Subgroups — #34000244

Order 18688.

286 conjugacy classes of subgroups.

The subgroup lattice has 216 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×C₄×#11924031	C₂	1	1	yes	maximal, characteristic
3	#22609796	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11924031	—	2	1	no	—
5	C₂²×#11924031	—	2	1	no	—
6	C₄⋊#11924031	—	2	1	no	—
7	C₄⋊#11924031	C₄	1	2	yes	complemented
8	C₄×#11924031	—	2	1	no	—
9	C₄×#11924031	C₄	1	2	yes	complemented
10	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₇₃	—	2	1	no	—
12	C₄×C₄⋊Dic₇₃	C₄	1	1	yes	characteristic
13	#17988341	—	2	1	no	—
14	C₂×C₄⋊C₈×C₇₃	C₄	1	1	yes	characteristic
15	C₂×#11924031	—	2	2	no	—
16	C₂×#11924031	—	2	2	no	—
17	C₂×#11924031	—	2	2	no	—
18	C₂×#11924031	—	2	2	no	—
19	C₂×#11924031	—	4	1	no	—
20	C₂×#11924031	—	4	1	no	—
21	C₂²×#10596831	D₄	1	1	yes	characteristic
22	C₂²×#10596831	D₄	1	1	yes	characteristic
23	C₂×C₄⋊Dic₇₃	C₈	1	2	yes	—
24	C₂×C₄×Dic₇₃	—	2	1	no	—
25	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
26	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
27	C₄×#10596831	D₄	1	1	yes	characteristic
28	C₄×#10596831	Q₈	1	1	yes	characteristic
29	C₈×Dic₇₃	—	4	1	no	—
30	C₈×Dic₇₃	—	4	1	no	—
31	#14556498	—	2	1	no	—
32	C₂²×C₈×C₇₃	—	2	1	no	—
33	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
34	C₄⋊C₈×C₇₃	—	2	1	no	—
35	C₄⋊C₈×C₇₃	C₈	1	2	yes	—
36	#11924031	—	2	4	no	—
37	#11924031	—	4	2	no	—
38	C₂²×Dic₇₃	—	2	2	no	—
39	C₂×#10596831	—	2	1	no	—
40	C₂×#10596831	—	2	1	no	—
41	C₂×#10596831	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×#10596831	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
44	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
45	C₂×#10596831	D₈	1	1	yes	characteristic
46	C₂×#10596831	SD₁₆	1	1	yes	characteristic
47	C₂×#10596831	SD₁₆	1	1	yes	characteristic
48	C₂×#10596831	Q₁₆	1	1	yes	characteristic
49	C₄⋊Dic₇₃	—	2	2	no	—
50	C₄⋊Dic₇₃	C₁₆	1	4	yes	complemented
51	C₄×Dic₇₃	—	4	1	no	—
52	C₄×Dic₇₃	—	4	1	no	—
53	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
55	C₂²×C₄×C₇₃	M₁₆	1	1	yes	characteristic
56	C₂×C₈×C₇₃	—	2	2	no	—
57	C₂×C₈×C₇₃	—	4	1	no	—
58	C₂×C₈×C₇₃	—	4	1	no	—
59	C₄²×C₇₃	C₄²	1	1	yes	characteristic
60	C₄²×C₇₃	C₄⋊C₄	1	1	yes	characteristic
61	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
62	C₄²×C₇₃	M₁₆	1	1	yes	characteristic
63	C₂³×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
64	C₂×C₄×C₇₃	—	2	2	no	—
65	C₂×C₄×C₇₃	#95	1	1	yes	characteristic
66	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
67	C₂×C₄×C₇₃	#97	1	1	yes	characteristic
68	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
69	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
70	C₂×C₄×C₇₃	#103	1	1	yes	characteristic
71	C₂×C₄×C₇₃	#104	1	1	yes	characteristic
72	C₂×C₄×C₇₃	#104	1	1	yes	characteristic
73	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
74	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
75	C₂×C₄×C₇₃	C₂×C₁₆	1	2	yes	—
76	C₂×C₄×C₇₃	M₃₂	1	2	yes	—
77	C₂×Dic₇₃	—	2	4	no	—
78	C₂×Dic₇₃	—	4	2	no	—
79	C₂×Dic₇₃	—	4	2	no	—
80	C₈×C₇₃	—	4	2	no	—
81	#10596831	—	2	1	no	—
82	#10596831	—	2	1	no	—
83	#10596831	#102	1	1	yes	characteristic
84	#10596831	#103	1	1	yes	characteristic
85	#10596831	#106	1	1	yes	characteristic
86	#10596831	#107	1	1	yes	characteristic
87	C₂²×C₇₃	D₄⋊C₈	1	1	yes	characteristic
88	C₂²×C₇₃	Q₈⋊C₈	1	1	yes	characteristic
89	C₂²×C₇₃	#336	1	1	yes	characteristic
90	C₂²×C₇₃	C₂²⋊C₁₆	1	2	yes	—
91	C₂²×C₇₃	#350	1	2	yes	—
92	C₄×C₇₃	—	2	1	no	—
93	C₄×C₇₃	—	2	1	no	—
94	C₄×C₇₃	D₄⋊C₈	1	1	yes	characteristic
95	C₄×C₇₃	Q₈⋊C₈	1	1	yes	characteristic
96	C₄×C₇₃	#339	1	1	yes	characteristic
97	C₄×C₇₃	#340	1	1	yes	characteristic
98	C₄×C₇₃	C₄×C₁₆	1	1	yes	characteristic
99	C₄×C₇₃	#346	1	1	yes	characteristic
100	C₄×C₇₃	C₄⋊C₁₆	1	2	yes	—
101	Dic₇₃	—	4	4	no	—
102	#7167	—	73	1	no	maximal, Sylow
103	C₂×C₇₃	#1279	1	1	yes	characteristic
104	C₂×C₇₃	D₄⋊C₁₆	1	2	yes	—
105	C₂×C₇₃	Q₈⋊C₁₆	1	2	yes	—
106	C₂×C₇₃	#1374	1	1	yes	characteristic
107	C₂×C₇₃	#1379	1	1	yes	characteristic
108	C₂×C₄⋊C₁₆	—	73	1	no	—
109	C₂×C₄×C₁₆	—	73	1	no	—
110	C₈×C₄⋊C₄	—	73	1	no	—
111	C₇₃	#7167	1	1	yes	characteristic, complemented, Sylow
112	C₂²×C₁₆	—	146	1	no	—
113	C₂²×C₁₆	—	146	1	no	—
114	C₂×C₄⋊C₈	#64408	1	1	yes	characteristic
115	C₂×C₄×C₈	—	73	1	no	—
116	C₂×C₄×C₈	—	146	1	no	—
117	C₄×C₄⋊C₄	—	73	1	no	—
118	C₄⋊C₁₆	—	73	2	no	—
119	C₄⋊C₁₆	—	146	1	no	—
120	C₄×C₁₆	—	73	2	no	—
121	C₄×C₁₆	—	146	1	no	—
122	#336	—	146	1	no	—
123	C₂²×C₈	—	2	1	no	—
124	C₂²×C₈	—	73	1	no	—
125	C₂²×C₈	—	73	1	no	—
126	C₂×C₄⋊C₄	—	73	2	no	—
127	C₂×C₄²	—	146	1	no	—
128	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
129	C₂×C₁₆	—	292	1	no	—
130	C₂×C₁₆	—	292	1	no	—
131	C₂×C₁₆	—	146	2	no	—
132	C₂×C₁₆	—	146	2	no	—
133	C₂×C₁₆	—	146	2	no	—
134	C₂×C₁₆	—	146	2	no	—
135	C₄⋊C₈	—	2	1	no	—
136	C₄⋊C₈	#10596833	1	2	yes	—
137	C₄×C₈	—	292	1	no	—
138	C₄×C₈	—	292	1	no	—
139	C₄×C₈	—	73	1	no	—
140	C₄×C₈	—	73	1	no	—
141	C₄×C₈	—	73	1	no	—
142	C₄×C₈	—	73	1	no	—
143	#95	—	146	1	no	—
144	C₂²×C₄	—	146	2	no	—
145	C₂²×C₄	C₂²⋊#64408	1	1	yes	characteristic
146	C₂²×C₄	#11924036	1	1	yes	characteristic
147	C₂²×C₄	#11924038	1	1	yes	characteristic
148	C₂×C₈	—	2	2	no	—
149	C₂×C₈	—	4	1	no	—
150	C₂×C₈	—	4	1	no	—
151	C₂×C₈	—	73	1	no	—
152	C₂×C₈	—	73	1	no	—
153	C₂×C₈	—	73	1	no	—
154	C₂×C₈	—	73	1	no	—
155	C₂×C₈	—	73	1	no	—
156	C₂×C₈	—	73	1	no	—
157	C₂×C₈	—	73	1	no	—
158	C₂×C₈	—	73	1	no	—
159	C₂×C₈	—	146	1	no	—
160	C₂×C₈	—	146	1	no	—
161	C₄⋊C₄	—	73	4	no	—
162	C₄⋊C₄	—	146	2	no	—
163	C₄²	—	292	1	no	—
164	C₄²	—	292	1	no	—
165	C₄²	C₄×#64408	1	1	yes	characteristic
166	C₄²	C₄⋊#64408	1	1	yes	characteristic
167	C₄²	C₂×#10596833	1	1	yes	characteristic
168	C₄²	#11924037	1	1	yes	characteristic
169	C₁₆	—	292	2	no	—
170	C₁₆	—	146	4	no	—
171	C₂³	#14556648	1	1	yes	characteristic
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	292	2	no	—
174	C₂×C₄	—	292	2	no	—
175	C₂×C₄	—	146	4	no	—
176	C₂×C₄	(C₂×#64408)⋊C₄	1	1	yes	characteristic
177	C₂×C₄	C₄×#10596833	1	1	yes	characteristic
178	C₂×C₄	#14556646	1	1	yes	characteristic
179	C₂×C₄	C₂²⋊#10596833	1	1	yes	characteristic
180	C₂×C₄	#14556653	1	1	yes	characteristic
181	C₂×C₄	#14556654	1	1	yes	characteristic
182	C₂×C₄	#14556655	1	1	yes	characteristic
183	C₂×C₄	#14556656	1	1	yes	characteristic
184	C₂×C₄	C₄⋊#10596833	1	1	yes	characteristic
185	C₂×C₄	#14556658	1	1	yes	characteristic
186	C₂×C₄	#14556664	1	2	yes	—
187	C₂×C₄	#14556666	1	2	yes	—
188	C₈	—	4	2	no	—
189	C₈	—	73	1	no	—
190	C₈	—	73	1	no	—
191	C₈	—	73	1	no	—
192	C₈	—	73	1	no	—
193	C₈	—	146	1	no	—
194	C₈	—	146	1	no	—
195	C₂²	#17989435	1	1	yes	characteristic
196	C₂²	#17989437	1	1	yes	characteristic
197	C₂²	#17989453	1	1	yes	characteristic
198	C₂²	#17989482	1	2	yes	—
199	C₂²	#17989486	1	2	yes	—
200	C₄	—	2	1	no	—
201	C₄	—	2	1	no	—
202	C₄	—	292	4	no	—
203	C₄	D₄⋊#10596833	1	1	yes	characteristic
204	C₄	Q₈⋊#10596833	1	1	yes	characteristic
205	C₄	#17989457	1	1	yes	characteristic
206	C₄	#17989460	1	1	yes	characteristic
207	C₄	#17989473	1	1	yes	characteristic
208	C₄	#17989476	1	1	yes	characteristic
209	C₄	#17989508	1	2	yes	—
210	C₂	#22624767	1	1	yes	characteristic
211	C₂	#22624882	1	2	yes	—
212	C₂	#22624898	1	2	yes	—
213	C₂	#22624970	1	1	yes	characteristic
214	C₂	#22624986	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²⋊#64408
commutator	C₄×C₇₃	C₄×C₁₆
Frattini	C₂×C₄²	C₂×#64408
Fitting	C₂×C₄⋊C₈×C₇₃	C₄
socle, prime radical	C₂³×C₇₃	C₂²⋊C₈
cosocle	C₂×C₄×#10596831	C₂²
hypercenter	C₂×C₄⋊C₈	#64408
hypocenter	C₇₃	#7167
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
7. C₄⋊#11924031	C₄	semidirect
9. C₄×#11924031	C₄	semidirect
50. C₄⋊Dic₇₃	C₁₆	semidirect
111. C₇₃	#7167	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×C₄×#11924031	2
#22609796	2
#7167	73