Subgroups — #33057484

Order 18688.

630 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	M₁₆×Dic₁₄₆	C₂	1	2	yes	maximal
2	C₂×C₄×Dic₂₉₂	C₂	1	1	yes	maximal, characteristic
3	#22608841	C₂	1	2	yes	maximal, complemented
4	#22608804	C₂	1	4	yes	maximal, complemented
5	#22608751	C₂	1	2	yes	maximal, complemented
6	#22608687	C₂	1	1	yes	maximal, characteristic
7	#22608361	C₂	1	2	yes	maximal
8	C₇₃×#1451	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₂₉₂	C₄	1	1	yes	characteristic
10	C₂×C₄×Dic₁₄₆	C₂²	1	2	yes	—
11	#17988717	C₂²	1	4	yes	—
12	#17988715	—	2	4	no	—
13	C₈×Dic₁₄₆	C₂²	1	4	yes	—
14	C₄×Dic₂₉₂	C₄	1	2	yes	—
15	C₄×Dic₂₉₂	C₄	1	1	yes	characteristic
16	C₄×Dic₂₉₂	C₄	1	1	yes	characteristic
17	C₄×Dic₂₉₂	C₂²	1	1	yes	characteristic
18	C₄×Dic₂₉₂	C₂²	1	2	yes	—
19	C₄×Dic₂₉₂	C₂²	1	1	yes	characteristic
20	C₂×#14556525	C₄	1	1	yes	characteristic
21	#17988611	C₂²	1	2	yes	—
22	#17988610	—	2	2	no	—
23	C₂×#14556516	C₄	1	2	yes	—
24	M₁₆×Dic₇₃	—	2	2	no	—
25	#17988338	C₂²	1	2	yes	—
26	#17988338	C₂²	1	2	yes	—
27	#17988319	C₂²	1	4	yes	—
28	#17988319	C₂²	1	4	yes	—
29	#431×C₇₃	C₂²	1	1	yes	characteristic
30	C₄×M₁₆×C₇₃	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
32	#322×C₇₃	C₂²	1	2	yes	—
33	#322×C₇₃	C₂²	1	2	yes	—
34	C₂²×Dic₁₄₆	C₂×C₄	1	2	yes	—
35	C₂×Dic₂₉₂	—	2	4	no	—
36	C₂×Dic₂₉₂	C₂×C₄	1	2	yes	—
37	C₂×Dic₂₉₂	C₂×C₄	1	1	yes	characteristic
38	C₂×Dic₂₉₂	C₂×C₄	1	1	yes	characteristic
39	C₂×#11924012	—	2	2	no	—
40	C₄×Dic₁₄₆	—	2	2	no	—
41	C₄×Dic₁₄₆	—	2	2	no	—
42	C₄×Dic₁₄₆	C₂×C₄	1	2	yes	—
43	C₄×Dic₁₄₆	C₂×C₄	1	2	yes	—
44	C₄×Dic₁₄₆	C₂³	1	2	yes	—
45	C₄×Dic₁₄₆	C₂³	1	2	yes	—
46	C₂×C₄⋊Dic₇₃	C₂×C₄	1	2	yes	—
47	C₂×#11924007	—	2	2	no	—
48	C₂×C₄×Dic₇₃	—	2	2	no	—
49	#14556525	C₂×C₄	1	2	yes	—
50	#14556525	C₂×C₄	1	1	yes	characteristic
51	#14556525	C₂×C₄	1	1	yes	characteristic
52	C₄⋊#10596831	—	2	2	no	—
53	C₄⋊#10596831	C₂³	1	4	yes	—
54	#14556520	—	2	4	no	—
55	#14556520	—	2	4	no	—
56	#14556516	C₂×C₄	1	2	yes	—
57	#14556516	C₂×C₄	1	2	yes	—
58	#14556516	C₂×C₄	1	2	yes	—
59	#14556516	C₂×C₄	1	2	yes	—
60	#14556501	—	2	4	no	—
61	C₈×Dic₇₃	—	2	4	no	—
62	C₂×M₁₆×C₇₃	D₄	1	2	yes	—
63	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
64	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
66	C₄⋊C₈×C₇₃	—	2	1	no	—
67	C₂²⋊C₈×C₇₃	—	2	1	no	—
68	#97×C₇₃	D₄	1	2	yes	—
69	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
70	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
71	C₄×C₈×C₇₃	C₂³	1	2	yes	—
72	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
73	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₇₃	C₂³	1	2	yes	—
75	C₂×Dic₁₄₆	—	2	4	no	—
76	C₂×Dic₁₄₆	—	2	2	no	—
77	C₂×Dic₁₄₆	—	4	2	no	—
78	C₂×Dic₁₄₆	C₂²×C₄	1	2	yes	—
79	C₂×Dic₁₄₆	C₂²×C₄	1	2	yes	—
80	C₂²×Dic₇₃	—	2	2	no	—
81	C₂²×Dic₇₃	—	2	2	no	—
82	Dic₂₉₂	—	2	4	no	—
83	Dic₂₉₂	—	2	2	no	—
84	Dic₂₉₂	M₁₆	1	4	yes	complemented
85	#11924012	—	4	4	no	—
86	C₂×#10596831	—	2	4	no	—
87	C₄⋊Dic₇₃	—	2	2	no	—
88	C₄⋊Dic₇₃	C₂²×C₄	1	2	yes	—
89	C₄⋊Dic₇₃	C₂²×C₄	1	2	yes	—
90	#11924007	—	2	2	no	—
91	#11924007	—	2	2	no	—
92	#11924007	—	2	2	no	—
93	#11924007	—	2	2	no	—
94	C₄×Dic₇₃	—	2	2	no	—
95	C₄×Dic₇₃	—	2	2	no	—
96	C₄×Dic₇₃	—	2	2	no	—
97	C₄×Dic₇₃	—	2	2	no	—
98	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
99	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
101	M₁₆×C₇₃	—	2	4	no	—
102	C₂×C₈×C₇₃	—	2	2	no	—
103	C₂×C₈×C₇₃	—	2	2	no	—
104	C₂×C₈×C₇₃	—	4	1	no	—
105	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₈×C₇₃	C₂×D₄	1	4	yes	—
110	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
111	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₇₃	#41	1	1	yes	characteristic
113	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
114	C₂³×C₇₃	C₄×D₄	1	1	yes	characteristic
115	C₂×C₄×C₇₃	—	2	1	no	—
116	C₂×C₄×C₇₃	—	2	1	no	—
117	C₂×C₄×C₇₃	—	4	1	no	—
118	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
119	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₇₃	#137	1	4	yes	—
122	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
123	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
125	C₂×Dic₇₃	—	2	2	no	—
126	C₂×Dic₇₃	—	2	2	no	—
127	C₂×Dic₇₃	—	2	2	no	—
128	C₂×Dic₇₃	—	2	2	no	—
129	C₂×Dic₇₃	—	4	4	no	—
130	C₂×Dic₇₃	—	8	2	no	—
131	Dic₁₄₆	—	2	2	no	—
132	Dic₁₄₆	—	2	2	no	—
133	Dic₁₄₆	—	2	2	no	—
134	Dic₁₄₆	—	4	4	no	—
135	Dic₁₄₆	C₂×M₁₆	1	4	yes	—
136	C₈×C₇₃	—	2	1	no	—
137	C₈×C₇₃	—	2	1	no	—
138	C₈×C₇₃	—	2	1	no	—
139	C₈×C₇₃	—	2	4	no	—
140	C₈×C₇₃	—	4	2	no	—
141	C₈×C₇₃	C₂×M₁₆	1	2	yes	—
142	#10596831	—	4	4	no	—
143	C₂²×C₇₃	—	2	1	no	—
144	C₂²×C₇₃	—	2	1	no	—
145	C₂²×C₇₃	#441	1	2	yes	—
146	C₂²×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
147	C₄×C₇₃	—	2	1	no	—
148	C₄×C₇₃	—	2	1	no	—
149	C₄×C₇₃	—	2	1	no	—
150	C₄×C₇₃	#441	1	2	yes	—
151	C₄×C₇₃	C₂²×M₁₆	1	1	yes	characteristic
152	C₄×C₇₃	#568	1	1	yes	characteristic
153	C₄×C₇₃	C₂×#137	1	2	yes	—
154	Dic₇₃	—	2	4	no	—
155	Dic₇₃	—	4	2	no	—
156	Dic₇₃	—	4	2	no	—
157	Dic₇₃	—	4	2	no	—
158	#18042	—	73	1	no	maximal, Sylow
159	C₂×C₇₃	—	2	2	no	—
160	C₂×C₇₃	D₄×M₁₆	1	1	yes	characteristic
161	C₂×C₇₃	C₂×#441	1	1	yes	characteristic
162	C₂×C₇₃	#2957	1	1	yes	characteristic
163	Q₈×M₁₆	—	73	2	no	—
164	C₂×C₄×Q₁₆	—	73	1	no	—
165	#1593	—	73	2	no	—
166	#1587	—	73	4	no	—
167	#1579	—	73	2	no	—
168	#1566	—	73	1	no	—
169	#1487	—	73	2	no	—
170	#1451	D₇₃	1	1	yes	characteristic
171	C₇₃	#18042	1	1	yes	characteristic, complemented, Sylow
172	C₂²×Q₁₆	—	73	1	no	—
173	C₂×C₄×Q₈	—	73	2	no	—
174	#446	—	73	4	no	—
175	#446	—	146	4	no	—
176	C₈×Q₈	—	73	4	no	—
177	C₄×Q₁₆	—	73	1	no	—
178	C₄×Q₁₆	—	73	2	no	—
179	C₄×Q₁₆	—	73	1	no	—
180	C₄×Q₁₆	—	73	1	no	—
181	C₄×Q₁₆	—	73	2	no	—
182	C₄×Q₁₆	—	73	1	no	—
183	#431	C₂×D₇₃	1	1	yes	characteristic
184	C₂×#107	—	73	1	no	—
185	#423	—	73	2	no	—
186	#423	—	146	2	no	—
187	C₂×#103	—	73	2	no	—
188	C₄×M₁₆	—	146	2	no	—
189	C₄×M₁₆	C₂×D₇₃	1	1	yes	characteristic
190	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
191	#334	—	73	2	no	—
192	#334	—	73	2	no	—
193	Q₈⋊C₈	—	73	4	no	—
194	Q₈⋊C₈	—	73	4	no	—
195	#322	C₂×D₇₃	1	2	yes	—
196	#322	C₂×D₇₃	1	2	yes	—
197	C₂²×Q₈	—	73	2	no	—
198	C₂×Q₁₆	—	73	1	no	—
199	C₂×Q₁₆	—	73	2	no	—
200	C₂×Q₁₆	—	73	1	no	—
201	C₂×Q₁₆	—	146	4	no	—
202	C₂×M₁₆	—	146	2	no	—
203	C₂×M₁₆	D₂₉₂	1	2	yes	—
204	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
205	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
206	C₄×Q₈	—	73	2	no	—
207	C₄×Q₈	—	73	2	no	—
208	C₄×Q₈	—	73	2	no	—
209	C₄×Q₈	—	73	2	no	—
210	C₄×Q₈	—	146	2	no	—
211	C₄×Q₈	—	146	2	no	—
212	C₂×C₄⋊C₄	—	73	2	no	—
213	C₂×C₄⋊C₄	—	146	2	no	—
214	C₂×C₄²	—	146	2	no	—
215	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
216	#107	—	73	2	no	—
217	#107	—	73	1	no	—
218	#107	—	73	1	no	—
219	C₄⋊C₈	—	2	1	no	—
220	C₄⋊C₈	—	73	4	no	—
221	C₄⋊C₈	—	146	4	no	—
222	C₄⋊C₈	—	146	4	no	—
223	C₄⋊C₈	—	146	2	no	—
224	#103	—	73	2	no	—
225	#103	—	73	2	no	—
226	#103	—	73	2	no	—
227	#103	—	73	2	no	—
228	C₂²⋊C₈	—	2	1	no	—
229	#97	—	146	4	no	—
230	#97	D₂₉₂	1	2	yes	—
231	C₄×C₈	—	146	4	no	—
232	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
233	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
234	C₄×C₈	C₂²×D₇₃	1	2	yes	—
235	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
236	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
237	C₄×C₈	C₂²×D₇₃	1	2	yes	—
238	C₂×Q₈	—	292	2	no	—
239	C₂×Q₈	—	73	2	no	—
240	C₂×Q₈	—	73	2	no	—
241	C₂×Q₈	—	146	4	no	—
242	C₂×Q₈	—	146	2	no	—
243	C₂²×C₄	—	146	2	no	—
244	C₂²×C₄	—	146	2	no	—
245	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
246	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
247	C₂²×C₄	#11924028	1	1	yes	characteristic
248	Q₁₆	—	73	4	no	—
249	Q₁₆	—	146	4	no	—
250	Q₁₆	—	146	2	no	—
251	M₁₆	—	2	4	no	—
252	M₁₆	—	292	4	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	4	1	no	—
256	C₂×C₈	—	146	4	no	—
257	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
258	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
259	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
260	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
261	C₂×C₈	C₂×D₂₉₂	1	4	yes	—
262	C₄⋊C₄	—	73	2	no	—
263	C₄⋊C₄	—	73	2	no	—
264	C₄⋊C₄	—	146	2	no	—
265	C₄⋊C₄	—	146	2	no	—
266	C₄⋊C₄	—	146	2	no	—
267	C₄⋊C₄	—	146	2	no	—
268	C₄⋊C₄	—	146	2	no	—
269	C₄²	—	146	2	no	—
270	C₄²	—	146	2	no	—
271	C₄²	—	146	2	no	—
272	C₄²	—	146	2	no	—
273	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
274	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
275	C₄²	#11924028	1	1	yes	characteristic
276	C₄²	C₂³×D₇₃	1	1	yes	characteristic
277	C₂³	C₄×D₂₉₂	1	1	yes	characteristic
278	Q₈	—	292	4	no	—
279	Q₈	—	73	4	no	—
280	Q₈	—	146	2	no	—
281	Q₈	—	146	2	no	—
282	Q₈	—	146	2	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	4	1	no	—
286	C₂×C₄	—	292	4	no	—
287	C₂×C₄	—	584	2	no	—
288	C₂×C₄	—	146	2	no	—
289	C₂×C₄	—	146	2	no	—
290	C₂×C₄	—	146	2	no	—
291	C₂×C₄	—	146	2	no	—
292	C₂×C₄	C₄×D₂₉₂	1	1	yes	characteristic
293	C₂×C₄	C₄×D₂₉₂	1	1	yes	characteristic
294	C₂×C₄	C₄×D₂₉₂	1	1	yes	characteristic
295	C₂×C₄	#14556624	1	4	yes	—
296	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
297	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
298	C₂×C₄	C₂×#11924028	1	1	yes	characteristic
299	C₈	—	2	4	no	—
300	C₈	—	2	1	no	—
301	C₈	—	2	1	no	—
302	C₈	—	2	1	no	—
303	C₈	—	4	2	no	—
304	C₈	—	292	4	no	—
305	C₈	M₁₆×D₇₃	1	2	yes	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	1	no	—
308	C₂²	#17988695	1	2	yes	—
309	C₂²	C₂×C₄×D₂₉₂	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	292	2	no	—
314	C₄	—	292	2	no	—
315	C₄	—	292	2	no	—
316	C₄	—	146	4	no	—
317	C₄	#17988695	1	2	yes	—
318	C₄	C₂×M₁₆×D₇₃	1	1	yes	characteristic
319	C₄	#17989353	1	1	yes	characteristic
320	C₄	C₂×#14556624	1	2	yes	—
321	C₂	—	2	2	no	—
322	C₂	M₁₆×D₂₉₂	1	1	yes	characteristic
323	C₂	C₂×#17988695	1	1	yes	characteristic
324	C₂	#22618849	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₉₂
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₄²	C₂³×D₇₃
Fitting	C₇₃×#1451	C₂
socle	C₂²×C₇₃	C₂×C₄×D₄
prime radical	C₂³×C₇₃	C₄×D₄
cosocle	C₄²×C₇₃	C₂⁴
hypercenter	#1451	D₇₃
hypocenter	C₇₃	#18042
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ #1451

Complements

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

N	H ≅ G/N	type
3. #22608841	C₂	semidirect
4. #22608804	C₂	semidirect
5. #22608751	C₂	semidirect
84. Dic₂₉₂	M₁₆	semidirect
171. C₇₃	#18042	semidirect

Maximal subgroups

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

subgroup	index
M₁₆×Dic₁₄₆	2
C₂×C₄×Dic₂₉₂	2
#22608841	2
#22608804	2
#22608751	2
#22608687	2
#22608361	2
C₇₃×#1451	2
#18042	73