Subgroups — #34030351

Order 18688.

508 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22626348	C₂	1	2	yes	maximal, complemented
2	#22626316	C₂	1	2	yes	maximal, complemented
3	#22626313	C₂	1	2	yes	maximal, complemented
4	#22626311	C₂	1	2	yes	maximal, complemented
5	#22626297	C₂	1	2	yes	maximal, complemented
6	#22626295	C₂	1	1	yes	maximal, characteristic, complemented
7	#22626142	C₂	1	2	yes	maximal, complemented
8	#22626138	C₂	1	1	yes	maximal, characteristic, complemented
9	#22618909	C₂	1	1	yes	maximal, characteristic
10	C₂×#14556666	C₂²	1	2	yes	—
11	C₂×#14556665	C₂²	1	2	yes	—
12	C₂²×#11924031	C₂²	1	2	yes	—
13	#17989508	C₂²	1	2	yes	—
14	#17989508	C₂²	1	2	yes	—
15	C₄⋊#11924031	C₂²	1	2	yes	—
16	#17989482	C₂²	1	1	yes	characteristic
17	#17989482	C₂²	1	1	yes	characteristic
18	C₂²⋊#11924031	C₂²	1	2	yes	—
19	C₂²⋊#11924031	C₂²	1	2	yes	—
20	#17989476	C₂²	1	2	yes	—
21	#17989476	C₂²	1	2	yes	—
22	#17989475	C₂²	1	2	yes	—
23	#17989473	C₂²	1	2	yes	—
24	#17989473	C₂²	1	1	yes	characteristic
25	#17989473	C₂²	1	1	yes	characteristic
26	#17989119	C₄	1	1	yes	characteristic
27	#17988717	C₄	1	1	yes	characteristic
28	#17988711	C₂²	1	1	yes	characteristic
29	#17988656	C₄	1	2	yes	—
30	D₄×#10596831	C₂²	1	1	yes	characteristic
31	#17988650	C₂²	1	2	yes	—
32	#17988645	C₄	1	2	yes	—
33	#17988643	C₄	1	1	yes	characteristic
34	#17988545	C₂²	1	2	yes	—
35	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
36	#436×C₇₃	C₄	1	1	yes	characteristic
37	#14556666	—	2	4	no	—
38	#14556665	—	2	4	no	—
39	C₂×#11924031	—	2	4	no	—
40	C₂×#11924031	C₂³	1	2	yes	—
41	C₂×#11924031	C₂³	1	2	yes	—
42	C₂×#11924031	C₂³	1	2	yes	—
43	C₂×#11924031	C₂³	1	2	yes	—
44	C₂²×#10596831	C₂³	1	2	yes	—
45	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
47	#14556584	C₈	1	2	yes	—
48	#14556582	C₈	1	1	yes	characteristic
49	#14556578	C₈	1	1	yes	characteristic
50	#14556573	C₈	1	2	yes	—
51	#14556571	C₈	1	1	yes	characteristic
52	#14556562	C₈	1	1	yes	characteristic
53	C₄×Dic₁₄₆	C₂×C₄	1	1	yes	characteristic
54	C₄×C₂²⋊D₇₃	C₂×C₄	1	2	yes	—
55	#14556550	C₂×C₄	1	2	yes	—
56	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
57	C₄⋊#10596831	C₂×C₄	1	1	yes	characteristic
58	#14556520	C₂×C₄	1	2	yes	—
59	#14556520	C₂×C₄	1	2	yes	—
60	C₂²⋊#10596831	C₂×C₄	1	2	yes	—
61	#14556503	C₂×C₄	1	2	yes	—
62	#14556501	C₂×C₄	1	2	yes	—
63	#14556501	C₂×C₄	1	2	yes	—
64	C₄×#10596831	C₂³	1	1	yes	characteristic
65	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
66	C₈×Dic₇₃	C₂³	1	2	yes	—
67	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
68	C₂×M₁₆×C₇₃	C₂×C₄	1	2	yes	—
69	C₄×D₄×C₇₃	C₂×C₄	1	1	yes	characteristic
70	C₄⋊C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
71	C₂²⋊C₈×C₇₃	C₂×C₄	1	2	yes	—
72	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
73	#11924031	—	2	2	no	—
74	#11924031	—	2	2	no	—
75	#11924031	—	2	2	no	—
76	#11924031	#41	1	4	yes	—
77	C₂×Dic₁₄₆	C₂×C₈	1	1	yes	characteristic
78	C₂×C₂²⋊D₇₃	C₂×C₈	1	2	yes	—
79	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
80	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
81	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
82	C₂×#10596831	—	2	2	no	—
83	C₂×#10596831	—	2	2	no	—
84	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
85	C₂×#10596831	C₂²×C₄	1	2	yes	—
86	C₂×#10596831	C₂⁴	1	1	yes	characteristic
87	C₈×D₇₃	—	2	2	no	—
88	C₈×D₇₃	—	2	2	no	—
89	C₄⋊Dic₇₃	C₂×C₈	1	2	yes	—
90	C₄⋊Dic₇₃	C₂×C₈	1	2	yes	—
91	C₄⋊Dic₇₃	C₂×C₈	1	1	yes	characteristic
92	#11924007	C₂×C₈	1	2	yes	—
93	#11924007	C₂×C₈	1	2	yes	—
94	C₂²⋊Dic₇₃	C₂×C₈	1	2	yes	—
95	C₂²⋊Dic₇₃	C₂×C₈	1	2	yes	—
96	C₂²⋊Dic₇₃	C₂×C₈	1	2	yes	—
97	#11924005	C₂×C₈	1	2	yes	—
98	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
99	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
100	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
101	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
102	C₂×D₄×C₇₃	C₂×C₈	1	1	yes	characteristic
103	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
104	M₁₆×C₇₃	—	2	4	no	—
105	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
107	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
108	C₄⋊C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
109	C₂²⋊C₄×C₇₃	C₂×C₈	1	2	yes	—
110	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
111	C₂³×C₇₃	C₂²×C₈	1	2	yes	—
112	C₂²×D₇₃	C₂²×C₈	1	1	yes	characteristic
113	D₄×C₇₃	—	4	1	no	—
114	C₂×C₄×C₇₃	—	2	2	no	—
115	C₂×C₄×C₇₃	—	2	2	no	—
116	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
117	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
118	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
119	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
120	C₂²⋊D₇₃	—	4	2	no	—
121	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
122	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
123	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
124	C₂×Dic₇₃	C₂²×C₈	1	2	yes	—
125	C₂×Dic₇₃	C₂²×C₈	1	2	yes	—
126	C₄×D₇₃	—	2	1	no	—
127	C₄×D₇₃	—	2	1	no	—
128	C₄×D₇₃	—	4	1	no	—
129	C₄×D₇₃	—	4	1	no	—
130	Dic₁₄₆	—	4	1	no	—
131	C₈×C₇₃	—	2	1	no	—
132	C₈×C₇₃	—	2	2	no	—
133	C₈×C₇₃	#131	1	2	yes	—
134	#10596831	—	2	2	no	—
135	#10596831	—	2	1	no	—
136	#10596831	C₂×#41	1	2	yes	—
137	C₂²×C₇₃	—	2	2	no	—
138	C₂²×C₇₃	—	2	2	no	—
139	C₂²×C₇₃	—	2	2	no	—
140	C₂²×C₇₃	C₂³×C₈	1	1	yes	characteristic
141	C₂×D₇₃	—	2	1	no	—
142	C₂×D₇₃	—	2	1	no	—
143	C₂×D₇₃	—	2	1	no	—
144	C₄×C₇₃	—	2	1	no	—
145	C₄×C₇₃	—	2	1	no	—
146	C₄×C₇₃	—	2	2	no	—
147	C₄×C₇₃	C₄×#41	1	1	yes	characteristic
148	C₄×C₇₃	C₂×#131	1	1	yes	characteristic
149	Dic₇₃	—	2	2	no	—
150	Dic₇₃	—	2	1	no	—
151	Dic₇₃	—	2	1	no	—
152	Dic₇₃	—	2	2	no	—
153	Dic₇₃	—	2	1	no	—
154	#33481	—	73	1	no	maximal, Sylow
155	C₂×C₇₃	—	4	2	no	—
156	C₂×C₇₃	C₈×#41	1	1	yes	characteristic
157	C₂×C₇₃	#3407	1	1	yes	characteristic
158	C₂×C₇₃	#3407	1	1	yes	characteristic
159	D₇₃	—	4	1	no	—
160	#2966	—	73	1	no	—
161	#2183	—	73	2	no	—
162	#2169	—	73	2	no	—
163	#2168	—	73	2	no	—
164	#2168	—	73	2	no	—
165	#2164	—	73	2	no	—
166	#2163	—	73	1	no	—
167	#2108	—	73	2	no	—
168	C₄×M₃₂	—	73	1	no	—
169	C₇₃	#33481	1	1	yes	characteristic, complemented, Sylow
170	#529	—	73	1	no	—
171	C₂×M₃₂	—	73	2	no	—
172	C₂×M₃₂	—	73	2	no	—
173	C₂²×C₁₆	—	73	2	no	—
174	#446	—	73	1	no	—
175	C₈×Q₈	—	73	1	no	—
176	#436	#64408	1	1	yes	characteristic
177	#435	—	73	2	no	—
178	C₈×D₄	—	73	1	no	—
179	C₈×D₄	—	73	2	no	—
180	#433	—	73	2	no	—
181	#432	—	73	1	no	—
182	#405	—	73	2	no	—
183	C₂×C₄×C₈	—	73	1	no	—
184	C₄⋊C₁₆	—	73	2	no	—
185	C₄⋊C₁₆	—	73	2	no	—
186	C₄⋊C₁₆	—	73	2	no	—
187	C₂²⋊C₁₆	—	73	2	no	—
188	C₂²⋊C₁₆	—	73	2	no	—
189	C₂²⋊C₁₆	—	73	1	no	—
190	C₂²⋊C₁₆	—	73	1	no	—
191	#346	—	73	2	no	—
192	#346	—	73	2	no	—
193	#346	—	73	2	no	—
194	C₄×C₁₆	—	73	1	no	—
195	C₄×C₁₆	—	73	1	no	—
196	C₄×C₁₆	—	73	2	no	—
197	C₂×M₁₆	C₂×#64408	1	2	yes	—
198	C₂²×C₈	—	73	2	no	—
199	C₂²×C₈	—	73	1	no	—
200	C₂²×C₈	—	73	1	no	—
201	#126	—	73	2	no	—
202	#125	—	73	1	no	—
203	#124	—	73	1	no	—
204	#123	—	73	2	no	—
205	C₂²⋊Q₈	—	73	1	no	—
206	#121	—	73	1	no	—
207	C₄×Q₈	—	73	1	no	—
208	C₄×D₄	—	73	2	no	—
209	C₄×D₄	C₂×#64408	1	1	yes	characteristic
210	#117	—	73	2	no	—
211	C₂×C₄²	—	73	1	no	—
212	M₃₂	—	146	4	no	—
213	M₃₂	—	146	4	no	—
214	C₂×C₁₆	—	73	2	no	—
215	C₂×C₁₆	—	73	2	no	—
216	C₂×C₁₆	—	73	2	no	—
217	C₂×C₁₆	—	73	2	no	—
218	C₂×C₁₆	—	146	4	no	—
219	C₄⋊C₈	—	73	2	no	—
220	C₄⋊C₈	—	73	2	no	—
221	C₄⋊C₈	—	73	1	no	—
222	C₄⋊C₈	C₂×#64408	1	1	yes	characteristic
223	C₂²⋊C₈	—	73	2	no	—
224	C₂²⋊C₈	—	73	2	no	—
225	C₂²⋊C₈	C₂×#64408	1	2	yes	—
226	#97	—	73	2	no	—
227	#97	—	73	2	no	—
228	C₄×C₈	—	73	1	no	—
229	C₄×C₈	—	73	1	no	—
230	C₄×C₈	—	73	2	no	—
231	C₄×C₈	—	73	1	no	—
232	C₄×C₈	C₂×#64408	1	1	yes	characteristic
233	C₂×Q₈	—	73	1	no	—
234	C₂×D₄	—	73	2	no	—
235	C₂×D₄	#11924036	1	1	yes	characteristic
236	C₂²×C₄	—	73	1	no	—
237	C₂²×C₄	—	73	1	no	—
238	C₂²×C₄	—	73	1	no	—
239	C₂²×C₄	C₂²×#64408	1	2	yes	—
240	M₁₆	—	2	4	no	—
241	C₂×C₈	—	73	1	no	—
242	C₂×C₈	—	73	1	no	—
243	C₂×C₈	—	73	2	no	—
244	C₂×C₈	—	146	2	no	—
245	C₂×C₈	—	146	2	no	—
246	C₂×C₈	—	146	2	no	—
247	C₂×C₈	—	146	2	no	—
248	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
249	C₂×C₈	C₂²×#64408	1	2	yes	—
250	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
251	C₄⋊C₄	—	73	2	no	—
252	C₄⋊C₄	—	73	2	no	—
253	C₄⋊C₄	—	73	2	no	—
254	C₄⋊C₄	—	73	2	no	—
255	C₄⋊C₄	—	73	1	no	—
256	C₄⋊C₄	#11924036	1	1	yes	characteristic
257	C₂²⋊C₄	—	73	2	no	—
258	C₂²⋊C₄	—	73	2	no	—
259	C₂²⋊C₄	—	73	2	no	—
260	C₂²⋊C₄	—	73	2	no	—
261	C₂²⋊C₄	#11924036	1	2	yes	—
262	C₄²	—	73	1	no	—
263	C₄²	—	73	1	no	—
264	C₄²	—	73	1	no	—
265	C₄²	—	73	2	no	—
266	C₄²	C₂²×#64408	1	1	yes	characteristic
267	C₁₆	—	73	4	no	—
268	C₁₆	—	146	2	no	—
269	C₁₆	—	146	2	no	—
270	C₁₆	—	146	2	no	—
271	C₂³	—	73	1	no	—
272	C₂³	C₂×#11924036	1	2	yes	—
273	Q₈	—	292	1	no	—
274	D₄	—	4	1	no	—
275	D₄	—	292	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	292	1	no	—
279	C₂×C₄	—	292	1	no	—
280	C₂×C₄	—	73	1	no	—
281	C₂×C₄	—	73	1	no	—
282	C₂×C₄	—	73	1	no	—
283	C₂×C₄	—	73	2	no	—
284	C₂×C₄	—	73	2	no	—
285	C₂×C₄	—	146	1	no	—
286	C₂×C₄	—	146	1	no	—
287	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
288	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
289	C₂×C₄	C₂×#11924036	1	2	yes	—
290	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
291	C₈	—	2	1	no	—
292	C₈	—	2	2	no	—
293	C₈	—	73	2	no	—
294	C₈	—	146	2	no	—
295	C₈	—	146	1	no	—
296	C₈	#14556678	1	2	yes	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	146	1	no	—
301	C₂²	—	146	1	no	—
302	C₂²	—	146	1	no	—
303	C₂²	C₂²×#11924036	1	1	yes	characteristic
304	C₄	—	2	1	no	—
305	C₄	—	2	1	no	—
306	C₄	—	2	2	no	—
307	C₄	—	146	1	no	—
308	C₄	—	146	2	no	—
309	C₄	—	146	1	no	—
310	C₄	—	146	2	no	—
311	C₄	—	146	1	no	—
312	C₄	#41×#64408	1	1	yes	characteristic
313	C₄	C₂×#14556678	1	1	yes	characteristic
314	C₂	—	4	2	no	—
315	C₂	—	292	1	no	—
316	C₂	#22626736	1	1	yes	characteristic
317	C₂	#22626899	1	1	yes	characteristic
318	C₂	#22626900	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#64408
commutator, socle	C₂²×C₇₃	C₂³×C₈
Fitting	#436×C₇₃	C₄
prime radical	#14556562	C₈
cosocle	C₂×#10596831	C₂⁴
hypercenter	#436	#64408
hypocenter	C₇₃	#33481
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ #436

Complements

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

N	H ≅ G/N	type
1. #22626348	C₂	semidirect
2. #22626316	C₂	semidirect
3. #22626313	C₂	semidirect
4. #22626311	C₂	semidirect
5. #22626297	C₂	semidirect
6. #22626295	C₂	semidirect
7. #22626142	C₂	semidirect
8. #22626138	C₂	semidirect
169. C₇₃	#33481	semidirect

Maximal subgroups

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

subgroup	index
#22626348	2
#22626316	2
#22626313	2
#22626311	2
#22626297	2
#22626295	2
#22626142	2
#22626138	2
#22618909	2
#33481	73