Subgroups — #32985230

Order 18688.

654 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988964	C₂	1	1	yes	maximal, characteristic
2	#22610858	C₂	1	2	yes	maximal, complemented
3	C₂×#17988340	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#17988330	C₂	1	2	yes	maximal, complemented
5	C₈×C₄⋊C₄×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₅₈₄	—	2	1	no	—
7	C₂²×D₅₈₄	—	2	1	no	—
8	C₂×#14556588	C₂²	1	2	yes	—
9	#17988964	—	2	2	no	—
10	#17988964	C₄	1	4	yes	complemented
11	C₂×C₄⋊#10596831	C₂²	1	2	yes	—
12	C₂×#14556511	—	2	4	no	—
13	#17988340	—	2	2	no	—
14	#17988330	—	2	4	no	—
15	C₂×C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₇₃	—	2	1	no	—
17	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
18	C₄×C₄⋊C₄×C₇₃	C₂²	1	1	yes	characteristic
19	#336×C₇₃	—	2	1	no	—
20	C₂²×D₂₉₂	—	2	2	no	—
21	C₂²×D₂₉₂	—	2	2	no	—
22	C₂²×D₂₉₂	—	2	2	no	—
23	C₂×D₅₈₄	—	2	4	no	—
24	C₂×D₅₈₄	—	2	4	no	—
25	C₂×D₅₈₄	—	4	2	no	—
26	C₂×D₅₈₄	—	4	2	no	—
27	C₂×D₅₈₄	—	4	2	no	—
28	C₂×D₅₈₄	—	4	2	no	—
29	C₂²×#10596831	—	2	2	no	—
30	#14556588	—	2	2	no	—
31	#14556588	—	2	2	no	—
32	#14556588	—	2	2	no	—
33	#14556588	C₂×C₄	1	4	yes	—
34	C₄⋊#10596831	—	2	2	no	—
35	C₄⋊#10596831	—	2	2	no	—
36	#14556511	—	4	8	no	—
37	C₂²×C₈×C₇₃	—	2	1	no	—
38	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
40	C₂×C₄⋊C₄×C₇₃	D₄	1	2	yes	—
41	C₂×C₄²×C₇₃	—	2	1	no	—
42	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
43	C₄⋊C₈×C₇₃	—	2	1	no	—
44	C₄⋊C₈×C₇₃	D₄	1	2	yes	—
45	C₄×C₈×C₇₃	—	4	2	no	—
46	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
47	C₄×C₈×C₇₃	D₄	1	2	yes	—
48	#95×C₇₃	—	2	1	no	—
49	C₂³×D₇₃	—	4	2	no	—
50	C₂×D₂₉₂	—	2	4	no	—
51	C₂×D₂₉₂	—	2	4	no	—
52	C₂×D₂₉₂	—	2	4	no	—
53	C₂×D₂₉₂	—	4	4	no	—
54	C₂×D₂₉₂	—	4	2	no	—
55	C₂×D₂₉₂	—	4	2	no	—
56	C₂×D₂₉₂	—	4	4	no	—
57	C₂×D₂₉₂	—	4	2	no	—
58	C₂×D₂₉₂	—	4	2	no	—
59	C₂×D₂₉₂	—	4	2	no	—
60	C₂×D₂₉₂	—	4	2	no	—
61	C₂×D₂₉₂	—	4	2	no	—
62	C₂×D₂₉₂	—	8	4	no	—
63	D₅₈₄	—	4	8	no	—
64	D₅₈₄	—	4	8	no	—
65	C₂×#10596831	—	4	2	no	—
66	C₂×#10596831	—	4	4	no	—
67	C₂²×C₄×C₇₃	—	2	2	no	—
68	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
71	C₂×C₈×C₇₃	—	2	1	no	—
72	C₂×C₈×C₇₃	—	2	1	no	—
73	C₂×C₈×C₇₃	—	2	2	no	—
74	C₂×C₈×C₇₃	—	4	1	no	—
75	C₂×C₈×C₇₃	—	4	1	no	—
76	C₂×C₈×C₇₃	C₂²⋊C₄	1	2	yes	—
77	C₂×C₈×C₇₃	C₂²⋊C₄	1	2	yes	—
78	C₂×C₈×C₇₃	D₈	1	2	yes	—
79	C₂×C₈×C₇₃	SD₁₆	1	2	yes	—
80	C₄⋊C₄×C₇₃	—	2	2	no	—
81	C₄⋊C₄×C₇₃	D₈	1	4	yes	complemented
82	C₄²×C₇₃	—	4	1	no	—
83	C₄²×C₇₃	—	4	1	no	—
84	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
85	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₇₃	#41	1	1	yes	characteristic
88	C₂³×C₇₃	#121	1	1	yes	characteristic
89	C₂²×D₇₃	—	4	4	no	—
90	C₂²×D₇₃	—	8	2	no	—
91	C₂²×D₇₃	—	8	2	no	—
92	C₂²×D₇₃	—	8	4	no	—
93	C₂²×D₇₃	—	8	4	no	—
94	C₂×C₄×C₇₃	—	2	2	no	—
95	C₂×C₄×C₇₃	—	2	4	no	—
96	C₂×C₄×C₇₃	—	4	2	no	—
97	C₂×C₄×C₇₃	—	4	2	no	—
98	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
100	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
101	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
102	C₂×C₄×C₇₃	#124	1	1	yes	characteristic
103	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
104	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
105	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
106	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
107	C₂×C₄×C₇₃	#135	1	2	yes	—
108	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
109	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
110	D₂₉₂	—	4	4	no	—
111	D₂₉₂	—	4	4	no	—
112	D₂₉₂	—	4	8	no	—
113	D₂₉₂	—	8	4	no	—
114	D₂₉₂	—	8	4	no	—
115	D₂₉₂	—	8	2	no	—
116	D₂₉₂	—	8	2	no	—
117	D₂₉₂	—	8	2	no	—
118	D₂₉₂	—	8	2	no	—
119	C₈×C₇₃	—	2	2	no	—
120	C₈×C₇₃	—	4	2	no	—
121	C₈×C₇₃	#102	1	4	yes	complemented
122	#10596831	—	8	2	no	—
123	C₂²×C₇₃	#390	1	1	yes	characteristic
124	C₂²×C₇₃	#459	1	1	yes	characteristic
125	C₂²×C₇₃	#460	1	1	yes	characteristic
126	C₂²×C₇₃	#466	1	2	yes	—
127	C₂²×C₇₃	#472	1	2	yes	—
128	C₂×D₇₃	—	8	4	no	—
129	C₂×D₇₃	—	8	4	no	—
130	C₂×D₇₃	—	8	4	no	—
131	C₂×D₇₃	—	16	4	no	—
132	C₂×D₇₃	—	16	4	no	—
133	C₄×C₇₃	—	2	1	no	—
134	C₄×C₇₃	—	2	1	no	—
135	C₄×C₇₃	—	4	4	no	—
136	C₄×C₇₃	C₂×#102	1	1	yes	characteristic
137	C₄×C₇₃	#418	1	1	yes	characteristic
138	C₄×C₇₃	C₄×D₈	1	2	yes	—
139	C₄×C₇₃	#459	1	1	yes	characteristic
140	C₄×C₇₃	#460	1	1	yes	characteristic
141	C₄×C₇₃	#493	1	1	yes	characteristic
142	C₄×C₇₃	#495	1	1	yes	characteristic
143	#9734	—	73	1	no	maximal, Sylow
144	C₂×C₇₃	#1667	1	2	yes	—
145	C₂×C₇₃	#1668	1	2	yes	—
146	C₂×C₇₃	#1970	1	1	yes	characteristic
147	C₂×C₇₃	#1978	1	1	yes	characteristic
148	C₂×C₇₃	#1982	1	1	yes	characteristic
149	D₇₃	—	16	4	no	—
150	C₂×#493	—	73	1	no	—
151	#1982	—	73	2	no	—
152	C₈×C₄⋊C₄	D₇₃	1	1	yes	characteristic, complemented
153	C₂×#335	—	73	1	no	—
154	C₂×#331	—	73	2	no	—
155	C₇₃	#9734	1	1	yes	characteristic, complemented, Sylow
156	C₂²×D₈	—	146	1	no	—
157	C₂²×D₈	—	146	1	no	—
158	C₂×#127	—	73	2	no	—
159	#493	—	73	4	no	—
160	#493	—	146	2	no	—
161	C₂×C₄⋊C₈	—	73	2	no	—
162	C₂×C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
163	C₂×#102	—	146	4	no	—
164	C₂×C₄×C₈	—	2	1	no	—
165	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
166	C₄×C₄⋊C₄	C₂×D₇₃	1	1	yes	characteristic
167	#336	—	2	1	no	—
168	#335	—	146	2	no	—
169	#331	—	146	4	no	—
170	C₂²×D₄	—	146	2	no	—
171	C₂²×D₄	—	146	2	no	—
172	C₂²×D₄	—	146	2	no	—
173	C₂×D₈	—	292	2	no	—
174	C₂×D₈	—	292	2	no	—
175	C₂×D₈	—	292	2	no	—
176	C₂×D₈	—	292	2	no	—
177	C₂×D₈	—	146	4	no	—
178	C₂×D₈	—	146	4	no	—
179	C₂²×C₈	—	2	1	no	—
180	C₂²×C₈	—	146	2	no	—
181	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
182	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
183	#127	—	73	4	no	—
184	#127	—	146	2	no	—
185	#127	—	146	2	no	—
186	#127	—	146	2	no	—
187	C₂×C₄⋊C₄	D₂₉₂	1	2	yes	—
188	C₂×C₄²	—	2	1	no	—
189	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
190	C₄⋊C₈	—	2	1	no	—
191	C₄⋊C₈	—	146	2	no	—
192	C₄⋊C₈	—	146	2	no	—
193	C₄⋊C₈	D₂₉₂	1	2	yes	—
194	#102	—	292	8	no	—
195	C₄×C₈	—	4	2	no	—
196	C₄×C₈	C₄×D₇₃	1	2	yes	—
197	C₄×C₈	D₂₉₂	1	2	yes	—
198	#95	—	2	1	no	—
199	C₂⁴	—	292	2	no	—
200	C₂×D₄	—	292	2	no	—
201	C₂×D₄	—	292	2	no	—
202	C₂×D₄	—	292	2	no	—
203	C₂×D₄	—	292	2	no	—
204	C₂×D₄	—	292	4	no	—
205	C₂×D₄	—	292	2	no	—
206	C₂×D₄	—	292	4	no	—
207	C₂×D₄	—	292	2	no	—
208	C₂×D₄	—	292	2	no	—
209	C₂×D₄	—	584	4	no	—
210	C₂×D₄	—	146	4	no	—
211	C₂×D₄	—	146	4	no	—
212	C₂×D₄	—	146	4	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
215	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
216	C₂²×C₄	#11924028	1	1	yes	characteristic
217	D₈	—	292	8	no	—
218	D₈	—	292	8	no	—
219	C₂×C₈	—	2	1	no	—
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	4	1	no	—
223	C₂×C₈	—	4	1	no	—
224	C₂×C₈	—	292	2	no	—
225	C₂×C₈	—	292	4	no	—
226	C₂×C₈	#11924005	1	2	yes	—
227	C₂×C₈	#11924005	1	2	yes	—
228	C₂×C₈	#11924013	1	2	yes	—
229	C₂×C₈	#11924015	1	2	yes	—
230	C₄⋊C₄	—	2	2	no	—
231	C₄⋊C₄	D₅₈₄	1	4	yes	complemented
232	C₄²	—	4	1	no	—
233	C₄²	—	4	1	no	—
234	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
235	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
236	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
237	C₄²	#11924028	1	1	yes	characteristic
238	C₂³	—	292	4	no	—
239	C₂³	—	584	2	no	—
240	C₂³	—	584	2	no	—
241	C₂³	—	584	4	no	—
242	C₂³	—	584	4	no	—
243	C₂³	#14556565	1	1	yes	characteristic
244	D₄	—	292	8	no	—
245	D₄	—	292	4	no	—
246	D₄	—	292	4	no	—
247	D₄	—	584	4	no	—
248	D₄	—	584	4	no	—
249	D₄	—	584	2	no	—
250	D₄	—	584	2	no	—
251	D₄	—	584	2	no	—
252	D₄	—	584	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	C₂×#11924005	1	1	yes	characteristic
258	C₂×C₄	C₄×D₂₉₂	1	1	yes	characteristic
259	C₂×C₄	C₄×D₂₉₂	1	1	yes	characteristic
260	C₂×C₄	#14556565	1	1	yes	characteristic
261	C₂×C₄	#14556579	1	1	yes	characteristic
262	C₂×C₄	#14556588	1	1	yes	characteristic
263	C₂×C₄	C₂×#11924013	1	1	yes	characteristic
264	C₂×C₄	C₂×D₅₈₄	1	2	yes	—
265	C₂×C₄	C₂×#11924015	1	1	yes	characteristic
266	C₂×C₄	#14556613	1	2	yes	—
267	C₂×C₄	#14556616	1	1	yes	characteristic
268	C₂×C₄	#14556616	1	1	yes	characteristic
269	C₈	—	2	2	no	—
270	C₈	—	4	2	no	—
271	C₈	—	584	2	no	—
272	C₈	#14556511	1	4	yes	complemented
273	C₂²	—	584	4	no	—
274	C₂²	—	584	4	no	—
275	C₂²	—	584	4	no	—
276	C₂²	—	1168	4	no	—
277	C₂²	—	1168	4	no	—
278	C₂²	#17988489	1	1	yes	characteristic
279	C₂²	#17988782	1	1	yes	characteristic
280	C₂²	#17988787	1	1	yes	characteristic
281	C₂²	#17988823	1	2	yes	—
282	C₂²	#17988857	1	2	yes	—
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	4	4	no	—
286	C₄	C₂×#14556511	1	1	yes	characteristic
287	C₄	#17988591	1	1	yes	characteristic
288	C₄	C₄×D₅₈₄	1	2	yes	—
289	C₄	#17988782	1	1	yes	characteristic
290	C₄	#17988787	1	1	yes	characteristic
291	C₄	#17988964	1	1	yes	characteristic
292	C₄	#17988971	1	1	yes	characteristic
293	C₂	—	1168	4	no	—
294	C₂	#22609286	1	2	yes	—
295	C₂	#22609291	1	2	yes	—
296	C₂	#22610797	1	1	yes	characteristic
297	C₂	#22610841	1	1	yes	characteristic
298	C₂	#22610858	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14556565
commutator	C₄²×C₇₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₈×C₄⋊C₄×C₇₃	C₂
socle	C₂³×C₇₃	#121
prime radical	C₂×#17988964	C₂
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	C₈×C₄⋊C₄	D₇₃
hypocenter	C₇₃	#9734
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. #22610858	C₂	semidirect
3. C₂×#17988340	C₂	semidirect
4. C₂×#17988330	C₂	semidirect
5. C₈×C₄⋊C₄×C₇₃	C₂	semidirect
10. #17988964	C₄	semidirect
81. C₄⋊C₄×C₇₃	D₈	semidirect
121. C₈×C₇₃	#102	semidirect
152. C₈×C₄⋊C₄	D₇₃	semidirect
155. C₇₃	#9734	semidirect
231. C₄⋊C₄	D₅₈₄	semidirect
272. C₈	#14556511	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17988964	2
#22610858	2
C₂×#17988340	2
C₂×#17988330	2
C₈×C₄⋊C₄×C₇₃	2
#9734	73