Subgroups — #33002930

Order 18688.

574 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988828	C₂	1	1	yes	maximal, characteristic
2	C₂²×#14556525	C₂	1	1	yes	maximal, characteristic
3	C₂×#17988392	C₂	1	2	yes	maximal, complemented
4	#22608023	C₂	1	2	yes	maximal, complemented
5	C₂×C₂²⋊C₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₉₂	C₂²	1	1	yes	characteristic
7	C₂×#14556570	—	2	1	no	—
8	C₂²×C₄⋊Dic₇₃	—	2	1	no	—
9	#17988828	—	2	2	no	—
10	#17988828	C₄	1	4	yes	complemented
11	C₂×#14556525	—	2	4	no	—
12	C₂×#14556525	—	2	1	no	—
13	C₂×#14556525	C₂²	1	1	yes	characteristic
14	C₂×#14556525	C₂²	1	2	yes	—
15	C₂×#14556516	—	2	1	no	—
16	#17988392	—	2	4	no	—
17	C₂³×C₈×C₇₃	C₂²	1	1	yes	characteristic
18	C₂²×C₁₆×C₇₃	C₂²	1	2	yes	—
19	C₂²⋊C₁₆×C₇₃	—	2	2	no	—
20	C₂²×Dic₁₄₆	—	2	1	no	—
21	C₂³×Dic₇₃	—	4	1	no	—
22	C₂×Dic₂₉₂	—	2	2	no	—
23	C₂×Dic₂₉₂	—	2	1	no	—
24	C₂×Dic₂₉₂	—	2	2	no	—
25	C₂×Dic₂₉₂	C₂×C₄	1	2	yes	complemented
26	#14556570	—	2	4	no	—
27	#14556570	—	4	2	no	—
28	C₂×C₄⋊Dic₇₃	—	2	4	no	—
29	C₂×C₄⋊Dic₇₃	—	2	2	no	—
30	C₂×C₄⋊Dic₇₃	—	2	1	no	—
31	C₂×C₄⋊Dic₇₃	—	4	2	no	—
32	C₂×C₄⋊Dic₇₃	—	4	1	no	—
33	C₂×#11924007	—	4	1	no	—
34	C₂×C₂²⋊Dic₇₃	—	4	1	no	—
35	#14556525	—	2	4	no	—
36	#14556525	—	2	2	no	—
37	#14556525	—	2	1	no	—
38	#14556525	C₂×C₄	1	2	yes	complemented
39	#14556525	D₄	1	4	yes	—
40	#14556516	—	2	4	no	—
41	C₂³×C₄×C₇₃	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₇₃	—	2	2	no	—
43	C₂²×C₈×C₇₃	—	2	2	no	—
44	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
45	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₇₃	D₄	1	2	yes	—
47	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
48	C₂×C₁₆×C₇₃	—	2	2	no	—
49	C₂×C₁₆×C₇₃	—	2	2	no	—
50	C₂×C₁₆×C₇₃	—	2	2	no	—
51	C₂×Dic₁₄₆	—	2	2	no	—
52	C₂×Dic₁₄₆	—	4	1	no	—
53	C₂×Dic₁₄₆	—	4	1	no	—
54	C₂×Dic₁₄₆	—	4	1	no	—
55	C₂×Dic₁₄₆	—	4	1	no	—
56	C₂×Dic₁₄₆	—	4	1	no	—
57	C₂²×Dic₇₃	—	4	4	no	—
58	C₂²×Dic₇₃	—	4	1	no	—
59	C₂²×Dic₇₃	—	4	1	no	—
60	C₂²×Dic₇₃	—	4	2	no	—
61	C₂²×Dic₇₃	—	8	2	no	—
62	C₂²×Dic₇₃	—	8	2	no	—
63	Dic₂₉₂	—	2	4	no	—
64	Dic₂₉₂	—	4	2	no	—
65	C₄⋊Dic₇₃	—	2	2	no	—
66	C₄⋊Dic₇₃	—	2	4	no	—
67	C₄⋊Dic₇₃	—	4	4	no	—
68	C₄⋊Dic₇₃	—	4	1	no	—
69	C₄⋊Dic₇₃	—	4	2	no	—
70	#11924007	—	4	4	no	—
71	C₂²⋊Dic₇₃	—	4	4	no	—
72	C₂⁴×C₇₃	D₈	1	1	yes	characteristic
73	C₂²×C₄×C₇₃	—	2	1	no	—
74	C₂²×C₄×C₇₃	—	2	1	no	—
75	C₂²×C₄×C₇₃	—	2	1	no	—
76	C₂²×C₄×C₇₃	—	2	1	no	—
77	C₂²×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
78	C₂²×C₄×C₇₃	D₈	1	1	yes	characteristic
79	C₂²×C₄×C₇₃	SD₁₆	1	2	yes	—
80	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₇₃	—	2	1	no	—
82	C₂×C₈×C₇₃	—	2	1	no	—
83	C₂×C₈×C₇₃	—	2	4	no	—
84	C₂×C₈×C₇₃	—	2	4	no	—
85	C₂×C₈×C₇₃	C₂²⋊C₄	1	2	yes	—
86	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₇₃	#41	1	2	yes	—
90	C₁₆×C₇₃	—	4	2	no	—
91	C₂³×C₇₃	—	2	1	no	—
92	C₂³×C₇₃	—	2	1	no	—
93	C₂³×C₇₃	—	2	1	no	—
94	C₂³×C₇₃	—	2	1	no	—
95	C₂³×C₇₃	#102	1	2	yes	—
96	C₂³×C₇₃	SD₃₂	1	2	yes	—
97	C₂³×C₇₃	Q₃₂	1	2	yes	—
98	C₂³×C₇₃	C₂×D₈	1	1	yes	characteristic
99	C₂×C₄×C₇₃	—	2	1	no	—
100	C₂×C₄×C₇₃	—	2	1	no	—
101	C₂×C₄×C₇₃	—	2	2	no	—
102	C₂×C₄×C₇₃	—	2	2	no	—
103	C₂×C₄×C₇₃	—	2	2	no	—
104	C₂×C₄×C₇₃	—	2	2	no	—
105	C₂×C₄×C₇₃	#102	1	2	yes	—
106	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₇₃	#120	1	1	yes	characteristic
108	C₂×C₄×C₇₃	#123	1	1	yes	characteristic
109	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
110	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
111	C₂×C₄×C₇₃	#137	1	1	yes	characteristic
112	C₂×Dic₇₃	—	4	2	no	—
113	C₂×Dic₇₃	—	4	2	no	—
114	C₂×Dic₇₃	—	4	4	no	—
115	C₂×Dic₇₃	—	8	8	no	—
116	C₂×Dic₇₃	—	8	1	no	—
117	C₂×Dic₇₃	—	8	1	no	—
118	C₂×Dic₇₃	—	8	2	no	—
119	C₂×Dic₇₃	—	8	1	no	—
120	C₂×Dic₇₃	—	8	2	no	—
121	C₂×Dic₇₃	—	8	1	no	—
122	Dic₁₄₆	—	4	2	no	—
123	Dic₁₄₆	—	4	2	no	—
124	Dic₁₄₆	—	8	1	no	—
125	Dic₁₄₆	—	8	1	no	—
126	C₈×C₇₃	—	2	2	no	—
127	C₈×C₇₃	C₄×D₄	1	2	yes	—
128	C₈×C₇₃	#121	1	2	yes	—
129	C₂²×C₇₃	—	2	2	no	—
130	C₂²×C₇₃	—	2	2	no	—
131	C₂²×C₇₃	—	2	2	no	—
132	C₂²×C₇₃	—	2	2	no	—
133	C₂²×C₇₃	—	2	2	no	—
134	C₂²×C₇₃	—	2	2	no	—
135	C₂²×C₇₃	#358	1	4	yes	complemented
136	C₂²×C₇₃	C₂×#102	1	1	yes	characteristic
137	C₂²×C₇₃	#447	1	1	yes	characteristic
138	C₂²×C₇₃	#480	1	1	yes	characteristic
139	C₂²×C₇₃	C₂×SD₃₂	1	1	yes	characteristic
140	C₂²×C₇₃	C₂×Q₃₂	1	1	yes	characteristic
141	C₂²×C₇₃	#509	1	1	yes	characteristic
142	C₂²×C₇₃	#510	1	1	yes	characteristic
143	C₄×C₇₃	—	2	2	no	—
144	C₄×C₇₃	#386	1	1	yes	characteristic
145	C₄×C₇₃	#417	1	1	yes	characteristic
146	C₄×C₇₃	#448	1	1	yes	characteristic
147	C₄×C₇₃	#481	1	1	yes	characteristic
148	Dic₇₃	—	8	2	no	—
149	Dic₇₃	—	8	2	no	—
150	Dic₇₃	—	8	4	no	—
151	#12993	—	73	1	no	maximal, Sylow
152	C₂×C₇₃	—	2	4	no	—
153	C₂×C₇₃	#1874	1	1	yes	characteristic
154	C₂×C₇₃	C₂×#358	1	1	yes	characteristic
155	C₂×C₇₃	#2139	1	1	yes	characteristic
156	C₂×C₇₃	#2187	1	1	yes	characteristic
157	C₂×C₇₃	#2188	1	1	yes	characteristic
158	C₂×C₇₃	#2235	1	1	yes	characteristic
159	C₂×C₇₃	#2236	1	1	yes	characteristic
160	C₂×#467	—	73	1	no	—
161	C₂²×#107	—	73	1	no	—
162	C₂×#358	—	73	2	no	—
163	C₂×C₂²⋊C₁₆	D₇₃	1	1	yes	characteristic
164	#1380	—	73	2	no	—
165	C₇₃	#12993	1	1	yes	characteristic, complemented, Sylow
166	C₂²×Q₁₆	—	73	1	no	—
167	C₂³×C₈	C₂×D₇₃	1	1	yes	characteristic
168	C₂×C₂²⋊Q₈	—	146	1	no	—
169	C₂²×C₄⋊C₄	—	146	1	no	—
170	C₂²×C₁₆	C₂×D₇₃	1	2	yes	—
171	#467	—	73	4	no	—
172	#467	—	146	2	no	—
173	C₂×#107	—	73	2	no	—
174	C₂×#107	—	73	1	no	—
175	C₂×#107	—	146	4	no	—
176	C₂×#107	—	146	1	no	—
177	C₂×#103	—	146	1	no	—
178	#358	—	146	4	no	—
179	C₂²⋊C₁₆	—	2	2	no	—
180	C₂²×Q₈	—	146	1	no	—
181	C₂³×C₄	—	292	1	no	—
182	C₂³×C₄	D₂₉₂	1	1	yes	characteristic
183	C₂×Q₁₆	—	73	2	no	—
184	C₂×Q₁₆	—	146	2	no	—
185	C₂×Q₁₆	—	146	1	no	—
186	C₂×Q₁₆	—	146	2	no	—
187	C₂²×C₈	—	2	2	no	—
188	C₂²×C₈	—	2	2	no	—
189	C₂²×C₈	C₄×D₇₃	1	2	yes	—
190	C₂²×C₈	D₂₉₂	1	1	yes	characteristic
191	C₂²×C₈	C₂²⋊D₇₃	1	2	yes	—
192	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
193	C₂²⋊Q₈	—	292	2	no	—
194	C₂²⋊Q₈	—	146	4	no	—
195	C₂×C₄⋊C₄	—	292	2	no	—
196	C₂×C₄⋊C₄	—	292	1	no	—
197	C₂×C₄⋊C₄	—	292	1	no	—
198	C₂×C₄⋊C₄	—	146	4	no	—
199	C₂×C₄⋊C₄	—	146	2	no	—
200	C₂×C₄⋊C₄	—	146	1	no	—
201	C₂×C₂²⋊C₄	—	292	1	no	—
202	C₂×C₁₆	—	2	2	no	—
203	C₂×C₁₆	—	2	2	no	—
204	C₂×C₁₆	—	2	2	no	—
205	#107	—	73	2	no	—
206	#107	—	73	4	no	—
207	#107	—	146	4	no	—
208	#107	—	146	2	no	—
209	#107	—	146	1	no	—
210	#103	—	146	4	no	—
211	C₂⁴	D₅₈₄	1	1	yes	characteristic
212	C₂×Q₈	—	292	1	no	—
213	C₂×Q₈	—	292	1	no	—
214	C₂×Q₈	—	292	1	no	—
215	C₂×Q₈	—	292	1	no	—
216	C₂×Q₈	—	292	1	no	—
217	C₂×Q₈	—	146	2	no	—
218	C₂²×C₄	—	2	1	no	—
219	C₂²×C₄	—	2	1	no	—
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	—	292	4	no	—
223	C₂²×C₄	—	292	1	no	—
224	C₂²×C₄	—	292	1	no	—
225	C₂²×C₄	—	292	2	no	—
226	C₂²×C₄	—	584	2	no	—
227	C₂²×C₄	—	584	2	no	—
228	C₂²×C₄	#11924005	1	2	yes	—
229	C₂²×C₄	D₅₈₄	1	1	yes	characteristic
230	C₂²×C₄	#11924017	1	2	yes	—
231	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
232	Q₁₆	—	292	2	no	—
233	Q₁₆	—	146	4	no	—
234	C₂×C₈	—	2	1	no	—
235	C₂×C₈	—	2	1	no	—
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	#11924005	1	2	yes	—
239	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
240	C₂×C₈	D₄×D₇₃	1	2	yes	—
241	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
242	C₂×C₈	#11924027	1	2	yes	—
243	C₄⋊C₄	—	292	4	no	—
244	C₄⋊C₄	—	292	4	no	—
245	C₄⋊C₄	—	292	1	no	—
246	C₄⋊C₄	—	292	2	no	—
247	C₄⋊C₄	—	146	2	no	—
248	C₄⋊C₄	—	146	4	no	—
249	C₂²⋊C₄	—	292	4	no	—
250	C₁₆	—	4	2	no	—
251	C₂³	—	2	1	no	—
252	C₂³	—	2	1	no	—
253	C₂³	—	2	1	no	—
254	C₂³	—	2	1	no	—
255	C₂³	#14556513	1	2	yes	—
256	C₂³	#14556536	1	2	yes	—
257	C₂³	Dic₅₈₄	1	2	yes	—
258	C₂³	C₂×D₅₈₄	1	1	yes	characteristic
259	Q₈	—	292	2	no	—
260	Q₈	—	292	2	no	—
261	Q₈	—	584	1	no	—
262	Q₈	—	584	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	292	4	no	—
270	C₂×C₄	—	292	2	no	—
271	C₂×C₄	—	292	2	no	—
272	C₂×C₄	—	584	8	no	—
273	C₂×C₄	—	584	1	no	—
274	C₂×C₄	—	584	1	no	—
275	C₂×C₄	—	584	2	no	—
276	C₂×C₄	—	584	2	no	—
277	C₂×C₄	—	584	1	no	—
278	C₂×C₄	—	584	1	no	—
279	C₂×C₄	#14556513	1	2	yes	—
280	C₂×C₄	C₂×#11924005	1	1	yes	characteristic
281	C₂×C₄	#14556561	1	1	yes	characteristic
282	C₂×C₄	#14556575	1	1	yes	characteristic
283	C₂×C₄	C₂×#11924017	1	1	yes	characteristic
284	C₂×C₄	#14556619	1	1	yes	characteristic
285	C₂×C₄	#14556624	1	1	yes	characteristic
286	C₈	—	2	2	no	—
287	C₈	#14556551	1	2	yes	—
288	C₈	#14556563	1	2	yes	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	2	2	no	—
293	C₂²	—	2	2	no	—
294	C₂²	—	2	2	no	—
295	C₂²	#17988392	1	4	yes	complemented
296	C₂²	C₂×#14556513	1	1	yes	characteristic
297	C₂²	#17988723	1	1	yes	characteristic
298	C₂²	#17988901	1	1	yes	characteristic
299	C₂²	C₂×#14556536	1	1	yes	characteristic
300	C₂²	C₂×Dic₅₈₄	1	1	yes	characteristic
301	C₂²	#17989025	1	1	yes	characteristic
302	C₂²	#17989030	1	1	yes	characteristic
303	C₄	—	2	2	no	—
304	C₄	—	584	2	no	—
305	C₄	—	584	4	no	—
306	C₄	—	584	2	no	—
307	C₄	#17988464	1	1	yes	characteristic
308	C₄	#17988589	1	1	yes	characteristic
309	C₄	#17988728	1	1	yes	characteristic
310	C₄	#17988906	1	1	yes	characteristic
311	C₂	—	2	4	no	—
312	C₂	#22610246	1	1	yes	characteristic
313	C₂	C₂×#17988392	1	1	yes	characteristic
314	C₂	#22611644	1	1	yes	characteristic
315	C₂	#22611862	1	1	yes	characteristic
316	C₂	#22611867	1	1	yes	characteristic
317	C₂	#22612129	1	1	yes	characteristic
318	C₂	#22612134	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₅₈₄
commutator	C₂×C₈×C₇₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₂×C₂²⋊C₁₆×C₇₃	C₂
socle	C₂³×C₇₃	C₂×D₈
prime radical	C₂⁴×C₇₃	D₈
cosocle	C₂²×C₈×C₇₃	C₂³
hypercenter	C₂×C₂²⋊C₁₆	D₇₃
hypocenter	C₇₃	#12993
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×C₂²⋊C₁₆

Complements

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

N	H ≅ G/N	type
3. C₂×#17988392	C₂	semidirect
4. #22608023	C₂	semidirect
10. #17988828	C₄	semidirect
25. C₂×Dic₂₉₂	C₂×C₄	semidirect
38. #14556525	C₂×C₄	semidirect
135. C₂²×C₇₃	#358	semidirect
165. C₇₃	#12993	semidirect
295. C₂²	#17988392	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17988828	2
C₂²×#14556525	2
C₂×#17988392	2
#22608023	2
C₂×C₂²⋊C₁₆×C₇₃	2
#12993	73