Subgroups — #34034316

Order 18688.

350 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#11924040	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#11924040	C₂	1	2	yes	maximal
3	C₂×C₄×#11924040	C₂	1	1	yes	maximal, characteristic
4	#22627001	C₂	1	2	yes	maximal
5	C₄⋊C₄×#10596833	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11924040	C₂²	1	2	yes	—
7	C₂²×#11924040	C₂²	1	1	yes	characteristic
8	C₂²×#11924040	C₂²	1	1	yes	characteristic
9	C₄⋊#11924040	—	2	2	no	—
10	C₄⋊#11924040	—	2	4	no	—
11	C₄×#11924040	—	2	2	no	—
12	C₂×C₄⋊#10596833	C₄	1	1	yes	characteristic
13	C₂×C₄×#10596833	C₂²	1	2	yes	—
14	C₂×C₄×#10596833	C₂²	1	1	yes	characteristic
15	#17989453	C₄	1	2	yes	—
16	C₄⋊C₄×Dic₇₃	C₄	1	1	yes	characteristic
17	C₂×#11924040	—	2	1	no	—
18	C₂×#11924040	—	2	1	no	—
19	C₂×#11924040	—	2	2	no	—
20	C₂×#11924040	—	2	2	no	—
21	C₂×#11924040	—	2	1	no	—
22	C₂×#11924040	—	2	1	no	—
23	C₂×#11924040	—	2	2	no	—
24	C₂×#11924040	D₄	1	2	yes	—
25	C₂×#11924040	Q₈	1	2	yes	—
26	C₂²×#10596833	C₂×C₄	1	2	yes	—
27	C₂²×#10596833	C₂×C₄	1	1	yes	characteristic
28	C₂²×#10596833	C₂³	1	1	yes	characteristic
29	C₄⋊#10596833	—	2	2	no	—
30	C₄×#10596833	—	2	2	no	—
31	C₄×#10596833	—	2	2	no	—
32	C₄×#10596833	D₄	1	2	yes	—
33	C₄×#10596833	Q₈	1	2	yes	—
34	C₂×C₄⋊Dic₇₃	C₈	1	1	yes	characteristic
35	C₂×C₄×Dic₇₃	C₂×C₄	1	2	yes	—
36	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
37	#14556498	C₈	1	2	yes	—
38	C₂×C₄⋊C₄×C₇₃	C₈	1	1	yes	characteristic
39	#11924040	—	2	2	no	—
40	#11924040	—	2	2	no	—
41	#11924040	—	4	2	no	—
42	C₂×#10596833	—	2	1	no	—
43	C₂×#10596833	—	2	1	no	—
44	C₂×#10596833	—	2	2	no	—
45	C₂×#10596833	—	2	2	no	—
46	C₂×#10596833	—	2	2	no	—
47	C₂×#10596833	C₄⋊C₄	1	2	yes	—
48	C₂×#10596833	C₂×D₄	1	1	yes	characteristic
49	C₂×#10596833	C₂×D₄	1	1	yes	characteristic
50	C₂×#10596833	C₂×Q₈	1	1	yes	characteristic
51	C₂×#10596833	C₂×Q₈	1	1	yes	characteristic
52	C₂×#10596833	#41	1	1	yes	characteristic
53	C₂×#10596833	#41	1	1	yes	characteristic
54	C₂²×Dic₇₃	C₂×C₈	1	2	yes	—
55	C₂²×Dic₇₃	C₂×C₈	1	1	yes	characteristic
56	C₂²×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
57	C₄⋊Dic₇₃	—	2	2	no	—
58	C₄×Dic₇₃	—	2	2	no	—
59	C₄×Dic₇₃	—	2	2	no	—
60	C₄×Dic₇₃	C₄⋊C₄	1	2	yes	—
61	C₄×Dic₇₃	M₁₆	1	2	yes	—
62	C₂²×C₄×C₇₃	C₂×C₈	1	2	yes	—
63	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
64	C₄⋊C₄×C₇₃	C₁₆	1	4	yes	complemented
65	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
66	C₂×C₄×C₇₃	—	2	2	no	—
67	C₂×C₄×C₇₃	—	2	2	no	—
68	C₂×C₄×C₇₃	C₄⋊C₈	1	2	yes	—
69	C₂×C₄×C₇₃	C₂×C₁₆	1	4	yes	—
70	C₂×C₄×C₇₃	C₂×C₁₆	1	2	yes	—
71	C₂×C₄×C₇₃	M₃₂	1	2	yes	—
72	C₂×Dic₇₃	—	2	1	no	—
73	C₂×Dic₇₃	—	2	1	no	—
74	C₂×Dic₇₃	—	2	2	no	—
75	C₂×Dic₇₃	—	2	2	no	—
76	C₂×Dic₇₃	—	2	2	no	—
77	C₂×Dic₇₃	C₄⋊C₈	1	2	yes	—
78	C₂×Dic₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂×Dic₇₃	C₄×D₄	1	1	yes	characteristic
80	C₂×Dic₇₃	C₄×Q₈	1	1	yes	characteristic
81	C₂×Dic₇₃	C₂×M₁₆	1	1	yes	characteristic
82	C₂×Dic₇₃	#131	1	1	yes	characteristic
83	C₂×Dic₇₃	#131	1	1	yes	characteristic
84	#10596833	—	2	2	no	—
85	#10596833	—	4	2	no	—
86	#10596833	#121	1	1	yes	characteristic
87	#10596833	C₂²⋊Q₈	1	1	yes	characteristic
88	#10596833	#125	1	1	yes	characteristic
89	#10596833	C₄⋊Q₈	1	1	yes	characteristic
90	C₂²×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
91	C₂²×C₇₃	C₈×D₄	1	1	yes	characteristic
92	C₂²×C₇₃	C₈×Q₈	1	1	yes	characteristic
93	C₂²×C₇₃	C₂²×C₁₆	1	1	yes	characteristic
94	C₂²×C₇₃	C₂×M₃₂	1	1	yes	characteristic
95	C₂²×C₇₃	#504	1	1	yes	characteristic
96	C₂²×C₇₃	#504	1	1	yes	characteristic
97	C₄×C₇₃	—	2	4	no	—
98	C₄×C₇₃	C₄⋊C₁₆	1	4	yes	complemented
99	Dic₇₃	—	2	2	no	—
100	Dic₇₃	—	4	2	no	—
101	Dic₇₃	#389	1	1	yes	characteristic
102	Dic₇₃	#424	1	1	yes	characteristic
103	Dic₇₃	#436	1	1	yes	characteristic
104	Dic₇₃	#446	1	1	yes	characteristic
105	#13159	—	73	1	no	maximal, Sylow
106	C₂×C₇₃	#1939	1	1	yes	characteristic
107	C₂×C₇₃	C₂×C₄⋊C₁₆	1	1	yes	characteristic
108	C₂×C₇₃	#2151	1	1	yes	characteristic
109	C₂×C₇₃	D₄×C₁₆	1	1	yes	characteristic
110	C₂×C₇₃	#2169	1	1	yes	characteristic
111	C₂×C₇₃	Q₈×C₁₆	1	1	yes	characteristic
112	C₂×C₇₃	#2183	1	1	yes	characteristic
113	C₂×C₄⋊C₁₆	—	73	1	no	—
114	C₂×C₄⋊C₁₆	—	73	2	no	—
115	C₂×C₄×C₁₆	—	73	1	no	—
116	C₈×C₄⋊C₄	—	73	1	no	—
117	#1374	—	73	2	no	—
118	C₇₃	#13159	1	1	yes	characteristic, complemented, Sylow
119	C₂²×C₁₆	—	73	1	no	—
120	C₂²×C₁₆	—	73	1	no	—
121	C₂²×C₁₆	—	73	2	no	—
122	C₂×C₄⋊C₈	—	73	1	no	—
123	C₂×C₄×C₈	—	73	1	no	—
124	C₂×C₄×C₈	—	73	2	no	—
125	C₄×C₄⋊C₄	—	73	1	no	—
126	C₄⋊C₁₆	—	146	2	no	—
127	C₄⋊C₁₆	—	146	4	no	—
128	C₄×C₁₆	—	146	2	no	—
129	#336	—	73	2	no	—
130	C₂²×C₈	—	73	1	no	—
131	C₂²×C₈	—	73	1	no	—
132	C₂²×C₈	—	73	2	no	—
133	C₂×C₄⋊C₄	—	73	1	no	—
134	C₂×C₄⋊C₄	#10596842	1	1	yes	characteristic
135	C₂×C₄²	—	73	1	no	—
136	C₂×C₄²	—	73	2	no	—
137	C₂×C₁₆	—	73	2	no	—
138	C₂×C₁₆	—	73	2	no	—
139	C₂×C₁₆	—	146	2	no	—
140	C₂×C₁₆	—	146	2	no	—
141	C₂×C₁₆	—	146	2	no	—
142	C₂×C₁₆	—	146	1	no	—
143	C₂×C₁₆	—	146	1	no	—
144	C₂×C₁₆	—	146	1	no	—
145	C₂×C₁₆	—	146	1	no	—
146	C₄⋊C₈	—	146	2	no	—
147	C₄×C₈	—	73	2	no	—
148	C₄×C₈	—	73	2	no	—
149	C₄×C₈	—	146	2	no	—
150	C₄×C₈	—	146	2	no	—
151	#95	—	73	2	no	—
152	C₂²×C₄	—	73	1	no	—
153	C₂²×C₄	—	73	1	no	—
154	C₂²×C₄	—	73	2	no	—
155	C₂²×C₄	C₂×#10596842	1	2	yes	—
156	C₂²×C₄	C₂×#10596842	1	1	yes	characteristic
157	C₂×C₈	—	73	1	no	—
158	C₂×C₈	—	73	1	no	—
159	C₂×C₈	—	73	1	no	—
160	C₂×C₈	—	73	1	no	—
161	C₂×C₈	—	73	1	no	—
162	C₂×C₈	—	73	1	no	—
163	C₂×C₈	—	73	2	no	—
164	C₂×C₈	—	146	2	no	—
165	C₂×C₈	—	146	2	no	—
166	C₂×C₈	—	146	2	no	—
167	C₂×C₈	—	146	1	no	—
168	C₂×C₈	—	146	1	no	—
169	C₄⋊C₄	—	146	2	no	—
170	C₄⋊C₄	#11924040	1	4	yes	complemented
171	C₄²	—	73	2	no	—
172	C₄²	—	73	2	no	—
173	C₄²	—	146	2	no	—
174	C₄²	—	146	2	no	—
175	C₁₆	—	292	2	no	—
176	C₁₆	—	146	2	no	—
177	C₁₆	—	146	2	no	—
178	C₂³	C₂²×#10596842	1	1	yes	characteristic
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	73	1	no	—
182	C₂×C₄	—	73	1	no	—
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	—	73	2	no	—
188	C₂×C₄	—	146	2	no	—
189	C₂×C₄	—	146	2	no	—
190	C₂×C₄	—	146	2	no	—
191	C₂×C₄	—	146	1	no	—
192	C₂×C₄	—	146	1	no	—
193	C₂×C₄	C₄⋊#10596842	1	2	yes	—
194	C₂×C₄	C₂×#11924040	1	4	yes	—
195	C₂×C₄	C₂×#11924040	1	2	yes	—
196	C₂×C₄	#14556688	1	2	yes	—
197	C₈	—	292	2	no	—
198	C₈	—	73	1	no	—
199	C₈	—	73	1	no	—
200	C₈	—	73	1	no	—
201	C₈	—	73	1	no	—
202	C₈	—	146	2	no	—
203	C₂²	C₂×C₄⋊#10596842	1	1	yes	characteristic
204	C₂²	D₄×#10596842	1	1	yes	characteristic
205	C₂²	Q₈×#10596842	1	1	yes	characteristic
206	C₂²	C₂²×#11924040	1	1	yes	characteristic
207	C₂²	C₂×#14556688	1	1	yes	characteristic
208	C₂²	#17989708	1	1	yes	characteristic
209	C₂²	#17989709	1	1	yes	characteristic
210	C₄	—	2	4	no	—
211	C₄	—	292	2	no	—
212	C₄	—	73	1	no	—
213	C₄	—	73	1	no	—
214	C₄	—	73	1	no	—
215	C₄	—	73	1	no	—
216	C₄	—	146	2	no	—
217	C₄	C₄⋊#11924040	1	4	yes	complemented
218	C₂	#22627080	1	1	yes	characteristic
219	C₂	C₂×C₄⋊#11924040	1	1	yes	characteristic
220	C₂	#22627123	1	1	yes	characteristic
221	C₂	D₄×#11924040	1	1	yes	characteristic
222	C₂	#22627146	1	1	yes	characteristic
223	C₂	Q₈×#11924040	1	1	yes	characteristic
224	C₂	#22627156	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×#10596842
commutator	C₂²×C₇₃	C₂²×C₁₆
Fitting	C₂×C₄⋊C₄×C₇₃	C₈
socle, prime radical	C₂³×C₇₃	C₂²×C₈
cosocle	C₂²×#10596833	C₂³
hypercenter	C₂×C₄⋊C₄	#10596842
hypocenter	C₇₃	#13159
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
64. C₄⋊C₄×C₇₃	C₁₆	semidirect
98. C₄×C₇₃	C₄⋊C₁₆	semidirect
118. C₇₃	#13159	semidirect
170. C₄⋊C₄	#11924040	semidirect
217. C₄	C₄⋊#11924040	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#11924040	2
C₂×C₄×#11924040	2
#22627001	2
C₄⋊C₄×#10596833	2
#13159	73