Subgroups — #34004059

Order 18688.

350 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	C₄×#14556661	C₂	1	1	yes	maximal, characteristic
2	#22625479	C₂	1	1	yes	maximal, characteristic
3	#22625141	C₂	1	2	yes	maximal
4	#22624763	C₂	1	2	yes	maximal
5	C₈²×D₇₃	C₂	1	1	yes	maximal, characteristic
6	#17989606	C₂²	1	2	yes	—
7	C₂×#14556661	C₄	1	2	yes	—
8	C₄×#11924038	C₂²	1	2	yes	—
9	#17989463	C₄	1	2	yes	—
10	#17989451	C₄	1	2	yes	—
11	#17989448	C₄	1	2	yes	—
12	#17989429	—	2	2	no	—
13	#17989428	—	2	2	no	—
14	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
15	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
16	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
17	C₈×#10596831	C₄	1	1	yes	characteristic
18	C₈×#10596831	C₄	1	1	yes	characteristic
19	C₈×#10596831	C₄	1	1	yes	characteristic
20	C₈²×C₇₃	C₄	1	1	yes	characteristic
21	C₂×#11924038	C₂×C₄	1	4	yes	—
22	#14556661	—	2	4	no	—
23	C₄⋊#10596833	C₂×C₄	1	4	yes	—
24	#14556648	—	2	2	no	—
25	#14556646	—	2	2	no	—
26	C₄×#10596833	—	2	2	no	—
27	C₄×#10596833	—	2	2	no	—
28	C₂×C₈×D₇₃	—	2	1	no	—
29	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈×D₇₃	Q₈	1	1	yes	characteristic
32	C₂×C₈×D₇₃	Q₈	1	1	yes	characteristic
33	C₄²×D₇₃	C₂³	1	1	yes	characteristic
34	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
35	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
36	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
37	C₈×Dic₇₃	—	2	1	no	—
38	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
39	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
40	C₈×Dic₇₃	D₄	1	1	yes	characteristic
41	C₈×Dic₇₃	D₄	1	1	yes	characteristic
42	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
45	#11924038	—	2	4	no	—
46	#11924038	—	2	4	no	—
47	C₂×#10596833	—	2	4	no	—
48	C₂×#10596833	—	2	4	no	—
49	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×D₇₃	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₄×D₇₃	#41	1	1	yes	characteristic
52	C₂×#10596831	—	2	1	no	—
53	C₂×#10596831	C₄²	1	1	yes	characteristic
54	C₂×#10596831	C₄²	1	1	yes	characteristic
55	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
56	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
57	C₈×D₇₃	—	2	4	no	—
58	C₈×D₇₃	—	2	2	no	—
59	C₈×D₇₃	—	2	2	no	—
60	C₈×D₇₃	—	2	2	no	—
61	C₈×D₇₃	M₁₆	1	4	yes	—
62	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
63	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
64	C₄×Dic₇₃	#41	1	1	yes	characteristic
65	C₂×C₈×C₇₃	—	2	1	no	—
66	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
67	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
68	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
70	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×D₇₃	C₄×Q₈	1	1	yes	characteristic
72	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
73	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
75	C₂×Dic₇₃	C₄×D₄	1	1	yes	characteristic
76	C₂×Dic₇₃	C₄×D₄	1	1	yes	characteristic
77	C₂×Dic₇₃	C₄×Q₈	1	1	yes	characteristic
78	C₄×D₇₃	—	2	1	no	—
79	C₄×D₇₃	—	2	1	no	—
80	C₄×D₇₃	—	2	2	no	—
81	C₄×D₇₃	C₂×M₁₆	1	2	yes	—
82	C₄×D₇₃	#131	1	2	yes	—
83	#10596833	—	4	4	no	—
84	#10596833	—	4	4	no	—
85	C₈×C₇₃	—	2	2	no	—
86	C₈×C₇₃	—	2	1	no	—
87	C₈×C₇₃	—	2	1	no	—
88	C₈×C₇₃	#97	1	2	yes	—
89	C₈×C₇₃	#108	1	2	yes	complemented
90	#10596831	—	2	2	no	—
91	#10596831	—	2	1	no	—
92	#10596831	—	2	1	no	—
93	#10596831	#97	1	2	yes	—
94	#10596831	#108	1	2	yes	complemented
95	C₂²×C₇₃	C₄×C₄⋊C₄	1	1	yes	characteristic
96	C₂×D₇₃	—	2	1	no	—
97	C₂×D₇₃	—	2	1	no	—
98	C₂×D₇₃	#446	1	2	yes	—
99	C₄×C₇₃	—	2	1	no	—
100	C₄×C₇₃	C₂×#97	1	1	yes	characteristic
101	C₄×C₇₃	#405	1	1	yes	characteristic
102	C₄×C₇₃	C₂×#108	1	1	yes	characteristic
103	C₄×C₇₃	C₂×#108	1	1	yes	characteristic
104	Dic₇₃	—	2	1	no	—
105	Dic₇₃	#436	1	2	yes	—
106	Dic₇₃	#443	1	2	yes	—
107	#8293	—	73	1	no	maximal, Sylow
108	C₂×C₇₃	#1770	1	1	yes	characteristic
109	C₂×C₇₃	#1772	1	1	yes	characteristic
110	C₂×C₇₃	C₄×#108	1	1	yes	characteristic
111	D₇₃	—	2	2	no	—
112	C₄×#108	—	73	1	no	—
113	#1564	—	73	1	no	—
114	#1451	—	73	2	no	—
115	C₂×C₈²	—	73	1	no	—
116	#1278	—	73	2	no	—
117	C₇₃	#8293	1	1	yes	characteristic, complemented, Sylow
118	#431	—	73	2	no	—
119	C₂×#108	—	73	2	no	—
120	C₄×M₁₆	—	73	2	no	—
121	C₂×C₄×C₈	—	73	1	no	—
122	C₂×C₄×C₈	—	73	1	no	—
123	C₂×C₄×C₈	—	73	1	no	—
124	#341	—	73	2	no	—
125	#335	—	73	2	no	—
126	#334	—	73	2	no	—
127	#322	—	146	2	no	—
128	#322	—	146	2	no	—
129	C₈²	—	73	1	no	—
130	C₈²	—	73	1	no	—
131	C₈²	—	73	1	no	—
132	C₈²	#64408	1	1	yes	characteristic
133	C₂×M₁₆	—	73	4	no	—
134	C₂²×C₈	—	73	1	no	—
135	C₂²×C₈	—	73	1	no	—
136	C₂²×C₈	—	73	1	no	—
137	C₂²×C₈	—	73	1	no	—
138	C₂²×C₈	—	146	1	no	—
139	C₂×C₄²	—	73	1	no	—
140	#108	—	146	4	no	—
141	C₄⋊C₈	—	73	4	no	—
142	C₂²⋊C₈	—	146	2	no	—
143	#97	—	146	2	no	—
144	C₄×C₈	—	73	1	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	—	146	1	no	—
152	C₄×C₈	—	146	2	no	—
153	C₄×C₈	—	146	2	no	—
154	C₄×C₈	C₂×#64408	1	1	yes	characteristic
155	C₄×C₈	C₂×#64408	1	1	yes	characteristic
156	C₄×C₈	C₂×#64408	1	1	yes	characteristic
157	C₂²×C₄	—	73	1	no	—
158	C₂²×C₄	—	73	1	no	—
159	C₂²×C₄	—	73	1	no	—
160	M₁₆	—	146	4	no	—
161	M₁₆	—	146	4	no	—
162	C₂×C₈	—	2	1	no	—
163	C₂×C₈	—	73	1	no	—
164	C₂×C₈	—	73	1	no	—
165	C₂×C₈	—	73	4	no	—
166	C₂×C₈	—	73	1	no	—
167	C₂×C₈	—	73	1	no	—
168	C₂×C₈	—	146	2	no	—
169	C₂×C₈	—	146	2	no	—
170	C₂×C₈	—	146	4	no	—
171	C₂×C₈	—	146	4	no	—
172	C₂×C₈	—	146	4	no	—
173	C₂×C₈	—	146	1	no	—
174	C₂×C₈	—	146	2	no	—
175	C₂×C₈	C₄×#64408	1	1	yes	characteristic
176	C₂×C₈	C₄×#64408	1	1	yes	characteristic
177	C₂×C₈	C₄⋊#64408	1	1	yes	characteristic
178	C₂×C₈	C₄⋊#64408	1	1	yes	characteristic
179	C₄²	—	73	1	no	—
180	C₄²	—	73	1	no	—
181	C₄²	—	73	1	no	—
182	C₄²	C₂²×#64408	1	1	yes	characteristic
183	C₂³	—	73	1	no	—
184	C₂×C₄	—	73	1	no	—
185	C₂×C₄	—	73	1	no	—
186	C₂×C₄	—	73	2	no	—
187	C₂×C₄	—	73	2	no	—
188	C₂×C₄	—	73	1	no	—
189	C₂×C₄	—	146	2	no	—
190	C₂×C₄	—	146	1	no	—
191	C₂×C₄	—	146	1	no	—
192	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
193	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
194	C₂×C₄	#14556670	1	1	yes	characteristic
195	C₈	—	2	2	no	—
196	C₈	—	2	1	no	—
197	C₈	—	2	1	no	—
198	C₈	—	292	4	no	—
199	C₈	—	292	4	no	—
200	C₈	—	73	2	no	—
201	C₈	—	73	2	no	—
202	C₈	—	146	2	no	—
203	C₈	—	146	1	no	—
204	C₈	—	146	1	no	—
205	C₈	#14556645	1	2	yes	—
206	C₈	#14556661	1	2	yes	complemented
207	C₂²	—	73	2	no	—
208	C₂²	—	146	1	no	—
209	C₂²	—	146	1	no	—
210	C₂²	C₄×C₄⋊#64408	1	1	yes	characteristic
211	C₄	—	2	1	no	—
212	C₄	—	73	2	no	—
213	C₄	—	73	2	no	—
214	C₄	—	146	1	no	—
215	C₄	C₂×#14556645	1	1	yes	characteristic
216	C₄	#17989550	1	1	yes	characteristic
217	C₄	C₂×#14556661	1	1	yes	characteristic
218	C₄	D₇₃⋊#108	1	1	yes	characteristic
219	C₂	—	146	2	no	—
220	C₂	#22625725	1	1	yes	characteristic
221	C₂	#22625733	1	1	yes	characteristic
222	C₂	C₄×#14556661	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄⋊#64408
commutator	C₂×C₄×C₇₃	C₂×C₄²
Frattini	C₄²	C₂²×#64408
Fitting	C₈²×C₇₃	C₄
socle	C₂²×C₇₃	C₄×C₄⋊C₄
prime radical	C₂²×D₇₃	C₄×Q₈
cosocle	C₄²×D₇₃	C₂³
hypercenter	C₈²	#64408
hypocenter	C₇₃	#8293
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈²

Complements

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

N	H ≅ G/N	type
89. C₈×C₇₃	#108	semidirect
94. #10596831	#108	semidirect
117. C₇₃	#8293	semidirect
206. C₈	#14556661	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14556661	2
#22625479	2
#22625141	2
#22624763	2
C₈²×D₇₃	2
#8293	73