Subgroups — #33104861

Order 18688.

730 conjugacy classes of subgroups.

The subgroup lattice has 336 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	#22610695	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×#14556523	C₂	1	1	yes	maximal, characteristic
9	C₂×#446×C₇₃	C₂	1	1	yes	maximal, characteristic
10	C₂×#14556525	C₂²	1	2	yes	—
11	C₂×#14556525	C₂²	1	2	yes	—
12	C₂×#14556523	C₂²	1	1	yes	characteristic
13	C₂×#14556523	C₂²	1	1	yes	characteristic
14	C₂×#14556523	C₂²	1	2	yes	—
15	C₂×#14556523	C₂²	1	1	yes	characteristic
16	C₂×#14556523	C₂²	1	1	yes	characteristic
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	4	no	—
46	#14556525	—	4	2	no	—
47	#14556523	—	2	2	no	—
48	#14556523	—	4	1	no	—
49	#14556523	—	4	1	no	—
50	#14556523	—	4	2	no	—
51	#14556523	Q₈	1	4	yes	complemented
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	1	yes	characteristic
61	C₂²×C₈×C₇₃	C₂³	1	2	yes	—
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	2	no	—
91	C₂×Q₈×C₇₃	C₄⋊C₄	1	4	yes	—
92	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
94	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
96	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
97	C₂×C₈×C₇₃	—	2	1	no	—
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	C₂²×C₄	1	2	yes	—
102	C₂×C₈×C₇₃	C₂²×C₄	1	4	yes	—
103	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
104	C₂×C₈×C₇₃	C₂×Q₈	1	2	yes	—
105	C₂×C₈×C₇₃	#41	1	2	yes	—
106	C₄⋊C₄×C₇₃	C₄⋊C₄	1	8	yes	—
107	C₄⋊C₄×C₇₃	C₄⋊C₄	1	4	yes	—
108	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
109	C₄²×C₇₃	C₂²×C₄	1	4	yes	—
110	C₄²×C₇₃	C₂×Q₈	1	2	yes	—
111	C₄²×C₇₃	C₂×Q₈	1	4	yes	—
112	C₂³×C₇₃	C₂²×D₄	1	1	yes	characteristic
113	Q₈×C₇₃	—	4	4	no	—
114	C₂×C₄×C₇₃	—	2	2	no	—
115	C₂×C₄×C₇₃	—	2	2	no	—
116	C₂×C₄×C₇₃	—	2	2	no	—
117	C₂×C₄×C₇₃	—	2	2	no	—
118	C₂×C₄×C₇₃	—	2	2	no	—
119	C₂×C₄×C₇₃	—	2	2	no	—
120	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
121	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	4	yes	—
122	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
123	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	4	yes	—
124	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
125	C₂×C₄×C₇₃	C₂²×Q₈	1	1	yes	characteristic
126	C₂×C₄×C₇₃	C₂²×Q₈	1	1	yes	characteristic
127	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
128	C₂×C₄×C₇₃	H(5,𝔽₂)	1	1	yes	characteristic
129	C₂×C₄×C₇₃	2₋¹⁺⁴	1	1	yes	characteristic
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	—	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	C₈×C₇₃	—	2	2	no	—
143	C₈×C₇₃	—	2	4	no	—
144	C₈×C₇₃	C₄×Q₈	1	4	yes	—
145	C₂²×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
146	C₂²×C₇₃	D₄×Q₈	1	1	yes	characteristic
147	C₂²×C₇₃	#550	1	1	yes	characteristic
148	C₂²×C₇₃	#575	1	2	yes	—
149	C₂²×C₇₃	#577	1	2	yes	—
150	C₄×C₇₃	—	2	2	no	—
151	C₄×C₇₃	—	2	4	no	—
152	C₄×C₇₃	—	2	2	no	—
153	C₄×C₇₃	—	2	4	no	—
154	C₄×C₇₃	C₂×C₄×Q₈	1	1	yes	characteristic
155	C₄×C₇₃	#520	1	1	yes	characteristic
156	C₄×C₇₃	#557	1	1	yes	characteristic
157	C₄×C₇₃	Q₈²	1	1	yes	characteristic
158	Dic₇₃	—	4	4	no	—
159	Dic₇₃	—	8	2	no	—
160	Dic₇₃	—	8	2	no	—
161	Dic₇₃	—	8	2	no	—
162	#22730	—	73	1	no	maximal, Sylow
163	C₂×C₇₃	Q₈×C₄⋊C₄	1	1	yes	characteristic
164	C₂×C₇₃	#2911	1	1	yes	characteristic
165	C₂×C₇₃	#2912	1	1	yes	characteristic
166	C₂×C₇₃	#3388	1	2	yes	—
167	C₂×C₇₃	#3401	1	2	yes	—
168	C₂×#446	D₇₃	1	1	yes	characteristic
169	#2355	—	73	2	no	—
170	#1949	—	73	2	no	—
171	#1949	—	73	2	no	—
172	#1948	—	73	2	no	—
173	#1847	—	73	2	no	—
174	#1846	—	73	1	no	—
175	#1776	—	73	2	no	—
176	C₄×#106	—	73	1	no	—
177	C₇₃	#22730	1	1	yes	characteristic, complemented, Sylow
178	C₂×C₄×Q₈	C₂×D₇₃	1	1	yes	characteristic
179	#446	Dic₇₃	1	8	yes	complemented
180	C₂×#107	—	73	2	no	—
181	C₂×#107	—	73	2	no	—
182	C₂×#106	—	73	1	no	—
183	C₂×#106	—	73	1	no	—
184	C₂×#106	—	73	2	no	—
185	C₂×#106	—	73	1	no	—
186	C₂×#106	—	73	1	no	—
187	C₂×C₄⋊C₈	C₂×D₇₃	1	2	yes	—
188	C₂×C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
189	C₂×#97	C₂×D₇₃	1	2	yes	—
190	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
191	#391	—	146	2	no	—
192	#387	—	146	2	no	—
193	#387	—	146	2	no	—
194	#384	—	73	2	no	—
195	#382	—	73	2	no	—
196	#382	—	73	2	no	—
197	C₄×C₄⋊C₄	—	73	2	no	—
198	C₄×C₄⋊C₄	—	73	2	no	—
199	C₄×C₄⋊C₄	—	73	2	no	—
200	C₄×C₄⋊C₄	—	146	2	no	—
201	#340	—	73	2	no	—
202	#340	—	73	2	no	—
203	#340	—	73	2	no	—
204	C₂²×Q₈	D₂₉₂	1	1	yes	characteristic
205	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
206	C₂²×C₈	C₂²×D₇₃	1	2	yes	—
207	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
208	C₄×Q₈	Dic₁₄₆	1	4	yes	—
209	C₄×Q₈	C₂×Dic₇₃	1	4	yes	—
210	C₂×C₄⋊C₄	—	73	2	no	—
211	C₂×C₄⋊C₄	—	73	2	no	—
212	C₂×C₄⋊C₄	—	73	2	no	—
213	C₂×C₄⋊C₄	—	73	2	no	—
214	C₂×C₄⋊C₄	—	146	2	no	—
215	C₂×C₄⋊C₄	—	146	2	no	—
216	C₂×C₄⋊C₄	D₂₉₂	1	2	yes	—
217	C₂×C₄⋊C₄	D₂₉₂	1	1	yes	characteristic
218	C₂×C₄²	—	146	2	no	—
219	C₂×C₄²	—	146	2	no	—
220	C₂×C₄²	—	146	2	no	—
221	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
222	C₂×C₄²	C₂²×D₇₃	1	2	yes	—
223	#107	—	292	2	no	—
224	#107	—	146	4	no	—
225	#106	—	292	1	no	—
226	#106	—	292	2	no	—
227	#106	—	292	1	no	—
228	#106	—	73	4	no	—
229	#106	—	146	2	no	—
230	C₄⋊C₈	C₂×Dic₇₃	1	4	yes	—
231	C₄⋊C₈	C₂×Dic₇₃	1	8	yes	—
232	#97	C₂×Dic₇₃	1	8	yes	—
233	C₄×C₈	C₂×Dic₇₃	1	4	yes	—
234	#95	—	146	2	no	—
235	#95	—	146	2	no	—
236	#95	—	146	2	no	—
237	#95	—	146	2	no	—
238	#95	—	146	2	no	—
239	#95	—	146	2	no	—
240	#95	—	146	2	no	—
241	C₂×Q₈	—	4	2	no	—
242	C₂×Q₈	C₄⋊Dic₇₃	1	4	yes	—
243	C₂²×C₄	—	146	2	no	—
244	C₂²×C₄	—	146	2	no	—
245	C₂²×C₄	—	146	2	no	—
246	C₂²×C₄	—	146	2	no	—
247	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
248	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
249	C₂²×C₄	C₂×D₂₉₂	1	2	yes	—
250	C₂²×C₄	C₂×D₂₉₂	1	2	yes	—
251	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
252	C₂×C₈	—	2	1	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	2	1	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	C₂²×Dic₇₃	1	2	yes	—
257	C₂×C₈	C₂²×Dic₇₃	1	4	yes	—
258	C₂×C₈	C₂²×Dic₇₃	1	2	yes	—
259	C₂×C₈	Q₈×D₇₃	1	2	yes	—
260	C₂×C₈	#11924029	1	2	yes	—
261	C₄⋊C₄	—	292	4	no	—
262	C₄⋊C₄	—	584	2	no	—
263	C₄⋊C₄	—	73	4	no	—
264	C₄⋊C₄	—	146	2	no	—
265	C₄⋊C₄	—	146	2	no	—
266	C₄⋊C₄	—	146	2	no	—
267	C₄⋊C₄	—	146	2	no	—
268	C₄⋊C₄	—	146	2	no	—
269	C₄⋊C₄	—	146	2	no	—
270	C₄⋊C₄	—	146	2	no	—
271	C₄⋊C₄	C₄⋊Dic₇₃	1	8	yes	—
272	C₄⋊C₄	C₄⋊Dic₇₃	1	4	yes	—
273	C₄²	—	292	4	no	—
274	C₄²	—	584	2	no	—
275	C₄²	—	584	2	no	—
276	C₄²	C₂²×Dic₇₃	1	2	yes	—
277	C₄²	C₂²×Dic₇₃	1	4	yes	—
278	C₄²	C₂×Dic₁₄₆	1	2	yes	—
279	C₄²	C₂×Dic₁₄₆	1	4	yes	—
280	C₂³	C₂²×D₂₉₂	1	1	yes	characteristic
281	Q₈	—	4	4	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	292	2	no	—
289	C₂×C₄	—	292	2	no	—
290	C₂×C₄	—	292	2	no	—
291	C₂×C₄	—	292	2	no	—
292	C₂×C₄	—	292	2	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	—	292	2	no	—
299	C₂×C₄	—	146	4	no	—
300	C₂×C₄	C₂×C₄⋊Dic₇₃	1	2	yes	—
301	C₂×C₄	C₂×C₄⋊Dic₇₃	1	4	yes	—
302	C₂×C₄	C₂×C₄⋊Dic₇₃	1	2	yes	—
303	C₂×C₄	C₂×C₄⋊Dic₇₃	1	4	yes	—
304	C₂×C₄	C₂³×Dic₇₃	1	1	yes	characteristic
305	C₂×C₄	C₂×Q₈×D₇₃	1	1	yes	characteristic
306	C₂×C₄	C₂²×Dic₁₄₆	1	1	yes	characteristic
307	C₂×C₄	C₂×#11924029	1	1	yes	characteristic
308	C₂×C₄	#14556637	1	1	yes	characteristic
309	C₂×C₄	#14556638	1	1	yes	characteristic
310	C₈	—	2	2	no	—
311	C₈	—	2	4	no	—
312	C₈	Q₈×Dic₇₃	1	4	yes	—
313	C₂²	C₂²×C₄⋊Dic₇₃	1	1	yes	characteristic
314	C₂²	Q₈×D₂₉₂	1	1	yes	characteristic
315	C₂²	#17989259	1	1	yes	characteristic
316	C₂²	#17989387	1	2	yes	—
317	C₂²	#17989399	1	2	yes	—
318	C₄	—	2	4	no	—
319	C₄	—	2	2	no	—
320	C₄	—	2	2	no	—
321	C₄	—	2	4	no	—
322	C₄	—	292	4	no	—
323	C₄	—	584	2	no	—
324	C₄	—	584	2	no	—
325	C₄	—	584	2	no	—
326	C₄	C₂×Q₈×Dic₇₃	1	1	yes	characteristic
327	C₄	#17989070	1	1	yes	characteristic
328	C₄	#17989316	1	1	yes	characteristic
329	C₄	Q₈×Dic₁₄₆	1	1	yes	characteristic
330	C₂	Q₈×C₄⋊Dic₇₃	1	1	yes	characteristic
331	C₂	#22618521	1	1	yes	characteristic
332	C₂	#22618526	1	1	yes	characteristic
333	C₂	#22622878	1	2	yes	—
334	C₂	#22622999	1	2	yes	—

	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	#22730
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
51. #14556523	Q₈	semidirect
177. C₇₃	#22730	semidirect
179. #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
#22610695	2
#22610099	2
#22610095	2
#22609828	2
C₄×#14556523	2
C₂×#446×C₇₃	2
#22730	73