Subgroups — #33002066

Order 18688.

410 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	C₂×#17988404	C₂	1	2	yes	maximal
3	#22610103	C₂	1	1	yes	maximal, characteristic
4	#22608040	C₂	1	2	yes	maximal
5	C₇₃×#2150	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₉₂	C₂²	1	1	yes	characteristic
7	C₂×C₄×Dic₁₄₆	—	2	1	no	—
8	C₄×Dic₂₉₂	—	2	1	no	—
9	C₄×Dic₂₉₂	—	2	1	no	—
10	C₄×Dic₂₉₂	C₄	1	2	yes	—
11	C₄×Dic₂₉₂	C₄	1	2	yes	—
12	C₂×#14556527	—	2	1	no	—
13	C₂×#14556527	C₂²	1	2	yes	—
14	C₂×#14556525	C₂²	1	1	yes	characteristic
15	#17988611	—	2	1	no	—
16	C₂×#14556516	—	2	1	no	—
17	#17988404	—	2	4	no	—
18	C₂×M₃₂×C₇₃	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
20	C₄⋊C₁₆×C₇₃	—	2	1	no	—
21	C₄⋊C₁₆×C₇₃	—	2	1	no	—
22	C₂²×Dic₁₄₆	—	2	1	no	—
23	C₂×Dic₂₉₂	—	2	1	no	—
24	C₂×Dic₂₉₂	—	2	1	no	—
25	C₂×Dic₂₉₂	—	2	1	no	—
26	C₂×Dic₂₉₂	C₂×C₄	1	2	yes	—
27	C₂×Dic₂₉₂	D₄	1	2	yes	—
28	C₂×Dic₂₉₂	Q₈	1	2	yes	—
29	C₂×#11924012	—	2	2	no	—
30	C₄×Dic₁₄₆	—	2	2	no	—
31	C₄×Dic₁₄₆	—	2	2	no	—
32	C₄×Dic₁₄₆	—	4	1	no	—
33	C₄×Dic₁₄₆	—	4	1	no	—
34	C₂×C₄⋊Dic₇₃	—	2	1	no	—
35	C₂×#11924007	—	4	1	no	—
36	C₂×C₄×Dic₇₃	—	4	1	no	—
37	#14556527	—	2	2	no	—
38	#14556527	—	2	2	no	—
39	#14556527	—	4	2	no	—
40	#14556525	—	2	1	no	—
41	#14556525	C₂×C₄	1	2	yes	—
42	C₄⋊#10596831	—	4	1	no	—
43	C₄⋊#10596831	—	4	1	no	—
44	#14556516	—	2	2	no	—
45	#14556516	—	2	2	no	—
46	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
49	M₃₂×C₇₃	—	2	2	no	—
50	M₃₂×C₇₃	—	2	2	no	—
51	C₂×C₁₆×C₇₃	—	2	2	no	—
52	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
55	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
56	C₂×Dic₁₄₆	—	2	2	no	—
57	C₂×Dic₁₄₆	—	2	2	no	—
58	C₂×Dic₁₄₆	—	2	2	no	—
59	C₂×Dic₁₄₆	—	4	1	no	—
60	C₂×Dic₁₄₆	—	4	1	no	—
61	C₂×Dic₁₄₆	—	4	1	no	—
62	C₂²×Dic₇₃	—	4	1	no	—
63	C₂²×Dic₇₃	—	4	1	no	—
64	Dic₂₉₂	—	2	2	no	—
65	Dic₂₉₂	—	2	2	no	—
66	Dic₂₉₂	—	2	2	no	—
67	Dic₂₉₂	C₄⋊C₄	1	4	yes	—
68	#11924012	—	4	2	no	—
69	#11924012	—	4	2	no	—
70	C₂×#10596831	—	4	2	no	—
71	C₄⋊Dic₇₃	—	2	2	no	—
72	C₄⋊Dic₇₃	—	4	1	no	—
73	#11924007	—	4	2	no	—
74	#11924007	—	4	2	no	—
75	C₄×Dic₇₃	—	4	2	no	—
76	C₄×Dic₇₃	—	4	2	no	—
77	C₂²×C₄×C₇₃	SD₁₆	1	1	yes	characteristic
78	C₂²×C₄×C₇₃	SD₁₆	1	1	yes	characteristic
79	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₇₃	—	2	1	no	—
81	C₂×C₈×C₇₃	—	2	1	no	—
82	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
88	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
89	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
90	C₄²×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
91	C₄²×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
92	C₄²×C₇₃	D₈	1	1	yes	characteristic
93	C₄²×C₇₃	D₈	1	1	yes	characteristic
94	C₁₆×C₇₃	—	4	2	no	—
95	C₂³×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
96	C₂×C₄×C₇₃	—	2	2	no	—
97	C₂×C₄×C₇₃	—	4	1	no	—
98	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
99	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
100	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
101	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
102	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
104	C₂×C₄×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
105	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
106	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
107	C₂×C₄×C₇₃	#137	1	1	yes	characteristic
108	C₂×Dic₇₃	—	4	2	no	—
109	C₂×Dic₇₃	—	4	2	no	—
110	C₂×Dic₇₃	—	4	2	no	—
111	C₂×Dic₇₃	—	4	2	no	—
112	C₂×Dic₇₃	—	8	2	no	—
113	Dic₁₄₆	—	2	4	no	—
114	Dic₁₄₆	—	4	2	no	—
115	Dic₁₄₆	—	4	2	no	—
116	Dic₁₄₆	—	4	2	no	—
117	C₈×C₇₃	—	2	1	no	—
118	C₈×C₇₃	—	2	1	no	—
119	C₈×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₈×C₇₃	#117	1	1	yes	characteristic
121	C₈×C₇₃	#121	1	1	yes	characteristic
122	C₈×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
123	#10596831	—	8	2	no	—
124	C₂²×C₇₃	—	2	1	no	—
125	C₂²×C₇₃	—	2	1	no	—
126	C₂²×C₇₃	C₂×#102	1	1	yes	characteristic
127	C₂²×C₇₃	#461	1	1	yes	characteristic
128	C₂²×C₇₃	#476	1	1	yes	characteristic
129	C₄×C₇₃	—	2	1	no	—
130	C₄×C₇₃	—	2	1	no	—
131	C₄×C₇₃	—	2	1	no	—
132	C₄×C₇₃	#362	1	2	yes	complemented
133	C₄×C₇₃	#380	1	1	yes	characteristic
134	C₄×C₇₃	#417	1	1	yes	characteristic
135	C₄×C₇₃	#459	1	1	yes	characteristic
136	C₄×C₇₃	#474	1	1	yes	characteristic
137	Dic₇₃	—	4	4	no	—
138	Dic₇₃	—	8	2	no	—
139	#12819	—	73	1	no	maximal, Sylow
140	C₂×C₇₃	—	2	1	no	—
141	C₂×C₇₃	—	2	1	no	—
142	C₂×C₇₃	#1802	1	1	yes	characteristic
143	C₂×C₇₃	C₂×#362	1	1	yes	characteristic
144	C₂×C₇₃	#2148	1	1	yes	characteristic
145	C₂×C₄×Q₁₆	—	73	1	no	—
146	#2150	D₇₃	1	1	yes	characteristic
147	C₂×#362	—	73	2	no	—
148	#1848	—	73	1	no	—
149	#1387	—	73	2	no	—
150	C₇₃	#12819	1	1	yes	characteristic, complemented, Sylow
151	C₂²×Q₁₆	—	73	1	no	—
152	C₂×C₄×Q₈	—	146	1	no	—
153	C₂×M₃₂	C₂×D₇₃	1	2	yes	—
154	C₄×Q₁₆	—	73	2	no	—
155	C₄×Q₁₆	—	73	2	no	—
156	C₄×Q₁₆	—	146	1	no	—
157	C₄×Q₁₆	—	146	1	no	—
158	C₂×#108	—	73	2	no	—
159	C₂×#108	—	146	1	no	—
160	C₂×#107	—	73	1	no	—
161	#423	—	146	1	no	—
162	C₂×#103	—	146	1	no	—
163	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
164	C₄⋊C₁₆	—	2	1	no	—
165	C₄⋊C₁₆	—	2	1	no	—
166	#362	—	146	4	no	—
167	C₂²×Q₈	—	146	1	no	—
168	C₂×Q₁₆	—	73	2	no	—
169	C₂×Q₁₆	—	73	2	no	—
170	C₂×Q₁₆	—	73	2	no	—
171	C₂×Q₁₆	—	146	1	no	—
172	C₂×Q₁₆	—	146	1	no	—
173	C₂×Q₁₆	—	146	1	no	—
174	C₂×M₁₆	—	146	2	no	—
175	C₂²×C₈	D₂₉₂	1	1	yes	characteristic
176	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
177	C₄×Q₈	—	292	1	no	—
178	C₄×Q₈	—	292	1	no	—
179	C₄×Q₈	—	146	2	no	—
180	C₄×Q₈	—	146	2	no	—
181	C₂×C₄⋊C₄	—	292	1	no	—
182	C₂×C₄⋊C₄	—	146	1	no	—
183	C₂×C₄²	—	292	1	no	—
184	C₂×C₄²	D₂₉₂	1	1	yes	characteristic
185	M₃₂	—	2	2	no	—
186	M₃₂	—	2	2	no	—
187	C₂×C₁₆	—	2	2	no	—
188	#108	—	292	2	no	—
189	#108	—	146	2	no	—
190	#108	—	146	2	no	—
191	#107	—	73	2	no	—
192	#107	—	146	1	no	—
193	C₄⋊C₈	—	292	1	no	—
194	C₄⋊C₈	—	292	1	no	—
195	#103	—	146	2	no	—
196	#103	—	146	2	no	—
197	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
198	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
199	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
200	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
201	C₂×Q₈	—	292	1	no	—
202	C₂×Q₈	—	292	1	no	—
203	C₂×Q₈	—	292	1	no	—
204	C₂×Q₈	—	146	2	no	—
205	C₂×Q₈	—	146	2	no	—
206	C₂×Q₈	—	146	2	no	—
207	C₂²×C₄	—	292	1	no	—
208	C₂²×C₄	—	292	1	no	—
209	C₂²×C₄	#11924017	1	1	yes	characteristic
210	C₂²×C₄	#11924017	1	1	yes	characteristic
211	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
212	Q₁₆	—	73	4	no	—
213	Q₁₆	—	146	2	no	—
214	Q₁₆	—	146	2	no	—
215	Q₁₆	—	146	2	no	—
216	M₁₆	—	292	2	no	—
217	M₁₆	—	292	2	no	—
218	C₂×C₈	—	2	1	no	—
219	C₂×C₈	—	2	1	no	—
220	C₂×C₈	—	292	2	no	—
221	C₂×C₈	#11924005	1	1	yes	characteristic
222	C₂×C₈	#11924005	1	1	yes	characteristic
223	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
224	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
225	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
226	C₂×C₈	Q₈×D₇₃	1	1	yes	characteristic
227	C₂×C₈	#11924027	1	1	yes	characteristic
228	C₂×C₈	#11924029	1	1	yes	characteristic
229	C₄⋊C₄	—	292	2	no	—
230	C₄⋊C₄	—	292	2	no	—
231	C₄⋊C₄	—	292	1	no	—
232	C₄⋊C₄	—	146	2	no	—
233	C₄²	—	292	2	no	—
234	C₄²	—	292	2	no	—
235	C₄²	#11924005	1	1	yes	characteristic
236	C₄²	#11924005	1	1	yes	characteristic
237	C₄²	D₅₈₄	1	1	yes	characteristic
238	C₄²	D₅₈₄	1	1	yes	characteristic
239	C₁₆	—	4	2	no	—
240	C₂³	C₂×#11924017	1	1	yes	characteristic
241	Q₈	—	292	2	no	—
242	Q₈	—	292	2	no	—
243	Q₈	—	292	2	no	—
244	Q₈	—	146	4	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	292	2	no	—
248	C₂×C₄	—	292	2	no	—
249	C₂×C₄	—	292	2	no	—
250	C₂×C₄	—	292	2	no	—
251	C₂×C₄	—	584	2	no	—
252	C₂×C₄	#14556513	1	1	yes	characteristic
253	C₂×C₄	#14556513	1	1	yes	characteristic
254	C₂×C₄	#14556513	1	1	yes	characteristic
255	C₂×C₄	#14556513	1	1	yes	characteristic
256	C₂×C₄	C₂×#11924005	1	1	yes	characteristic
257	C₂×C₄	#14556566	1	1	yes	characteristic
258	C₂×C₄	#14556571	1	1	yes	characteristic
259	C₂×C₄	C₂×D₅₈₄	1	1	yes	characteristic
260	C₂×C₄	#14556619	1	1	yes	characteristic
261	C₂×C₄	#14556624	1	1	yes	characteristic
262	C₈	—	2	1	no	—
263	C₈	—	2	1	no	—
264	C₈	—	584	2	no	—
265	C₈	C₄⋊C₄×D₇₃	1	1	yes	characteristic
266	C₈	#14556549	1	1	yes	characteristic
267	C₈	#14556562	1	1	yes	characteristic
268	C₈	#14556567	1	1	yes	characteristic
269	C₂²	—	2	1	no	—
270	C₂²	—	2	1	no	—
271	C₂²	C₂×#14556513	1	1	yes	characteristic
272	C₂²	#17988793	1	1	yes	characteristic
273	C₂²	#17988878	1	1	yes	characteristic
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	292	4	no	—
278	C₄	—	584	2	no	—
279	C₄	#17988404	1	2	yes	complemented
280	C₄	#17988444	1	1	yes	characteristic
281	C₄	#17988589	1	1	yes	characteristic
282	C₄	#17988783	1	1	yes	characteristic
283	C₄	#17988868	1	1	yes	characteristic
284	C₂	—	2	1	no	—
285	C₂	—	2	1	no	—
286	C₂	#22609932	1	1	yes	characteristic
287	C₂	C₂×#17988404	1	1	yes	characteristic
288	C₂	#22611681	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14556513
commutator	C₂×C₈×C₇₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₇₃×#2150	C₂
socle	C₂²×C₇₃	C₂×#102
prime radical	C₂³×C₇₃	C₂×SD₁₆
cosocle	C₂²×C₈×C₇₃	C₂³
hypercenter	#2150	D₇₃
hypocenter	C₇₃	#12819
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2150

Complements

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

N	H ≅ G/N	type
132. C₄×C₇₃	#362	semidirect
150. C₇₃	#12819	semidirect
279. C₄	#17988404	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
C₂×#17988404	2
#22610103	2
#22608040	2
C₇₃×#2150	2
#12819	73