Subgroups — #33004569

Order 18688.

358 conjugacy classes of subgroups.

The subgroup lattice has 238 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	#22608038	C₂	1	4	yes	maximal
4	C₄×#106×C₇₃	C₂	1	1	yes	maximal, characteristic
5	#17989144	—	2	1	no	—
6	C₂×#14556531	C₂²	1	2	yes	—
7	#17988992	C₄	1	1	yes	characteristic, complemented
8	#17988992	C₄	1	1	yes	characteristic, complemented
9	#17988989	—	2	1	no	—
10	#17988989	C₄	1	2	yes	complemented
11	#17988985	—	2	1	no	—
12	#17988625	—	2	1	no	—
13	#17988625	C₂²	1	2	yes	—
14	C₄⋊#11924003	—	2	1	no	—
15	C₄⋊#11924003	—	2	1	no	—
16	C₂×#106×C₇₃	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
18	C₄×C₄⋊C₄×C₇₃	—	2	1	no	—
19	#340×C₇₃	—	2	1	no	—
20	#14556591	—	2	1	no	—
21	#14556591	—	2	1	no	—
22	#14556583	—	2	2	no	—
23	#14556570	—	4	1	no	—
24	#14556570	—	4	1	no	—
25	C₄×Dic₁₄₆	—	4	1	no	—
26	C₄×Dic₁₄₆	—	4	1	no	—
27	#14556550	—	2	2	no	—
28	#14556531	—	2	4	no	—
29	C₂×#11924003	—	2	2	no	—
30	#14556525	—	2	1	no	—
31	#14556525	—	2	1	no	—
32	#14556525	C₂×C₄	1	2	yes	—
33	#14556525	C₂×C₄	1	2	yes	—
34	#14556523	—	2	2	no	—
35	#14556523	—	2	2	no	—
36	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
38	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
39	C₂×C₄²×C₇₃	—	4	1	no	—
40	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
41	#106×C₇₃	—	2	2	no	—
42	#106×C₇₃	Q₈	1	4	yes	complemented
43	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
46	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
47	#95×C₇₃	—	4	1	no	—
48	C₂×Dic₁₄₆	—	4	1	no	—
49	C₂×Dic₁₄₆	—	4	1	no	—
50	C₄⋊Dic₇₃	—	2	2	no	—
51	C₄⋊Dic₇₃	—	2	2	no	—
52	C₄⋊Dic₇₃	—	4	1	no	—
53	C₄⋊Dic₇₃	—	4	1	no	—
54	#11924007	—	4	2	no	—
55	#11924007	—	4	2	no	—
56	C₂²⋊Dic₇₃	—	4	2	no	—
57	C₄×Dic₇₃	—	4	2	no	—
58	#11924003	—	4	2	no	—
59	C₂²×C₄×C₇₃	—	4	2	no	—
60	C₂²×C₄×C₇₃	D₈	1	1	yes	characteristic
61	C₂²×C₄×C₇₃	D₈	1	1	yes	characteristic
62	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₇₃	—	2	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	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₇₃	C₂×Q₈	1	2	yes	—
69	C₂×C₈×C₇₃	#41	1	2	yes	—
70	C₄⋊C₄×C₇₃	—	2	4	no	—
71	C₄⋊C₄×C₇₃	—	4	2	no	—
72	C₄²×C₇₃	—	8	2	no	—
73	C₄²×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
74	C₄²×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
75	C₄²×C₇₃	SD₁₆	1	1	yes	characteristic
76	C₄²×C₇₃	SD₁₆	1	1	yes	characteristic
77	C₂³×C₇₃	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄×C₇₃	—	2	2	no	—
79	C₂×C₄×C₇₃	—	4	1	no	—
80	C₂×C₄×C₇₃	—	4	4	no	—
81	C₂×C₄×C₇₃	—	8	2	no	—
82	C₂×C₄×C₇₃	—	8	2	no	—
83	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
84	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
85	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
86	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
87	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
89	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
90	C₂×C₄×C₇₃	#137	1	2	yes	—
91	C₂×Dic₇₃	—	4	2	no	—
92	C₂×Dic₇₃	—	4	2	no	—
93	Dic₁₄₆	—	8	2	no	—
94	Dic₁₄₆	—	8	2	no	—
95	C₈×C₇₃	—	2	1	no	—
96	C₈×C₇₃	—	2	1	no	—
97	C₈×C₇₃	C₄×Q₈	1	2	yes	—
98	C₈×C₇₃	#124	1	1	yes	characteristic
99	C₈×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
100	C₂²×C₇₃	—	2	1	no	—
101	C₂²×C₇₃	—	2	1	no	—
102	C₂²×C₇₃	C₂×#102	1	1	yes	characteristic
103	C₂²×C₇₃	#474	1	2	yes	—
104	C₄×C₇₃	—	2	1	no	—
105	C₄×C₇₃	—	2	1	no	—
106	C₄×C₇₃	—	2	1	no	—
107	C₄×C₇₃	—	8	4	no	—
108	C₄×C₇₃	#360	1	2	yes	—
109	C₄×C₇₃	#391	1	1	yes	characteristic
110	C₄×C₇₃	#419	1	1	yes	characteristic
111	C₄×C₇₃	#475	1	2	yes	—
112	Dic₇₃	—	8	2	no	—
113	Dic₇₃	—	8	2	no	—
114	#13278	—	73	1	no	maximal, Sylow
115	C₂×C₇₃	—	2	2	no	—
116	C₂×C₇₃	#1988	1	1	yes	characteristic
117	C₂×C₇₃	C₂×#360	1	1	yes	characteristic
118	C₂×C₇₃	#2145	1	1	yes	characteristic
119	#3160	—	73	1	no	—
120	#2150	—	73	1	no	—
121	C₄×#106	D₇₃	1	1	yes	characteristic
122	#1386	—	73	4	no	—
123	C₇₃	#13278	1	1	yes	characteristic, complemented, Sylow
124	#533	—	146	1	no	—
125	C₂×M₃₂	—	73	2	no	—
126	#500	—	73	1	no	—
127	#500	—	73	1	no	—
128	#499	—	73	2	no	—
129	#499	—	146	1	no	—
130	#498	—	146	1	no	—
131	#427	—	73	2	no	—
132	#427	—	146	1	no	—
133	C₂×#106	C₂×D₇₃	1	2	yes	—
134	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
135	C₄×C₄⋊C₄	—	2	1	no	—
136	C₄⋊C₁₆	—	146	1	no	—
137	C₄⋊C₁₆	—	146	1	no	—
138	#340	—	2	1	no	—
139	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
140	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
141	C₄⋊Q₈	—	146	1	no	—
142	C₄⋊Q₈	—	146	1	no	—
143	#125	—	146	2	no	—
144	C₂²⋊Q₈	—	292	1	no	—
145	C₂²⋊Q₈	—	292	1	no	—
146	C₄×Q₈	—	292	1	no	—
147	C₄×Q₈	—	292	1	no	—
148	#117	—	146	2	no	—
149	C₂×C₄⋊C₄	—	2	2	no	—
150	C₂×C₄²	—	4	1	no	—
151	C₂×C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
152	M₃₂	—	146	4	no	—
153	C₂×C₁₆	—	146	2	no	—
154	#107	—	73	2	no	—
155	#107	—	73	2	no	—
156	#107	—	146	1	no	—
157	#107	—	146	1	no	—
158	#106	—	2	2	no	—
159	#106	—	146	2	no	—
160	#106	—	146	2	no	—
161	#106	Dic₁₄₆	1	4	yes	complemented
162	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
163	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
164	C₄×C₈	D₂₉₂	1	1	yes	characteristic
165	C₄×C₈	D₂₉₂	1	1	yes	characteristic
166	#95	—	4	1	no	—
167	C₂×Q₈	—	292	1	no	—
168	C₂×Q₈	—	292	1	no	—
169	C₂²×C₄	—	4	2	no	—
170	C₂²×C₄	#11924013	1	1	yes	characteristic
171	C₂²×C₄	#11924013	1	1	yes	characteristic
172	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	#11924005	1	1	yes	characteristic
175	C₂×C₈	#11924005	1	1	yes	characteristic
176	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
177	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
178	C₂×C₈	C₂×Dic₁₄₆	1	2	yes	—
179	C₂×C₈	#11924028	1	2	yes	—
180	C₄⋊C₄	—	2	4	no	—
181	C₄⋊C₄	—	4	2	no	—
182	C₄⋊C₄	—	292	2	no	—
183	C₄⋊C₄	—	292	2	no	—
184	C₄⋊C₄	—	292	1	no	—
185	C₄⋊C₄	—	292	1	no	—
186	C₄⋊C₄	—	146	2	no	—
187	C₄⋊C₄	—	146	2	no	—
188	C₂²⋊C₄	—	292	2	no	—
189	C₄²	—	8	2	no	—
190	C₄²	—	292	2	no	—
191	C₄²	#11924005	1	1	yes	characteristic
192	C₄²	#11924005	1	1	yes	characteristic
193	C₄²	#11924015	1	1	yes	characteristic
194	C₄²	#11924015	1	1	yes	characteristic
195	C₁₆	—	292	2	no	—
196	C₂³	C₂×#11924013	1	1	yes	characteristic
197	Q₈	—	584	2	no	—
198	Q₈	—	584	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	4	1	no	—
201	C₂×C₄	—	4	4	no	—
202	C₂×C₄	—	8	2	no	—
203	C₂×C₄	—	8	2	no	—
204	C₂×C₄	—	292	2	no	—
205	C₂×C₄	—	292	2	no	—
206	C₂×C₄	#14556511	1	1	yes	characteristic
207	C₂×C₄	#14556511	1	1	yes	characteristic
208	C₂×C₄	#14556511	1	1	yes	characteristic
209	C₂×C₄	#14556511	1	1	yes	characteristic
210	C₂×C₄	C₂×#11924005	1	1	yes	characteristic
211	C₂×C₄	#14556570	1	2	yes	—
212	C₂×C₄	C₂×#11924015	1	1	yes	characteristic
213	C₂×C₄	#14556621	1	2	yes	—
214	C₈	—	2	1	no	—
215	C₈	—	2	1	no	—
216	C₈	C₄×Dic₁₄₆	1	2	yes	—
217	C₈	#14556579	1	1	yes	characteristic
218	C₈	#14556591	1	1	yes	characteristic
219	C₂²	—	2	1	no	—
220	C₂²	—	2	1	no	—
221	C₂²	C₂×#14556511	1	1	yes	characteristic
222	C₂²	#17988867	1	2	yes	—
223	C₄	—	2	1	no	—
224	C₄	—	2	1	no	—
225	C₄	—	2	1	no	—
226	C₄	—	8	4	no	—
227	C₄	—	584	2	no	—
228	C₄	—	584	2	no	—
229	C₄	#17988396	1	2	yes	—
230	C₄	#17988493	1	1	yes	characteristic
231	C₄	#17988595	1	1	yes	characteristic
232	C₄	#17988872	1	2	yes	—
233	C₂	—	2	2	no	—
234	C₂	#22610883	1	1	yes	characteristic
235	C₂	C₂×#17988396	1	1	yes	characteristic
236	C₂	#22611666	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14556511
commutator	C₂×C₈×C₇₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₄×#106×C₇₃	C₂
socle	C₂²×C₇₃	C₂×#102
prime radical	C₂³×C₇₃	C₂×D₈
cosocle	C₂²×C₈×C₇₃	C₂³
hypercenter	C₄×#106	D₇₃
hypocenter	C₇₃	#13278
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#106

Complements

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

N	H ≅ G/N	type
7. #17988992	C₄	semidirect
8. #17988992	C₄	semidirect
10. #17988989	C₄	semidirect
42. #106×C₇₃	Q₈	semidirect
123. C₇₃	#13278	semidirect
161. #106	Dic₁₄₆	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
#22608038	2
C₄×#106×C₇₃	2
#13278	73