Subgroups — #32984601

Order 18688.

306 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	#22608683	C₂	1	2	yes	maximal
2	C₂×#17988340	C₂	1	1	yes	maximal, characteristic
3	#22608214	C₂	1	1	yes	maximal, characteristic
4	#22607782	C₂	1	2	yes	maximal
5	C₄×#108×C₇₃	C₂	1	1	yes	maximal, characteristic
6	#17988640	C₂²	1	1	yes	characteristic
7	#17988640	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊#10596831	C₂²	1	2	yes	—
9	#17988340	—	2	2	no	—
10	#17988340	—	2	1	no	—
11	#17988340	—	2	1	no	—
12	#17988338	—	2	2	no	—
13	#17988308	—	2	1	no	—
14	C₈×#10596831	—	2	1	no	—
15	C₂×#108×C₇₃	C₄	1	1	yes	characteristic
16	C₂×#108×C₇₃	C₄	1	1	yes	characteristic
17	C₄×M₁₆×C₇₃	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
19	#341×C₇₃	C₄	1	2	yes	—
20	C₂²×#10596831	—	2	2	no	—
21	C₄⋊#10596831	—	2	2	no	—
22	C₄⋊#10596831	—	2	2	no	—
23	C₄⋊#10596831	—	2	1	no	—
24	C₄⋊#10596831	D₄	1	1	yes	characteristic
25	C₄⋊#10596831	D₄	1	1	yes	characteristic
26	C₂²⋊#10596831	—	2	1	no	—
27	C₂²⋊#10596831	Q₈	1	2	yes	—
28	C₄×#10596831	—	2	1	no	—
29	C₄×#10596831	—	2	1	no	—
30	C₂×M₁₆×C₇₃	C₂×C₄	1	2	yes	—
31	C₂×M₁₆×C₇₃	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
35	#108×C₇₃	—	2	2	no	—
36	#108×C₇₃	C₈	1	4	yes	complemented
37	#97×C₇₃	—	2	2	no	—
38	C₄×C₈×C₇₃	—	2	2	no	—
39	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
40	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
41	C₄×C₈×C₇₃	Q₈	1	1	yes	characteristic
42	C₄×C₈×C₇₃	Q₈	1	1	yes	characteristic
43	C₂×#10596831	—	2	1	no	—
44	C₂×#10596831	—	2	1	no	—
45	C₂×#10596831	—	2	2	no	—
46	C₂×#10596831	—	4	2	no	—
47	C₂×#10596831	—	8	2	no	—
48	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
50	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
51	M₁₆×C₇₃	—	2	2	no	—
52	M₁₆×C₇₃	—	2	2	no	—
53	M₁₆×C₇₃	—	2	2	no	—
54	M₁₆×C₇₃	C₂×C₈	1	4	yes	—
55	C₂×C₈×C₇₃	—	2	2	no	—
56	C₂×C₈×C₇₃	—	2	1	no	—
57	C₂×C₈×C₇₃	—	2	1	no	—
58	C₂×C₈×C₇₃	—	2	2	no	—
59	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
64	C₂×C₈×C₇₃	M₁₆	1	2	yes	—
65	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
66	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
67	C₄²×C₇₃	C₂×Q₈	1	1	yes	characteristic
68	C₄²×C₇₃	#41	1	1	yes	characteristic
69	C₂³×C₇₃	C₄×Q₈	1	1	yes	characteristic
70	C₂×C₄×C₇₃	—	2	1	no	—
71	C₂×C₄×C₇₃	—	2	1	no	—
72	C₂×C₄×C₇₃	—	4	1	no	—
73	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
75	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
76	C₂×C₄×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
77	C₂×C₄×C₇₃	#125	1	1	yes	characteristic
78	C₂×C₄×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
79	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
80	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
81	C₂×C₄×C₇₃	#131	1	1	yes	characteristic
82	C₂×C₄×C₇₃	#131	1	1	yes	characteristic
83	C₈×C₇₃	—	2	1	no	—
84	C₈×C₇₃	—	2	1	no	—
85	C₈×C₇₃	—	2	4	no	—
86	C₈×C₇₃	—	4	2	no	—
87	C₈×C₇₃	C₄⋊C₈	1	2	yes	—
88	C₈×C₇₃	C₄⋊C₈	1	2	yes	—
89	#10596831	—	4	2	no	—
90	#10596831	—	4	1	no	—
91	#10596831	—	4	1	no	—
92	#10596831	—	8	2	no	—
93	C₂²×C₇₃	—	2	1	no	—
94	C₂²×C₇₃	—	2	1	no	—
95	C₂²×C₇₃	#389	1	1	yes	characteristic
96	C₂²×C₇₃	C₈×Q₈	1	1	yes	characteristic
97	C₂²×C₇₃	#446	1	1	yes	characteristic
98	C₄×C₇₃	—	2	1	no	—
99	C₄×C₇₃	—	2	1	no	—
100	C₄×C₇₃	—	2	1	no	—
101	C₄×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
102	C₄×C₇₃	#424	1	1	yes	characteristic
103	C₄×C₇₃	C₈×D₄	1	1	yes	characteristic
104	C₄×C₇₃	#436	1	1	yes	characteristic
105	C₄×C₇₃	#472	1	1	yes	characteristic
106	C₄×C₇₃	#473	1	1	yes	characteristic
107	#9608	—	73	1	no	maximal, Sylow
108	C₂×C₇₃	—	2	1	no	—
109	C₂×C₇₃	—	2	1	no	—
110	C₂×C₇₃	#1939	1	1	yes	characteristic
111	C₂×C₇₃	#1952	1	1	yes	characteristic
112	C₂×C₇₃	#1953	1	1	yes	characteristic
113	C₄×#108	D₇₃	1	1	yes	characteristic
114	#1565	—	73	2	no	—
115	C₂×#335	—	73	1	no	—
116	#1450	—	73	1	no	—
117	#1283	—	73	2	no	—
118	C₇₃	#9608	1	1	yes	characteristic, complemented, Sylow
119	#431	—	73	1	no	—
120	#431	—	73	1	no	—
121	C₂×#108	Dic₇₃	1	1	yes	characteristic
122	C₂×#108	Dic₇₃	1	1	yes	characteristic
123	C₂×C₄⋊C₈	—	73	2	no	—
124	C₄×M₁₆	C₂×D₇₃	1	2	yes	—
125	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
126	#341	Dic₇₃	1	2	yes	—
127	#335	—	146	2	no	—
128	#335	—	146	1	no	—
129	#335	—	146	1	no	—
130	#334	—	146	2	no	—
131	#322	—	146	1	no	—
132	C₈²	—	146	1	no	—
133	C₂×M₁₆	C₂×Dic₇₃	1	2	yes	—
134	C₂×M₁₆	C₂×Dic₇₃	1	2	yes	—
135	C₂²×C₈	—	146	2	no	—
136	C₂²×C₈	C₂×Dic₇₃	1	1	yes	characteristic
137	C₂²×C₈	C₂×Dic₇₃	1	1	yes	characteristic
138	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
139	#108	—	2	2	no	—
140	#108	#10596831	1	4	yes	complemented
141	C₄⋊C₈	—	73	1	no	—
142	C₄⋊C₈	—	73	1	no	—
143	C₄⋊C₈	—	146	2	no	—
144	C₄⋊C₈	—	146	2	no	—
145	C₄⋊C₈	—	146	1	no	—
146	C₂²⋊C₈	—	73	2	no	—
147	C₂²⋊C₈	—	146	1	no	—
148	#97	—	2	2	no	—
149	C₄×C₈	—	2	2	no	—
150	C₄×C₈	—	146	1	no	—
151	C₄×C₈	—	146	1	no	—
152	C₄×C₈	Dic₁₄₆	1	1	yes	characteristic
153	C₄×C₈	Dic₁₄₆	1	1	yes	characteristic
154	C₄×C₈	D₂₉₂	1	1	yes	characteristic
155	C₄×C₈	D₂₉₂	1	1	yes	characteristic
156	C₂²×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
157	C₂²×C₄	Q₈×D₇₃	1	1	yes	characteristic
158	C₂²×C₄	#11924029	1	1	yes	characteristic
159	M₁₆	—	2	2	no	—
160	M₁₆	—	2	2	no	—
161	M₁₆	—	2	2	no	—
162	M₁₆	C₂×#10596831	1	4	yes	—
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	292	2	no	—
168	C₂×C₈	—	584	2	no	—
169	C₂×C₈	—	146	1	no	—
170	C₂×C₈	—	146	1	no	—
171	C₂×C₈	—	146	2	no	—
172	C₂×C₈	C₄⋊Dic₇₃	1	1	yes	characteristic
173	C₂×C₈	C₄⋊Dic₇₃	1	1	yes	characteristic
174	C₂×C₈	C₄⋊Dic₇₃	1	1	yes	characteristic
175	C₂×C₈	C₄⋊Dic₇₃	1	1	yes	characteristic
176	C₂×C₈	C₂×#10596831	1	2	yes	—
177	C₂×C₈	#11924012	1	2	yes	—
178	C₄²	D₄×D₇₃	1	1	yes	characteristic
179	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
180	C₄²	C₂×Dic₁₄₆	1	1	yes	characteristic
181	C₄²	#11924027	1	1	yes	characteristic
182	C₂³	Q₈×Dic₇₃	1	1	yes	characteristic
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	4	1	no	—
186	C₂×C₄	C₂×C₄⋊Dic₇₃	1	1	yes	characteristic
187	C₂×C₄	D₄×Dic₇₃	1	1	yes	characteristic
188	C₂×C₄	#14556566	1	1	yes	characteristic
189	C₂×C₄	#14556571	1	1	yes	characteristic
190	C₂×C₄	#14556582	1	1	yes	characteristic
191	C₂×C₄	#14556590	1	1	yes	characteristic
192	C₂×C₄	C₂²×#10596831	1	1	yes	characteristic
193	C₂×C₄	C₂×#11924012	1	1	yes	characteristic
194	C₂×C₄	#14556598	1	1	yes	characteristic
195	C₂×C₄	#14556598	1	1	yes	characteristic
196	C₈	—	2	1	no	—
197	C₈	—	2	1	no	—
198	C₈	—	2	4	no	—
199	C₈	—	4	2	no	—
200	C₈	—	292	2	no	—
201	C₈	—	292	1	no	—
202	C₈	—	292	1	no	—
203	C₈	—	584	2	no	—
204	C₈	C₄⋊#10596831	1	2	yes	—
205	C₈	C₄⋊#10596831	1	2	yes	—
206	C₂²	—	2	1	no	—
207	C₂²	—	2	1	no	—
208	C₂²	#17988486	1	1	yes	characteristic
209	C₂²	Q₈×#10596831	1	1	yes	characteristic
210	C₂²	#17988718	1	1	yes	characteristic
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	C₂×C₄⋊#10596831	1	1	yes	characteristic
215	C₄	#17988614	1	1	yes	characteristic
216	C₄	D₄×#10596831	1	1	yes	characteristic
217	C₄	#17988665	1	1	yes	characteristic
218	C₄	#17988858	1	1	yes	characteristic
219	C₄	#17988863	1	1	yes	characteristic
220	C₂	—	2	1	no	—
221	C₂	—	2	1	no	—
222	C₂	#22610650	1	1	yes	characteristic
223	C₂	#22610717	1	1	yes	characteristic
224	C₂	#22610722	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14556566
commutator	C₂×C₄×C₇₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₄×#108×C₇₃	C₂
socle	C₂²×C₇₃	#389
prime radical	C₂³×C₇₃	C₄×Q₈
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	C₄×#108	D₇₃
hypocenter	C₇₃	#9608
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#108

Complements

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

N	H ≅ G/N	type
36. #108×C₇₃	C₈	semidirect
118. C₇₃	#9608	semidirect
140. #108	#10596831	semidirect

Maximal subgroups

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

subgroup	index
#22608683	2
C₂×#17988340	2
#22608214	2
#22607782	2
C₄×#108×C₇₃	2
#9608	73