Subgroups — #33104870

Order 18688.

730 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22612888	C₂	1	2	yes	maximal
2	#22610701	C₂	1	2	yes	maximal
3	#22610701	C₂	1	2	yes	maximal
4	#22610690	C₂	1	2	yes	maximal
5	#22610099	C₂	1	2	yes	maximal
6	#22610095	C₂	1	1	yes	maximal, characteristic
7	#22609828	C₂	1	2	yes	maximal
8	C₄×#14556525	C₂	1	1	yes	maximal, characteristic
9	C₂×#446×C₇₃	C₂	1	1	yes	maximal, characteristic
10	C₂×#14556525	C₂²	1	1	yes	characteristic
11	C₂×#14556525	C₂²	1	1	yes	characteristic
12	C₂×#14556525	C₂²	1	2	yes	—
13	C₂×#14556525	C₂²	1	1	yes	characteristic
14	C₂×#14556525	C₂²	1	1	yes	characteristic
15	C₂×#14556523	C₂²	1	2	yes	—
16	C₂×#14556523	C₂²	1	2	yes	—
17	#17988494	—	2	2	no	—
18	#17988473	—	2	2	no	—
19	#17988473	—	2	2	no	—
20	#17988457	C₂²	1	2	yes	—
21	#17988453	C₂²	1	2	yes	—
22	#17988453	C₂²	1	2	yes	—
23	C₄⋊C₄×Dic₇₃	—	2	2	no	—
24	C₄×C₄⋊Dic₇₃	C₂²	1	2	yes	—
25	C₄×C₄⋊Dic₇₃	C₂²	1	2	yes	—
26	C₄×C₄⋊Dic₇₃	C₂²	1	2	yes	—
27	#17988350	C₂²	1	2	yes	—
28	#17988350	C₂²	1	2	yes	—
29	#17988350	C₂²	1	2	yes	—
30	C₂×C₄×Q₈×C₇₃	C₂²	1	1	yes	characteristic
31	#446×C₇₃	C₄	1	8	yes	complemented
32	C₂×C₄⋊C₈×C₇₃	C₂²	1	2	yes	—
33	C₂×C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
34	C₂×#97×C₇₃	C₂²	1	2	yes	—
35	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
36	C₂×C₄⋊Dic₇₃	—	2	2	no	—
37	C₂×C₄⋊Dic₇₃	—	2	2	no	—
38	C₂×C₄⋊Dic₇₃	C₂³	1	2	yes	—
39	C₂×C₄⋊Dic₇₃	C₂³	1	2	yes	—
40	C₂×C₄⋊Dic₇₃	C₂³	1	2	yes	—
41	C₂×C₄⋊Dic₇₃	C₂³	1	2	yes	—
42	C₂×C₄×Dic₇₃	—	2	2	no	—
43	C₂×C₄×Dic₇₃	—	2	2	no	—
44	C₂×C₄×Dic₇₃	—	2	2	no	—
45	#14556525	—	2	2	no	—
46	#14556525	—	4	2	no	—
47	#14556525	—	4	1	no	—
48	#14556525	—	4	1	no	—
49	#14556525	Q₈	1	4	yes	complemented
50	#14556523	—	2	4	no	—
51	#14556523	—	4	2	no	—
52	#14556498	—	2	2	no	—
53	#14556498	—	2	2	no	—
54	#14556498	—	2	2	no	—
55	#14556498	—	2	2	no	—
56	#14556498	—	2	2	no	—
57	#14556498	—	2	2	no	—
58	#14556498	—	2	2	no	—
59	C₂²×Q₈×C₇₃	D₄	1	1	yes	characteristic
60	C₂²×C₈×C₇₃	C₂³	1	2	yes	—
61	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
62	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
63	C₄×Q₈×C₇₃	C₂×C₄	1	4	yes	—
64	C₄×Q₈×C₇₃	Q₈	1	4	yes	—
65	C₂×C₄⋊C₄×C₇₃	D₄	1	2	yes	—
66	C₂×C₄⋊C₄×C₇₃	D₄	1	1	yes	characteristic
67	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
68	C₂×C₄²×C₇₃	C₂³	1	2	yes	—
69	C₄⋊C₈×C₇₃	C₂×C₄	1	4	yes	—
70	C₄⋊C₈×C₇₃	C₂×C₄	1	8	yes	—
71	#97×C₇₃	C₂×C₄	1	8	yes	—
72	C₄×C₈×C₇₃	C₂×C₄	1	4	yes	—
73	C₂²×Dic₇₃	—	2	2	no	—
74	C₂²×Dic₇₃	—	2	2	no	—
75	C₂²×Dic₇₃	—	2	2	no	—
76	C₂²×Dic₇₃	—	2	2	no	—
77	C₄⋊Dic₇₃	—	2	2	no	—
78	C₄⋊Dic₇₃	—	2	2	no	—
79	C₄⋊Dic₇₃	—	2	2	no	—
80	C₄⋊Dic₇₃	—	2	2	no	—
81	C₄⋊Dic₇₃	—	2	2	no	—
82	C₄⋊Dic₇₃	—	2	2	no	—
83	C₄⋊Dic₇₃	—	2	2	no	—
84	C₄⋊Dic₇₃	—	4	4	no	—
85	C₄⋊Dic₇₃	—	8	2	no	—
86	C₄⋊Dic₇₃	C₂×Q₈	1	4	yes	—
87	C₄×Dic₇₃	—	4	4	no	—
88	C₄×Dic₇₃	—	8	2	no	—
89	C₄×Dic₇₃	—	8	2	no	—
90	C₂×Q₈×C₇₃	—	4	1	no	—
91	C₂×Q₈×C₇₃	—	4	1	no	—
92	C₂×Q₈×C₇₃	C₄⋊C₄	1	4	yes	—
93	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
95	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
97	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
98	C₂×C₈×C₇₃	—	2	2	no	—
99	C₂×C₈×C₇₃	—	2	1	no	—
100	C₂×C₈×C₇₃	—	2	2	no	—
101	C₂×C₈×C₇₃	—	2	1	no	—
102	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
103	C₂×C₈×C₇₃	C₂²×C₄	1	4	yes	—
104	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
105	C₂×C₈×C₇₃	C₂×Q₈	1	2	yes	—
106	C₂×C₈×C₇₃	#41	1	2	yes	—
107	C₄⋊C₄×C₇₃	C₄⋊C₄	1	8	yes	—
108	C₄⋊C₄×C₇₃	C₄⋊C₄	1	4	yes	—
109	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
110	C₄²×C₇₃	C₂²×C₄	1	4	yes	—
111	C₄²×C₇₃	C₂×Q₈	1	2	yes	—
112	C₄²×C₇₃	C₂×Q₈	1	4	yes	—
113	C₂³×C₇₃	C₂²×D₄	1	1	yes	characteristic
114	Q₈×C₇₃	—	4	4	no	—
115	C₂×C₄×C₇₃	—	2	1	no	—
116	C₂×C₄×C₇₃	—	2	1	no	—
117	C₂×C₄×C₇₃	—	2	2	no	—
118	C₂×C₄×C₇₃	—	2	2	no	—
119	C₂×C₄×C₇₃	—	2	1	no	—
120	C₂×C₄×C₇₃	—	2	2	no	—
121	C₂×C₄×C₇₃	—	2	1	no	—
122	C₂×C₄×C₇₃	—	2	2	no	—
123	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
124	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	4	yes	—
125	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
126	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	4	yes	—
127	C₂×C₄×C₇₃	C₂³×C₄	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×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	4	no	—
134	C₂×Dic₇₃	—	4	2	no	—
135	C₂×Dic₇₃	—	4	2	no	—
136	C₂×Dic₇₃	—	4	2	no	—
137	C₂×Dic₇₃	—	4	2	no	—
138	C₂×Dic₇₃	—	4	2	no	—
139	C₂×Dic₇₃	—	4	2	no	—
140	C₂×Dic₇₃	—	4	2	no	—
141	C₂×Dic₇₃	—	4	2	no	—
142	C₂×Dic₇₃	—	4	2	no	—
143	C₂×Dic₇₃	—	4	2	no	—
144	C₂×Dic₇₃	—	4	2	no	—
145	C₈×C₇₃	—	2	4	no	—
146	C₈×C₇₃	—	2	2	no	—
147	C₈×C₇₃	C₄×Q₈	1	4	yes	—
148	C₂²×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
149	C₂²×C₇₃	D₄×Q₈	1	1	yes	characteristic
150	C₂²×C₇₃	#550	1	1	yes	characteristic
151	C₂²×C₇₃	#575	1	1	yes	characteristic
152	C₂²×C₇₃	#575	1	1	yes	characteristic
153	C₂²×C₇₃	#576	1	1	yes	characteristic
154	C₂²×C₇₃	#578	1	1	yes	characteristic
155	C₄×C₇₃	—	2	2	no	—
156	C₄×C₇₃	—	2	4	no	—
157	C₄×C₇₃	—	2	2	no	—
158	C₄×C₇₃	—	2	4	no	—
159	C₄×C₇₃	C₂×C₄×Q₈	1	1	yes	characteristic
160	C₄×C₇₃	#520	1	1	yes	characteristic
161	C₄×C₇₃	#557	1	1	yes	characteristic
162	C₄×C₇₃	Q₈²	1	1	yes	characteristic
163	Dic₇₃	—	4	4	no	—
164	Dic₇₃	—	8	2	no	—
165	Dic₇₃	—	8	2	no	—
166	Dic₇₃	—	8	2	no	—
167	#22731	—	73	1	no	maximal, Sylow
168	C₂×C₇₃	Q₈×C₄⋊C₄	1	1	yes	characteristic
169	C₂×C₇₃	#2911	1	1	yes	characteristic
170	C₂×C₇₃	#2913	1	1	yes	characteristic
171	C₂×C₇₃	#3389	1	1	yes	characteristic
172	C₂×C₇₃	#3390	1	1	yes	characteristic
173	C₂×C₇₃	#3402	1	1	yes	characteristic
174	C₂×C₇₃	#3403	1	1	yes	characteristic
175	C₂×#446	D₇₃	1	1	yes	characteristic
176	#2355	—	73	2	no	—
177	#1949	—	73	2	no	—
178	#1949	—	73	2	no	—
179	#1947	—	73	2	no	—
180	#1847	—	73	2	no	—
181	#1846	—	73	1	no	—
182	#1776	—	73	2	no	—
183	C₄×#107	—	73	1	no	—
184	C₇₃	#22731	1	1	yes	characteristic, complemented, Sylow
185	C₂×C₄×Q₈	C₂×D₇₃	1	1	yes	characteristic
186	#446	Dic₇₃	1	8	yes	complemented
187	C₂×#107	—	73	1	no	—
188	C₂×#107	—	73	1	no	—
189	C₂×#107	—	73	2	no	—
190	C₂×#107	—	73	1	no	—
191	C₂×#107	—	73	1	no	—
192	C₂×#106	—	73	2	no	—
193	C₂×#106	—	73	2	no	—
194	C₂×C₄⋊C₈	C₂×D₇₃	1	2	yes	—
195	C₂×C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
196	C₂×#97	C₂×D₇₃	1	2	yes	—
197	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
198	#391	—	146	2	no	—
199	#387	—	146	2	no	—
200	#387	—	146	2	no	—
201	#384	—	73	2	no	—
202	#382	—	73	2	no	—
203	#382	—	73	2	no	—
204	C₄×C₄⋊C₄	—	73	2	no	—
205	C₄×C₄⋊C₄	—	73	2	no	—
206	C₄×C₄⋊C₄	—	73	2	no	—
207	C₄×C₄⋊C₄	—	146	2	no	—
208	#340	—	73	2	no	—
209	#340	—	73	2	no	—
210	#340	—	73	2	no	—
211	C₂²×Q₈	D₂₉₂	1	1	yes	characteristic
212	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
213	C₂²×C₈	C₂²×D₇₃	1	2	yes	—
214	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
215	C₄×Q₈	Dic₁₄₆	1	4	yes	—
216	C₄×Q₈	C₂×Dic₇₃	1	4	yes	—
217	C₂×C₄⋊C₄	—	73	2	no	—
218	C₂×C₄⋊C₄	—	73	2	no	—
219	C₂×C₄⋊C₄	—	73	2	no	—
220	C₂×C₄⋊C₄	—	73	2	no	—
221	C₂×C₄⋊C₄	—	146	2	no	—
222	C₂×C₄⋊C₄	—	146	2	no	—
223	C₂×C₄⋊C₄	D₂₉₂	1	2	yes	—
224	C₂×C₄⋊C₄	D₂₉₂	1	1	yes	characteristic
225	C₂×C₄²	—	146	2	no	—
226	C₂×C₄²	—	146	2	no	—
227	C₂×C₄²	—	146	2	no	—
228	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
229	C₂×C₄²	C₂²×D₇₃	1	2	yes	—
230	#107	—	292	1	no	—
231	#107	—	292	2	no	—
232	#107	—	292	1	no	—
233	#107	—	73	4	no	—
234	#107	—	146	2	no	—
235	#106	—	292	2	no	—
236	#106	—	146	4	no	—
237	C₄⋊C₈	C₂×Dic₇₃	1	4	yes	—
238	C₄⋊C₈	C₂×Dic₇₃	1	8	yes	—
239	#97	C₂×Dic₇₃	1	8	yes	—
240	C₄×C₈	C₂×Dic₇₃	1	4	yes	—
241	#95	—	146	2	no	—
242	#95	—	146	2	no	—
243	#95	—	146	2	no	—
244	#95	—	146	2	no	—
245	#95	—	146	2	no	—
246	#95	—	146	2	no	—
247	#95	—	146	2	no	—
248	C₂×Q₈	—	4	1	no	—
249	C₂×Q₈	—	4	1	no	—
250	C₂×Q₈	C₄⋊Dic₇₃	1	4	yes	—
251	C₂²×C₄	—	146	2	no	—
252	C₂²×C₄	—	146	2	no	—
253	C₂²×C₄	—	146	2	no	—
254	C₂²×C₄	—	146	2	no	—
255	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
256	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
257	C₂²×C₄	C₂×D₂₉₂	1	2	yes	—
258	C₂²×C₄	C₂×D₂₉₂	1	2	yes	—
259	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	1	no	—
262	C₂×C₈	—	2	1	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	C₂²×Dic₇₃	1	2	yes	—
265	C₂×C₈	C₂²×Dic₇₃	1	4	yes	—
266	C₂×C₈	C₂²×Dic₇₃	1	2	yes	—
267	C₂×C₈	Q₈×D₇₃	1	2	yes	—
268	C₂×C₈	#11924029	1	2	yes	—
269	C₄⋊C₄	—	292	4	no	—
270	C₄⋊C₄	—	584	2	no	—
271	C₄⋊C₄	—	73	4	no	—
272	C₄⋊C₄	—	146	2	no	—
273	C₄⋊C₄	—	146	2	no	—
274	C₄⋊C₄	—	146	2	no	—
275	C₄⋊C₄	—	146	2	no	—
276	C₄⋊C₄	—	146	2	no	—
277	C₄⋊C₄	—	146	2	no	—
278	C₄⋊C₄	—	146	2	no	—
279	C₄⋊C₄	C₄⋊Dic₇₃	1	8	yes	—
280	C₄⋊C₄	C₄⋊Dic₇₃	1	4	yes	—
281	C₄²	—	292	4	no	—
282	C₄²	—	584	2	no	—
283	C₄²	—	584	2	no	—
284	C₄²	C₂²×Dic₇₃	1	2	yes	—
285	C₄²	C₂²×Dic₇₃	1	4	yes	—
286	C₄²	C₂×Dic₁₄₆	1	2	yes	—
287	C₄²	C₂×Dic₁₄₆	1	4	yes	—
288	C₂³	C₂²×D₂₉₂	1	1	yes	characteristic
289	Q₈	—	4	4	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	292	2	no	—
299	C₂×C₄	—	292	2	no	—
300	C₂×C₄	—	292	2	no	—
301	C₂×C₄	—	292	2	no	—
302	C₂×C₄	—	292	2	no	—
303	C₂×C₄	—	292	2	no	—
304	C₂×C₄	—	292	2	no	—
305	C₂×C₄	—	292	2	no	—
306	C₂×C₄	—	292	2	no	—
307	C₂×C₄	—	292	2	no	—
308	C₂×C₄	—	292	2	no	—
309	C₂×C₄	—	146	4	no	—
310	C₂×C₄	C₂×C₄⋊Dic₇₃	1	2	yes	—
311	C₂×C₄	C₂×C₄⋊Dic₇₃	1	4	yes	—
312	C₂×C₄	C₂×C₄⋊Dic₇₃	1	2	yes	—
313	C₂×C₄	C₂×C₄⋊Dic₇₃	1	4	yes	—
314	C₂×C₄	C₂³×Dic₇₃	1	1	yes	characteristic
315	C₂×C₄	C₂×Q₈×D₇₃	1	1	yes	characteristic
316	C₂×C₄	C₂²×Dic₁₄₆	1	1	yes	characteristic
317	C₂×C₄	C₂×#11924029	1	1	yes	characteristic
318	C₂×C₄	#14556637	1	1	yes	characteristic
319	C₂×C₄	#14556638	1	1	yes	characteristic
320	C₈	—	2	4	no	—
321	C₈	—	2	2	no	—
322	C₈	Q₈×Dic₇₃	1	4	yes	—
323	C₂²	C₂²×C₄⋊Dic₇₃	1	1	yes	characteristic
324	C₂²	Q₈×D₂₉₂	1	1	yes	characteristic
325	C₂²	#17989259	1	1	yes	characteristic
326	C₂²	#17989387	1	1	yes	characteristic
327	C₂²	#17989387	1	1	yes	characteristic
328	C₂²	#17989392	1	1	yes	characteristic
329	C₂²	#17989404	1	1	yes	characteristic
330	C₄	—	2	4	no	—
331	C₄	—	2	2	no	—
332	C₄	—	2	2	no	—
333	C₄	—	2	4	no	—
334	C₄	—	292	4	no	—
335	C₄	—	584	2	no	—
336	C₄	—	584	2	no	—
337	C₄	—	584	2	no	—
338	C₄	C₂×Q₈×Dic₇₃	1	1	yes	characteristic
339	C₄	#17989070	1	1	yes	characteristic
340	C₄	#17989316	1	1	yes	characteristic
341	C₄	Q₈×Dic₁₄₆	1	1	yes	characteristic
342	C₂	Q₈×C₄⋊Dic₇₃	1	1	yes	characteristic
343	C₂	#22618521	1	1	yes	characteristic
344	C₂	#22618531	1	1	yes	characteristic
345	C₂	#22622891	1	1	yes	characteristic
346	C₂	#22622900	1	1	yes	characteristic
347	C₂	#22623012	1	1	yes	characteristic
348	C₂	#22623021	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₂₉₂
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₃
Fitting	C₂×#446×C₇₃	C₂
socle, prime radical	C₂³×C₇₃	C₂²×D₄
cosocle	C₂²×C₄×C₇₃	C₂⁴
hypercenter	C₂×#446	D₇₃
hypocenter	C₇₃	#22731
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×#446

Complements

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

N	H ≅ G/N	type
31. #446×C₇₃	C₄	semidirect
49. #14556525	Q₈	semidirect
184. C₇₃	#22731	semidirect
186. #446	Dic₇₃	semidirect

Maximal subgroups

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

subgroup	index
#22612888	2
#22610701	2
#22610690	2
#22610099	2
#22610095	2
#22609828	2
C₄×#14556525	2
C₂×#446×C₇₃	2
#22731	73