Subgroups — #34030355

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	#22626346	C₂	1	2	yes	maximal, complemented
2	#22626317	C₂	1	2	yes	maximal, complemented
3	#22626314	C₂	1	2	yes	maximal, complemented
4	#22626314	C₂	1	2	yes	maximal, complemented
5	#22626299	C₂	1	2	yes	maximal, complemented
6	#22626295	C₂	1	1	yes	maximal, characteristic, complemented
7	#22626141	C₂	1	2	yes	maximal, complemented
8	#22626138	C₂	1	1	yes	maximal, characteristic, complemented
9	#22618905	C₂	1	1	yes	maximal, characteristic
10	C₂×#14556666	C₂²	1	2	yes	—
11	C₂×#14556666	C₂²	1	2	yes	—
12	C₂×#14556664	C₂²	1	2	yes	—
13	C₄⋊#11924031	C₂²	1	2	yes	—
14	C₄⋊#11924031	C₂²	1	2	yes	—
15	C₄⋊#11924031	C₂²	1	2	yes	—
16	#17989482	C₂²	1	1	yes	characteristic
17	#17989482	C₂²	1	2	yes	—
18	#17989482	C₂²	1	2	yes	—
19	#17989482	C₂²	1	1	yes	characteristic
20	#17989476	C₂²	1	2	yes	—
21	#17989475	C₂²	1	2	yes	—
22	#17989475	C₂²	1	2	yes	—
23	#17989473	C₂²	1	1	yes	characteristic
24	#17989473	C₂²	1	1	yes	characteristic
25	C₄×#11924031	C₂²	1	2	yes	—
26	#17989124	C₄	1	1	yes	characteristic
27	Q₈×#10596831	C₂²	1	1	yes	characteristic
28	#17988666	C₄	1	1	yes	characteristic
29	#17988660	C₄	1	2	yes	—
30	#17988652	C₂²	1	1	yes	characteristic
31	#17988652	C₂²	1	2	yes	—
32	#17988647	C₄	1	2	yes	—
33	#17988643	C₄	1	1	yes	characteristic
34	#17988544	C₂²	1	2	yes	—
35	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
36	#446×C₇₃	C₄	1	1	yes	characteristic
37	#14556666	—	2	4	no	—
38	#14556666	—	2	4	no	—
39	#14556664	—	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	C₂×#11924011	C₂×C₄	1	2	yes	—
45	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₈×D₇₃	C₂³	1	2	yes	—
47	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
48	#14556586	C₈	1	2	yes	—
49	#14556582	C₈	1	1	yes	characteristic
50	#14556580	C₈	1	1	yes	characteristic
51	#14556576	C₈	1	2	yes	—
52	#14556567	C₈	1	1	yes	characteristic
53	#14556566	C₈	1	1	yes	characteristic
54	C₄×D₂₉₂	C₂×C₄	1	2	yes	—
55	C₄×D₂₉₂	C₂×C₄	1	1	yes	characteristic
56	#14556548	C₂×C₄	1	2	yes	—
57	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
58	C₄⋊#10596831	C₂×C₄	1	1	yes	characteristic
59	C₄⋊#10596831	C₂×C₄	1	2	yes	—
60	#14556503	C₂×C₄	1	2	yes	—
61	#14556503	C₂×C₄	1	2	yes	—
62	#14556503	C₂×C₄	1	2	yes	—
63	#14556501	C₂×C₄	1	2	yes	—
64	C₄×#10596831	C₂³	1	2	yes	—
65	C₄×#10596831	C₂³	1	1	yes	characteristic
66	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
67	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
68	C₄×Q₈×C₇₃	C₂×C₄	1	1	yes	characteristic
69	C₄⋊C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
70	C₄⋊C₈×C₇₃	C₂×C₄	1	2	yes	—
71	#97×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×D₂₉₂	C₂×C₈	1	2	yes	—
78	C₂×D₂₉₂	C₂×C₈	1	1	yes	characteristic
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×C₄×D₇₃	C₂²×C₄	1	2	yes	—
83	#11924011	—	2	4	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	C₈×D₇₃	—	2	2	no	—
90	C₈×D₇₃	—	2	2	no	—
91	C₄⋊Dic₇₃	C₂×C₈	1	1	yes	characteristic
92	C₄⋊Dic₇₃	C₂×C₈	1	2	yes	—
93	#11924007	C₂×C₈	1	2	yes	—
94	#11924007	C₂×C₈	1	2	yes	—
95	#11924005	C₂×C₈	1	2	yes	—
96	#11924005	C₂×C₈	1	2	yes	—
97	#11924005	C₂×C₈	1	2	yes	—
98	#11924005	C₂×C₈	1	2	yes	—
99	#11924005	C₂×C₈	1	2	yes	—
100	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
101	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
102	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
103	C₂×Q₈×C₇₃	C₂×C₈	1	1	yes	characteristic
104	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
106	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
107	C₄⋊C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
108	C₄⋊C₄×C₇₃	C₂×C₈	1	2	yes	—
109	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
110	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
111	C₂²×D₇₃	C₂²×C₈	1	2	yes	—
112	C₂²×D₇₃	C₂²×C₈	1	1	yes	characteristic
113	Q₈×C₇₃	—	4	1	no	—
114	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
115	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
116	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
117	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
118	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
119	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
120	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
121	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
122	C₂×Dic₇₃	C₂²×C₈	1	2	yes	—
123	D₂₉₂	—	4	1	no	—
124	D₂₉₂	—	4	2	no	—
125	C₄×D₇₃	—	2	2	no	—
126	C₄×D₇₃	—	2	1	no	—
127	C₄×D₇₃	—	2	1	no	—
128	C₄×D₇₃	—	2	2	no	—
129	C₄×D₇₃	—	4	1	no	—
130	C₄×D₇₃	—	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	1	no	—
135	#10596831	—	2	2	no	—
136	#10596831	C₂×#41	1	2	yes	—
137	C₂²×C₇₃	C₂³×C₈	1	1	yes	characteristic
138	C₂×D₇₃	—	2	1	no	—
139	C₂×D₇₃	—	2	1	no	—
140	C₂×D₇₃	—	2	2	no	—
141	C₂×D₇₃	—	2	2	no	—
142	C₂×D₇₃	—	2	1	no	—
143	C₂×D₇₃	—	2	2	no	—
144	C₄×C₇₃	—	2	1	no	—
145	C₄×C₇₃	—	2	1	no	—
146	C₄×C₇₃	—	2	2	no	—
147	C₄×C₇₃	—	2	2	no	—
148	C₄×C₇₃	C₄×#41	1	1	yes	characteristic
149	C₄×C₇₃	C₂×#131	1	1	yes	characteristic
150	Dic₇₃	—	2	1	no	—
151	Dic₇₃	—	2	1	no	—
152	Dic₇₃	—	2	1	no	—
153	Dic₇₃	—	2	2	no	—
154	#33481	—	73	1	no	maximal, Sylow
155	C₂×C₇₃	C₈×#41	1	1	yes	characteristic
156	C₂×C₇₃	#3407	1	1	yes	characteristic
157	C₂×C₇₃	#3407	1	1	yes	characteristic
158	D₇₃	—	4	1	no	—
159	D₇₃	—	4	2	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	#64408	1	1	yes	characteristic
175	C₈×Q₈	—	73	1	no	—
176	#436	—	73	1	no	—
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	1	no	—
188	C₂²⋊C₁₆	—	73	1	no	—
189	C₂²⋊C₁₆	—	73	2	no	—
190	C₂²⋊C₁₆	—	73	2	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	—	73	2	no	—
198	C₂²×C₈	—	73	1	no	—
199	C₂²×C₈	—	73	2	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	C₂×#64408	1	1	yes	characteristic
208	C₄×D₄	—	73	2	no	—
209	C₄×D₄	—	73	1	no	—
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	1	no	—
220	C₄⋊C₈	—	73	2	no	—
221	C₄⋊C₈	C₂×#64408	1	1	yes	characteristic
222	C₄⋊C₈	C₂×#64408	1	2	yes	—
223	C₂²⋊C₈	—	73	2	no	—
224	C₂²⋊C₈	—	73	2	no	—
225	C₂²⋊C₈	—	73	2	no	—
226	#97	—	73	2	no	—
227	#97	C₂×#64408	1	2	yes	—
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	#11924036	1	1	yes	characteristic
234	C₂×D₄	—	73	2	no	—
235	C₂×D₄	—	73	1	no	—
236	C₂²×C₄	—	73	1	no	—
237	C₂²×C₄	—	73	1	no	—
238	C₂²×C₄	—	73	1	no	—
239	C₂²×C₄	—	73	2	no	—
240	M₁₆	—	146	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	1	no	—
254	C₄⋊C₄	—	73	2	no	—
255	C₄⋊C₄	#11924036	1	1	yes	characteristic
256	C₄⋊C₄	#11924036	1	2	yes	—
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	—	73	2	no	—
262	C₄²	—	73	1	no	—
263	C₄²	—	73	1	no	—
264	C₄²	—	73	1	no	—
265	C₄²	C₂²×#64408	1	1	yes	characteristic
266	C₄²	C₂²×#64408	1	2	yes	—
267	C₁₆	—	73	4	no	—
268	C₁₆	—	146	2	no	—
269	C₁₆	—	146	2	no	—
270	C₁₆	—	146	2	no	—
271	C₂³	—	73	2	no	—
272	C₂³	—	73	1	no	—
273	Q₈	—	4	1	no	—
274	D₄	—	292	2	no	—
275	D₄	—	292	1	no	—
276	C₂×C₄	—	292	1	no	—
277	C₂×C₄	—	292	1	no	—
278	C₂×C₄	—	73	1	no	—
279	C₂×C₄	—	73	1	no	—
280	C₂×C₄	—	73	1	no	—
281	C₂×C₄	—	73	2	no	—
282	C₂×C₄	—	146	2	no	—
283	C₂×C₄	—	146	1	no	—
284	C₂×C₄	—	146	1	no	—
285	C₂×C₄	—	146	2	no	—
286	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
287	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
288	C₂×C₄	C₂×#11924036	1	2	yes	—
289	C₂×C₄	C₂×#11924036	1	2	yes	—
290	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
291	C₈	—	2	2	no	—
292	C₈	—	2	1	no	—
293	C₈	—	73	2	no	—
294	C₈	—	146	1	no	—
295	C₈	—	146	2	no	—
296	C₈	#14556679	1	2	yes	—
297	C₂²	—	146	1	no	—
298	C₂²	—	146	2	no	—
299	C₂²	—	146	1	no	—
300	C₂²	—	146	2	no	—
301	C₂²	—	146	1	no	—
302	C₂²	—	146	2	no	—
303	C₂²	C₂²×#11924036	1	1	yes	characteristic
304	C₄	—	2	1	no	—
305	C₄	—	2	2	no	—
306	C₄	—	2	1	no	—
307	C₄	—	2	2	no	—
308	C₄	—	146	1	no	—
309	C₄	—	146	1	no	—
310	C₄	—	146	1	no	—
311	C₄	—	146	2	no	—
312	C₄	#41×#64408	1	1	yes	characteristic
313	C₄	C₂×#14556679	1	1	yes	characteristic
314	C₂	—	292	1	no	—
315	C₂	—	292	2	no	—
316	C₂	#22626737	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	#446×C₇₃	C₄
prime radical	#14556566	C₈
cosocle	C₂×#10596831	C₂⁴
hypercenter	#446	#64408
hypocenter	C₇₃	#33481
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ #446

Complements

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

N	H ≅ G/N	type
1. #22626346	C₂	semidirect
2. #22626317	C₂	semidirect
3. #22626314	C₂	semidirect
4. #22626314	C₂	semidirect
5. #22626299	C₂	semidirect
6. #22626295	C₂	semidirect
7. #22626141	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
#22626346	2
#22626317	2
#22626314	2
#22626299	2
#22626295	2
#22626141	2
#22626138	2
#22618905	2
#33481	73