Subgroups — #34007472

Order 18688.

398 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22626028	C₂	1	2	yes	maximal
2	C₂×#17989448	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×#10596833	C₂	1	1	yes	maximal, characteristic
4	#22624767	C₂	1	2	yes	maximal
5	#106×Dic₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#10596833	—	2	2	no	—
7	C₂×C₄⋊#10596833	C₂²	1	2	yes	—
8	C₂×C₄×#10596833	C₂²	1	1	yes	characteristic
9	C₂×C₄×#10596833	C₂²	1	1	yes	characteristic
10	#17989453	—	2	2	no	—
11	#17989448	—	2	2	no	—
12	C₈×#10596833	—	2	2	no	—
13	C₂×#14556523	C₄	1	1	yes	characteristic
14	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×Dic₇₃	C₂²	1	2	yes	—
16	#17988349	C₄	1	2	yes	—
17	C₂×#106×C₇₃	C₄	1	1	yes	characteristic
18	C₂²×#10596833	—	2	2	no	—
19	C₂²×#10596833	—	2	1	no	—
20	C₂²×#10596833	D₄	1	2	yes	—
21	C₄⋊#10596833	—	2	2	no	—
22	C₄⋊#10596833	—	2	2	no	—
23	C₄⋊#10596833	—	4	4	no	—
24	C₄×#10596833	—	2	1	no	—
25	C₄×#10596833	—	2	1	no	—
26	C₄×#10596833	—	2	1	no	—
27	C₄×#10596833	Q₈	1	2	yes	—
28	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄⋊Dic₇₃	C₂×C₄	1	2	yes	—
30	C₂×C₄×Dic₇₃	—	2	2	no	—
31	C₂×C₄×Dic₇₃	C₂³	1	1	yes	characteristic
32	#14556523	—	2	2	no	—
33	C₈×Dic₇₃	D₄	1	2	yes	—
34	C₈×Dic₇₃	Q₈	1	2	yes	—
35	#14556498	—	2	2	no	—
36	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
38	#106×C₇₃	C₈	1	4	yes	complemented
39	C₂×#10596833	—	2	4	no	—
40	C₂×#10596833	—	2	2	no	—
41	C₂×#10596833	—	2	2	no	—
42	C₂×#10596833	—	2	2	no	—
43	C₂×#10596833	—	4	2	no	—
44	C₂×#10596833	—	4	2	no	—
45	C₂×#10596833	—	4	2	no	—
46	C₂×#10596833	SD₁₆	1	4	yes	—
47	C₂²×Dic₇₃	—	2	2	no	—
48	C₂²×Dic₇₃	C₂×D₄	1	1	yes	characteristic
49	C₂²×Dic₇₃	#41	1	1	yes	characteristic
50	C₂×#10596831	—	2	1	no	—
51	C₂×#10596831	—	2	1	no	—
52	C₂×#10596831	C₄⋊C₄	1	2	yes	—
53	C₄⋊Dic₇₃	—	2	2	no	—
54	C₄⋊Dic₇₃	—	2	2	no	—
55	C₄×Dic₇₃	—	4	2	no	—
56	C₄×Dic₇₃	—	4	2	no	—
57	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
58	C₄×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
59	C₄×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
60	C₄×Dic₇₃	#41	1	1	yes	characteristic
61	C₂²×C₄×C₇₃	—	2	2	no	—
62	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
64	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
65	C₂×C₈×C₇₃	M₁₆	1	2	yes	—
66	C₄⋊C₄×C₇₃	—	2	2	no	—
67	C₄⋊C₄×C₇₃	C₂×C₈	1	4	yes	—
68	C₂³×C₇₃	C₄×D₄	1	1	yes	characteristic
69	C₂×C₄×C₇₃	—	2	4	no	—
70	C₂×C₄×C₇₃	—	4	2	no	—
71	C₂×C₄×C₇₃	—	4	2	no	—
72	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₇₃	C₄×Q₈	1	1	yes	characteristic
74	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
75	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
76	C₂×C₄×C₇₃	#131	1	1	yes	characteristic
77	C₂×C₄×C₇₃	#131	1	1	yes	characteristic
78	C₂×Dic₇₃	—	2	2	no	—
79	C₂×Dic₇₃	—	4	2	no	—
80	C₂×Dic₇₃	—	4	2	no	—
81	C₂×Dic₇₃	—	4	2	no	—
82	C₂×Dic₇₃	#121	1	1	yes	characteristic
83	C₂×Dic₇₃	C₂²⋊Q₈	1	1	yes	characteristic
84	C₂×Dic₇₃	#125	1	1	yes	characteristic
85	C₂×Dic₇₃	C₄⋊Q₈	1	1	yes	characteristic
86	C₂×Dic₇₃	C₂×SD₁₆	1	2	yes	—
87	C₂×Dic₇₃	#135	1	2	yes	—
88	#10596833	—	2	4	no	—
89	#10596833	—	2	4	no	—
90	#10596833	—	8	2	no	—
91	C₈×C₇₃	C₄⋊C₈	1	4	yes	complemented
92	#10596831	—	2	2	no	—
93	C₂²×C₇₃	#389	1	1	yes	characteristic
94	C₂²×C₇₃	C₈×D₄	1	1	yes	characteristic
95	C₂²×C₇₃	#436	1	1	yes	characteristic
96	C₂²×C₇₃	C₄×SD₁₆	1	2	yes	—
97	C₂²×C₇₃	#443	1	2	yes	—
98	C₄×C₇₃	—	4	4	no	—
99	C₄×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
100	C₄×C₇₃	#424	1	1	yes	characteristic
101	C₄×C₇₃	C₈×Q₈	1	1	yes	characteristic
102	C₄×C₇₃	#446	1	1	yes	characteristic
103	Dic₇₃	—	2	1	no	—
104	Dic₇₃	—	2	1	no	—
105	Dic₇₃	—	8	2	no	—
106	Dic₇₃	#465	1	2	yes	—
107	Dic₇₃	#498	1	1	yes	characteristic
108	Dic₇₃	#499	1	1	yes	characteristic
109	#9593	—	73	1	no	maximal, Sylow
110	C₂×C₇₃	C₈×SD₁₆	1	2	yes	—
111	C₂×C₇₃	#1590	1	2	yes	—
112	C₂×C₇₃	#1939	1	1	yes	characteristic
113	C₂×C₇₃	#1941	1	1	yes	characteristic
114	C₂×C₇₃	#1945	1	1	yes	characteristic
115	#1939	—	73	2	no	—
116	C₄×#106	—	73	1	no	—
117	C₂×#334	—	73	1	no	—
118	C₂×C₈²	—	73	1	no	—
119	#1279	—	73	2	no	—
120	C₇₃	#9593	1	1	yes	characteristic, complemented, Sylow
121	C₂×#106	—	73	1	no	—
122	C₂×#106	#64408	1	1	yes	characteristic
123	C₂×C₄⋊C₈	—	73	2	no	—
124	C₂×C₄⋊C₈	—	146	2	no	—
125	C₂×C₄×C₈	—	73	1	no	—
126	C₂×C₄×C₈	—	73	1	no	—
127	C₂×C₄×C₈	—	73	1	no	—
128	C₄×C₄⋊C₄	—	73	2	no	—
129	#340	—	73	2	no	—
130	#336	—	146	2	no	—
131	#334	—	146	2	no	—
132	C₈²	—	146	2	no	—
133	C₂²×C₈	—	73	2	no	—
134	C₂²×C₈	—	73	1	no	—
135	C₂²×C₈	—	146	2	no	—
136	C₂²×C₈	—	146	1	no	—
137	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
138	C₂×C₄⋊C₄	—	73	2	no	—
139	C₂×C₄⋊C₄	C₂×#64408	1	2	yes	—
140	C₂×C₄²	—	73	1	no	—
141	C₂×C₄²	—	146	2	no	—
142	#106	—	146	2	no	—
143	#106	#10596833	1	4	yes	complemented
144	C₄⋊C₈	—	292	4	no	—
145	C₄⋊C₈	—	146	2	no	—
146	C₄⋊C₈	—	146	2	no	—
147	C₄×C₈	—	73	2	no	—
148	C₄×C₈	—	73	2	no	—
149	C₄×C₈	—	73	2	no	—
150	C₄×C₈	—	146	1	no	—
151	C₄×C₈	—	146	1	no	—
152	C₄×C₈	—	146	1	no	—
153	#95	—	146	2	no	—
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	—	73	1	no	—
156	C₂²×C₄	—	73	1	no	—
157	C₂²×C₄	—	146	2	no	—
158	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
159	C₂×C₈	—	292	2	no	—
160	C₂×C₈	—	292	2	no	—
161	C₂×C₈	—	292	2	no	—
162	C₂×C₈	—	73	4	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	4	no	—
168	C₂×C₈	—	146	1	no	—
169	C₂×C₈	—	146	1	no	—
170	C₂×C₈	C₄⋊#64408	1	2	yes	—
171	C₂×C₈	C₂×#10596833	1	2	yes	—
172	C₂×C₈	#11924038	1	2	yes	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	146	2	no	—
175	C₄⋊C₄	—	146	2	no	—
176	C₄⋊C₄	C₂×#10596833	1	4	yes	—
177	C₄²	—	292	2	no	—
178	C₄²	—	292	2	no	—
179	C₄²	—	73	1	no	—
180	C₄²	—	73	1	no	—
181	C₄²	—	73	1	no	—
182	C₄²	—	73	1	no	—
183	C₂³	D₄×#64408	1	1	yes	characteristic
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	—	4	2	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	—	292	2	no	—
188	C₂×C₄	—	292	2	no	—
189	C₂×C₄	—	292	2	no	—
190	C₂×C₄	—	73	1	no	—
191	C₂×C₄	—	73	1	no	—
192	C₂×C₄	—	73	2	no	—
193	C₂×C₄	—	73	2	no	—
194	C₂×C₄	—	73	1	no	—
195	C₂×C₄	—	73	1	no	—
196	C₂×C₄	—	146	2	no	—
197	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
198	C₂×C₄	Q₈×#64408	1	1	yes	characteristic
199	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
200	C₂×C₄	C₂×#11924038	1	1	yes	characteristic
201	C₂×C₄	#14556678	1	1	yes	characteristic
202	C₂×C₄	#14556679	1	1	yes	characteristic
203	C₈	—	584	2	no	—
204	C₈	—	146	4	no	—
205	C₈	—	146	4	no	—
206	C₈	—	146	2	no	—
207	C₈	C₄⋊#10596833	1	4	yes	complemented
208	C₂²	#17989538	1	1	yes	characteristic
209	C₂²	D₄×#10596833	1	1	yes	characteristic
210	C₂²	#17989621	1	1	yes	characteristic
211	C₂²	SD₁₆×#64408	1	2	yes	—
212	C₂²	#17989630	1	2	yes	—
213	C₄	—	4	4	no	—
214	C₄	—	584	2	no	—
215	C₄	—	73	2	no	—
216	C₄	—	73	1	no	—
217	C₄	—	73	1	no	—
218	C₄	—	146	1	no	—
219	C₄	—	146	1	no	—
220	C₄	C₂×C₄⋊#10596833	1	1	yes	characteristic
221	C₄	#17989594	1	1	yes	characteristic
222	C₄	Q₈×#10596833	1	1	yes	characteristic
223	C₄	#17989637	1	1	yes	characteristic
224	C₂	SD₁₆×#10596833	1	2	yes	—
225	C₂	#22625566	1	2	yes	—
226	C₂	#22626028	1	1	yes	characteristic
227	C₂	#22626034	1	1	yes	characteristic
228	C₂	#22626038	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₄×#64408
commutator	C₂×C₄×C₇₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#64408
Fitting	C₂×#106×C₇₃	C₄
socle, prime radical	C₂³×C₇₃	C₄×D₄
cosocle	C₂×C₄×Dic₇₃	C₂³
hypercenter	C₂×#106	#64408
hypocenter	C₇₃	#9593
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

Complements

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

N	H ≅ G/N	type
38. #106×C₇₃	C₈	semidirect
91. C₈×C₇₃	C₄⋊C₈	semidirect
120. C₇₃	#9593	semidirect
143. #106	#10596833	semidirect
207. C₈	C₄⋊#10596833	semidirect

Maximal subgroups

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

subgroup	index
#22626028	2
C₂×#17989448	2
C₂×C₈×#10596833	2
#22624767	2
#106×Dic₇₃	2
#9593	73