Subgroups — #33086428

Order 18688.

1070 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988630	C₂	1	2	yes	maximal
2	C₂²×#14556523	C₂	1	1	yes	maximal, characteristic
3	C₂×#17988542	C₂	1	1	yes	maximal, characteristic
4	#22612551	C₂	1	2	yes	maximal
5	#22610098	C₂	1	4	yes	maximal, complemented
6	#22610092	C₂	1	2	yes	maximal, complemented
7	#22610086	C₂	1	2	yes	maximal, complemented
8	C₂×#427×C₇₃	C₂	1	1	yes	maximal, characteristic
9	C₂²×#11924012	C₂²	1	1	yes	characteristic
10	C₂×#14556550	C₂²	1	1	yes	characteristic
11	C₂²×C₄⋊Dic₇₃	C₂²	1	2	yes	—
12	#17988630	—	2	8	no	—
13	C₂×#14556524	C₂²	1	4	yes	—
14	C₂×#14556523	—	2	4	no	—
15	C₂×#14556523	C₂²	1	2	yes	—
16	C₂×#14556523	C₂²	1	2	yes	—
17	C₂×#14556522	C₂²	1	4	yes	—
18	#17988542	C₄	1	8	yes	complemented
19	C₂×#14556501	C₂²	1	2	yes	—
20	C₂×C₈×Dic₇₃	C₂²	1	2	yes	—
21	#17988485	—	2	4	no	—
22	#17988459	—	2	4	no	—
23	#17988456	C₂²	1	4	yes	—
24	#17988452	C₂²	1	4	yes	—
25	C₂³×C₈×C₇₃	C₂²	1	1	yes	characteristic
26	C₂×#117×C₇₃	C₂²	1	2	yes	—
27	#427×C₇₃	—	2	4	no	—
28	C₂×#107×C₇₃	C₂²	1	2	yes	—
29	C₂×#106×C₇₃	C₂²	1	1	yes	characteristic
30	C₂×#106×C₇₃	C₂²	1	1	yes	characteristic
31	C₂³×Dic₇₃	—	2	2	no	—
32	C₂×#11924012	C₂×C₄	1	4	yes	—
33	C₂×#11924012	D₄	1	4	yes	—
34	C₂×#11924012	Q₈	1	4	yes	—
35	C₂²×#10596831	C₂³	1	2	yes	—
36	#14556550	C₂×C₄	1	4	yes	—
37	#14556550	Q₈	1	4	yes	—
38	C₂×C₄⋊Dic₇₃	—	2	4	no	—
39	C₂×C₄⋊Dic₇₃	—	2	4	no	—
40	C₂×C₄⋊Dic₇₃	—	2	2	no	—
41	C₂×C₄⋊Dic₇₃	D₄	1	2	yes	—
42	C₂×C₄⋊Dic₇₃	C₂³	1	4	yes	—
43	C₂×#11924007	—	2	8	no	—
44	C₂×C₂²⋊Dic₇₃	D₄	1	2	yes	—
45	C₂×C₄×Dic₇₃	C₂³	1	2	yes	—
46	#14556524	—	2	8	no	—
47	#14556523	—	2	4	no	—
48	#14556523	—	2	4	no	—
49	#14556522	—	2	8	no	—
50	#14556501	C₂×C₄	1	8	yes	complemented
51	C₈×Dic₇₃	C₂×C₄	1	8	yes	complemented
52	#14556498	—	2	8	no	—
53	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₇₃	C₂×C₄	1	4	yes	—
55	C₂²×C₈×C₇₃	D₄	1	4	yes	—
56	C₂²×C₈×C₇₃	Q₈	1	4	yes	—
57	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
58	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
59	#117×C₇₃	—	2	4	no	—
60	#117×C₇₃	—	2	4	no	—
61	C₂×C₄⋊C₄×C₇₃	C₂³	1	2	yes	—
62	C₂×C₄⋊C₄×C₇₃	C₂³	1	2	yes	—
63	C₂×C₂²⋊C₄×C₇₃	—	2	2	no	—
64	C₂×C₄²×C₇₃	—	2	2	no	—
65	#107×C₇₃	—	2	4	no	—
66	#106×C₇₃	—	2	2	no	—
67	#106×C₇₃	—	2	2	no	—
68	C₂²×Dic₇₃	—	2	4	no	—
69	C₂²×Dic₇₃	—	4	4	no	—
70	C₂²×Dic₇₃	—	8	4	no	—
71	C₂²×Dic₇₃	C₂×D₄	1	4	yes	—
72	#11924012	C₄⋊C₄	1	16	yes	complemented
73	C₂×#10596831	C₂²×C₄	1	4	yes	—
74	C₂×#10596831	C₂×D₄	1	4	yes	—
75	C₂×#10596831	C₂×Q₈	1	4	yes	—
76	C₄⋊Dic₇₃	—	2	4	no	—
77	C₄⋊Dic₇₃	—	2	4	no	—
78	C₄⋊Dic₇₃	—	4	4	no	—
79	C₄⋊Dic₇₃	C₄⋊C₄	1	8	yes	—
80	#11924007	—	8	8	no	—
81	C₂²⋊Dic₇₃	C₄⋊C₄	1	8	yes	—
82	C₄×Dic₇₃	C₂²×C₄	1	4	yes	—
83	C₄×Dic₇₃	C₂×Q₈	1	4	yes	—
84	C₂⁴×C₇₃	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₇₃	—	2	2	no	—
86	C₂²×C₄×C₇₃	—	2	2	no	—
87	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
88	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
90	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
91	C₂²×C₄×C₇₃	C₂×Q₈	1	2	yes	—
92	C₂²×C₄×C₇₃	C₂×Q₈	1	2	yes	—
93	C₂²×C₄×C₇₃	C₂×Q₈	1	2	yes	—
94	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
95	C₂×C₈×C₇₃	C₄⋊C₄	1	16	yes	complemented
96	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
97	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
98	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
99	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
100	C₂×C₈×C₇₃	C₂×Q₈	1	2	yes	—
101	C₂×C₈×C₇₃	C₂×Q₈	1	2	yes	—
102	C₄⋊C₄×C₇₃	—	2	2	no	—
103	C₄⋊C₄×C₇₃	—	2	2	no	—
104	C₄⋊C₄×C₇₃	—	2	2	no	—
105	C₄⋊C₄×C₇₃	—	2	2	no	—
106	C₂²⋊C₄×C₇₃	—	4	4	no	—
107	C₄²×C₇₃	—	4	4	no	—
108	C₂³×C₇₃	—	2	4	no	—
109	C₂³×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
110	C₂³×C₇₃	#127	1	2	yes	—
111	C₂³×C₇₃	C₄⋊Q₈	1	2	yes	—
112	C₂³×C₇₃	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₇₃	—	4	2	no	—
114	C₂×C₄×C₇₃	—	4	2	no	—
115	C₂×C₄×C₇₃	—	4	4	no	—
116	C₂×C₄×C₇₃	—	4	4	no	—
117	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	4	yes	—
118	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
119	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	4	yes	—
120	C₂×C₄×C₇₃	#127	1	2	yes	—
121	C₂×C₄×C₇₃	C₄⋊Q₈	1	4	yes	—
122	C₂×C₄×C₇₃	C₄⋊Q₈	1	4	yes	—
123	C₂×C₄×C₇₃	C₄⋊Q₈	1	2	yes	—
124	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
125	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₇₃	C₂²×Q₈	1	1	yes	characteristic
128	C₂×C₄×C₇₃	C₂²×Q₈	1	1	yes	characteristic
129	C₂×C₄×C₇₃	C₂²×Q₈	1	1	yes	characteristic
130	C₂×Dic₇₃	—	2	8	no	—
131	C₂×Dic₇₃	—	4	4	no	—
132	C₂×Dic₇₃	—	4	8	no	—
133	C₂×Dic₇₃	—	8	8	no	—
134	C₂×Dic₇₃	C₂×C₄⋊C₄	1	8	yes	—
135	C₈×C₇₃	C₂×C₄⋊C₄	1	4	yes	—
136	C₈×C₇₃	C₂×C₄⋊C₄	1	4	yes	—
137	#10596831	C₂×C₄⋊C₄	1	8	yes	—
138	C₂²×C₇₃	—	2	4	no	—
139	C₂²×C₇₃	—	2	4	no	—
140	C₂²×C₇₃	—	2	4	no	—
141	C₂²×C₇₃	#384	1	4	yes	—
142	C₂²×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
143	C₂²×C₇₃	C₂×#127	1	1	yes	characteristic
144	C₂²×C₇₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
145	C₂²×C₇₃	#577	1	4	yes	—
146	C₄×C₇₃	—	8	2	no	—
147	C₄×C₇₃	—	8	2	no	—
148	C₄×C₇₃	#384	1	4	yes	—
149	C₄×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
150	C₄×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
151	C₄×C₇₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
152	C₄×C₇₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
153	Dic₇₃	—	2	8	no	—
154	Dic₇₃	—	8	4	no	—
155	#20802	—	73	1	no	maximal, Sylow
156	C₂×C₇₃	—	2	4	no	—
157	C₂×C₇₃	C₂×#384	1	1	yes	characteristic
158	C₂×C₇₃	#2912	1	2	yes	—
159	C₂×C₇₃	#3154	1	2	yes	—
160	C₂×C₇₃	#3163	1	2	yes	—
161	C₂×#428	—	73	2	no	—
162	C₂×#427	D₇₃	1	1	yes	characteristic
163	C₂²×#106	—	73	1	no	—
164	C₂×#405	—	73	1	no	—
165	#2317	—	73	2	no	—
166	#1847	—	73	4	no	—
167	#1846	—	73	2	no	—
168	#1844	—	73	2	no	—
169	C₇₃	#20802	1	1	yes	characteristic, complemented, Sylow
170	C₂²×M₁₆	—	73	1	no	—
171	C₂³×C₈	C₂×D₇₃	1	1	yes	characteristic
172	C₂×#117	—	73	1	no	—
173	C₂×#117	C₂×D₇₃	1	2	yes	—
174	C₂²×C₄⋊C₄	—	73	2	no	—
175	#428	—	146	8	no	—
176	#427	—	2	4	no	—
177	C₂×#107	—	73	4	no	—
178	C₂×#107	C₂×D₇₃	1	2	yes	—
179	C₂×#106	—	73	2	no	—
180	C₂×#106	—	73	2	no	—
181	C₂×#106	—	73	4	no	—
182	C₂×#106	—	146	4	no	—
183	C₂×#106	C₂×D₇₃	1	1	yes	characteristic
184	C₂×#106	C₂×D₇₃	1	1	yes	characteristic
185	#405	—	73	8	no	—
186	C₂×#97	—	73	2	no	—
187	C₂×C₄×C₈	—	73	2	no	—
188	#389	—	146	4	no	—
189	#385	—	146	4	no	—
190	#384	—	73	4	no	—
191	#382	—	73	4	no	—
192	C₂³×C₄	—	146	2	no	—
193	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
194	C₂×M₁₆	—	73	4	no	—
195	C₂×M₁₆	—	73	4	no	—
196	C₂×M₁₆	—	73	4	no	—
197	C₂²×C₈	—	73	2	no	—
198	C₂²×C₈	Dic₁₄₆	1	4	yes	—
199	C₂²×C₈	C₄×D₇₃	1	4	yes	—
200	C₂²×C₈	C₂²⋊D₇₃	1	4	yes	—
201	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
202	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
203	#117	—	2	4	no	—
204	#117	—	2	4	no	—
205	#117	—	73	4	no	—
206	#117	—	73	4	no	—
207	C₂×C₄⋊C₄	—	73	2	no	—
208	C₂×C₄⋊C₄	—	73	4	no	—
209	C₂×C₄⋊C₄	—	146	2	no	—
210	C₂×C₄⋊C₄	—	146	4	no	—
211	C₂×C₄⋊C₄	—	146	4	no	—
212	C₂×C₄⋊C₄	—	146	8	no	—
213	C₂×C₄⋊C₄	C₂²×D₇₃	1	2	yes	—
214	C₂×C₄⋊C₄	C₂²×D₇₃	1	2	yes	—
215	C₂×C₂²⋊C₄	—	2	2	no	—
216	C₂×C₂²⋊C₄	—	73	2	no	—
217	C₂×C₄²	—	2	2	no	—
218	C₂×C₄²	—	73	2	no	—
219	#107	—	2	4	no	—
220	#107	—	146	8	no	—
221	#106	—	2	2	no	—
222	#106	—	2	2	no	—
223	#106	—	146	8	no	—
224	#106	—	146	4	no	—
225	#106	—	146	4	no	—
226	#97	—	73	8	no	—
227	C₄×C₈	—	73	8	no	—
228	#95	—	146	8	no	—
229	C₂⁴	D₄×D₇₃	1	1	yes	characteristic
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	292	4	no	—
233	C₂²×C₄	—	584	4	no	—
234	C₂²×C₄	—	73	4	no	—
235	C₂²×C₄	—	146	4	no	—
236	C₂²×C₄	C₂×C₄×D₇₃	1	2	yes	—
237	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
238	C₂²×C₄	D₄×D₇₃	1	2	yes	—
239	C₂²×C₄	C₂×C₂²⋊D₇₃	1	2	yes	—
240	C₂²×C₄	Q₈×D₇₃	1	2	yes	—
241	C₂²×C₄	Q₈×D₇₃	1	2	yes	—
242	C₂²×C₄	C₂×Dic₁₄₆	1	2	yes	—
243	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
244	M₁₆	—	73	16	no	—
245	C₂×C₈	—	73	4	no	—
246	C₂×C₈	—	73	4	no	—
247	C₂×C₈	—	73	4	no	—
248	C₂×C₈	#11924007	1	16	yes	complemented
249	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
250	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
251	C₂×C₈	C₂×C₂²⋊D₇₃	1	2	yes	—
252	C₂×C₈	C₂×C₂²⋊D₇₃	1	2	yes	—
253	C₂×C₈	C₂×Dic₁₄₆	1	2	yes	—
254	C₂×C₈	C₂×Dic₁₄₆	1	2	yes	—
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	—	292	4	no	—
260	C₄⋊C₄	—	584	8	no	—
261	C₄⋊C₄	—	73	8	no	—
262	C₄⋊C₄	—	146	4	no	—
263	C₄⋊C₄	—	146	4	no	—
264	C₂²⋊C₄	—	4	4	no	—
265	C₂²⋊C₄	—	73	8	no	—
266	C₄²	—	4	4	no	—
267	C₄²	—	73	4	no	—
268	C₄²	—	73	4	no	—
269	C₂³	—	2	4	no	—
270	C₂³	C₄⋊C₄×D₇₃	1	2	yes	—
271	C₂³	#14556587	1	2	yes	—
272	C₂³	#14556590	1	2	yes	—
273	C₂³	C₂×D₄×D₇₃	1	1	yes	characteristic
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	4	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	—	292	4	no	—
279	C₂×C₄	—	292	8	no	—
280	C₂×C₄	—	584	8	no	—
281	C₂×C₄	—	73	8	no	—
282	C₂×C₄	—	146	8	no	—
283	C₂×C₄	C₄⋊C₄×D₇₃	1	2	yes	—
284	C₂×C₄	C₄⋊C₄×D₇₃	1	4	yes	—
285	C₂×C₄	C₂×#11924007	1	4	yes	—
286	C₂×C₄	#14556587	1	2	yes	—
287	C₂×C₄	#14556589	1	4	yes	—
288	C₂×C₄	#14556590	1	4	yes	—
289	C₂×C₄	#14556590	1	2	yes	—
290	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
291	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
292	C₂×C₄	C₂²×C₂²⋊D₇₃	1	1	yes	characteristic
293	C₂×C₄	C₂×Q₈×D₇₃	1	1	yes	characteristic
294	C₂×C₄	C₂×Q₈×D₇₃	1	1	yes	characteristic
295	C₂×C₄	C₂²×Dic₁₄₆	1	1	yes	characteristic
296	C₈	—	73	8	no	—
297	C₈	C₂×#11924007	1	4	yes	—
298	C₈	C₂×#11924007	1	4	yes	—
299	C₂²	—	2	4	no	—
300	C₂²	—	2	4	no	—
301	C₂²	—	2	4	no	—
302	C₂²	#17988456	1	4	yes	—
303	C₂²	C₂×C₄⋊C₄×D₇₃	1	1	yes	characteristic
304	C₂²	C₂×#14556587	1	1	yes	characteristic
305	C₂²	C₂×#14556590	1	1	yes	characteristic
306	C₂²	#17989394	1	4	yes	—
307	C₄	—	8	2	no	—
308	C₄	—	8	2	no	—
309	C₄	—	584	4	no	—
310	C₄	—	146	8	no	—
311	C₄	#17988456	1	4	yes	—
312	C₄	C₂×C₄⋊C₄×D₇₃	1	1	yes	characteristic
313	C₄	C₂²×#11924007	1	1	yes	characteristic
314	C₄	C₂×#14556589	1	1	yes	characteristic
315	C₄	C₂×#14556590	1	1	yes	characteristic
316	C₂	—	2	4	no	—
317	C₂	C₂×#17988456	1	1	yes	characteristic
318	C₂	#22618523	1	2	yes	—
319	C₂	#22620408	1	2	yes	—
320	C₂	#22620463	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₇₃
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₃
Fitting	C₂×#427×C₇₃	C₂
socle	C₂³×C₇₃	C₂²×D₄
prime radical	C₂⁴×C₇₃	C₂×D₄
cosocle	C₂²×C₄×C₇₃	C₂⁴
hypercenter	C₂×#427	D₇₃
hypocenter	C₇₃	#20802
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#427

Complements

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

N	H ≅ G/N	type
5. #22610098	C₂	semidirect
6. #22610092	C₂	semidirect
7. #22610086	C₂	semidirect
18. #17988542	C₄	semidirect
50. #14556501	C₂×C₄	semidirect
51. C₈×Dic₇₃	C₂×C₄	semidirect
72. #11924012	C₄⋊C₄	semidirect
95. C₂×C₈×C₇₃	C₄⋊C₄	semidirect
169. C₇₃	#20802	semidirect
248. C₂×C₈	#11924007	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17988630	2
C₂²×#14556523	2
C₂×#17988542	2
#22612551	2
#22610098	2
#22610092	2
#22610086	2
C₂×#427×C₇₃	2
#20802	73