Subgroups — #33244075

Order 18688.

504 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22622920	C₂	1	2	yes	maximal, complemented
2	#22612202	C₂	1	2	yes	maximal, complemented
3	#22612192	C₂	1	1	yes	maximal, characteristic
4	#22611994	C₂	1	2	yes	maximal, complemented
5	#22611987	C₂	1	2	yes	maximal
6	#22611982	C₂	1	2	yes	maximal, complemented
7	#22611797	C₂	1	2	yes	maximal, complemented
8	C₄×Dic₅₈₄	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇₃×#2183	C₂	1	1	yes	maximal, characteristic, complemented
10	#17989310	—	2	2	no	—
11	C₂×Dic₅₈₄	C₂²	1	1	yes	characteristic
12	C₂×Dic₅₈₄	C₂²	1	1	yes	characteristic
13	C₂×Dic₅₈₄	C₂²	1	2	yes	—
14	C₂×#14556536	C₂²	1	2	yes	—
15	C₂×#14556536	C₂²	1	2	yes	—
16	C₂×D₁₁₆₈	C₂²	1	1	yes	characteristic
17	#17988971	C₂²	1	2	yes	—
18	#17988971	C₂²	1	2	yes	—
19	#17988967	C₂²	1	2	yes	—
20	#17988810	—	2	2	no	—
21	#17988810	—	2	2	no	—
22	#17988793	—	2	2	no	—
23	C₄×Dic₂₉₂	C₂²	1	2	yes	—
24	C₄×Dic₂₉₂	C₂²	1	2	yes	—
25	C₄×D₅₈₄	C₂²	1	2	yes	—
26	#17988413	C₂²	1	1	yes	characteristic
27	#17988392	C₂²	1	2	yes	—
28	#17988392	C₂²	1	2	yes	—
29	#17988389	C₂²	1	2	yes	—
30	C₈×Q₈×C₇₃	C₂²	1	1	yes	characteristic
31	C₄⋊C₁₆×C₇₃	C₂²	1	2	yes	—
32	C₄⋊C₁₆×C₇₃	C₂²	1	1	yes	characteristic
33	#346×C₇₃	C₂²	1	2	yes	—
34	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
35	C₂×Dic₂₉₂	—	2	2	no	—
36	C₂×Dic₂₉₂	C₂³	1	2	yes	—
37	C₂×Dic₂₉₂	C₂³	1	2	yes	—
38	C₂×#11924017	—	2	2	no	—
39	C₂×#11924017	—	2	2	no	—
40	C₂×D₅₈₄	C₂³	1	2	yes	—
41	#14556591	—	2	2	no	—
42	#14556586	—	4	2	no	—
43	#14556586	—	4	2	no	—
44	#14556579	—	2	2	no	—
45	#14556579	—	2	2	no	—
46	#14556567	—	4	2	no	—
47	C₄×Dic₁₄₆	—	2	2	no	—
48	C₄×Dic₁₄₆	—	2	2	no	—
49	C₄×D₂₉₂	—	2	2	no	—
50	#14556549	—	4	2	no	—
51	Dic₅₈₄	—	2	2	no	—
52	Dic₅₈₄	—	4	1	no	—
53	Dic₅₈₄	—	4	2	no	—
54	#14556536	—	2	4	no	—
55	#14556536	—	4	2	no	—
56	D₁₁₆₈	—	4	1	no	—
57	#14556525	C₂³	1	2	yes	—
58	#14556516	—	2	2	no	—
59	#14556516	—	2	2	no	—
60	#14556513	—	2	2	no	—
61	C₄×Q₈×C₇₃	D₄	1	1	yes	characteristic
62	C₂×C₁₆×C₇₃	C₂³	1	2	yes	—
63	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
64	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
65	C₄⋊C₈×C₇₃	D₄	1	2	yes	—
66	C₄⋊C₈×C₇₃	D₄	1	1	yes	characteristic
67	C₄×C₈×C₇₃	C₂³	1	2	yes	—
68	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
69	C₂×Dic₁₄₆	—	2	2	no	—
70	C₂×Dic₁₄₆	—	2	2	no	—
71	C₂×D₂₉₂	—	2	2	no	—
72	C₂×C₄×D₇₃	—	4	2	no	—
73	Dic₂₉₂	—	2	2	no	—
74	Dic₂₉₂	—	2	2	no	—
75	Dic₂₉₂	—	2	2	no	—
76	Dic₂₉₂	—	2	2	no	—
77	Dic₂₉₂	—	8	2	no	—
78	#11924017	—	8	2	no	—
79	#11924017	—	8	2	no	—
80	D₅₈₄	—	2	2	no	—
81	D₅₈₄	—	2	2	no	—
82	C₄⋊Dic₇₃	—	2	2	no	—
83	C₄⋊Dic₇₃	—	4	2	no	—
84	C₄⋊Dic₇₃	—	4	2	no	—
85	#11924007	—	4	2	no	—
86	#11924007	—	4	2	no	—
87	#11924005	—	4	2	no	—
88	#11924005	—	4	2	no	—
89	#11924005	—	4	2	no	—
90	#11924005	—	4	2	no	—
91	#11924005	—	4	2	no	—
92	C₄×Dic₇₃	—	4	2	no	—
93	C₄×Dic₇₃	—	4	2	no	—
94	C₂×Q₈×C₇₃	D₈	1	1	yes	characteristic
95	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
96	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
98	C₄⋊C₄×C₇₃	D₈	1	2	yes	—
99	C₄⋊C₄×C₇₃	D₈	1	1	yes	characteristic
100	C₄²×C₇₃	C₂×D₄	1	2	yes	—
101	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
102	C₁₆×C₇₃	—	2	1	no	—
103	C₁₆×C₇₃	—	2	2	no	—
104	C₁₆×C₇₃	#41	1	2	yes	—
105	C₂²×D₇₃	—	4	2	no	—
106	Q₈×C₇₃	—	4	1	no	—
107	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
108	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
109	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
110	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
111	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
112	C₂×Dic₇₃	—	4	2	no	—
113	C₂×Dic₇₃	—	4	2	no	—
114	C₂×Dic₇₃	—	4	2	no	—
115	D₂₉₂	—	4	2	no	—
116	D₂₉₂	—	4	2	no	—
117	C₄×D₇₃	—	8	4	no	—
118	Dic₁₄₆	—	4	2	no	—
119	Dic₁₄₆	—	4	2	no	—
120	Dic₁₄₆	—	4	2	no	—
121	Dic₁₄₆	—	4	2	no	—
122	C₈×C₇₃	—	2	1	no	—
123	C₈×C₇₃	—	2	2	no	—
124	C₈×C₇₃	C₂×#41	1	1	yes	characteristic
125	C₈×C₇₃	H(5,𝔽₂)	1	1	yes	characteristic
126	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
127	C₂×D₇₃	—	8	2	no	—
128	C₂×D₇₃	—	8	2	no	—
129	C₂×D₇₃	—	8	2	no	—
130	C₄×C₇₃	—	2	2	no	—
131	C₄×C₇₃	—	2	1	no	—
132	C₄×C₇₃	—	2	1	no	—
133	C₄×C₇₃	—	2	2	no	—
134	C₄×C₇₃	#550	1	1	yes	characteristic
135	C₄×C₇₃	#576	1	1	yes	characteristic
136	Dic₇₃	—	8	2	no	—
137	Dic₇₃	—	8	2	no	—
138	Dic₇₃	—	8	2	no	—
139	#33969	—	73	1	no	maximal, Sylow
140	C₂×C₇₃	#3391	1	1	yes	characteristic
141	C₂×C₇₃	#3414	1	1	yes	characteristic
142	C₂×C₇₃	#3417	1	1	yes	characteristic
143	D₇₃	—	16	2	no	—
144	#3393	—	73	2	no	—
145	#2251	—	73	2	no	—
146	#2249	—	73	1	no	—
147	#2211	—	73	2	no	—
148	#2210	—	73	2	no	—
149	#2209	—	73	2	no	—
150	#2183	D₇₃	1	1	yes	characteristic, complemented
151	#2175	—	73	2	no	—
152	C₄×Q₃₂	—	73	1	no	—
153	C₇₃	#33969	1	1	yes	characteristic, complemented, Sylow
154	#556	—	146	2	no	—
155	C₂×Q₃₂	—	73	1	no	—
156	C₂×Q₃₂	—	73	2	no	—
157	C₂×Q₃₂	—	73	1	no	—
158	C₂×SD₃₂	—	73	2	no	—
159	C₂×SD₃₂	—	73	2	no	—
160	C₂×D₁₆	—	73	1	no	—
161	#495	—	73	2	no	—
162	#495	—	73	2	no	—
163	#494	—	73	2	no	—
164	#464	—	146	2	no	—
165	#464	—	146	2	no	—
166	#461	—	146	2	no	—
167	C₈×Q₈	C₂×D₇₃	1	1	yes	characteristic
168	C₄×Q₁₆	—	73	2	no	—
169	C₄×Q₁₆	—	73	2	no	—
170	C₄×D₈	—	73	2	no	—
171	#366	—	73	1	no	—
172	C₄⋊C₁₆	C₂×D₇₃	1	2	yes	—
173	C₄⋊C₁₆	C₂×D₇₃	1	1	yes	characteristic
174	#358	—	73	2	no	—
175	#358	—	73	2	no	—
176	#357	—	73	2	no	—
177	#346	C₂×D₇₃	1	2	yes	—
178	C₄×C₁₆	C₂×D₇₃	1	1	yes	characteristic
179	C₂×Q₁₆	—	73	2	no	—
180	C₂×Q₁₆	—	73	2	no	—
181	C₂×Q₁₆	—	146	2	no	—
182	C₂×SD₁₆	—	146	2	no	—
183	C₂×SD₁₆	—	146	2	no	—
184	C₂×D₈	—	73	2	no	—
185	C₄⋊Q₈	—	146	2	no	—
186	#126	—	292	2	no	—
187	#126	—	292	2	no	—
188	#124	—	146	2	no	—
189	#124	—	146	2	no	—
190	C₂²⋊Q₈	—	292	2	no	—
191	C₄×Q₈	—	146	2	no	—
192	C₄×Q₈	—	146	2	no	—
193	C₄×Q₈	D₂₉₂	1	1	yes	characteristic
194	C₄×D₄	—	146	2	no	—
195	#117	—	292	2	no	—
196	Q₃₂	—	292	1	no	—
197	Q₃₂	—	292	2	no	—
198	Q₃₂	—	146	2	no	—
199	SD₃₂	—	292	2	no	—
200	SD₃₂	—	146	4	no	—
201	D₁₆	—	292	1	no	—
202	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
203	C₂×C₁₆	C₂²×D₇₃	1	2	yes	—
204	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
205	#107	—	73	2	no	—
206	C₄⋊C₈	D₂₉₂	1	2	yes	—
207	C₄⋊C₈	D₂₉₂	1	1	yes	characteristic
208	#103	—	146	2	no	—
209	#103	—	146	2	no	—
210	#102	—	146	2	no	—
211	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
212	C₄×C₈	C₂²×D₇₃	1	2	yes	—
213	C₂×Q₈	—	146	2	no	—
214	C₂×Q₈	—	146	2	no	—
215	C₂×Q₈	D₅₈₄	1	1	yes	characteristic
216	C₂×D₄	—	146	2	no	—
217	C₂²×C₄	—	292	2	no	—
218	Q₁₆	—	584	2	no	—
219	Q₁₆	—	146	2	no	—
220	Q₁₆	—	146	2	no	—
221	Q₁₆	—	146	2	no	—
222	Q₁₆	—	146	2	no	—
223	SD₁₆	—	584	2	no	—
224	SD₁₆	—	584	2	no	—
225	D₈	—	146	2	no	—
226	D₈	—	146	2	no	—
227	C₂×C₈	C₂×D₂₉₂	1	2	yes	—
228	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
229	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
230	C₄⋊C₄	—	292	2	no	—
231	C₄⋊C₄	—	292	2	no	—
232	C₄⋊C₄	—	292	2	no	—
233	C₄⋊C₄	—	292	2	no	—
234	C₄⋊C₄	—	146	2	no	—
235	C₄⋊C₄	D₅₈₄	1	2	yes	—
236	C₄⋊C₄	D₅₈₄	1	1	yes	characteristic
237	C₂²⋊C₄	—	292	2	no	—
238	C₂²⋊C₄	—	292	2	no	—
239	C₂²⋊C₄	—	292	2	no	—
240	C₂²⋊C₄	—	292	2	no	—
241	C₂²⋊C₄	—	292	2	no	—
242	C₄²	—	292	2	no	—
243	C₄²	—	292	2	no	—
244	C₄²	C₂×D₂₉₂	1	2	yes	—
245	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
246	C₁₆	—	2	1	no	—
247	C₁₆	—	2	2	no	—
248	C₁₆	#11924029	1	2	yes	—
249	C₂³	—	292	2	no	—
250	Q₈	—	4	1	no	—
251	Q₈	—	292	2	no	—
252	Q₈	—	292	2	no	—
253	Q₈	—	292	2	no	—
254	Q₈	—	292	2	no	—
255	D₄	—	292	2	no	—
256	D₄	—	292	2	no	—
257	C₂×C₄	—	292	2	no	—
258	C₂×C₄	—	292	2	no	—
259	C₂×C₄	—	292	2	no	—
260	C₂×C₄	—	584	4	no	—
261	C₂×C₄	C₂×D₅₈₄	1	1	yes	characteristic
262	C₂×C₄	C₂×D₅₈₄	1	2	yes	—
263	C₂×C₄	C₂×D₅₈₄	1	1	yes	characteristic
264	C₂×C₄	C₂×D₅₈₄	1	2	yes	—
265	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
266	C₈	—	2	1	no	—
267	C₈	—	2	2	no	—
268	C₈	C₂×#11924029	1	1	yes	characteristic
269	C₈	#14556637	1	1	yes	characteristic
270	C₂²	—	584	2	no	—
271	C₂²	—	584	2	no	—
272	C₂²	—	584	2	no	—
273	C₂²	C₂²×D₅₈₄	1	1	yes	characteristic
274	C₄	—	2	2	no	—
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	2	2	no	—
278	C₄	—	584	2	no	—
279	C₄	—	584	2	no	—
280	C₄	—	584	2	no	—
281	C₄	#17989259	1	1	yes	characteristic
282	C₄	#17989392	1	1	yes	characteristic
283	C₂	—	1168	2	no	—
284	C₂	#22622906	1	1	yes	characteristic
285	C₂	#22623067	1	1	yes	characteristic
286	C₂	#22623084	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₅₈₄
commutator, cosocle	C₂×C₈×C₇₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₇₃×#2183	C₂
socle	C₂²×C₇₃	C₂²×D₈
prime radical	C₂×D₁₁₆₈	C₂²
hypercenter	#2183	D₇₃
hypocenter	C₇₃	#33969
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ #2183

Complements

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

N	H ≅ G/N	type
1. #22622920	C₂	semidirect
2. #22612202	C₂	semidirect
4. #22611994	C₂	semidirect
6. #22611982	C₂	semidirect
7. #22611797	C₂	semidirect
8. C₄×Dic₅₈₄	C₂	semidirect
9. C₇₃×#2183	C₂	semidirect
150. #2183	D₇₃	semidirect
153. C₇₃	#33969	semidirect

Maximal subgroups

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

subgroup	index
#22622920	2
#22612202	2
#22612192	2
#22611994	2
#22611987	2
#22611982	2
#22611797	2
C₄×Dic₅₈₄	2
C₇₃×#2183	2
#33969	73