Subgroups — #33004573

Order 18688.

366 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×#17988992	C₂	1	1	yes	maximal, characteristic
2	#22611687	C₂	1	1	yes	maximal, characteristic
3	#22608041	C₂	1	4	yes	maximal
4	C₄×#108×C₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556591	—	2	1	no	—
6	C₂×#14556531	C₂²	1	2	yes	—
7	#17988992	—	2	1	no	—
8	#17988992	—	2	1	no	—
9	#17988992	C₄	1	2	yes	—
10	#17988992	C₄	1	2	yes	—
11	C₂×#14556525	—	2	1	no	—
12	C₂×#14556525	C₂²	1	2	yes	—
13	C₄⋊#11924003	—	2	1	no	—
14	C₄⋊#11924003	—	2	1	no	—
15	C₂×#108×C₇₃	C₂²	1	2	yes	—
16	C₄×M₁₆×C₇₃	—	2	1	no	—
17	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
18	#341×C₇₃	—	2	1	no	—
19	C₂²×Dic₁₄₆	—	4	1	no	—
20	#14556591	—	2	2	no	—
21	#14556591	—	2	2	no	—
22	#14556591	—	4	1	no	—
23	#14556591	—	4	1	no	—
24	C₂×C₄⋊Dic₇₃	—	2	2	no	—
25	C₂×C₄⋊Dic₇₃	—	4	1	no	—
26	#14556531	—	2	4	no	—
27	C₂×#11924003	—	2	2	no	—
28	#14556525	—	2	2	no	—
29	#14556525	—	2	2	no	—
30	#14556525	—	2	2	no	—
31	#14556525	C₂×C₄	1	4	yes	—
32	C₂×M₁₆×C₇₃	—	2	2	no	—
33	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
35	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
36	#108×C₇₃	—	2	2	no	—
37	#108×C₇₃	Q₈	1	4	yes	complemented
38	#97×C₇₃	—	4	1	no	—
39	C₄×C₈×C₇₃	—	4	1	no	—
40	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
41	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
43	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
44	C₂×Dic₁₄₆	—	4	2	no	—
45	C₂×Dic₁₄₆	—	4	2	no	—
46	C₂×Dic₁₄₆	—	8	4	no	—
47	C₂²×Dic₇₃	—	4	2	no	—
48	C₄⋊Dic₇₃	—	2	4	no	—
49	C₄⋊Dic₇₃	—	4	2	no	—
50	C₄⋊Dic₇₃	—	4	2	no	—
51	C₄⋊Dic₇₃	—	4	2	no	—
52	#11924003	—	4	2	no	—
53	C₂²×C₄×C₇₃	SD₁₆	1	1	yes	characteristic
54	C₂²×C₄×C₇₃	SD₁₆	1	1	yes	characteristic
55	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
56	M₁₆×C₇₃	—	2	4	no	—
57	M₁₆×C₇₃	—	4	2	no	—
58	C₂×C₈×C₇₃	—	2	2	no	—
59	C₂×C₈×C₇₃	—	4	2	no	—
60	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
64	C₂×C₈×C₇₃	C₂×Q₈	1	2	yes	—
65	C₂×C₈×C₇₃	#41	1	2	yes	—
66	C₄²×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₄²×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
68	C₄²×C₇₃	D₈	1	1	yes	characteristic
69	C₄²×C₇₃	D₈	1	1	yes	characteristic
70	C₂³×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
71	C₂×C₄×C₇₃	—	2	2	no	—
72	C₂×C₄×C₇₃	—	2	2	no	—
73	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
74	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
75	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
76	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
77	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
79	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
80	C₂×C₄×C₇₃	#137	1	2	yes	—
81	C₂×Dic₇₃	—	4	4	no	—
82	C₂×Dic₇₃	—	8	4	no	—
83	Dic₁₄₆	—	8	4	no	—
84	Dic₁₄₆	—	8	4	no	—
85	C₈×C₇₃	—	2	1	no	—
86	C₈×C₇₃	—	2	1	no	—
87	C₈×C₇₃	—	4	4	no	—
88	C₈×C₇₃	C₄×Q₈	1	2	yes	—
89	C₈×C₇₃	#124	1	1	yes	characteristic
90	C₈×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
91	C₂²×C₇₃	—	2	1	no	—
92	C₂²×C₇₃	—	2	1	no	—
93	C₂²×C₇₃	C₂×#102	1	1	yes	characteristic
94	C₂²×C₇₃	#475	1	2	yes	—
95	C₄×C₇₃	—	2	1	no	—
96	C₄×C₇₃	—	2	1	no	—
97	C₄×C₇₃	#361	1	2	yes	—
98	C₄×C₇₃	#362	1	2	yes	—
99	C₄×C₇₃	#391	1	1	yes	characteristic
100	C₄×C₇₃	#419	1	1	yes	characteristic
101	C₄×C₇₃	#474	1	2	yes	—
102	Dic₇₃	—	8	4	no	—
103	#13279	—	73	1	no	maximal, Sylow
104	C₂×C₇₃	—	2	2	no	—
105	C₂×C₇₃	#1988	1	1	yes	characteristic
106	C₂×C₇₃	C₂×#361	1	1	yes	characteristic
107	C₂×C₇₃	C₂×#362	1	1	yes	characteristic
108	C₂×#500	—	73	1	no	—
109	#2150	—	73	1	no	—
110	C₄×#108	D₇₃	1	1	yes	characteristic
111	#1387	—	73	4	no	—
112	C₇₃	#13279	1	1	yes	characteristic, complemented, Sylow
113	C₂×C₄⋊Q₈	—	146	1	no	—
114	C₂×M₃₂	—	73	2	no	—
115	#500	—	73	2	no	—
116	#500	—	73	2	no	—
117	#500	—	146	1	no	—
118	#500	—	146	1	no	—
119	C₂×#108	C₂×D₇₃	1	2	yes	—
120	C₂×#107	—	73	2	no	—
121	C₂×#107	—	146	1	no	—
122	C₄×M₁₆	—	2	1	no	—
123	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
124	C₄⋊C₁₆	—	146	1	no	—
125	C₄⋊C₁₆	—	146	1	no	—
126	#341	—	2	1	no	—
127	C₂²×Q₈	—	292	1	no	—
128	C₂×M₁₆	—	2	2	no	—
129	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
130	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
131	C₄⋊Q₈	—	292	1	no	—
132	C₄⋊Q₈	—	292	1	no	—
133	C₄⋊Q₈	—	146	2	no	—
134	C₄⋊Q₈	—	146	2	no	—
135	C₂×C₄⋊C₄	—	292	1	no	—
136	C₂×C₄⋊C₄	—	146	2	no	—
137	C₂×C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
138	M₃₂	—	146	4	no	—
139	C₂×C₁₆	—	146	2	no	—
140	#108	—	2	2	no	—
141	#108	Dic₁₄₆	1	4	yes	complemented
142	#107	—	73	4	no	—
143	#107	—	146	2	no	—
144	#107	—	146	2	no	—
145	#107	—	146	2	no	—
146	#97	—	4	1	no	—
147	C₄×C₈	—	4	1	no	—
148	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
149	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
150	C₄×C₈	D₂₉₂	1	1	yes	characteristic
151	C₄×C₈	D₂₉₂	1	1	yes	characteristic
152	C₂×Q₈	—	292	2	no	—
153	C₂×Q₈	—	292	2	no	—
154	C₂×Q₈	—	584	4	no	—
155	C₂²×C₄	—	292	2	no	—
156	C₂²×C₄	#11924015	1	1	yes	characteristic
157	C₂²×C₄	#11924015	1	1	yes	characteristic
158	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
159	M₁₆	—	2	4	no	—
160	M₁₆	—	4	2	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	4	2	no	—
163	C₂×C₈	#11924005	1	1	yes	characteristic
164	C₂×C₈	#11924005	1	1	yes	characteristic
165	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
166	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
167	C₂×C₈	C₂×Dic₁₄₆	1	2	yes	—
168	C₂×C₈	#11924028	1	2	yes	—
169	C₄⋊C₄	—	292	2	no	—
170	C₄⋊C₄	—	292	2	no	—
171	C₄⋊C₄	—	292	2	no	—
172	C₄⋊C₄	—	146	4	no	—
173	C₄²	#11924005	1	1	yes	characteristic
174	C₄²	#11924005	1	1	yes	characteristic
175	C₄²	#11924013	1	1	yes	characteristic
176	C₄²	#11924013	1	1	yes	characteristic
177	C₁₆	—	292	2	no	—
178	C₂³	C₂×#11924015	1	1	yes	characteristic
179	Q₈	—	584	4	no	—
180	Q₈	—	584	4	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	292	4	no	—
184	C₂×C₄	—	584	4	no	—
185	C₂×C₄	#14556511	1	1	yes	characteristic
186	C₂×C₄	#14556511	1	1	yes	characteristic
187	C₂×C₄	#14556511	1	1	yes	characteristic
188	C₂×C₄	#14556511	1	1	yes	characteristic
189	C₂×C₄	C₂×#11924005	1	1	yes	characteristic
190	C₂×C₄	#14556570	1	2	yes	—
191	C₂×C₄	C₂×#11924013	1	1	yes	characteristic
192	C₂×C₄	#14556621	1	2	yes	—
193	C₈	—	2	1	no	—
194	C₈	—	2	1	no	—
195	C₈	—	4	4	no	—
196	C₈	C₄×Dic₁₄₆	1	2	yes	—
197	C₈	#14556579	1	1	yes	characteristic
198	C₈	#14556591	1	1	yes	characteristic
199	C₂²	—	2	1	no	—
200	C₂²	—	2	1	no	—
201	C₂²	C₂×#14556511	1	1	yes	characteristic
202	C₂²	#17988872	1	2	yes	—
203	C₄	—	2	1	no	—
204	C₄	—	2	1	no	—
205	C₄	—	584	4	no	—
206	C₄	#17988399	1	2	yes	—
207	C₄	#17988402	1	2	yes	—
208	C₄	#17988493	1	1	yes	characteristic
209	C₄	#17988595	1	1	yes	characteristic
210	C₄	#17988867	1	2	yes	—
211	C₂	—	2	2	no	—
212	C₂	#22610883	1	1	yes	characteristic
213	C₂	C₂×#17988399	1	1	yes	characteristic
214	C₂	C₂×#17988402	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14556511
commutator	C₂×C₈×C₇₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₄×#108×C₇₃	C₂
socle	C₂²×C₇₃	C₂×#102
prime radical	C₂³×C₇₃	C₂×SD₁₆
cosocle	C₂²×C₈×C₇₃	C₂³
hypercenter	C₄×#108	D₇₃
hypocenter	C₇₃	#13279
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#108

Complements

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

N	H ≅ G/N	type
37. #108×C₇₃	Q₈	semidirect
112. C₇₃	#13279	semidirect
141. #108	Dic₁₄₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17988992	2
#22611687	2
#22608041	2
C₄×#108×C₇₃	2
#13279	73