Subgroups — #33104081

Order 18688.

1126 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₂₉₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#106×D₇₃	C₂	1	2	yes	maximal, complemented
3	#22612831	C₂	1	2	yes	maximal
4	#22610692	C₂	1	2	yes	maximal, complemented
5	#22610122	C₂	1	4	yes	maximal, complemented
6	#22609953	C₂	1	2	yes	maximal, complemented
7	C₄×#14556523	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇₃×#1844	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₈×D₇₃	C₂²	1	2	yes	—
10	C₂×C₄×D₂₉₂	C₂²	1	1	yes	characteristic
11	C₂×#14556553	—	2	2	no	—
12	C₂×C₄⋊C₄×D₇₃	C₂²	1	4	yes	—
13	C₈×D₂₉₂	C₄	1	8	yes	complemented
14	C₂×#14556523	C₂²	1	1	yes	characteristic
15	C₂×#14556523	C₂²	1	1	yes	characteristic
16	C₂×#14556522	C₂²	1	4	yes	—
17	#106×D₇₃	—	2	8	no	—
18	C₂×C₄⋊#10596831	C₂²	1	1	yes	characteristic
19	C₂×#14556503	C₂²	1	2	yes	—
20	#17988483	—	2	2	no	—
21	#17988458	—	2	4	no	—
22	#17988444	C₂²	1	4	yes	—
23	C₄×C₄⋊Dic₇₃	C₂²	1	2	yes	—
24	#17988350	C₂²	1	2	yes	—
25	#17988349	C₂²	1	4	yes	—
26	C₂×#106×C₇₃	C₂²	1	2	yes	—
27	C₂×#106×C₇₃	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
29	#384×C₇₃	C₂²	1	2	yes	—
30	C₂²×D₂₉₂	D₄	1	1	yes	characteristic
31	C₂²×C₄×D₇₃	—	2	4	no	—
32	C₂²×C₄×D₇₃	C₂³	1	2	yes	—
33	C₂²×#10596831	C₂³	1	2	yes	—
34	C₂×C₈×D₇₃	—	2	4	no	—
35	C₂×C₈×D₇₃	—	2	4	no	—
36	C₂×C₈×D₇₃	C₂×C₄	1	8	yes	complemented
37	C₄×D₂₉₂	C₂×C₄	1	4	yes	—
38	C₄×D₂₉₂	Q₈	1	4	yes	—
39	#14556553	—	4	8	no	—
40	C₂×C₄⋊Dic₇₃	D₄	1	1	yes	characteristic
41	C₂×C₄⋊Dic₇₃	C₂³	1	2	yes	—
42	C₂×C₄⋊Dic₇₃	C₂³	1	2	yes	—
43	C₂×#11924007	—	2	4	no	—
44	C₄⋊C₄×D₇₃	—	2	8	no	—
45	C₄⋊C₄×D₇₃	—	4	4	no	—
46	C₂×#11924005	—	2	4	no	—
47	C₂×#11924005	D₄	1	2	yes	—
48	C₂×C₄×Dic₇₃	—	2	2	no	—
49	#14556523	—	2	2	no	—
50	#14556523	D₄	1	4	yes	complemented
51	#14556522	—	2	8	no	—
52	C₄⋊#10596831	C₂×C₄	1	4	yes	complemented
53	#14556503	C₂×C₄	1	8	yes	complemented
54	#14556498	—	2	4	no	—
55	#14556498	—	2	2	no	—
56	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
57	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
58	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
59	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
60	C₂×C₄⋊C₄×C₇₃	C₂³	1	4	yes	—
61	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
62	#106×C₇₃	—	2	4	no	—
63	#106×C₇₃	—	4	2	no	—
64	C₄×C₈×C₇₃	C₂×C₄	1	4	yes	complemented
65	C₂³×D₇₃	C₂×D₄	1	2	yes	—
66	C₂×D₂₉₂	C₄⋊C₄	1	4	yes	—
67	C₂×D₂₉₂	SD₁₆	1	8	yes	—
68	C₂²×Dic₇₃	—	2	2	no	—
69	C₂²×Dic₇₃	—	2	2	no	—
70	C₂²×Dic₇₃	C₂×D₄	1	2	yes	—
71	C₂×C₄×D₇₃	—	2	2	no	—
72	C₂×C₄×D₇₃	—	2	2	no	—
73	C₂×C₄×D₇₃	—	2	2	no	—
74	C₂×C₄×D₇₃	—	2	2	no	—
75	C₂×C₄×D₇₃	—	4	8	no	—
76	C₂×C₄×D₇₃	—	4	8	no	—
77	C₂×C₄×D₇₃	—	8	4	no	—
78	C₂×C₄×D₇₃	C₂²×C₄	1	4	yes	—
79	C₂×C₄×D₇₃	C₂×Q₈	1	4	yes	—
80	C₂×#10596831	—	2	2	no	—
81	C₂×#10596831	—	2	2	no	—
82	C₂×#10596831	C₂²×C₄	1	4	yes	—
83	C₈×D₇₃	—	2	16	no	—
84	C₄⋊Dic₇₃	—	2	2	no	—
85	C₄⋊Dic₇₃	—	2	2	no	—
86	C₄⋊Dic₇₃	—	2	2	no	—
87	C₄⋊Dic₇₃	C₄⋊C₄	1	4	yes	—
88	C₄⋊Dic₇₃	C₂×D₄	1	4	yes	—
89	#11924007	—	4	8	no	—
90	#11924005	—	4	8	no	—
91	#11924005	C₄⋊C₄	1	8	yes	—
92	C₄×Dic₇₃	—	4	4	no	—
93	C₂²×C₄×C₇₃	—	2	4	no	—
94	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
97	C₂×C₈×C₇₃	—	2	1	no	—
98	C₂×C₈×C₇₃	—	2	1	no	—
99	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
100	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
101	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
102	C₂×C₈×C₇₃	#41	1	2	yes	—
103	C₄⋊C₄×C₇₃	—	2	4	no	—
104	C₄⋊C₄×C₇₃	—	2	4	no	—
105	C₄⋊C₄×C₇₃	—	4	4	no	—
106	C₄⋊C₄×C₇₃	—	8	2	no	—
107	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
108	C₄²×C₇₃	C₂×Q₈	1	2	yes	—
109	C₂³×C₇₃	C₂²×D₄	1	1	yes	characteristic
110	C₂²×D₇₃	—	2	2	no	—
111	C₂²×D₇₃	—	2	2	no	—
112	C₂²×D₇₃	—	2	4	no	—
113	C₂²×D₇₃	C₂×C₄⋊C₄	1	4	yes	—
114	C₂²×D₇₃	C₂×SD₁₆	1	8	yes	—
115	C₂×C₄×C₇₃	—	2	2	no	—
116	C₂×C₄×C₇₃	—	4	4	no	—
117	C₂×C₄×C₇₃	—	4	4	no	—
118	C₂×C₄×C₇₃	—	4	4	no	—
119	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
120	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
121	C₂×C₄×C₇₃	C₂×SD₁₆	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	C₂²×Q₈	1	1	yes	characteristic
125	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
126	C₂×C₄×C₇₃	H(5,𝔽₂)	1	1	yes	characteristic
127	C₂×C₄×C₇₃	2₋¹⁺⁴	1	1	yes	characteristic
128	C₂×Dic₇₃	—	2	2	no	—
129	C₂×Dic₇₃	—	2	2	no	—
130	C₂×Dic₇₃	—	2	4	no	—
131	C₂×Dic₇₃	—	4	2	no	—
132	C₂×Dic₇₃	—	4	2	no	—
133	C₂×Dic₇₃	—	4	2	no	—
134	C₂×Dic₇₃	—	4	2	no	—
135	C₂×Dic₇₃	—	4	2	no	—
136	C₂×Dic₇₃	C₂×C₄⋊C₄	1	4	yes	—
137	D₂₉₂	#106	1	16	yes	complemented
138	C₄×D₇₃	—	2	4	no	—
139	C₄×D₇₃	—	2	4	no	—
140	C₄×D₇₃	—	2	4	no	—
141	C₄×D₇₃	—	2	4	no	—
142	C₄×D₇₃	—	8	16	no	—
143	C₈×C₇₃	—	2	2	no	—
144	C₈×C₇₃	C₄×D₄	1	4	yes	complemented
145	#10596831	—	2	4	no	—
146	C₂²×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
147	C₂²×C₇₃	D₄²	1	1	yes	characteristic
148	C₂²×C₇₃	#547	1	1	yes	characteristic
149	C₂²×C₇₃	C₂²×SD₁₆	1	2	yes	—
150	C₂²×C₇₃	#577	1	2	yes	—
151	C₂×D₇₃	—	2	8	no	—
152	C₂×D₇₃	—	2	4	no	—
153	C₂×D₇₃	—	2	4	no	—
154	C₂×D₇₃	—	2	8	no	—
155	C₂×D₇₃	C₂×#106	1	8	yes	complemented
156	C₄×C₇₃	—	2	2	no	—
157	C₄×C₇₃	—	8	4	no	—
158	C₄×C₇₃	C₂×#106	1	4	yes	complemented
159	C₄×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
160	C₄×C₇₃	#520	1	1	yes	characteristic
161	C₄×C₇₃	D₄×Q₈	1	1	yes	characteristic
162	C₄×C₇₃	#554	1	1	yes	characteristic
163	Dic₇₃	—	2	4	no	—
164	Dic₇₃	—	4	4	no	—
165	Dic₇₃	—	8	2	no	—
166	D₄×#106	—	73	1	no	maximal, Sylow
167	C₂×C₇₃	D₄×C₄⋊C₄	1	1	yes	characteristic
168	C₂×C₇₃	C₂²×#106	1	1	yes	characteristic
169	C₂×C₇₃	#2912	1	1	yes	characteristic
170	C₂×C₇₃	D₄×SD₁₆	1	2	yes	—
171	C₂×C₇₃	#3335	1	2	yes	—
172	D₇₃	—	2	8	no	—
173	C₂×C₈×D₄	—	73	1	no	—
174	C₂²×#106	—	73	2	no	—
175	D₄×C₄⋊C₄	—	73	2	no	—
176	#1948	—	73	2	no	—
177	#1853	—	73	4	no	—
178	#1844	D₇₃	1	1	yes	characteristic, complemented
179	#1807	—	73	2	no	—
180	C₄×#106	—	73	1	no	—
181	C₇₃	D₄×#106	1	1	yes	characteristic, complemented, Sylow
182	C₂³×C₈	—	73	2	no	—
183	C₂×C₄×D₄	—	73	1	no	—
184	C₂×C₄×D₄	—	146	2	no	—
185	C₂²×C₄⋊C₄	—	73	4	no	—
186	C₈×D₄	—	73	8	no	—
187	C₂×#106	—	73	1	no	—
188	C₂×#106	—	73	1	no	—
189	C₂×#106	—	73	4	no	—
190	C₂×#106	—	146	8	no	—
191	C₂×#106	C₂×D₇₃	1	2	yes	—
192	C₂×#106	C₂×D₇₃	1	2	yes	—
193	C₂×C₄⋊C₈	—	73	1	no	—
194	C₂×C₂²⋊C₈	—	73	2	no	—
195	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
196	#389	—	146	2	no	—
197	#385	—	146	4	no	—
198	#384	C₂×D₇₃	1	2	yes	—
199	#380	—	73	4	no	—
200	C₄×C₄⋊C₄	—	73	2	no	—
201	#340	—	73	2	no	—
202	#340	—	73	4	no	—
203	C₂²×D₄	—	73	1	no	—
204	C₂³×C₄	—	73	2	no	—
205	C₂³×C₄	—	146	4	no	—
206	C₂²×C₈	—	73	2	no	—
207	C₂²×C₈	—	73	8	no	—
208	C₂²×C₈	—	146	4	no	—
209	C₂²×C₈	—	146	4	no	—
210	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
211	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
212	C₄×D₄	—	292	8	no	—
213	C₄×D₄	—	73	4	no	—
214	C₄×D₄	—	73	4	no	—
215	C₂×C₄⋊C₄	—	2	2	no	—
216	C₂×C₄⋊C₄	—	2	2	no	—
217	C₂×C₄⋊C₄	—	292	4	no	—
218	C₂×C₄⋊C₄	—	73	1	no	—
219	C₂×C₄⋊C₄	—	73	2	no	—
220	C₂×C₄⋊C₄	—	73	2	no	—
221	C₂×C₄⋊C₄	—	146	8	no	—
222	C₂×C₄⋊C₄	—	146	4	no	—
223	C₂×C₄⋊C₄	C₂²×D₇₃	1	4	yes	—
224	C₂×C₂²⋊C₄	—	73	2	no	—
225	C₂×C₂²⋊C₄	—	146	4	no	—
226	C₂×C₄²	—	146	2	no	—
227	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
228	#106	—	2	4	no	—
229	#106	—	4	2	no	—
230	#106	—	73	4	no	—
231	#106	—	146	8	no	—
232	#106	—	146	2	no	—
233	C₄⋊C₈	—	73	4	no	—
234	C₂²⋊C₈	—	73	8	no	—
235	C₄×C₈	C₄×D₇₃	1	4	yes	complemented
236	#95	—	146	4	no	—
237	#95	—	146	2	no	—
238	C₂⁴	—	73	2	no	—
239	C₂×D₄	—	73	4	no	—
240	C₂×D₄	—	73	8	no	—
241	C₂²×C₄	—	2	4	no	—
242	C₂²×C₄	—	292	8	no	—
243	C₂²×C₄	—	292	8	no	—
244	C₂²×C₄	—	584	4	no	—
245	C₂²×C₄	—	73	2	no	—
246	C₂²×C₄	—	73	4	no	—
247	C₂²×C₄	—	73	4	no	—
248	C₂²×C₄	—	146	2	no	—
249	C₂²×C₄	—	146	2	no	—
250	C₂²×C₄	—	146	2	no	—
251	C₂²×C₄	—	146	2	no	—
252	C₂²×C₄	—	146	2	no	—
253	C₂²×C₄	—	146	2	no	—
254	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
255	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
256	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
257	C₂×C₈	—	2	1	no	—
258	C₂×C₈	—	2	1	no	—
259	C₂×C₈	—	73	4	no	—
260	C₂×C₈	—	146	2	no	—
261	C₂×C₈	—	146	2	no	—
262	C₂×C₈	—	146	16	no	—
263	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
264	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
265	C₂×C₈	D₄×D₇₃	1	2	yes	—
266	C₂×C₈	#11924029	1	2	yes	—
267	C₄⋊C₄	—	2	4	no	—
268	C₄⋊C₄	—	2	4	no	—
269	C₄⋊C₄	—	4	4	no	—
270	C₄⋊C₄	—	8	2	no	—
271	C₄⋊C₄	—	292	8	no	—
272	C₄⋊C₄	—	73	4	no	—
273	C₄⋊C₄	—	73	4	no	—
274	C₄⋊C₄	—	146	2	no	—
275	C₄⋊C₄	—	146	2	no	—
276	C₄⋊C₄	—	146	2	no	—
277	C₂²⋊C₄	—	292	8	no	—
278	C₂²⋊C₄	—	73	8	no	—
279	C₄²	—	292	4	no	—
280	C₄²	C₂×C₄×D₇₃	1	2	yes	—
281	C₄²	Q₈×D₇₃	1	2	yes	—
282	C₂³	—	73	4	no	—
283	C₂³	—	73	8	no	—
284	C₂³	—	146	4	no	—
285	C₂³	—	146	2	no	—
286	C₂³	—	146	2	no	—
287	C₂³	C₂×D₄×D₇₃	1	1	yes	characteristic
288	D₄	—	73	16	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	4	4	no	—
291	C₂×C₄	—	4	4	no	—
292	C₂×C₄	—	4	4	no	—
293	C₂×C₄	—	292	2	no	—
294	C₂×C₄	—	292	2	no	—
295	C₂×C₄	—	292	2	no	—
296	C₂×C₄	—	292	2	no	—
297	C₂×C₄	—	292	2	no	—
298	C₂×C₄	—	584	16	no	—
299	C₂×C₄	—	73	4	no	—
300	C₂×C₄	—	146	4	no	—
301	C₂×C₄	—	146	2	no	—
302	C₂×C₄	—	146	2	no	—
303	C₂×C₄	—	146	4	no	—
304	C₂×C₄	—	146	4	no	—
305	C₂×C₄	—	146	4	no	—
306	C₂×C₄	—	146	4	no	—
307	C₂×C₄	C₄⋊C₄×D₇₃	1	2	yes	—
308	C₂×C₄	C₄⋊C₄×D₇₃	1	2	yes	—
309	C₂×C₄	SD₁₆×D₇₃	1	4	yes	—
310	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
311	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
312	C₂×C₄	C₂×Q₈×D₇₃	1	1	yes	characteristic
313	C₂×C₄	C₂×#11924029	1	1	yes	characteristic
314	C₂×C₄	#14556636	1	1	yes	characteristic
315	C₂×C₄	#14556639	1	1	yes	characteristic
316	C₈	—	2	2	no	—
317	C₈	—	146	4	no	—
318	C₈	#14556553	1	4	yes	complemented
319	C₂²	—	73	8	no	—
320	C₂²	—	146	8	no	—
321	C₂²	—	146	4	no	—
322	C₂²	—	146	4	no	—
323	C₂²	—	146	8	no	—
324	C₂²	C₂×C₄⋊C₄×D₇₃	1	1	yes	characteristic
325	C₂²	#17989222	1	1	yes	characteristic
326	C₂²	#17989240	1	1	yes	characteristic
327	C₂²	C₂×SD₁₆×D₇₃	1	2	yes	—
328	C₂²	#17989393	1	2	yes	—
329	C₄	—	2	2	no	—
330	C₄	—	8	4	no	—
331	C₄	—	292	4	no	—
332	C₄	—	584	2	no	—
333	C₄	—	146	4	no	—
334	C₄	#106×D₇₃	1	4	yes	complemented
335	C₄	C₂×#14556553	1	1	yes	characteristic
336	C₄	#17989068	1	1	yes	characteristic
337	C₄	#17989252	1	1	yes	characteristic
338	C₄	#17989298	1	1	yes	characteristic
339	C₂	—	146	8	no	—
340	C₂	#22612831	1	1	yes	characteristic
341	C₂	C₂×#106×D₇₃	1	1	yes	characteristic
342	C₂	#22618522	1	1	yes	characteristic
343	C₂	#22621626	1	2	yes	—
344	C₂	#22622287	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₇₃
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₃
Fitting	C₇₃×#1844	C₂
socle	C₂³×C₇₃	C₂²×D₄
prime radical	C₂²×D₂₉₂	D₄
cosocle	C₂²×C₄×C₇₃	C₂⁴
hypercenter	#1844	D₇₃
hypocenter	C₇₃	D₄×#106
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

Complements

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

N	H ≅ G/N	type
2. C₂×#106×D₇₃	C₂	semidirect
4. #22610692	C₂	semidirect
5. #22610122	C₂	semidirect
6. #22609953	C₂	semidirect
7. C₄×#14556523	C₂	semidirect
8. C₇₃×#1844	C₂	semidirect
13. C₈×D₂₉₂	C₄	semidirect
36. C₂×C₈×D₇₃	C₂×C₄	semidirect
50. #14556523	D₄	semidirect
52. C₄⋊#10596831	C₂×C₄	semidirect
53. #14556503	C₂×C₄	semidirect
64. C₄×C₈×C₇₃	C₂×C₄	semidirect
137. D₂₉₂	#106	semidirect
144. C₈×C₇₃	C₄×D₄	semidirect
155. C₂×D₇₃	C₂×#106	semidirect
158. C₄×C₇₃	C₂×#106	semidirect
178. #1844	D₇₃	semidirect
181. C₇₃	D₄×#106	semidirect
235. C₄×C₈	C₄×D₇₃	semidirect
318. C₈	#14556553	semidirect
334. C₄	#106×D₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₂₉₂	2
C₂×#106×D₇₃	2
#22612831	2
#22610692	2
#22610122	2
#22609953	2
C₄×#14556523	2
C₇₃×#1844	2
D₄×#106	73