Subgroups — #34012666

Order 18688.

398 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#11924031	C₂	1	1	yes	maximal, characteristic
2	C₂×#17989473	C₂	1	2	yes	maximal
3	C₂×C₄×#11924031	C₂	1	1	yes	maximal, characteristic
4	#22624970	C₂	1	2	yes	maximal
5	#22609796	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11924031	C₂²	1	1	yes	characteristic
7	C₂²×#11924031	C₂²	1	2	yes	—
8	C₂²×#11924031	C₂²	1	1	yes	characteristic
9	C₄⋊#11924031	C₄	1	4	yes	complemented
10	#17989473	—	2	4	no	—
11	C₄×#11924031	C₄	1	4	yes	complemented
12	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
13	C₂×C₈×Dic₇₃	C₂²	1	2	yes	—
14	C₄×C₄⋊Dic₇₃	C₄	1	1	yes	characteristic
15	#17988341	C₄	1	2	yes	—
16	C₂×C₄⋊C₈×C₇₃	C₄	1	1	yes	characteristic
17	C₂×#11924031	—	2	2	no	—
18	C₂×#11924031	—	2	1	no	—
19	C₂×#11924031	—	2	2	no	—
20	C₂×#11924031	—	2	1	no	—
21	C₂×#11924031	C₂×C₄	1	2	yes	—
22	C₂×#11924031	C₂×C₄	1	4	yes	—
23	C₂×#11924031	C₂×C₄	1	2	yes	—
24	C₂×#11924031	D₄	1	2	yes	—
25	C₂×#11924031	Q₈	1	2	yes	—
26	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
27	C₂²×#10596831	C₂³	1	1	yes	characteristic
28	C₂×C₄⋊Dic₇₃	C₈	1	2	yes	—
29	C₂×C₄×Dic₇₃	C₂×C₄	1	2	yes	—
30	C₄×#10596831	C₂×C₄	1	2	yes	—
31	C₄×#10596831	C₂×C₄	1	2	yes	—
32	C₈×Dic₇₃	—	2	2	no	—
33	C₈×Dic₇₃	—	2	2	no	—
34	#14556498	C₈	1	2	yes	—
35	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
37	C₄⋊C₈×C₇₃	C₈	1	4	yes	—
38	#11924031	—	2	2	no	—
39	#11924031	—	2	4	no	—
40	#11924031	C₄⋊C₄	1	4	yes	complemented
41	C₂²×Dic₇₃	C₂×C₈	1	4	yes	—
42	C₂×#10596831	—	2	1	no	—
43	C₂×#10596831	—	2	1	no	—
44	C₂×#10596831	C₄²	1	2	yes	—
45	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
46	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
47	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
48	C₂×#10596831	C₂×Q₈	1	1	yes	characteristic
49	C₂×#10596831	#41	1	1	yes	characteristic
50	C₂×#10596831	#41	1	1	yes	characteristic
51	C₄⋊Dic₇₃	C₁₆	1	8	yes	complemented
52	C₄×Dic₇₃	—	2	2	no	—
53	C₄×Dic₇₃	—	2	2	no	—
54	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
55	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
56	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₇₃	—	2	2	no	—
58	C₂×C₈×C₇₃	—	2	2	no	—
59	C₂×C₈×C₇₃	C₂×C₈	1	4	yes	—
60	C₄²×C₇₃	C₄²	1	2	yes	—
61	C₄²×C₇₃	C₂×C₈	1	2	yes	—
62	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
63	C₂×C₄×C₇₃	—	2	2	no	—
64	C₂×C₄×C₇₃	C₄×C₈	1	2	yes	—
65	C₂×C₄×C₇₃	C₄×C₈	1	2	yes	—
66	C₂×C₄×C₇₃	C₂×C₁₆	1	4	yes	—
67	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
68	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
69	C₂×C₄×C₇₃	C₄×Q₈	1	1	yes	characteristic
70	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
71	C₂×C₄×C₇₃	#131	1	1	yes	characteristic
72	C₂×C₄×C₇₃	#131	1	1	yes	characteristic
73	C₂×Dic₇₃	—	2	4	no	—
74	C₂×Dic₇₃	—	2	4	no	—
75	C₂×Dic₇₃	C₂×C₁₆	1	8	yes	—
76	C₈×C₇₃	—	2	4	no	—
77	#10596831	—	2	2	no	—
78	#10596831	C₂×C₄⋊C₄	1	1	yes	characteristic
79	#10596831	#117	1	1	yes	characteristic
80	#10596831	C₄×D₄	1	1	yes	characteristic
81	#10596831	C₄×Q₈	1	1	yes	characteristic
82	C₂²×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
83	C₂²×C₇₃	C₈×D₄	1	1	yes	characteristic
84	C₂²×C₇₃	C₈×Q₈	1	1	yes	characteristic
85	C₂²×C₇₃	C₂²×C₁₆	1	2	yes	—
86	C₂²×C₇₃	#504	1	2	yes	—
87	C₄×C₇₃	—	2	2	no	—
88	C₄×C₇₃	C₄×C₁₆	1	4	yes	complemented
89	C₄×C₇₃	C₄×C₄⋊C₄	1	1	yes	characteristic
90	C₄×C₇₃	#405	1	1	yes	characteristic
91	C₄×C₇₃	C₈×D₄	1	1	yes	characteristic
92	C₄×C₇₃	C₈×Q₈	1	1	yes	characteristic
93	Dic₇₃	—	2	8	no	—
94	C₁₆×C₄⋊C₄	—	73	1	no	maximal, Sylow
95	C₂×C₇₃	C₈×C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₇₃	C₂×C₄×C₁₆	1	1	yes	characteristic
97	C₂×C₇₃	#2108	1	1	yes	characteristic
98	C₂×C₇₃	D₄×C₁₆	1	2	yes	—
99	C₂×C₇₃	Q₈×C₁₆	1	2	yes	—
100	C₂×C₄⋊C₁₆	—	73	1	no	—
101	C₂×C₄×C₁₆	—	73	1	no	—
102	C₂×C₄×C₁₆	—	73	2	no	—
103	C₈×C₄⋊C₄	—	73	1	no	—
104	#1374	—	73	2	no	—
105	C₇₃	C₁₆×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
106	C₂²×C₁₆	—	73	1	no	—
107	C₂²×C₁₆	—	73	1	no	—
108	C₂²×C₁₆	—	73	2	no	—
109	C₂×C₄⋊C₈	#64408	1	1	yes	characteristic
110	C₂×C₄×C₈	—	73	1	no	—
111	C₂×C₄×C₈	—	73	2	no	—
112	C₄×C₄⋊C₄	—	73	1	no	—
113	C₄⋊C₁₆	—	73	4	no	—
114	C₄×C₁₆	—	73	4	no	—
115	C₄×C₁₆	—	146	4	no	—
116	#336	—	73	2	no	—
117	C₂²×C₈	—	73	1	no	—
118	C₂²×C₈	—	73	1	no	—
119	C₂²×C₈	C₂×#64408	1	2	yes	—
120	C₂×C₄⋊C₄	—	73	2	no	—
121	C₂×C₄²	—	73	2	no	—
122	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
123	C₂×C₁₆	—	73	2	no	—
124	C₂×C₁₆	—	73	2	no	—
125	C₂×C₁₆	—	73	2	no	—
126	C₂×C₁₆	—	73	2	no	—
127	C₂×C₁₆	—	73	4	no	—
128	C₂×C₁₆	—	146	2	no	—
129	C₂×C₁₆	—	146	1	no	—
130	C₂×C₁₆	—	146	2	no	—
131	C₂×C₁₆	—	146	1	no	—
132	C₄⋊C₈	#10596833	1	4	yes	—
133	C₄×C₈	—	73	2	no	—
134	C₄×C₈	—	73	2	no	—
135	C₄×C₈	—	146	2	no	—
136	C₄×C₈	—	146	2	no	—
137	#95	—	73	2	no	—
138	C₂²×C₄	—	73	4	no	—
139	C₂²×C₄	#11924036	1	1	yes	characteristic
140	C₂²×C₄	#11924036	1	1	yes	characteristic
141	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	73	1	no	—
145	C₂×C₈	—	73	1	no	—
146	C₂×C₈	—	73	1	no	—
147	C₂×C₈	—	73	1	no	—
148	C₂×C₈	—	73	1	no	—
149	C₂×C₈	—	73	1	no	—
150	C₂×C₈	—	73	2	no	—
151	C₂×C₈	—	146	1	no	—
152	C₂×C₈	—	146	1	no	—
153	C₂×C₈	C₂×#10596833	1	4	yes	—
154	C₄⋊C₄	—	73	8	no	—
155	C₄²	—	146	2	no	—
156	C₄²	—	146	2	no	—
157	C₄²	C₄×#64408	1	2	yes	—
158	C₄²	C₂×#10596833	1	2	yes	—
159	C₁₆	—	73	4	no	—
160	C₁₆	—	146	4	no	—
161	C₁₆	—	146	2	no	—
162	C₂³	C₂×#11924036	1	1	yes	characteristic
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	73	8	no	—
165	C₂×C₄	—	146	4	no	—
166	C₂×C₄	—	146	4	no	—
167	C₂×C₄	C₄×#10596833	1	2	yes	—
168	C₂×C₄	C₄×#10596833	1	2	yes	—
169	C₂×C₄	#14556664	1	4	yes	—
170	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
171	C₂×C₄	D₄×#64408	1	1	yes	characteristic
172	C₂×C₄	Q₈×#64408	1	1	yes	characteristic
173	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
174	C₂×C₄	#14556678	1	1	yes	characteristic
175	C₂×C₄	#14556679	1	1	yes	characteristic
176	C₈	—	2	4	no	—
177	C₈	—	73	1	no	—
178	C₈	—	73	1	no	—
179	C₈	—	73	1	no	—
180	C₈	—	73	1	no	—
181	C₈	—	146	2	no	—
182	C₂²	C₂×C₄×#10596833	1	1	yes	characteristic
183	C₂²	#17989615	1	1	yes	characteristic
184	C₂²	#17989636	1	1	yes	characteristic
185	C₂²	C₂×#14556664	1	2	yes	—
186	C₂²	#17989646	1	2	yes	—
187	C₄	—	2	2	no	—
188	C₄	—	146	8	no	—
189	C₄	#17989473	1	4	yes	complemented
190	C₄	C₄⋊C₄×#64408	1	1	yes	characteristic
191	C₄	#17989552	1	1	yes	characteristic
192	C₄	D₄×#10596833	1	1	yes	characteristic
193	C₄	Q₈×#10596833	1	1	yes	characteristic
194	C₂	C₄⋊C₄×#10596833	1	1	yes	characteristic
195	C₂	C₂×#17989473	1	1	yes	characteristic
196	C₂	#22626142	1	1	yes	characteristic
197	C₂	#22626310	1	2	yes	—
198	C₂	#22626345	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#64408
commutator	C₂×C₇₃	C₂×C₄×C₁₆
Fitting	C₂×C₄⋊C₈×C₇₃	C₄
socle, prime radical	C₂³×C₇₃	C₂²×C₈
cosocle	C₂²×#10596831	C₂³
hypercenter	C₂×C₄⋊C₈	#64408
hypocenter	C₇₃	C₁₆×C₄⋊C₄
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
9. C₄⋊#11924031	C₄	semidirect
11. C₄×#11924031	C₄	semidirect
40. #11924031	C₄⋊C₄	semidirect
51. C₄⋊Dic₇₃	C₁₆	semidirect
88. C₄×C₇₃	C₄×C₁₆	semidirect
105. C₇₃	C₁₆×C₄⋊C₄	semidirect
189. C₄	#17989473	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#11924031	2
C₂×#17989473	2
C₂×C₄×#11924031	2
#22624970	2
#22609796	2
C₁₆×C₄⋊C₄	73