Subgroups — #34030285

Order 18688.

642 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22626711	C₂	1	2	yes	maximal, complemented
2	#22626334	C₂	1	4	yes	maximal, complemented
3	#22626332	C₂	1	4	yes	maximal, complemented
4	D₇₃⋊#368	C₂	1	1	yes	maximal, characteristic, complemented
5	#22626247	C₂	1	2	yes	maximal, complemented
6	#22626139	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#17989020	C₂	1	1	yes	maximal, characteristic
8	#17989667	—	2	2	no	—
9	#17989643	C₂²	1	1	yes	characteristic
10	#17989633	—	2	4	no	—
11	#17989631	C₂²	1	8	yes	complemented
12	D₇₃⋊#108	C₂²	1	2	yes	—
13	#17989586	—	2	2	no	—
14	#17989550	C₂²	1	1	yes	characteristic
15	#17989516	C₂²	1	2	yes	complemented
16	#17989515	C₂²	1	2	yes	complemented
17	#17989502	C₂²	1	8	yes	complemented
18	#17989474	C₂²	1	2	yes	complemented
19	C₁₆×#64408	C₂²	1	2	yes	complemented
20	C₂×#14556614	C₂²	1	2	yes	—
21	#17989020	C₄	1	2	yes	complemented
22	#17989020	C₄	1	2	yes	complemented
23	#17989020	C₂²	1	2	yes	—
24	#17989020	C₂²	1	2	yes	—
25	C₂×#14556534	C₄	1	2	yes	complemented
26	C₂×D₁₁₆₈	C₄	1	1	yes	characteristic, complemented
27	C₂×C₁₆×D₇₃	C₂²	1	1	yes	characteristic
28	C₂×Q₃₂×C₇₃	C₄	1	1	yes	characteristic, complemented
29	#14556679	—	2	8	no	—
30	#14556679	—	4	4	no	—
31	D₇₃⋊M₁₆	—	2	2	no	—
32	D₇₃⋊M₁₆	—	4	4	no	—
33	D₇₃⋊M₁₆	D₄	1	1	yes	characteristic
34	#14556670	D₄	1	1	yes	characteristic
35	#14556666	—	2	1	no	—
36	#14556665	—	2	1	no	—
37	#14556664	C₂³	1	2	yes	—
38	#14556662	C₂³	1	4	yes	—
39	#14556661	—	2	2	no	—
40	#14556656	—	2	8	no	—
41	#14556645	D₄	1	2	yes	—
42	C₈×#64408	C₂³	1	2	yes	—
43	C₂×#11924029	—	2	2	no	—
44	#14556614	—	2	2	no	—
45	#14556614	—	2	2	no	—
46	#14556614	C₂×C₄	1	4	yes	—
47	#14556614	C₂³	1	4	yes	—
48	C₂×#11924015	—	2	2	no	—
49	C₂×D₅₈₄	C₂×C₄	1	2	yes	—
50	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
51	#14556534	C₂×C₄	1	4	yes	—
52	#14556534	C₂×C₄	1	4	yes	—
53	D₁₁₆₈	C₂×C₄	1	2	yes	—
54	D₁₁₆₈	C₂×C₄	1	2	yes	—
55	C₂×#11924003	C₂×C₄	1	1	yes	characteristic
56	C₁₆×D₇₃	C₂×C₄	1	1	yes	characteristic
57	C₁₆×D₇₃	C₂×C₄	1	1	yes	characteristic
58	C₁₆×D₇₃	C₂³	1	1	yes	characteristic
59	C₁₆×D₇₃	C₂³	1	1	yes	characteristic
60	C₂×Q₁₆×C₇₃	C₂×C₄	1	2	yes	—
61	Q₃₂×C₇₃	C₂×C₄	1	2	yes	—
62	Q₃₂×C₇₃	C₂×C₄	1	2	yes	—
63	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
64	#11924038	—	2	4	no	—
65	#11924038	—	4	8	no	—
66	#11924038	D₈	1	2	yes	—
67	#11924037	—	4	2	no	—
68	#11924037	D₈	1	2	yes	—
69	#11924036	—	4	2	no	—
70	#11924036	—	4	8	no	—
71	#11924036	C₂×D₄	1	2	yes	—
72	C₄⋊#64408	—	2	1	no	—
73	C₂²⋊#64408	—	2	1	no	—
74	C₄×#64408	C₂×D₄	1	2	yes	—
75	#11924031	—	2	2	no	—
76	#11924029	—	2	4	no	—
77	#11924029	—	2	4	no	—
78	#11924029	—	4	2	no	—
79	#11924029	—	4	2	no	—
80	C₂×D₂₉₂	—	2	2	no	—
81	C₂×D₂₉₂	—	4	2	no	—
82	C₂×C₄×D₇₃	—	4	2	no	—
83	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
84	#11924015	—	2	4	no	—
85	#11924015	—	2	4	no	—
86	D₅₈₄	—	2	2	no	—
87	D₅₈₄	C₂²×C₄	1	4	yes	—
88	C₂×#10596831	#41	1	1	yes	characteristic
89	C₈×D₇₃	C₂²×C₄	1	1	yes	characteristic
90	C₈×D₇₃	C₂×D₄	1	1	yes	characteristic
91	C₈×D₇₃	#41	1	1	yes	characteristic
92	C₈×D₇₃	C₂⁴	1	1	yes	characteristic
93	#11924003	C₂²×C₄	1	1	yes	characteristic
94	#11924003	C₂²×C₄	1	1	yes	characteristic
95	C₂×Q₈×C₇₃	—	2	2	no	—
96	Q₁₆×C₇₃	—	2	2	no	—
97	Q₁₆×C₇₃	C₂²×C₄	1	4	yes	—
98	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
99	C₁₆×C₇₃	C₂²×C₄	1	1	yes	characteristic
100	C₁₆×C₇₃	C₂²×C₄	1	1	yes	characteristic
101	C₂²×D₇₃	—	4	2	no	—
102	C₂²×D₇₃	#135	1	1	yes	characteristic
103	C₂×#64408	—	2	2	no	—
104	Q₈×C₇₃	—	2	4	no	—
105	Q₈×C₇₃	—	4	2	no	—
106	C₂×C₄×C₇₃	—	4	2	no	—
107	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
108	C₂×Dic₇₃	C₂×D₈	1	1	yes	characteristic
109	D₂₉₂	—	2	4	no	—
110	D₂₉₂	—	4	4	no	—
111	D₂₉₂	—	4	4	no	—
112	D₂₉₂	—	4	2	no	—
113	C₄×D₇₃	—	4	4	no	—
114	C₄×D₇₃	—	4	4	no	—
115	C₄×D₇₃	C₄×D₄	1	1	yes	characteristic
116	C₄×D₇₃	C₂×D₈	1	1	yes	characteristic
117	C₄×D₇₃	#135	1	1	yes	characteristic
118	C₄×D₇₃	C₂²×D₄	1	1	yes	characteristic
119	#10596833	—	4	4	no	—
120	#10596833	C₂×D₈	1	4	yes	—
121	C₈×C₇₃	C₄×D₄	1	1	yes	characteristic
122	C₈×C₇₃	C₂³×C₄	1	1	yes	characteristic
123	#10596831	C₄×D₄	1	1	yes	characteristic
124	#10596831	C₂×#41	1	1	yes	characteristic
125	C₂²×C₇₃	C₄×D₈	1	1	yes	characteristic
126	C₂×D₇₃	—	4	1	no	—
127	C₂×D₇₃	—	4	4	no	—
128	C₂×D₇₃	—	16	2	no	—
129	C₂×D₇₃	C₄×D₈	1	1	yes	characteristic
130	C₂×D₇₃	C₂×#135	1	1	yes	characteristic
131	#64408	—	4	2	no	—
132	C₄×C₇₃	—	4	4	no	—
133	C₄×C₇₃	C₄×D₈	1	1	yes	characteristic
134	C₄×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
135	Dic₇₃	C₄×D₈	1	1	yes	characteristic
136	Dic₇₃	C₂²×D₈	1	1	yes	characteristic
137	#33465	—	73	1	no	maximal, Sylow
138	C₂×C₇₃	—	2	1	no	—
139	C₂×C₇₃	C₂×C₄×D₈	1	1	yes	characteristic
140	D₇₃	—	2	1	no	—
141	D₇₃	—	2	1	no	—
142	D₇₃	—	8	4	no	—
143	C₂×#508	—	73	1	no	—
144	#2957	—	73	2	no	—
145	#2179	—	73	4	no	—
146	#2178	—	73	4	no	—
147	C₂×#368	—	73	1	no	—
148	#2143	—	73	2	no	—
149	#2108	—	73	1	no	—
150	C₇₃	#33465	1	1	yes	characteristic, complemented, Sylow
151	C₂×#135	—	73	2	no	—
152	#568	—	146	2	no	—
153	#508	—	73	2	no	—
154	#508	—	73	2	no	—
155	#508	—	73	2	no	—
156	#508	—	73	2	no	—
157	C₂×Q₃₂	#64408	1	1	yes	characteristic, complemented
158	C₂×SD₃₂	—	73	2	no	—
159	C₂×D₁₆	—	73	1	no	—
160	C₂×M₃₂	—	73	1	no	—
161	C₂²×C₁₆	—	73	1	no	—
162	#444	—	146	4	no	—
163	#443	—	73	8	no	—
164	C₂×#108	—	73	2	no	—
165	#421	—	146	2	no	—
166	#405	—	73	1	no	—
167	#368	—	73	2	no	—
168	#368	—	73	2	no	—
169	#359	—	73	8	no	—
170	#346	—	73	2	no	—
171	C₄×C₁₆	—	73	2	no	—
172	C₂×#41	—	146	2	no	—
173	#135	—	73	4	no	—
174	#135	—	73	4	no	—
175	#135	—	146	2	no	—
176	#135	—	146	2	no	—
177	C₂×Q₁₆	C₂×#64408	1	2	yes	—
178	C₂×SD₁₆	—	146	2	no	—
179	C₂×D₈	—	73	2	no	—
180	#131	—	292	4	no	—
181	#131	—	146	8	no	—
182	C₂×M₁₆	—	292	4	no	—
183	C₂×M₁₆	—	73	1	no	—
184	C₂×M₁₆	—	146	2	no	—
185	C₂²×C₈	—	73	1	no	—
186	#117	—	73	1	no	—
187	Q₃₂	C₂×#64408	1	2	yes	—
188	Q₃₂	C₂×#64408	1	2	yes	—
189	SD₃₂	—	73	4	no	—
190	SD₃₂	—	73	4	no	—
191	D₁₆	—	73	2	no	—
192	D₁₆	—	73	2	no	—
193	M₃₂	—	146	1	no	—
194	M₃₂	—	146	1	no	—
195	C₂×C₁₆	—	73	1	no	—
196	C₂×C₁₆	—	73	1	no	—
197	C₂×C₁₆	—	73	1	no	—
198	C₂×C₁₆	—	73	1	no	—
199	C₂×C₁₆	—	73	1	no	—
200	C₂×C₁₆	—	73	2	no	—
201	C₂×C₁₆	C₂×#64408	1	1	yes	characteristic
202	#108	—	73	4	no	—
203	#108	—	146	2	no	—
204	#104	—	146	8	no	—
205	#97	—	73	2	no	—
206	C₄×C₈	—	73	2	no	—
207	#41	—	292	2	no	—
208	#41	—	292	2	no	—
209	#41	—	146	4	no	—
210	#41	—	146	4	no	—
211	C₂×Q₈	—	2	2	no	—
212	C₂×D₄	—	292	2	no	—
213	C₂×D₄	—	146	2	no	—
214	C₂²×C₄	—	292	2	no	—
215	C₂²×C₄	—	73	1	no	—
216	Q₁₆	—	2	2	no	—
217	Q₁₆	C₂²×#64408	1	4	yes	—
218	SD₁₆	—	146	4	no	—
219	SD₁₆	—	146	4	no	—
220	D₈	—	73	4	no	—
221	D₈	—	146	2	no	—
222	M₁₆	—	292	2	no	—
223	M₁₆	—	292	8	no	—
224	M₁₆	—	73	2	no	—
225	M₁₆	—	73	2	no	—
226	M₁₆	—	146	4	no	—
227	C₂×C₈	—	292	2	no	—
228	C₂×C₈	—	292	8	no	—
229	C₂×C₈	—	73	2	no	—
230	C₂×C₈	—	73	1	no	—
231	C₂×C₈	—	73	1	no	—
232	C₂×C₈	—	73	1	no	—
233	C₂×C₈	—	73	1	no	—
234	C₂×C₈	—	73	1	no	—
235	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
236	C₄⋊C₄	—	146	1	no	—
237	C₂²⋊C₄	—	146	1	no	—
238	C₄²	—	73	2	no	—
239	C₁₆	—	73	1	no	—
240	C₁₆	—	73	1	no	—
241	C₁₆	—	146	2	no	—
242	C₁₆	C₂²×#64408	1	1	yes	characteristic
243	C₁₆	C₂²×#64408	1	1	yes	characteristic
244	C₂³	—	292	2	no	—
245	C₂³	—	73	1	no	—
246	Q₈	—	2	4	no	—
247	Q₈	—	4	2	no	—
248	D₄	—	292	4	no	—
249	D₄	—	292	4	no	—
250	D₄	—	292	2	no	—
251	D₄	—	146	4	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	292	4	no	—
254	C₂×C₄	—	292	4	no	—
255	C₂×C₄	—	73	1	no	—
256	C₂×C₄	—	73	1	no	—
257	C₂×C₄	—	73	1	no	—
258	C₂×C₄	—	73	1	no	—
259	C₂×C₄	—	73	1	no	—
260	C₂×C₄	—	146	2	no	—
261	C₂×C₄	D₄×#64408	1	1	yes	characteristic
262	C₈	—	292	4	no	—
263	C₈	—	73	4	no	—
264	C₈	—	73	1	no	—
265	C₈	—	73	1	no	—
266	C₈	D₄×#64408	1	1	yes	characteristic
267	C₈	C₂³×#64408	1	1	yes	characteristic
268	C₂²	—	292	1	no	—
269	C₂²	—	292	4	no	—
270	C₂²	—	73	1	no	—
271	C₂²	—	73	1	no	—
272	C₂²	—	1168	2	no	—
273	C₂²	D₈×#64408	1	1	yes	characteristic
274	C₄	—	4	4	no	—
275	C₄	—	292	2	no	—
276	C₄	—	73	1	no	—
277	C₄	—	73	1	no	—
278	C₄	D₈×#64408	1	1	yes	characteristic
279	C₄	C₂×D₄×#64408	1	1	yes	characteristic
280	C₂	—	2	1	no	—
281	C₂	—	584	4	no	—
282	C₂	—	146	1	no	—
283	C₂	—	146	1	no	—
284	C₂	C₂×D₈×#64408	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₈×#64408
commutator	C₈×C₇₃	C₂³×C₄
Frattini	C₈	C₂³×#64408
Fitting	C₂×Q₃₂×C₇₃	C₄
socle	C₂×C₇₃	C₂×C₄×D₈
prime radical	C₂×#17989020	C₂
cosocle	C₈×D₇₃	C₂⁴
hypercenter	C₂×Q₃₂	#64408
hypocenter	C₇₃	#33465
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

Complements

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

N	H ≅ G/N	type
1. #22626711	C₂	semidirect
2. #22626334	C₂	semidirect
3. #22626332	C₂	semidirect
4. D₇₃⋊#368	C₂	semidirect
5. #22626247	C₂	semidirect
6. #22626139	C₂	semidirect
11. #17989631	C₂²	semidirect
15. #17989516	C₂²	semidirect
16. #17989515	C₂²	semidirect
17. #17989502	C₂²	semidirect
18. #17989474	C₂²	semidirect
19. C₁₆×#64408	C₂²	semidirect
21. #17989020	C₄	semidirect
22. #17989020	C₄	semidirect
25. C₂×#14556534	C₄	semidirect
26. C₂×D₁₁₆₈	C₄	semidirect
28. C₂×Q₃₂×C₇₃	C₄	semidirect
150. C₇₃	#33465	semidirect
157. C₂×Q₃₂	#64408	semidirect

Maximal subgroups

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

subgroup	index
#22626711	2
#22626334	2
#22626332	2
D₇₃⋊#368	2
#22626247	2
#22626139	2
C₂×#17989020	2
#33465	73