₂×C₈	—	73	2	no	—
165	C₂×C₈	—	73	1	no	—
166	C₂×C₈	—	73	1	no	—
167	C₂×C₈	—	146	4	no	—
168	C₂×C₈	—	146	4	no	—
169	C₂×C₈	—	146	2	no	—
170	C₂×C₈	—	146	4	no	—
171	C₂×C₈	—	146	1	no	—
172	C₂×C₈	—	146	1	no	—
173	C₂×C₈	—	146	2	no	—
174	C₂×C₈	C₄×#64408	1	1	yes	characteristic
175	C₂×C₈	C₄×#64408	1	1	yes	characteristic
176	C₂×C₈	C₂×#10596833	1	2	yes	—
177	C₂×C₈	#11924036	1	2	yes	—
178	C₄²	—	73	1	no	—
179	C₄²	—	73	1	no	—
180	C₄²	—	73	1	no	—
181	C₄²	—	73	1	no	—
182	C₂³	Q₈×#64408	1	1	yes	characteristic
183	C₂×C₄	—	73	1	no	—
184	C₂×C₄	—	73	1	no	—
185	C₂×C₄	—	73	1	no	—
186	C₂×C₄	—	73	1	no	—
187	C₂×C₄	—	146	2	no	—
188	C₂×C₄	—	146	2	no	—
189	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
190	C₂×C₄	D₄×#64408	1	1	yes	characteristic
191	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
192	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
193	C₂×C₄	#14556678	1	1	yes	characteristic
194	C₂×C₄	#14556679	1	1	yes	characteristic
195	C₈	—	2	4	no	—
196	C₈	—	292	4	no	—
197	C₈	—	73	4	no	—
198	C₈	—	146	2	no	—
199	C₈	—	146	4	no	—
200	C₈	—	146	1	no	—
201	C₈	—	146	1	no	—
202	C₈	—	146	4	no	—
203	C₈	C₄×#10596833	1	2	yes	—
204	C₈	C₄×#10596833	1	2	yes	—
205	C₂²	—	2	1	no	—
206	C₂²	—	2	1	no	—
207	C₂²	C₄⋊C₄×#64408	1	1	yes	characteristic
208	C₂²	Q₈×#10596833	1	1	yes	characteristic
209	C₂²	#17989636	1	1	yes	characteristic
210	C₄	—	73	2	no	—
211	C₄	—	73	2	no	—
212	C₄	—	146	1	no	—
213	C₄	—	146	1	no	—
214	C₄	C₂×C₄×#10596833	1	1	yes	characteristic
215	C₄	#17989552	1	1	yes	characteristic
216	C₄	D₄×#10596833	1	1	yes	characteristic
217	C₄	#17989615	1	1	yes	characteristic
218	C₂	—	2	1	no	—
219	C₂	—	2	1	no	—
220	C₂	C₄⋊C₄×#10596833	1	1	yes	characteristic
221	C₂	#22625736	1	1	yes	characteristic
222	C₂	#108×#64408	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×#64408
commutator	C₄×C₇₃	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×#64408
Fitting	C₂×#108×C₇₃	C₄
socle	C₂²×C₇₃	C₄×C₄⋊C₄
prime radical	C₂³×C₇₃	C₄×Q₈
cosocle	C₂×C₄×Dic₇₃	C₂³
hypercenter	C₂×#108	#64408
hypocenter	C₇₃	C₈×#108
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

Complements

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

N	H ≅ G/N	type
33. #14556527	C₈	semidirect
42. #108×C₇₃	C₈	semidirect
86. #10596833	#108	direct (N × H)
117. C₇₃	C₈×#108	semidirect
140. #108	#10596833	direct (N × H)

Maximal subgroups

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

subgroup	index
#22625478	2
M₁₆×#10596833	2
C₂×C₈×#10596833	2
#22624764	2
#108×Dic₇₃	2
C₈×#108	73