Subgroups — #33104376

Order 18688.

1134 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₅₈₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#17988669	C₂	1	2	yes	maximal
3	#22612831	C₂	1	2	yes	maximal, complemented
4	#22610662	C₂	1	2	yes	maximal, complemented
5	#22610175	C₂	1	4	yes	maximal, complemented
6	#22609928	C₂	1	2	yes	maximal, complemented
7	C₈×C₄⋊Dic₇₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇₃×#1845	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₅₈₄	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₂₉₂	C₂²	1	2	yes	—
11	C₂×#14556553	C₂²	1	4	yes	—
12	C₂×C₄⋊C₄×D₇₃	—	2	2	no	—
13	C₄×D₅₈₄	C₄	1	8	yes	complemented
14	#17988669	—	2	8	no	—
15	C₂×#14556525	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊#10596831	C₂²	1	1	yes	characteristic
17	C₂×#14556513	C₂²	1	2	yes	—
18	C₂×#14556511	C₂²	1	4	yes	—
19	C₂×C₈×Dic₇₃	C₂²	1	2	yes	—
20	#17988483	C₂²	1	4	yes	—
21	#17988458	—	2	4	no	—
22	#17988444	—	2	2	no	—
23	C₄×C₄⋊Dic₇₃	C₂²	1	1	yes	characteristic
24	#17988349	C₂²	1	4	yes	—
25	#17988341	C₂²	1	2	yes	—
26	C₂×#107×C₇₃	C₂²	1	2	yes	—
27	C₂×#107×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	C₂³	1	2	yes	—
31	C₂²×C₄×D₇₃	—	2	4	no	—
32	C₂²×C₄×D₇₃	—	2	2	no	—
33	C₂×D₅₈₄	C₂×C₄	1	4	yes	—
34	C₂×D₅₈₄	D₄	1	4	yes	—
35	C₂×D₅₈₄	Q₈	1	4	yes	—
36	C₂²×#10596831	C₂³	1	2	yes	—
37	C₄×D₂₉₂	—	2	4	no	—
38	C₄×D₂₉₂	C₂×C₄	1	8	yes	complemented
39	#14556553	—	2	8	no	—
40	#14556553	—	4	4	no	—
41	C₂×C₄⋊Dic₇₃	D₄	1	1	yes	characteristic
42	C₂×C₄⋊Dic₇₃	D₄	1	1	yes	characteristic
43	C₂×C₄⋊Dic₇₃	C₂³	1	2	yes	—
44	C₂×#11924007	—	2	4	no	—
45	C₄⋊C₄×D₇₃	—	4	8	no	—
46	C₂×#11924005	—	2	4	no	—
47	C₂×#11924005	—	2	2	no	—
48	C₂×C₄×Dic₇₃	C₂³	1	2	yes	—
49	#14556525	C₂×C₄	1	4	yes	complemented
50	C₄⋊#10596831	—	2	2	no	—
51	#14556513	C₂×C₄	1	8	yes	complemented
52	#14556511	—	2	8	no	—
53	C₈×Dic₇₃	—	2	4	no	—
54	#14556498	—	2	4	no	—
55	#14556498	D₄	1	2	yes	—
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	#107×C₇₃	—	2	4	no	—
63	#107×C₇₃	—	4	2	no	—
64	C₄×C₈×C₇₃	C₂×C₄	1	4	yes	complemented
65	C₂³×D₇₃	—	2	2	no	—
66	C₂×D₂₉₂	—	2	2	no	—
67	C₂×D₂₉₂	—	2	2	no	—
68	C₂×D₂₉₂	—	2	2	no	—
69	C₂×D₂₉₂	C₂²×C₄	1	4	yes	—
70	C₂×D₂₉₂	C₂×D₄	1	4	yes	—
71	C₂×D₂₉₂	C₂×Q₈	1	4	yes	—
72	C₂²×Dic₇₃	—	2	2	no	—
73	C₂²×Dic₇₃	C₂×D₄	1	2	yes	—
74	C₂²×Dic₇₃	C₂×D₄	1	2	yes	—
75	C₂×C₄×D₇₃	—	2	8	no	—
76	C₂×C₄×D₇₃	—	4	4	no	—
77	C₂×C₄×D₇₃	—	4	4	no	—
78	C₂×C₄×D₇₃	—	4	8	no	—
79	C₂×C₄×D₇₃	—	4	8	no	—
80	C₂×C₄×D₇₃	—	8	4	no	—
81	D₅₈₄	C₄⋊C₄	1	16	yes	complemented
82	C₂×#10596831	—	2	2	no	—
83	C₂×#10596831	—	2	2	no	—
84	C₂×#10596831	—	2	2	no	—
85	C₄⋊Dic₇₃	—	2	2	no	—
86	C₄⋊Dic₇₃	—	2	2	no	—
87	C₄⋊Dic₇₃	D₈	1	4	yes	complemented
88	C₄⋊Dic₇₃	C₂²×C₄	1	4	yes	—
89	#11924007	—	4	8	no	—
90	#11924005	—	2	8	no	—
91	#11924005	—	4	8	no	—
92	C₄×Dic₇₃	—	2	2	no	—
93	C₄×Dic₇₃	—	2	2	no	—
94	C₂²×C₄×C₇₃	—	2	4	no	—
95	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
98	C₂×C₈×C₇₃	—	2	1	no	—
99	C₂×C₈×C₇₃	—	2	1	no	—
100	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
101	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
102	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
103	C₂×C₈×C₇₃	C₂×Q₈	1	2	yes	—
104	C₄⋊C₄×C₇₃	—	2	4	no	—
105	C₄⋊C₄×C₇₃	—	2	4	no	—
106	C₄⋊C₄×C₇₃	—	4	4	no	—
107	C₄⋊C₄×C₇₃	—	8	2	no	—
108	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
109	C₄²×C₇₃	#41	1	2	yes	—
110	C₂³×C₇₃	C₂²×D₄	1	1	yes	characteristic
111	C₂²×D₇₃	—	2	4	no	—
112	C₂²×D₇₃	—	2	4	no	—
113	C₂²×D₇₃	—	2	4	no	—
114	C₂²×D₇₃	—	4	2	no	—
115	C₂²×D₇₃	—	4	2	no	—
116	C₂²×D₇₃	—	4	2	no	—
117	C₂²×D₇₃	—	4	2	no	—
118	C₂×C₄×C₇₃	—	2	1	no	—
119	C₂×C₄×C₇₃	—	2	1	no	—
120	C₂×C₄×C₇₃	—	4	4	no	—
121	C₂×C₄×C₇₃	—	4	4	no	—
122	C₂×C₄×C₇₃	—	4	4	no	—
123	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
124	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
125	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
126	C₂×C₄×C₇₃	#135	1	2	yes	—
127	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
128	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
129	C₂×C₄×C₇₃	C₂²×Q₈	1	1	yes	characteristic
130	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
131	C₂×C₄×C₇₃	H(5,𝔽₂)	1	1	yes	characteristic
132	C₂×C₄×C₇₃	2₋¹⁺⁴	1	1	yes	characteristic
133	C₂×Dic₇₃	—	2	2	no	—
134	C₂×Dic₇₃	—	2	2	no	—
135	C₂×Dic₇₃	—	2	4	no	—
136	C₂×Dic₇₃	—	2	2	no	—
137	C₂×Dic₇₃	—	2	2	no	—
138	C₂×Dic₇₃	—	2	2	no	—
139	C₂×Dic₇₃	—	4	2	no	—
140	C₂×Dic₇₃	—	4	2	no	—
141	C₂×Dic₇₃	C₂×D₈	1	4	yes	—
142	D₂₉₂	—	2	4	no	—
143	D₂₉₂	—	2	4	no	—
144	D₂₉₂	—	2	4	no	—
145	D₂₉₂	C₂×C₄⋊C₄	1	8	yes	complemented
146	C₄×D₇₃	—	4	16	no	—
147	C₄×D₇₃	—	8	16	no	—
148	C₈×C₇₃	—	2	2	no	—
149	C₈×C₇₃	C₂×C₄⋊C₄	1	4	yes	complemented
150	#10596831	—	4	2	no	—
151	C₂²×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
152	C₂²×C₇₃	D₄²	1	1	yes	characteristic
153	C₂²×C₇₃	D₄×Q₈	1	1	yes	characteristic
154	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
155	C₂²×C₇₃	C₂×#135	1	1	yes	characteristic
156	C₂²×C₇₃	#576	1	1	yes	characteristic
157	C₂²×C₇₃	#578	1	1	yes	characteristic
158	C₂×D₇₃	—	2	8	no	—
159	C₂×D₇₃	—	4	4	no	—
160	C₂×D₇₃	—	4	4	no	—
161	C₂×D₇₃	—	4	4	no	—
162	C₂×D₇₃	—	4	4	no	—
163	C₂×D₇₃	—	4	4	no	—
164	C₂×D₇₃	—	4	4	no	—
165	C₄×C₇₃	—	2	2	no	—
166	C₄×C₇₃	—	8	4	no	—
167	C₄×C₇₃	C₄×D₈	1	4	yes	complemented
168	C₄×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
169	C₄×C₇₃	#520	1	1	yes	characteristic
170	C₄×C₇₃	#547	1	1	yes	characteristic
171	C₄×C₇₃	#554	1	1	yes	characteristic
172	Dic₇₃	—	2	4	no	—
173	Dic₇₃	—	4	4	no	—
174	Dic₇₃	—	4	2	no	—
175	C₄⋊C₄×D₈	—	73	1	no	maximal, Sylow
176	C₂×C₇₃	D₄×C₄⋊C₄	1	1	yes	characteristic
177	C₂×C₇₃	C₂×C₄×D₈	1	1	yes	characteristic
178	C₂×C₇₃	#2951	1	1	yes	characteristic
179	C₂×C₇₃	D₄×D₈	1	1	yes	characteristic
180	C₂×C₇₃	#3283	1	1	yes	characteristic
181	C₂×C₇₃	Q₈×D₈	1	1	yes	characteristic
182	C₂×C₇₃	#3380	1	1	yes	characteristic
183	D₇₃	—	4	8	no	—
184	C₂×C₄×D₈	—	73	2	no	—
185	C₂×C₄×D₈	—	73	1	no	—
186	D₄×C₄⋊C₄	—	73	2	no	—
187	#1942	—	73	2	no	—
188	#1862	—	73	4	no	—
189	#1845	D₇₃	1	1	yes	characteristic, complemented
190	#1802	—	73	2	no	—
191	C₈×C₄⋊C₄	—	73	1	no	—
192	C₇₃	C₄⋊C₄×D₈	1	1	yes	characteristic, complemented, Sylow
193	C₂²×D₈	—	73	1	no	—
194	C₂×C₄×D₄	—	73	4	no	—
195	C₂×C₄×D₄	—	73	2	no	—
196	C₂²×C₄⋊C₄	—	146	2	no	—
197	C₄×D₈	—	73	8	no	—
198	C₄×D₈	—	146	8	no	—
199	C₂×#107	—	73	1	no	—
200	C₂×#107	C₂×D₇₃	1	2	yes	—
201	C₂×#107	C₂×D₇₃	1	2	yes	—
202	C₂×C₄⋊C₈	—	73	1	no	—
203	C₂×#102	—	73	2	no	—
204	C₂×#102	—	73	4	no	—
205	C₂×C₄×C₈	—	73	2	no	—
206	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
207	#389	—	73	4	no	—
208	#385	—	146	4	no	—
209	#384	C₂×D₇₃	1	2	yes	—
210	#380	—	146	2	no	—
211	C₄×C₄⋊C₄	—	73	1	no	—
212	#340	—	73	4	no	—
213	#336	—	73	2	no	—
214	C₂²×D₄	—	73	2	no	—
215	C₂³×C₄	—	146	4	no	—
216	C₂³×C₄	—	146	2	no	—
217	C₂×D₈	—	73	4	no	—
218	C₂×D₈	—	73	4	no	—
219	C₂×D₈	—	73	4	no	—
220	C₂²×C₈	—	73	2	no	—
221	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
222	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
223	C₄×D₄	—	292	4	no	—
224	C₄×D₄	—	73	8	no	—
225	C₄×D₄	—	146	8	no	—
226	C₄×D₄	—	146	4	no	—
227	C₂×C₄⋊C₄	—	2	2	no	—
228	C₂×C₄⋊C₄	—	2	2	no	—
229	C₂×C₄⋊C₄	—	292	8	no	—
230	C₂×C₄⋊C₄	—	73	1	no	—
231	C₂×C₄⋊C₄	—	73	2	no	—
232	C₂×C₄⋊C₄	—	73	1	no	—
233	C₂×C₄⋊C₄	—	146	4	no	—
234	C₂×C₄⋊C₄	C₂²×D₇₃	1	4	yes	—
235	C₂×C₂²⋊C₄	—	146	4	no	—
236	C₂×C₂²⋊C₄	—	146	2	no	—
237	C₂×C₄²	—	73	2	no	—
238	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
239	#107	—	2	4	no	—
240	#107	—	4	2	no	—
241	#107	—	73	4	no	—
242	C₄⋊C₈	—	146	2	no	—
243	#102	—	73	8	no	—
244	#102	—	146	8	no	—
245	C₄×C₈	—	146	4	no	—
246	C₄×C₈	C₄×D₇₃	1	4	yes	complemented
247	#95	—	73	2	no	—
248	#95	—	146	4	no	—
249	C₂⁴	—	146	2	no	—
250	C₂×D₄	—	73	4	no	—
251	C₂×D₄	—	73	4	no	—
252	C₂×D₄	—	73	4	no	—
253	C₂×D₄	—	146	2	no	—
254	C₂×D₄	—	146	2	no	—
255	C₂×D₄	—	146	2	no	—
256	C₂²×C₄	—	2	4	no	—
257	C₂²×C₄	—	292	8	no	—
258	C₂²×C₄	—	292	8	no	—
259	C₂²×C₄	—	292	4	no	—
260	C₂²×C₄	—	292	4	no	—
261	C₂²×C₄	—	584	4	no	—
262	C₂²×C₄	—	73	2	no	—
263	C₂²×C₄	—	73	2	no	—
264	C₂²×C₄	—	146	2	no	—
265	C₂²×C₄	—	146	8	no	—
266	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
267	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
268	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
269	D₈	—	73	16	no	—
270	C₂×C₈	—	2	1	no	—
271	C₂×C₈	—	2	1	no	—
272	C₂×C₈	—	146	2	no	—
273	C₂×C₈	—	146	2	no	—
274	C₂×C₈	—	146	2	no	—
275	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
276	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
277	C₂×C₈	D₄×D₇₃	1	2	yes	—
278	C₂×C₈	Q₈×D₇₃	1	2	yes	—
279	C₄⋊C₄	—	2	4	no	—
280	C₄⋊C₄	—	2	4	no	—
281	C₄⋊C₄	—	4	4	no	—
282	C₄⋊C₄	—	8	2	no	—
283	C₄⋊C₄	—	292	8	no	—
284	C₄⋊C₄	—	73	4	no	—
285	C₄⋊C₄	—	73	4	no	—
286	C₄⋊C₄	—	146	2	no	—
287	C₄⋊C₄	—	146	2	no	—
288	C₂²⋊C₄	—	292	8	no	—
289	C₂²⋊C₄	—	146	8	no	—
290	C₄²	—	146	2	no	—
291	C₄²	—	146	2	no	—
292	C₄²	C₂×C₄×D₇₃	1	2	yes	—
293	C₄²	#11924029	1	2	yes	—
294	C₂³	—	292	2	no	—
295	C₂³	—	292	2	no	—
296	C₂³	—	292	2	no	—
297	C₂³	—	292	2	no	—
298	C₂³	—	146	4	no	—
299	C₂³	—	146	4	no	—
300	C₂³	—	146	4	no	—
301	C₂³	C₂×D₄×D₇₃	1	1	yes	characteristic
302	D₄	—	73	8	no	—
303	D₄	—	146	4	no	—
304	D₄	—	146	4	no	—
305	D₄	—	146	4	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	4	4	no	—
309	C₂×C₄	—	4	4	no	—
310	C₂×C₄	—	4	4	no	—
311	C₂×C₄	—	292	2	no	—
312	C₂×C₄	—	292	2	no	—
313	C₂×C₄	—	292	16	no	—
314	C₂×C₄	—	584	16	no	—
315	C₂×C₄	—	73	4	no	—
316	C₂×C₄	—	146	2	no	—
317	C₂×C₄	—	146	2	no	—
318	C₂×C₄	—	146	4	no	—
319	C₂×C₄	—	146	2	no	—
320	C₂×C₄	—	146	2	no	—
321	C₂×C₄	—	146	2	no	—
322	C₂×C₄	#14556553	1	2	yes	—
323	C₂×C₄	#14556553	1	2	yes	—
324	C₂×C₄	D₈×D₇₃	1	2	yes	—
325	C₂×C₄	#14556614	1	2	yes	—
326	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
327	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
328	C₂×C₄	C₂×Q₈×D₇₃	1	1	yes	characteristic
329	C₂×C₄	C₂×#11924029	1	1	yes	characteristic
330	C₂×C₄	#14556636	1	1	yes	characteristic
331	C₂×C₄	#14556639	1	1	yes	characteristic
332	C₈	—	2	2	no	—
333	C₈	—	292	2	no	—
334	C₈	C₄⋊C₄×D₇₃	1	4	yes	complemented
335	C₂²	—	292	4	no	—
336	C₂²	—	292	4	no	—
337	C₂²	—	292	4	no	—
338	C₂²	—	292	4	no	—
339	C₂²	—	292	4	no	—
340	C₂²	—	292	4	no	—
341	C₂²	—	146	8	no	—
342	C₂²	C₂×#14556553	1	1	yes	characteristic
343	C₂²	#17989222	1	1	yes	characteristic
344	C₂²	#17989252	1	1	yes	characteristic
345	C₂²	C₂×D₈×D₇₃	1	1	yes	characteristic
346	C₂²	C₂×#14556614	1	1	yes	characteristic
347	C₂²	#17989388	1	1	yes	characteristic
348	C₂²	#17989400	1	1	yes	characteristic
349	C₄	—	2	2	no	—
350	C₄	—	8	4	no	—
351	C₄	—	292	4	no	—
352	C₄	—	292	2	no	—
353	C₄	—	146	4	no	—
354	C₄	#17988669	1	4	yes	complemented
355	C₄	C₂×C₄⋊C₄×D₇₃	1	1	yes	characteristic
356	C₄	#17989068	1	1	yes	characteristic
357	C₄	#17989240	1	1	yes	characteristic
358	C₄	#17989298	1	1	yes	characteristic
359	C₂	—	292	8	no	—
360	C₂	#22612831	1	1	yes	characteristic
361	C₂	C₂×#17988669	1	1	yes	characteristic
362	C₂	#22618805	1	1	yes	characteristic
363	C₂	#22621606	1	1	yes	characteristic
364	C₂	#22621651	1	1	yes	characteristic
365	C₂	#22622783	1	1	yes	characteristic
366	C₂	#22622800	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₄×D₇₃
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₃
Fitting	C₇₃×#1845	C₂
socle	C₂³×C₇₃	C₂²×D₄
prime radical	C₂²×D₅₈₄	C₂²
cosocle	C₂²×C₄×C₇₃	C₂⁴
hypercenter	#1845	D₇₃
hypocenter	C₇₃	C₄⋊C₄×D₈
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

Complements

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

N	H ≅ G/N	type
3. #22612831	C₂	semidirect
4. #22610662	C₂	semidirect
5. #22610175	C₂	semidirect
6. #22609928	C₂	semidirect
7. C₈×C₄⋊Dic₇₃	C₂	semidirect
8. C₇₃×#1845	C₂	semidirect
13. C₄×D₅₈₄	C₄	semidirect
38. C₄×D₂₉₂	C₂×C₄	semidirect
49. #14556525	C₂×C₄	semidirect
51. #14556513	C₂×C₄	semidirect
64. C₄×C₈×C₇₃	C₂×C₄	semidirect
81. D₅₈₄	C₄⋊C₄	semidirect
87. C₄⋊Dic₇₃	D₈	semidirect
145. D₂₉₂	C₂×C₄⋊C₄	semidirect
149. C₈×C₇₃	C₂×C₄⋊C₄	semidirect
167. C₄×C₇₃	C₄×D₈	semidirect
189. #1845	D₇₃	semidirect
192. C₇₃	C₄⋊C₄×D₈	semidirect
246. C₄×C₈	C₄×D₇₃	semidirect
334. C₈	C₄⋊C₄×D₇₃	semidirect
354. C₄	#17988669	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₅₈₄	2
C₂×#17988669	2
#22612831	2
#22610662	2
#22610175	2
#22609928	2
C₈×C₄⋊Dic₇₃	2
C₇₃×#1845	2
C₄⋊C₄×D₈	73