Subgroups — #33104935

Order 18688.

822 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Dic₁₄₆	C₂	1	1	yes	maximal, characteristic
2	C₄⋊C₄×Dic₁₄₆	C₂	1	2	yes	maximal
3	#22610689	C₂	1	2	yes	maximal
4	#22610686	C₂	1	4	yes	maximal
5	#22610059	C₂	1	1	yes	maximal, characteristic
6	#22610057	C₂	1	2	yes	maximal
7	#22609823	C₂	1	2	yes	maximal
8	C₄×#107×C₇₃	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Dic₁₄₆	—	2	2	no	—
10	C₂×C₄×Dic₁₄₆	C₂²	1	1	yes	characteristic
11	C₈×Dic₁₄₆	C₄	1	8	yes	complemented
12	C₂×#14556524	C₂²	1	4	yes	—
13	C₂×C₄⋊#10596831	C₂²	1	1	yes	characteristic
14	C₂×#14556520	C₂²	1	2	yes	—
15	C₂×C₈×Dic₇₃	C₂²	1	2	yes	—
16	#17988485	—	2	2	no	—
17	#17988484	—	2	4	no	—
18	#17988457	C₂²	1	2	yes	—
19	#17988456	C₂²	1	4	yes	—
20	C₄⋊C₄×Dic₇₃	C₂²	1	4	yes	—
21	#17988350	C₂²	1	2	yes	—
22	#17988350	C₂²	1	2	yes	—
23	#17988349	C₂²	1	4	yes	—
24	C₂×#107×C₇₃	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
26	C₄×C₄⋊C₄×C₇₃	C₂²	1	2	yes	—
27	#340×C₇₃	C₂²	1	1	yes	characteristic
28	#340×C₇₃	C₂²	1	1	yes	characteristic
29	C₂²×Dic₁₄₆	D₄	1	1	yes	characteristic
30	C₂²×#10596831	C₂³	1	2	yes	—
31	C₄×Dic₁₄₆	—	4	8	no	—
32	C₄×Dic₁₄₆	C₂×C₄	1	4	yes	—
33	C₄×Dic₁₄₆	Q₈	1	4	yes	—
34	C₂×C₄⋊Dic₇₃	—	2	2	no	—
35	C₂×C₄⋊Dic₇₃	—	2	2	no	—
36	C₂×C₄⋊Dic₇₃	D₄	1	1	yes	characteristic
37	C₂×C₄⋊Dic₇₃	C₂³	1	4	yes	—
38	C₂×#11924007	—	2	4	no	—
39	C₂×#11924007	—	2	4	no	—
40	C₂×#11924007	D₄	1	2	yes	—
41	C₂×C₄×Dic₇₃	—	2	4	no	—
42	C₂×C₄×Dic₇₃	C₂³	1	2	yes	—
43	#14556524	—	2	8	no	—
44	C₄⋊#10596831	C₂×C₄	1	4	yes	—
45	#14556520	C₂×C₄	1	8	yes	—
46	C₈×Dic₇₃	C₂×C₄	1	8	yes	—
47	#14556498	—	2	4	no	—
48	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
50	C₂×C₄⋊C₄×C₇₃	C₂³	1	2	yes	—
51	C₂×C₄⋊C₄×C₇₃	C₂³	1	2	yes	—
52	C₂×C₄²×C₇₃	—	2	2	no	—
53	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
54	#107×C₇₃	—	2	4	no	—
55	C₄×C₈×C₇₃	C₂×C₄	1	4	yes	—
56	#95×C₇₃	—	2	2	no	—
57	C₂×Dic₁₄₆	C₄⋊C₄	1	4	yes	—
58	C₂×Dic₁₄₆	D₈	1	4	yes	—
59	C₂×Dic₁₄₆	Q₁₆	1	4	yes	—
60	C₂²×Dic₇₃	—	2	4	no	—
61	C₂²×Dic₇₃	C₂×D₄	1	2	yes	—
62	C₂²×Dic₇₃	C₂×D₄	1	2	yes	—
63	C₂×#10596831	—	2	4	no	—
64	C₂×#10596831	C₂²×C₄	1	4	yes	—
65	C₄⋊Dic₇₃	—	2	4	no	—
66	C₄⋊Dic₇₃	—	2	4	no	—
67	C₄⋊Dic₇₃	—	4	4	no	—
68	C₄⋊Dic₇₃	—	8	2	no	—
69	C₄⋊Dic₇₃	C₄⋊C₄	1	4	yes	—
70	#11924007	—	4	8	no	—
71	#11924007	—	8	4	no	—
72	#11924007	C₄⋊C₄	1	8	yes	—
73	C₄×Dic₇₃	—	4	8	no	—
74	C₄×Dic₇₃	C₂²×C₄	1	4	yes	—
75	C₄×Dic₇₃	C₂×Q₈	1	4	yes	—
76	C₂²×C₄×C₇₃	—	2	2	no	—
77	C₂²×C₄×C₇₃	—	2	2	no	—
78	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
81	C₂×C₈×C₇₃	—	2	2	no	—
82	C₂×C₈×C₇₃	—	2	2	no	—
83	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
84	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
85	C₄⋊C₄×C₇₃	—	2	2	no	—
86	C₄⋊C₄×C₇₃	—	2	2	no	—
87	C₄⋊C₄×C₇₃	—	2	2	no	—
88	C₄⋊C₄×C₇₃	#41	1	4	yes	—
89	C₄²×C₇₃	—	4	4	no	—
90	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
91	C₄²×C₇₃	C₂×Q₈	1	2	yes	—
92	C₂³×C₇₃	C₂²×D₄	1	1	yes	characteristic
93	C₂×C₄×C₇₃	—	2	2	no	—
94	C₂×C₄×C₇₃	—	2	4	no	—
95	C₂×C₄×C₇₃	—	4	2	no	—
96	C₂×C₄×C₇₃	—	4	2	no	—
97	C₂×C₄×C₇₃	—	4	2	no	—
98	C₂×C₄×C₇₃	—	4	2	no	—
99	C₂×C₄×C₇₃	—	4	2	no	—
100	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
101	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
102	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
103	C₂×C₄×C₇₃	C₂×Q₁₆	1	2	yes	—
104	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
105	C₂×C₄×C₇₃	C₂²×Q₈	1	1	yes	characteristic
106	C₂×C₄×C₇₃	C₂×#41	1	2	yes	—
107	C₂×C₄×C₇₃	2₋¹⁺⁴	1	2	yes	—
108	C₂×Dic₇₃	—	2	4	no	—
109	C₂×Dic₇₃	—	4	4	no	—
110	C₂×Dic₇₃	—	4	4	no	—
111	C₂×Dic₇₃	—	4	4	no	—
112	C₂×Dic₇₃	C₂×C₄⋊C₄	1	4	yes	—
113	C₂×Dic₇₃	C₂×C₄⋊C₄	1	4	yes	—
114	C₂×Dic₇₃	C₂×D₈	1	4	yes	—
115	C₂×Dic₇₃	C₂×Q₁₆	1	4	yes	—
116	Dic₁₄₆	#107	1	16	yes	complemented
117	C₈×C₇₃	—	2	2	no	—
118	C₈×C₇₃	—	2	2	no	—
119	#10596831	—	2	4	no	—
120	C₂²×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
121	C₂²×C₇₃	#548	1	2	yes	—
122	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
123	C₂²×C₇₃	C₂²×Q₁₆	1	1	yes	characteristic
124	C₂²×C₇₃	#577	1	2	yes	—
125	C₄×C₇₃	—	2	2	no	—
126	C₄×C₇₃	—	4	4	no	—
127	C₄×C₇₃	—	8	2	no	—
128	C₄×C₇₃	C₂×#107	1	4	yes	—
129	C₄×C₇₃	C₄×#41	1	1	yes	characteristic
130	C₄×C₇₃	#519	1	1	yes	characteristic
131	C₄×C₇₃	#557	1	2	yes	—
132	Dic₇₃	—	2	4	no	—
133	Dic₇₃	—	8	4	no	—
134	Dic₇₃	C₂×#107	1	8	yes	—
135	#22740	—	73	1	no	maximal, Sylow
136	C₂×C₇₃	#2351	1	1	yes	characteristic
137	C₂×C₇₃	C₂²×#107	1	1	yes	characteristic
138	C₂×C₇₃	#2912	1	1	yes	characteristic
139	C₂×C₇₃	#3358	1	2	yes	—
140	C₂×C₇₃	#3363	1	2	yes	—
141	C₂×C₈×Q₈	—	73	1	no	—
142	Q₈×C₄⋊C₄	—	73	2	no	—
143	#1947	—	73	4	no	—
144	#1947	—	73	2	no	—
145	#1834	—	73	1	no	—
146	#1834	—	73	2	no	—
147	C₄×#107	—	73	2	no	—
148	C₄×#107	D₇₃	1	1	yes	characteristic
149	C₇₃	#22740	1	1	yes	characteristic, complemented, Sylow
150	C₂×C₄×Q₈	—	73	1	no	—
151	C₂×C₄×Q₈	—	146	2	no	—
152	C₈×Q₈	—	73	8	no	—
153	C₂×#107	—	73	4	no	—
154	C₂×#107	C₂×D₇₃	1	2	yes	—
155	C₂×C₄⋊C₈	—	73	2	no	—
156	C₂×C₄⋊C₈	—	73	1	no	—
157	C₂×C₄×C₈	—	73	2	no	—
158	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
159	#389	—	146	4	no	—
160	#389	—	146	2	no	—
161	#384	—	73	4	no	—
162	#384	—	73	2	no	—
163	C₄×C₄⋊C₄	—	73	4	no	—
164	C₄×C₄⋊C₄	C₂×D₇₃	1	2	yes	—
165	#340	—	73	2	no	—
166	#340	—	73	2	no	—
167	#340	—	73	4	no	—
168	#340	C₂×D₇₃	1	1	yes	characteristic
169	#340	C₂×D₇₃	1	1	yes	characteristic
170	C₂²×Q₈	—	73	1	no	—
171	C₂²×C₈	—	73	2	no	—
172	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
173	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
174	C₄×Q₈	—	292	8	no	—
175	C₄×Q₈	—	73	4	no	—
176	C₄×Q₈	—	73	4	no	—
177	C₂×C₄⋊C₄	—	73	2	no	—
178	C₂×C₄⋊C₄	—	73	1	no	—
179	C₂×C₄⋊C₄	—	73	4	no	—
180	C₂×C₄⋊C₄	—	146	4	no	—
181	C₂×C₄⋊C₄	—	146	2	no	—
182	C₂×C₄⋊C₄	—	146	4	no	—
183	C₂×C₄⋊C₄	—	146	2	no	—
184	C₂×C₄⋊C₄	C₂²×D₇₃	1	2	yes	—
185	C₂×C₄⋊C₄	C₂²×D₇₃	1	2	yes	—
186	C₂×C₄²	—	2	2	no	—
187	C₂×C₄²	—	73	2	no	—
188	C₂×C₄²	—	146	4	no	—
189	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
190	#107	—	2	4	no	—
191	#107	—	146	8	no	—
192	C₄⋊C₈	—	73	8	no	—
193	C₄⋊C₈	—	73	4	no	—
194	C₄×C₈	—	73	8	no	—
195	C₄×C₈	C₄×D₇₃	1	4	yes	—
196	#95	—	2	2	no	—
197	#95	—	146	4	no	—
198	C₂×Q₈	—	73	4	no	—
199	C₂×Q₈	—	73	4	no	—
200	C₂×Q₈	—	73	4	no	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	—	73	2	no	—
204	C₂²×C₄	—	73	2	no	—
205	C₂²×C₄	—	146	4	no	—
206	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
207	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
208	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	73	4	no	—
212	C₂×C₈	—	146	4	no	—
213	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
214	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
215	C₄⋊C₄	—	2	2	no	—
216	C₄⋊C₄	—	2	2	no	—
217	C₄⋊C₄	—	2	2	no	—
218	C₄⋊C₄	—	292	8	no	—
219	C₄⋊C₄	—	292	4	no	—
220	C₄⋊C₄	—	584	2	no	—
221	C₄⋊C₄	—	584	4	no	—
222	C₄⋊C₄	—	73	8	no	—
223	C₄⋊C₄	—	73	4	no	—
224	C₄⋊C₄	—	146	4	no	—
225	C₄⋊C₄	—	146	4	no	—
226	C₄⋊C₄	#11924028	1	4	yes	—
227	C₄²	—	4	4	no	—
228	C₄²	—	292	8	no	—
229	C₄²	—	73	4	no	—
230	C₄²	—	73	4	no	—
231	C₄²	C₂×C₄×D₇₃	1	2	yes	—
232	C₄²	Q₈×D₇₃	1	2	yes	—
233	C₂³	C₂×D₄×D₇₃	1	1	yes	characteristic
234	Q₈	—	73	16	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	292	4	no	—
243	C₂×C₄	—	292	4	no	—
244	C₂×C₄	—	292	4	no	—
245	C₂×C₄	—	73	4	no	—
246	C₂×C₄	—	73	4	no	—
247	C₂×C₄	—	73	4	no	—
248	C₂×C₄	—	73	4	no	—
249	C₂×C₄	—	146	4	no	—
250	C₂×C₄	C₄⋊C₄×D₇₃	1	2	yes	—
251	C₂×C₄	C₄⋊C₄×D₇₃	1	2	yes	—
252	C₂×C₄	D₈×D₇₃	1	2	yes	—
253	C₂×C₄	Q₁₆×D₇₃	1	2	yes	—
254	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
255	C₂×C₄	C₂×Q₈×D₇₃	1	1	yes	characteristic
256	C₂×C₄	C₂×#11924028	1	2	yes	—
257	C₂×C₄	#14556638	1	2	yes	—
258	C₈	—	2	2	no	—
259	C₈	—	2	2	no	—
260	C₈	—	146	4	no	—
261	C₂²	C₂×C₄⋊C₄×D₇₃	1	1	yes	characteristic
262	C₂²	#17989246	1	2	yes	—
263	C₂²	C₂×D₈×D₇₃	1	1	yes	characteristic
264	C₂²	C₂×Q₁₆×D₇₃	1	1	yes	characteristic
265	C₂²	#17989394	1	2	yes	—
266	C₄	—	2	2	no	—
267	C₄	—	4	4	no	—
268	C₄	—	8	2	no	—
269	C₄	—	584	4	no	—
270	C₄	—	73	8	no	—
271	C₄	—	146	4	no	—
272	C₄	#107×D₇₃	1	4	yes	—
273	C₄	C₄×#11924028	1	1	yes	characteristic
274	C₄	#17989064	1	1	yes	characteristic
275	C₄	#17989315	1	2	yes	—
276	C₂	#22612851	1	1	yes	characteristic
277	C₂	C₂×#107×D₇₃	1	1	yes	characteristic
278	C₂	#22618523	1	1	yes	characteristic
279	C₂	#22622564	1	2	yes	—
280	C₂	#22622607	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₇₃
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₃
Fitting	C₄×#107×C₇₃	C₂
socle, prime radical	C₂³×C₇₃	C₂²×D₄
cosocle	C₂²×C₄×C₇₃	C₂⁴
hypercenter	C₄×#107	D₇₃
hypocenter	C₇₃	#22740
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×#107

Complements

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

N	H ≅ G/N	type
11. C₈×Dic₁₄₆	C₄	semidirect
116. Dic₁₄₆	#107	semidirect
149. C₇₃	#22740	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Dic₁₄₆	2
C₄⋊C₄×Dic₁₄₆	2
#22610689	2
#22610686	2
#22610059	2
#22610057	2
#22609823	2
C₄×#107×C₇₃	2
#22740	73