Subgroups — #33105798

Order 18688.

754 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₂₉₂	C₂	1	1	yes	maximal, characteristic
2	#22612894	C₂	1	2	yes	maximal
3	#22610802	C₂	1	2	yes	maximal
4	#22610573	C₂	1	4	yes	maximal
5	#22610095	C₂	1	1	yes	maximal, characteristic
6	#22609947	C₂	1	2	yes	maximal
7	#22609828	C₂	1	2	yes	maximal
8	C₇₃×#1770	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₂₉₂	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₁₄₆	C₂²	1	2	yes	—
11	C₄×Dic₂₉₂	C₄	1	8	yes	complemented
12	C₂×#14556525	C₂²	1	2	yes	—
13	C₂×#14556525	C₂²	1	1	yes	characteristic
14	C₂×#14556523	C₂²	1	2	yes	—
15	C₂×#14556516	C₂²	1	4	yes	—
16	C₂×#14556516	C₂²	1	2	yes	—
17	#17988493	C₂²	1	4	yes	—
18	#17988468	—	2	4	no	—
19	#17988454	—	2	2	no	—
20	#17988453	C₂²	1	2	yes	—
21	C₄×C₄⋊Dic₇₃	C₂²	1	4	yes	—
22	#17988432	—	2	2	no	—
23	#17988350	C₂²	1	4	yes	—
24	C₂×C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
25	C₂×#97×C₇₃	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
27	C₄×C₄⋊C₄×C₇₃	C₂²	1	1	yes	characteristic
28	#336×C₇₃	C₂²	1	2	yes	—
29	C₂²×Dic₁₄₆	C₂³	1	2	yes	—
30	C₂×Dic₂₉₂	—	2	2	no	—
31	C₂×Dic₂₉₂	—	2	2	no	—
32	C₂×Dic₂₉₂	C₂×C₄	1	4	yes	—
33	C₄×Dic₁₄₆	—	2	4	no	—
34	C₄×Dic₁₄₆	C₂×C₄	1	8	yes	—
35	C₂×C₄⋊Dic₇₃	C₂³	1	4	yes	—
36	C₂×C₄⋊Dic₇₃	C₂³	1	2	yes	—
37	C₂×#11924007	—	2	2	no	—
38	C₂×C₄×Dic₇₃	—	2	2	no	—
39	C₂×C₄×Dic₇₃	—	2	4	no	—
40	#14556525	—	4	2	no	—
41	#14556525	C₂×C₄	1	4	yes	—
42	#14556523	—	2	4	no	—
43	#14556516	—	4	4	no	—
44	#14556516	C₂×C₄	1	8	yes	—
45	#14556498	—	2	2	no	—
46	#14556498	—	2	4	no	—
47	#14556498	—	2	4	no	—
48	#14556498	—	2	2	no	—
49	#14556498	—	2	4	no	—
50	C₂²×C₈×C₇₃	C₂³	1	2	yes	—
51	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₇₃	D₄	1	2	yes	—
54	C₂×C₄²×C₇₃	C₂³	1	2	yes	—
55	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
56	C₄⋊C₈×C₇₃	—	2	2	no	—
57	#97×C₇₃	—	2	4	no	—
58	C₄×C₈×C₇₃	C₂×C₄	1	4	yes	—
59	#95×C₇₃	D₄	1	2	yes	—
60	C₂×Dic₁₄₆	—	2	2	no	—
61	C₂×Dic₁₄₆	—	2	2	no	—
62	C₂×Dic₁₄₆	—	2	2	no	—
63	C₂×Dic₁₄₆	—	2	2	no	—
64	C₂×Dic₁₄₆	—	2	2	no	—
65	C₂×Dic₁₄₆	C₂²×C₄	1	4	yes	—
66	C₂²×Dic₇₃	—	2	2	no	—
67	C₂²×Dic₇₃	—	2	2	no	—
68	C₂²×Dic₇₃	—	2	4	no	—
69	Dic₂₉₂	—	2	8	no	—
70	C₄⋊Dic₇₃	—	2	2	no	—
71	C₄⋊Dic₇₃	—	2	4	no	—
72	C₄⋊Dic₇₃	—	2	4	no	—
73	C₄⋊Dic₇₃	C₂²×C₄	1	4	yes	—
74	#11924007	—	2	8	no	—
75	C₄×Dic₇₃	—	2	8	no	—
76	C₄×Dic₇₃	—	4	8	no	—
77	C₂²×C₄×C₇₃	C₂×D₄	1	4	yes	—
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	1	no	—
82	C₂×C₈×C₇₃	—	2	1	no	—
83	C₂×C₈×C₇₃	—	2	2	no	—
84	C₂×C₈×C₇₃	—	2	2	no	—
85	C₂×C₈×C₇₃	—	2	2	no	—
86	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
87	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
88	C₂×C₈×C₇₃	#41	1	2	yes	—
89	C₂×C₈×C₇₃	#41	1	2	yes	—
90	C₄⋊C₄×C₇₃	—	4	2	no	—
91	C₄²×C₇₃	—	2	2	no	—
92	C₄²×C₇₃	—	2	2	no	—
93	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
94	C₄²×C₇₃	#41	1	2	yes	—
95	C₂³×C₇₃	C₂²×D₄	1	1	yes	characteristic
96	C₂×C₄×C₇₃	—	2	2	no	—
97	C₂×C₄×C₇₃	—	2	2	no	—
98	C₂×C₄×C₇₃	—	2	4	no	—
99	C₂×C₄×C₇₃	—	2	4	no	—
100	C₂×C₄×C₇₃	—	2	4	no	—
101	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
102	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
103	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
104	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
105	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
106	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
107	C₂×C₄×C₇₃	H(5,𝔽₂)	1	1	yes	characteristic
108	C₂×C₄×C₇₃	2₋¹⁺⁴	1	1	yes	characteristic
109	C₂×Dic₇₃	—	2	4	no	—
110	C₂×Dic₇₃	—	2	4	no	—
111	C₂×Dic₇₃	—	4	2	no	—
112	C₂×Dic₇₃	—	4	2	no	—
113	C₂×Dic₇₃	—	4	4	no	—
114	C₂×Dic₇₃	—	4	4	no	—
115	C₂×Dic₇₃	—	4	4	no	—
116	C₂×Dic₇₃	—	4	4	no	—
117	Dic₁₄₆	—	2	4	no	—
118	Dic₁₄₆	—	2	4	no	—
119	Dic₁₄₆	—	4	2	no	—
120	Dic₁₄₆	—	4	2	no	—
121	C₈×C₇₃	—	2	2	no	—
122	C₈×C₇₃	—	4	2	no	—
123	C₈×C₇₃	#117	1	4	yes	—
124	C₂²×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
125	C₂²×C₇₃	#547	1	1	yes	characteristic
126	C₂²×C₇₃	#550	1	1	yes	characteristic
127	C₂²×C₇₃	#575	1	2	yes	—
128	C₂²×C₇₃	#578	1	2	yes	—
129	C₄×C₇₃	—	2	2	no	—
130	C₄×C₇₃	—	2	2	no	—
131	C₄×C₇₃	—	4	4	no	—
132	C₄×C₇₃	C₂×#117	1	1	yes	characteristic
133	C₄×C₇₃	#520	1	1	yes	characteristic
134	C₄×C₇₃	#555	1	1	yes	characteristic
135	C₄×C₇₃	#556	1	1	yes	characteristic
136	Dic₇₃	—	4	4	no	—
137	Dic₇₃	—	4	4	no	—
138	Dic₇₃	—	8	4	no	—
139	#22815	—	73	1	no	maximal, Sylow
140	C₂×C₇₃	#2354	1	1	yes	characteristic
141	C₂×C₇₃	#2943	1	1	yes	characteristic
142	C₂×C₇₃	#2950	1	1	yes	characteristic
143	C₂×C₇₃	#3352	1	2	yes	—
144	C₂×C₇₃	#3402	1	2	yes	—
145	C₂×C₄×Q₁₆	—	73	1	no	—
146	#2356	—	73	2	no	—
147	#1971	—	73	2	no	—
148	#1928	—	73	4	no	—
149	#1846	—	73	1	no	—
150	#1805	—	73	2	no	—
151	#1776	—	73	2	no	—
152	#1770	D₇₃	1	1	yes	characteristic
153	C₇₃	#22815	1	1	yes	characteristic, complemented, Sylow
154	C₂²×Q₁₆	—	73	1	no	—
155	C₂×C₄×Q₈	—	73	2	no	—
156	C₄×Q₁₆	—	73	8	no	—
157	C₂×#107	—	73	2	no	—
158	C₂×#107	—	73	1	no	—
159	C₂×#106	—	73	2	no	—
160	C₂×C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
161	C₂×#103	—	73	4	no	—
162	C₂×#103	—	73	2	no	—
163	C₂×#97	C₂×D₇₃	1	2	yes	—
164	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
165	#391	—	73	4	no	—
166	#387	—	146	4	no	—
167	#383	—	146	2	no	—
168	#382	—	73	2	no	—
169	C₄×C₄⋊C₄	—	73	4	no	—
170	C₄×C₄⋊C₄	C₂×D₇₃	1	1	yes	characteristic
171	#376	—	146	2	no	—
172	#340	—	73	4	no	—
173	#336	C₂×D₇₃	1	2	yes	—
174	C₂²×Q₈	—	73	2	no	—
175	C₂×Q₁₆	—	73	4	no	—
176	C₂×Q₁₆	—	146	2	no	—
177	C₂×Q₁₆	—	146	2	no	—
178	C₂²×C₈	C₂²×D₇₃	1	2	yes	—
179	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
180	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
181	C₄×Q₈	—	73	8	no	—
182	C₄×Q₈	—	146	4	no	—
183	C₂×C₄⋊C₄	—	73	2	no	—
184	C₂×C₄⋊C₄	—	73	4	no	—
185	C₂×C₄⋊C₄	—	146	2	no	—
186	C₂×C₄⋊C₄	D₂₉₂	1	2	yes	—
187	C₂×C₄²	—	146	4	no	—
188	C₂×C₄²	—	146	2	no	—
189	C₂×C₄²	C₂²×D₇₃	1	2	yes	—
190	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
191	#107	—	292	2	no	—
192	#107	—	73	4	no	—
193	#106	—	146	4	no	—
194	C₄⋊C₈	—	2	2	no	—
195	#103	—	292	4	no	—
196	#103	—	73	8	no	—
197	#97	—	2	4	no	—
198	C₄×C₈	C₄×D₇₃	1	4	yes	—
199	#95	—	146	2	no	—
200	#95	—	146	4	no	—
201	#95	—	146	4	no	—
202	#95	—	146	2	no	—
203	#95	—	146	4	no	—
204	#95	D₂₉₂	1	2	yes	—
205	C₂×Q₈	—	73	4	no	—
206	C₂×Q₈	—	146	2	no	—
207	C₂×Q₈	—	146	2	no	—
208	C₂×Q₈	—	146	2	no	—
209	C₂×Q₈	—	146	2	no	—
210	C₂×Q₈	—	146	2	no	—
211	C₂²×C₄	—	146	4	no	—
212	C₂²×C₄	—	146	2	no	—
213	C₂²×C₄	—	146	2	no	—
214	C₂²×C₄	C₂×D₂₉₂	1	4	yes	—
215	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
216	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
217	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
218	Q₁₆	—	146	8	no	—
219	C₂×C₈	—	2	1	no	—
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
225	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
226	C₂×C₈	#11924027	1	2	yes	—
227	C₂×C₈	#11924029	1	2	yes	—
228	C₄⋊C₄	—	4	2	no	—
229	C₄⋊C₄	—	73	4	no	—
230	C₄⋊C₄	—	146	4	no	—
231	C₄⋊C₄	—	146	4	no	—
232	C₄⋊C₄	—	146	8	no	—
233	C₄⋊C₄	—	146	2	no	—
234	C₄²	—	2	2	no	—
235	C₄²	—	2	2	no	—
236	C₄²	—	292	8	no	—
237	C₄²	—	146	8	no	—
238	C₄²	C₂×C₄×D₇₃	1	2	yes	—
239	C₄²	#11924028	1	2	yes	—
240	C₂³	C₂²×D₂₉₂	1	1	yes	characteristic
241	Q₈	—	292	2	no	—
242	Q₈	—	292	2	no	—
243	Q₈	—	146	4	no	—
244	Q₈	—	146	4	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	292	4	no	—
251	C₂×C₄	—	292	2	no	—
252	C₂×C₄	—	292	2	no	—
253	C₂×C₄	—	292	4	no	—
254	C₂×C₄	—	292	4	no	—
255	C₂×C₄	—	292	4	no	—
256	C₂×C₄	—	146	4	no	—
257	C₂×C₄	—	146	4	no	—
258	C₂×C₄	C₄×D₂₉₂	1	2	yes	—
259	C₂×C₄	C₄×D₂₉₂	1	2	yes	—
260	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
261	C₂×C₄	C₂×#11924027	1	1	yes	characteristic
262	C₂×C₄	C₂×#11924028	1	1	yes	characteristic
263	C₂×C₄	C₂×#11924029	1	1	yes	characteristic
264	C₂×C₄	#14556637	1	1	yes	characteristic
265	C₂×C₄	#14556638	1	1	yes	characteristic
266	C₈	—	2	2	no	—
267	C₈	—	4	2	no	—
268	C₈	#14556549	1	4	yes	—
269	C₂²	C₂×C₄×D₂₉₂	1	1	yes	characteristic
270	C₂²	#17989241	1	1	yes	characteristic
271	C₂²	#17989259	1	1	yes	characteristic
272	C₂²	#17989387	1	2	yes	—
273	C₂²	#17989404	1	2	yes	—
274	C₄	—	2	2	no	—
275	C₄	—	2	2	no	—
276	C₄	—	4	4	no	—
277	C₄	—	292	4	no	—
278	C₄	—	292	4	no	—
279	C₄	—	584	4	no	—
280	C₄	C₂×#14556549	1	1	yes	characteristic
281	C₄	#17989067	1	1	yes	characteristic
282	C₄	#17989302	1	1	yes	characteristic
283	C₄	#17989310	1	1	yes	characteristic
284	C₂	#22612878	1	1	yes	characteristic
285	C₂	#22618733	1	1	yes	characteristic
286	C₂	#22618800	1	1	yes	characteristic
287	C₂	#22622505	1	2	yes	—
288	C₂	#22623012	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₂₉₂
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₃
Fitting	C₇₃×#1770	C₂
socle, prime radical	C₂³×C₇₃	C₂²×D₄
cosocle	C₂²×C₄×C₇₃	C₂⁴
hypercenter	#1770	D₇₃
hypocenter	C₇₃	#22815
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ #1770

Complements

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

N	H ≅ G/N	type
11. C₄×Dic₂₉₂	C₄	semidirect
153. C₇₃	#22815	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₂₉₂	2
#22612894	2
#22610802	2
#22610573	2
#22610095	2
#22609947	2
#22609828	2
C₇₃×#1770	2
#22815	73