Subgroups — #32997285

Order 18688.

310 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22620445	C₂	1	1	yes	maximal, characteristic
2	#22611687	C₂	1	1	yes	maximal, characteristic
3	#22607990	C₂	1	2	yes	maximal, complemented
4	#22607987	C₂	1	2	yes	maximal, complemented
5	C₂×#335×C₇₃	C₂	1	1	yes	maximal, characteristic
6	#17989144	—	2	1	no	—
7	C₂×#14556531	—	2	1	no	—
8	#17988992	C₂²	1	1	yes	characteristic
9	#17988992	C₂²	1	1	yes	characteristic
10	#17988989	—	2	1	no	—
11	#17988989	C₄	1	2	yes	—
12	#17988985	—	2	1	no	—
13	#17988625	—	2	1	no	—
14	#17988625	C₄	1	2	yes	—
15	C₄⋊#11924003	—	2	1	no	—
16	C₄⋊#11924003	C₂²	1	2	yes	—
17	C₂×C₄⋊C₈×C₇₃	—	2	1	no	—
18	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
19	#335×C₇₃	—	2	1	no	—
20	#335×C₇₃	C₂²	1	2	yes	—
21	#14556591	—	2	1	no	—
22	#14556591	—	2	1	no	—
23	#14556583	—	2	2	no	—
24	#14556570	—	4	1	no	—
25	#14556570	—	4	1	no	—
26	C₄×Dic₁₄₆	—	4	1	no	—
27	C₄×Dic₁₄₆	—	4	1	no	—
28	#14556550	—	2	2	no	—
29	#14556531	—	4	2	no	—
30	C₂×#11924003	—	2	2	no	—
31	#14556525	—	2	1	no	—
32	#14556525	—	2	1	no	—
33	#14556525	C₂×C₄	1	2	yes	—
34	#14556525	C₂×C₄	1	2	yes	—
35	#14556523	—	2	2	no	—
36	#14556523	—	2	2	no	—
37	C₂²×C₈×C₇₃	—	4	1	no	—
38	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
41	C₄⋊C₈×C₇₃	—	2	2	no	—
42	C₄⋊C₈×C₇₃	—	4	1	no	—
43	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
45	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
46	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
47	C₂×Dic₁₄₆	—	4	1	no	—
48	C₂×Dic₁₄₆	—	4	1	no	—
49	C₄⋊Dic₇₃	—	2	2	no	—
50	C₄⋊Dic₇₃	—	2	2	no	—
51	C₄⋊Dic₇₃	—	4	1	no	—
52	C₄⋊Dic₇₃	—	4	1	no	—
53	#11924007	—	4	2	no	—
54	#11924007	—	4	2	no	—
55	C₂²⋊Dic₇₃	—	4	2	no	—
56	C₄×Dic₇₃	—	4	2	no	—
57	#11924003	—	4	2	no	—
58	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₇₃	D₈	1	1	yes	characteristic
60	C₂²×C₄×C₇₃	SD₁₆	1	1	yes	characteristic
61	C₂×C₈×C₇₃	—	2	2	no	—
62	C₂×C₈×C₇₃	—	4	2	no	—
63	C₂×C₈×C₇₃	—	8	2	no	—
64	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₇₃	SD₁₆	1	2	yes	—
67	C₂×C₈×C₇₃	Q₁₆	1	2	yes	—
68	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
70	C₄²×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
71	C₄²×C₇₃	D₈	1	1	yes	characteristic
72	C₄²×C₇₃	SD₁₆	1	1	yes	characteristic
73	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
74	C₂³×C₇₃	#102	1	1	yes	characteristic
75	C₂×C₄×C₇₃	—	2	1	no	—
76	C₂×C₄×C₇₃	—	2	1	no	—
77	C₂×C₄×C₇₃	—	4	1	no	—
78	C₂×C₄×C₇₃	#101	1	2	yes	—
79	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
80	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
81	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
82	C₂×C₄×C₇₃	#103	1	2	yes	—
83	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
85	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
86	C₂×Dic₇₃	—	4	2	no	—
87	C₂×Dic₇₃	—	4	2	no	—
88	Dic₁₄₆	—	8	2	no	—
89	Dic₁₄₆	—	8	1	no	—
90	Dic₁₄₆	—	8	1	no	—
91	C₈×C₇₃	—	2	1	no	—
92	C₈×C₇₃	—	2	1	no	—
93	C₈×C₇₃	—	8	2	no	—
94	C₈×C₇₃	#103	1	2	yes	—
95	C₈×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
96	C₈×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
97	C₂²×C₇₃	—	2	1	no	—
98	C₂²×C₇₃	—	2	1	no	—
99	C₂²×C₇₃	#332	1	2	yes	—
100	C₂²×C₇₃	C₂×#102	1	1	yes	characteristic
101	C₄×C₇₃	—	2	1	no	—
102	C₄×C₇₃	—	2	1	no	—
103	C₄×C₇₃	—	2	1	no	—
104	C₄×C₇₃	#332	1	2	yes	—
105	C₄×C₇₃	C₂×#101	1	1	yes	characteristic
106	C₄×C₇₃	C₂×#103	1	1	yes	characteristic
107	C₄×C₇₃	#509	1	1	yes	characteristic
108	C₄×C₇₃	#510	1	1	yes	characteristic
109	Dic₇₃	—	8	2	no	—
110	Dic₇₃	—	8	2	no	—
111	#11942	—	73	1	no	maximal, Sylow
112	C₂×C₇₃	—	2	2	no	—
113	C₂×C₇₃	C₂×#332	1	1	yes	characteristic
114	C₂×C₇₃	#2139	1	1	yes	characteristic
115	C₂×C₇₃	#2148	1	1	yes	characteristic
116	#3160	—	73	1	no	—
117	#2150	—	73	1	no	—
118	C₂×#335	D₇₃	1	1	yes	characteristic
119	#1365	—	73	2	no	—
120	#1364	—	73	2	no	—
121	C₇₃	#11942	1	1	yes	characteristic, complemented, Sylow
122	#533	—	146	1	no	—
123	C₂×M₃₂	—	146	1	no	—
124	#500	—	73	1	no	—
125	#500	—	73	1	no	—
126	#499	—	73	2	no	—
127	#499	—	146	1	no	—
128	#498	—	146	1	no	—
129	#427	—	73	2	no	—
130	#427	—	146	1	no	—
131	C₂×C₄⋊C₈	—	2	1	no	—
132	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
133	C₄⋊C₁₆	—	73	2	no	—
134	C₄⋊C₁₆	—	146	1	no	—
135	#335	—	2	1	no	—
136	#335	C₂×D₇₃	1	2	yes	—
137	C₂²×C₈	—	4	1	no	—
138	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
139	C₂²×C₈	D₂₉₂	1	1	yes	characteristic
140	C₄⋊Q₈	—	146	1	no	—
141	C₄⋊Q₈	—	146	1	no	—
142	#125	—	146	2	no	—
143	C₂²⋊Q₈	—	292	1	no	—
144	C₂²⋊Q₈	—	292	1	no	—
145	C₄×Q₈	—	292	1	no	—
146	C₄×Q₈	—	292	1	no	—
147	#117	—	146	2	no	—
148	C₂×C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
149	M₃₂	—	292	2	no	—
150	C₂×C₁₆	—	146	2	no	—
151	#107	—	73	2	no	—
152	#107	—	73	2	no	—
153	#107	—	146	1	no	—
154	#107	—	146	1	no	—
155	#106	—	146	2	no	—
156	#106	—	146	2	no	—
157	C₄⋊C₈	—	2	2	no	—
158	C₄⋊C₈	—	4	1	no	—
159	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
160	C₄×C₈	D₂₉₂	1	1	yes	characteristic
161	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
162	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
163	C₂×Q₈	—	292	1	no	—
164	C₂×Q₈	—	292	1	no	—
165	C₂²×C₄	#11924005	1	1	yes	characteristic
166	C₂²×C₄	#11924013	1	1	yes	characteristic
167	C₂²×C₄	#11924015	1	1	yes	characteristic
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	4	2	no	—
170	C₂×C₈	—	8	2	no	—
171	C₂×C₈	#11924005	1	1	yes	characteristic
172	C₂×C₈	#11924005	1	1	yes	characteristic
173	C₂×C₈	#11924017	1	2	yes	—
174	C₂×C₈	Dic₂₉₂	1	2	yes	—
175	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
176	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
177	C₄⋊C₄	—	292	2	no	—
178	C₄⋊C₄	—	292	2	no	—
179	C₄⋊C₄	—	292	1	no	—
180	C₄⋊C₄	—	292	1	no	—
181	C₄⋊C₄	—	146	2	no	—
182	C₄⋊C₄	—	146	2	no	—
183	C₂²⋊C₄	—	292	2	no	—
184	C₄²	—	292	2	no	—
185	C₄²	#11924005	1	1	yes	characteristic
186	C₄²	#11924013	1	1	yes	characteristic
187	C₄²	#11924015	1	1	yes	characteristic
188	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
189	C₁₆	—	292	2	no	—
190	C₂³	#14556511	1	1	yes	characteristic
191	Q₈	—	584	2	no	—
192	Q₈	—	584	1	no	—
193	Q₈	—	584	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	4	1	no	—
197	C₂×C₄	—	292	2	no	—
198	C₂×C₄	—	292	2	no	—
199	C₂×C₄	#14556509	1	2	yes	—
200	C₂×C₄	#14556511	1	1	yes	characteristic
201	C₂×C₄	#14556511	1	1	yes	characteristic
202	C₂×C₄	#14556511	1	1	yes	characteristic
203	C₂×C₄	#14556516	1	2	yes	—
204	C₂×C₄	C₂×#11924005	1	1	yes	characteristic
205	C₂×C₄	C₂×#11924013	1	1	yes	characteristic
206	C₂×C₄	C₂×#11924015	1	1	yes	characteristic
207	C₈	—	2	1	no	—
208	C₈	—	2	1	no	—
209	C₈	—	8	2	no	—
210	C₈	#14556516	1	2	yes	—
211	C₈	C₂×#11924017	1	1	yes	characteristic
212	C₈	C₂×Dic₂₉₂	1	1	yes	characteristic
213	C₂²	—	2	1	no	—
214	C₂²	—	2	1	no	—
215	C₂²	#17988334	1	2	yes	—
216	C₂²	C₂×#14556511	1	1	yes	characteristic
217	C₄	—	2	1	no	—
218	C₄	—	2	1	no	—
219	C₄	—	2	1	no	—
220	C₄	—	584	2	no	—
221	C₄	—	584	2	no	—
222	C₄	#17988334	1	2	yes	—
223	C₄	C₂×#14556509	1	1	yes	characteristic
224	C₄	C₂×#14556516	1	1	yes	characteristic
225	C₄	#17989023	1	1	yes	characteristic
226	C₄	#17989028	1	1	yes	characteristic
227	C₂	—	2	2	no	—
228	C₂	C₂×#17988334	1	1	yes	characteristic
229	C₂	#22611641	1	1	yes	characteristic
230	C₂	#22611678	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14556511
commutator	C₂×C₈×C₇₃	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₇₃
Fitting	C₂×#335×C₇₃	C₂
socle	C₂²×C₇₃	C₂×#102
prime radical	C₂³×C₇₃	#102
cosocle	C₄×C₈×C₇₃	C₂³
hypercenter	C₂×#335	D₇₃
hypocenter	C₇₃	#11942
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
3. #22607990	C₂	semidirect
4. #22607987	C₂	semidirect
121. C₇₃	#11942	semidirect

Maximal subgroups

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

subgroup	index
#22620445	2
#22611687	2
#22607990	2
#22607987	2
C₂×#335×C₇₃	2
#11942	73