Subgroups — #33999902

Order 18688.

430 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22625785	C₂	1	2	yes	maximal
2	C₂×#17988341	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊#10596833	—	2	2	no	—
4	C₂×C₂²⋊#10596833	—	2	2	no	—
5	#17989453	C₄	1	4	yes	—
6	C₂³×#10596831	C₄	1	1	yes	characteristic
7	C₂²×C₄×Dic₇₃	C₂²	1	1	yes	characteristic
8	#17988341	—	2	2	no	—
9	C₂³×C₈×C₇₃	C₄	1	1	yes	characteristic
10	C₂²×#10596833	—	2	4	no	—
11	C₄⋊#10596833	—	4	2	no	—
12	C₄⋊#10596833	—	4	2	no	—
13	C₂²⋊#10596833	—	4	2	no	—
14	C₂²⋊#10596833	—	4	2	no	—
15	C₂³×Dic₇₃	D₄	1	1	yes	characteristic
16	C₂³×Dic₇₃	Q₈	1	1	yes	characteristic
17	C₂²×#10596831	—	2	1	no	—
18	C₂²×#10596831	—	2	1	no	—
19	C₂²×#10596831	—	4	1	no	—
20	C₂²×#10596831	—	4	1	no	—
21	C₂²×#10596831	C₈	1	2	yes	complemented
22	C₂×C₄×Dic₇₃	—	4	1	no	—
23	C₂×C₄×Dic₇₃	—	4	1	no	—
24	C₂×C₄×Dic₇₃	C₂×C₄	1	2	yes	—
25	C₂×C₄×Dic₇₃	D₄	1	2	yes	—
26	C₂³×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₇₃	—	2	1	no	—
28	C₂²×C₈×C₇₃	—	2	1	no	—
29	C₂²×C₈×C₇₃	—	4	1	no	—
30	C₂²×C₈×C₇₃	—	4	1	no	—
31	C₂²×C₈×C₇₃	C₈	1	2	yes	complemented
32	C₂×#10596833	—	4	4	no	—
33	C₂×#10596833	—	4	4	no	—
34	C₂×#10596833	—	4	4	no	—
35	C₂²×Dic₇₃	—	2	1	no	—
36	C₂²×Dic₇₃	—	2	1	no	—
37	C₂²×Dic₇₃	—	4	1	no	—
38	C₂²×Dic₇₃	—	4	1	no	—
39	C₂²×Dic₇₃	—	4	1	no	—
40	C₂²×Dic₇₃	—	4	1	no	—
41	C₂²×Dic₇₃	C₂²⋊C₄	1	2	yes	—
42	C₂²×Dic₇₃	C₄⋊C₄	1	2	yes	—
43	C₂²×Dic₇₃	D₈	1	1	yes	characteristic
44	C₂²×Dic₇₃	SD₁₆	1	2	yes	—
45	C₂²×Dic₇₃	Q₁₆	1	1	yes	characteristic
46	C₂×#10596831	—	2	2	no	—
47	C₂×#10596831	—	4	2	no	—
48	C₂×#10596831	—	4	1	no	—
49	C₂×#10596831	—	4	1	no	—
50	C₂×#10596831	—	4	2	no	—
51	C₄×Dic₇₃	—	2	2	no	—
52	C₄×Dic₇₃	—	2	2	no	—
53	C₄×Dic₇₃	—	4	2	no	—
54	C₂⁴×C₇₃	C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₇₃	—	2	1	no	—
56	C₂²×C₄×C₇₃	—	2	1	no	—
57	C₂²×C₄×C₇₃	—	2	1	no	—
58	C₂²×C₄×C₇₃	—	2	1	no	—
59	C₂²×C₄×C₇₃	C₄²	1	1	yes	characteristic
60	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
62	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
63	C₂²×C₄×C₇₃	M₁₆	1	1	yes	characteristic
64	C₂²×C₄×C₇₃	M₁₆	1	1	yes	characteristic
65	C₂×C₈×C₇₃	—	2	2	no	—
66	C₂×C₈×C₇₃	—	4	2	no	—
67	C₂×C₈×C₇₃	—	4	1	no	—
68	C₂×C₈×C₇₃	—	4	1	no	—
69	C₂×C₈×C₇₃	—	4	2	no	—
70	C₂³×C₇₃	—	2	1	no	—
71	C₂³×C₇₃	—	2	1	no	—
72	C₂³×C₇₃	—	2	1	no	—
73	C₂³×C₇₃	—	2	1	no	—
74	C₂³×C₇₃	#95	1	1	yes	characteristic
75	C₂³×C₇₃	C₄⋊C₈	1	1	yes	characteristic
76	C₂³×C₇₃	C₄⋊C₈	1	1	yes	characteristic
77	C₂³×C₇₃	#106	1	1	yes	characteristic
78	C₂³×C₇₃	#107	1	1	yes	characteristic
79	C₂³×C₇₃	#108	1	1	yes	characteristic
80	C₂³×C₇₃	#108	1	1	yes	characteristic
81	C₂×C₄×C₇₃	—	2	1	no	—
82	C₂×C₄×C₇₃	—	2	1	no	—
83	C₂×C₄×C₇₃	—	2	1	no	—
84	C₂×C₄×C₇₃	—	2	1	no	—
85	C₂×C₄×C₇₃	—	2	1	no	—
86	C₂×C₄×C₇₃	—	2	1	no	—
87	C₂×C₄×C₇₃	—	2	1	no	—
88	C₂×C₄×C₇₃	—	2	1	no	—
89	C₂×C₄×C₇₃	—	2	1	no	—
90	C₂×C₄×C₇₃	—	2	1	no	—
91	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
92	C₂×C₄×C₇₃	#97	1	1	yes	characteristic
93	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
94	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
95	C₂×C₄×C₇₃	#99	1	1	yes	characteristic
96	C₂×C₄×C₇₃	#99	1	1	yes	characteristic
97	C₂×C₄×C₇₃	#100	1	1	yes	characteristic
98	C₂×C₄×C₇₃	#101	1	1	yes	characteristic
99	C₂×Dic₇₃	—	2	2	no	—
100	C₂×Dic₇₃	—	2	2	no	—
101	C₂×Dic₇₃	—	2	2	no	—
102	C₂×Dic₇₃	—	2	2	no	—
103	C₂×Dic₇₃	—	2	2	no	—
104	C₂×Dic₇₃	—	4	1	no	—
105	C₂×Dic₇₃	—	4	2	no	—
106	C₂×Dic₇₃	—	4	1	no	—
107	C₂×Dic₇₃	—	4	2	no	—
108	C₂×Dic₇₃	—	4	2	no	—
109	C₂×Dic₇₃	#102	1	2	yes	—
110	C₂×Dic₇₃	#103	1	2	yes	—
111	#10596833	—	8	4	no	—
112	C₈×C₇₃	—	4	2	no	—
113	#10596831	—	4	2	no	—
114	C₂²×C₇₃	—	2	1	no	—
115	C₂²×C₇₃	—	2	1	no	—
116	C₂²×C₇₃	—	2	1	no	—
117	C₂²×C₇₃	—	2	1	no	—
118	C₂²×C₇₃	—	2	1	no	—
119	C₂²×C₇₃	—	2	1	no	—
120	C₂²×C₇₃	—	2	1	no	—
121	C₂²×C₇₃	—	2	1	no	—
122	C₂²×C₇₃	—	2	1	no	—
123	C₂²×C₇₃	—	2	1	no	—
124	C₂²×C₇₃	—	2	1	no	—
125	C₂²×C₇₃	—	2	1	no	—
126	C₂²×C₇₃	#334	1	1	yes	characteristic
127	C₂²×C₇₃	#334	1	1	yes	characteristic
128	C₂²×C₇₃	#335	1	1	yes	characteristic
129	C₂²×C₇₃	#335	1	1	yes	characteristic
130	C₂²×C₇₃	#336	1	1	yes	characteristic
131	C₂²×C₇₃	H(3,ℤ/4)	1	1	yes	characteristic
132	C₂²×C₇₃	#338	1	1	yes	characteristic
133	C₂²×C₇₃	#340	1	1	yes	characteristic
134	C₂²×C₇₃	#341	1	1	yes	characteristic
135	C₂²×C₇₃	#342	1	1	yes	characteristic
136	C₂²×C₇₃	#343	1	1	yes	characteristic
137	C₄×C₇₃	—	2	1	no	—
138	C₄×C₇₃	—	2	1	no	—
139	C₄×C₇₃	#323	1	1	yes	characteristic
140	C₄×C₇₃	#323	1	1	yes	characteristic
141	C₄×C₇₃	#324	1	1	yes	characteristic
142	C₄×C₇₃	#324	1	1	yes	characteristic
143	Dic₇₃	—	2	2	no	—
144	Dic₇₃	—	2	2	no	—
145	Dic₇₃	—	4	2	no	—
146	#7020	—	73	1	no	maximal, Sylow
147	C₂×C₇₃	—	2	1	no	—
148	C₂×C₇₃	—	2	1	no	—
149	C₂×C₇₃	—	2	1	no	—
150	C₂×C₇₃	—	2	1	no	—
151	C₂×C₇₃	#1275	1	1	yes	characteristic
152	C₂×C₇₃	#1277	1	1	yes	characteristic
153	C₂×C₇₃	#1280	1	1	yes	characteristic
154	C₂×C₇₃	#1289	1	1	yes	characteristic
155	C₂×C₇₃	#1290	1	1	yes	characteristic
156	C₂×C₇₃	#1292	1	1	yes	characteristic
157	C₂×C₇₃	#1293	1	1	yes	characteristic
158	#1797	—	73	2	no	—
159	C₂×#336	—	73	1	no	—
160	C₇₃	#7020	1	1	yes	characteristic, complemented, Sylow
161	C₂³×C₈	—	73	1	no	—
162	C₂³×C₈	#64408	1	1	yes	characteristic
163	C₂²×C₄²	—	73	1	no	—
164	C₂×C₄⋊C₈	—	146	2	no	—
165	C₂×C₂²⋊C₈	—	146	2	no	—
166	#336	—	73	4	no	—
167	#336	—	146	2	no	—
168	C₂³×C₄	—	73	1	no	—
169	C₂³×C₄	—	73	1	no	—
170	C₂³×C₄	C₂×#64408	1	1	yes	characteristic
171	C₂²×C₈	—	2	1	no	—
172	C₂²×C₈	—	2	1	no	—
173	C₂²×C₈	—	4	1	no	—
174	C₂²×C₈	—	4	1	no	—
175	C₂²×C₈	—	292	1	no	—
176	C₂²×C₈	—	292	1	no	—
177	C₂²×C₈	—	73	2	no	—
178	C₂²×C₈	—	146	4	no	—
179	C₂²×C₈	—	146	1	no	—
180	C₂²×C₈	—	146	1	no	—
181	C₂²×C₈	#10596833	1	2	yes	complemented
182	C₂×C₄²	—	292	1	no	—
183	C₂×C₄²	—	292	1	no	—
184	C₂×C₄²	—	73	2	no	—
185	C₂×C₄²	—	73	2	no	—
186	C₄⋊C₈	—	292	2	no	—
187	C₄⋊C₈	—	292	2	no	—
188	C₂²⋊C₈	—	292	2	no	—
189	C₂²⋊C₈	—	292	2	no	—
190	C₂⁴	C₄⋊#64408	1	1	yes	characteristic
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	—	2	1	no	—
193	C₂²×C₄	—	2	1	no	—
194	C₂²×C₄	—	2	1	no	—
195	C₂²×C₄	—	292	1	no	—
196	C₂²×C₄	—	292	1	no	—
197	C₂²×C₄	—	292	1	no	—
198	C₂²×C₄	—	292	1	no	—
199	C₂²×C₄	—	73	2	no	—
200	C₂²×C₄	—	73	1	no	—
201	C₂²×C₄	—	73	1	no	—
202	C₂²×C₄	—	73	2	no	—
203	C₂²×C₄	—	73	2	no	—
204	C₂²×C₄	—	146	1	no	—
205	C₂²×C₄	—	146	1	no	—
206	C₂²×C₄	C₄×#64408	1	1	yes	characteristic
207	C₂²×C₄	C₂²⋊#64408	1	1	yes	characteristic
208	C₂²×C₄	C₂×#10596833	1	1	yes	characteristic
209	C₂²×C₄	#11924036	1	1	yes	characteristic
210	C₂²×C₄	#11924037	1	1	yes	characteristic
211	C₂²×C₄	#11924038	1	1	yes	characteristic
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	4	2	no	—
214	C₂×C₈	—	4	1	no	—
215	C₂×C₈	—	4	1	no	—
216	C₂×C₈	—	4	2	no	—
217	C₂×C₈	—	292	2	no	—
218	C₂×C₈	—	292	1	no	—
219	C₂×C₈	—	292	1	no	—
220	C₂×C₈	—	292	2	no	—
221	C₂×C₈	—	292	4	no	—
222	C₂×C₈	—	292	4	no	—
223	C₂×C₈	—	292	4	no	—
224	C₂×C₈	—	146	2	no	—
225	C₄²	—	292	2	no	—
226	C₄²	—	146	2	no	—
227	C₄²	—	146	2	no	—
228	C₂³	—	2	1	no	—
229	C₂³	—	2	1	no	—
230	C₂³	—	2	1	no	—
231	C₂³	—	2	1	no	—
232	C₂³	(C₂×#64408)⋊C₄	1	1	yes	characteristic
233	C₂³	C₄⋊#10596833	1	1	yes	characteristic
234	C₂³	#14556658	1	1	yes	characteristic
235	C₂³	#14556659	1	1	yes	characteristic
236	C₂³	#14556660	1	1	yes	characteristic
237	C₂³	#14556661	1	1	yes	characteristic
238	C₂³	#14556662	1	1	yes	characteristic
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	292	2	no	—
250	C₂×C₄	—	292	2	no	—
251	C₂×C₄	—	292	1	no	—
252	C₂×C₄	—	292	2	no	—
253	C₂×C₄	—	292	1	no	—
254	C₂×C₄	—	73	2	no	—
255	C₂×C₄	—	73	2	no	—
256	C₂×C₄	—	146	2	no	—
257	C₂×C₄	—	146	2	no	—
258	C₂×C₄	—	146	2	no	—
259	C₂×C₄	—	146	2	no	—
260	C₂×C₄	—	146	2	no	—
261	C₂×C₄	C₄×#10596833	1	1	yes	characteristic
262	C₂×C₄	#14556646	1	1	yes	characteristic
263	C₂×C₄	C₂²⋊#10596833	1	1	yes	characteristic
264	C₂×C₄	#14556648	1	1	yes	characteristic
265	C₂×C₄	#14556649	1	1	yes	characteristic
266	C₂×C₄	#14556650	1	1	yes	characteristic
267	C₂×C₄	#14556651	1	1	yes	characteristic
268	C₂×C₄	#14556652	1	1	yes	characteristic
269	C₈	—	4	2	no	—
270	C₈	—	292	2	no	—
271	C₈	—	584	4	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	2	1	no	—
276	C₂²	—	2	1	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	2	1	no	—
280	C₂²	—	2	1	no	—
281	C₂²	—	2	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	2	1	no	—
284	C₂²	#17989448	1	1	yes	characteristic
285	C₂²	#17989449	1	1	yes	characteristic
286	C₂²	#17989450	1	1	yes	characteristic
287	C₂²	#17989451	1	1	yes	characteristic
288	C₂²	#17989453	1	1	yes	characteristic
289	C₂²	#17989455	1	1	yes	characteristic
290	C₂²	#17989456	1	1	yes	characteristic
291	C₂²	#17989461	1	1	yes	characteristic
292	C₂²	#17989463	1	1	yes	characteristic
293	C₂²	#17989465	1	1	yes	characteristic
294	C₂²	#17989467	1	1	yes	characteristic
295	C₄	—	2	1	no	—
296	C₄	—	2	1	no	—
297	C₄	—	292	2	no	—
298	C₄	—	146	2	no	—
299	C₄	—	146	2	no	—
300	C₄	#17989430	1	1	yes	characteristic
301	C₄	#17989431	1	1	yes	characteristic
302	C₄	#17989432	1	1	yes	characteristic
303	C₄	#17989433	1	1	yes	characteristic
304	C₂	—	2	1	no	—
305	C₂	—	2	1	no	—
306	C₂	—	2	1	no	—
307	C₂	—	2	1	no	—
308	C₂	#22624757	1	1	yes	characteristic
309	C₂	#22624761	1	1	yes	characteristic
310	C₂	#22624769	1	1	yes	characteristic
311	C₂	#22624794	1	1	yes	characteristic
312	C₂	#22624796	1	1	yes	characteristic
313	C₂	#22624801	1	1	yes	characteristic
314	C₂	#22624803	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	(C₂×#64408)⋊C₄
commutator	C₂×C₄×C₇₃	C₄×C₈
Frattini	C₂³×C₄	C₂×#64408
Fitting	C₂³×C₈×C₇₃	C₄
socle	C₂³×C₇₃	#95
prime radical	C₂⁴×C₇₃	C₄⋊C₄
cosocle	C₂²×C₄×Dic₇₃	C₂²
hypercenter	C₂³×C₈	#64408
hypocenter	C₇₃	#7020
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
21. C₂²×#10596831	C₈	semidirect
31. C₂²×C₈×C₇₃	C₈	semidirect
160. C₇₃	#7020	semidirect
181. C₂²×C₈	#10596833	semidirect

Maximal subgroups

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

subgroup	index
#22625785	2
C₂×#17988341	2
#7020	73