Subgroups — #364×#64408 #34012637

Order 18688.

398 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22626270	C₂	1	1	yes	maximal, characteristic
2	M₃₂×#64408	C₂	1	2	yes	maximal
3	C₄×C₈×#64408	C₂	1	1	yes	maximal, characteristic
4	#22624974	C₂	1	2	yes	maximal
5	#364×D₇₃	C₂	1	1	yes	maximal, characteristic
6	#17989643	C₂²	1	2	yes	—
7	C₄×#11924036	C₄	1	1	yes	characteristic
8	C₂×C₈×#64408	C₄	1	1	yes	characteristic
9	C₂×C₈×#64408	C₄	1	1	yes	characteristic
10	C₂×C₈×#64408	C₂²	1	1	yes	characteristic
11	C₂×C₈×#64408	C₂²	1	1	yes	characteristic
12	C₄²×#64408	C₄	1	1	yes	characteristic
13	#17989510	C₄	1	2	yes	complemented
14	#17989509	C₄	1	2	yes	complemented
15	#17989474	—	2	2	no	—
16	C₁₆×#64408	—	2	2	no	—
17	M₃₂×D₇₃	C₂²	1	2	yes	—
18	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
19	#17988408	C₄	1	1	yes	characteristic, complemented
20	#17988407	C₄	1	2	yes	complemented
21	#364×C₇₃	C₄	1	1	yes	characteristic, complemented
22	C₂×#11924036	C₂×C₄	1	1	yes	characteristic
23	C₂×#11924036	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×#64408	—	2	2	no	—
25	C₂×C₄×#64408	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×#64408	C₂×C₄	1	1	yes	characteristic
27	#14556666	—	2	2	no	—
28	#14556665	C₂×C₄	1	4	yes	—
29	#14556664	—	2	2	no	—
30	C₄×#10596833	—	2	2	no	—
31	C₈×#64408	—	2	1	no	—
32	C₈×#64408	—	2	1	no	—
33	C₈×#64408	C₈	1	4	yes	—
34	C₈×#64408	C₈	1	4	yes	—
35	C₈×#64408	D₄	1	1	yes	characteristic
36	C₈×#64408	D₄	1	1	yes	characteristic
37	C₈×#64408	Q₈	1	1	yes	characteristic
38	C₈×#64408	Q₈	1	1	yes	characteristic
39	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
41	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
42	#14556531	C₂×C₄	1	2	yes	—
43	#14556530	—	2	2	no	—
44	C₁₆×D₇₃	—	2	2	no	—
45	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
46	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
47	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
48	M₃₂×C₇₃	C₂×C₄	1	2	yes	—
49	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
50	C₂²×#64408	—	2	1	no	—
51	C₂²×#64408	M₁₆	1	2	yes	—
52	#11924036	—	2	1	no	—
53	#11924036	C₄⋊C₄	1	1	yes	characteristic
54	#11924036	C₄⋊C₄	1	1	yes	characteristic
55	#11924036	C₂×C₈	1	2	yes	—
56	#11924036	C₂×C₈	1	2	yes	—
57	C₂×#10596833	—	2	1	no	—
58	C₂×#10596833	M₁₆	1	2	yes	—
59	C₄×#64408	—	2	8	no	—
60	C₄×#64408	—	2	1	no	—
61	C₄×#64408	C₄⋊C₄	1	1	yes	characteristic
62	C₄×#64408	C₄⋊C₄	1	1	yes	characteristic
63	C₄×#64408	C₂×C₈	1	2	yes	—
64	C₄×#64408	C₂×C₈	1	2	yes	—
65	#11924031	—	2	4	no	—
66	C₂×C₄×D₇₃	—	2	1	no	—
67	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
68	C₂×#10596831	C₄²	1	1	yes	characteristic
69	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
70	C₈×D₇₃	C₂×C₈	1	2	yes	—
71	C₈×D₇₃	C₂×C₈	1	2	yes	—
72	C₈×D₇₃	C₂×D₄	1	1	yes	characteristic
73	C₈×D₇₃	C₂×Q₈	1	1	yes	characteristic
74	C₈×D₇₃	#41	1	1	yes	characteristic
75	C₈×D₇₃	#41	1	1	yes	characteristic
76	C₄×Dic₇₃	—	2	1	no	—
77	C₄×Dic₇₃	C₄²	1	1	yes	characteristic
78	#11924003	—	2	2	no	—
79	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
80	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
81	C₄²×C₇₃	C₄²	1	1	yes	characteristic
82	C₁₆×C₇₃	—	2	2	no	—
83	C₂²×D₇₃	C₂×M₁₆	1	1	yes	characteristic
84	C₂×#64408	—	2	4	no	—
85	C₂×#64408	—	2	2	no	—
86	C₂×#64408	—	2	4	no	—
87	C₂×#64408	C₄⋊C₈	1	2	yes	—
88	C₂×#64408	C₄⋊C₈	1	2	yes	—
89	C₂×C₄×C₇₃	—	2	1	no	—
90	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
91	C₂×Dic₇₃	—	2	1	no	—
92	C₂×Dic₇₃	C₂×M₁₆	1	1	yes	characteristic
93	C₄×D₇₃	—	2	4	no	—
94	C₄×D₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₄×D₇₃	#117	1	1	yes	characteristic
96	C₄×D₇₃	C₂²×C₈	1	1	yes	characteristic
97	C₄×D₇₃	C₂²×C₈	1	1	yes	characteristic
98	#10596833	—	2	2	no	—
99	#10596833	C₄⋊C₈	1	2	yes	—
100	#10596833	C₄⋊C₈	1	2	yes	—
101	C₈×C₇₃	C₄×C₈	1	2	yes	—
102	C₈×C₇₃	C₄×D₄	1	1	yes	characteristic
103	C₈×C₇₃	C₄×Q₈	1	1	yes	characteristic
104	#10596831	C₄×C₈	1	2	yes	—
105	#10596831	C₄×D₄	1	1	yes	characteristic
106	#10596831	C₄×Q₈	1	1	yes	characteristic
107	C₂²×C₇₃	C₄×M₁₆	1	1	yes	characteristic
108	C₂×D₇₃	—	2	2	no	—
109	C₂×D₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
110	C₂×D₇₃	#431	1	1	yes	characteristic
111	#64408	—	2	4	no	—
112	#64408	—	2	2	no	—
113	#64408	#364	1	4	yes	direct product factor
114	C₄×C₇₃	—	2	2	no	—
115	C₄×C₇₃	C₄×C₄⋊C₄	1	1	yes	characteristic
116	C₄×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
117	Dic₇₃	—	2	2	no	—
118	Dic₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
119	Dic₇₃	#431	1	1	yes	characteristic
120	C₄×#364	—	73	1	no	maximal, Sylow
121	C₂×C₇₃	—	2	1	no	—
122	C₂×C₇₃	C₄×C₄⋊C₈	1	1	yes	characteristic
123	D₇₃	—	2	1	no	—
124	D₇₃	C₂×#364	1	2	yes	—
125	C₂×#364	—	73	1	no	—
126	C₂×#364	—	73	1	no	—
127	C₄×M₃₂	—	73	2	no	—
128	C₄²×C₈	—	73	1	no	—
129	#1376	—	73	2	no	—
130	C₇₃	C₄×#364	1	1	yes	characteristic, complemented, Sylow
131	C₂×M₃₂	—	73	2	no	—
132	C₂×M₃₂	—	73	2	no	—
133	C₂×C₄×C₈	—	73	1	no	—
134	C₂×C₄×C₈	—	73	1	no	—
135	C₂×C₄×C₈	—	73	1	no	—
136	C₂×C₄×C₈	—	73	1	no	—
137	C₂×C₄×C₈	—	73	1	no	—
138	C₂×C₄×C₈	—	73	1	no	—
139	C₄³	—	73	1	no	—
140	#364	—	73	2	no	—
141	#364	—	73	2	no	—
142	#364	—	73	2	no	—
143	#364	—	73	1	no	—
144	#364	#64408	1	1	yes	characteristic, direct product factor
145	#346	—	146	2	no	—
146	C₄×C₁₆	—	146	2	no	—
147	C₂²×C₈	—	73	1	no	—
148	C₂²×C₈	—	73	1	no	—
149	C₂²×C₈	—	73	1	no	—
150	C₂²×C₈	—	73	1	no	—
151	C₂×C₄²	—	73	1	no	—
152	C₂×C₄²	—	73	1	no	—
153	C₂×C₄²	—	73	1	no	—
154	C₂×C₄²	—	146	2	no	—
155	M₃₂	—	73	4	no	—
156	M₃₂	—	73	2	no	—
157	M₃₂	—	146	2	no	—
158	M₃₂	—	146	2	no	—
159	M₃₂	C₂×#64408	1	2	yes	—
160	C₂×C₁₆	—	146	2	no	—
161	C₂×C₁₆	—	146	2	no	—
162	C₄×C₈	—	73	1	no	—
163	C₄×C₈	—	73	1	no	—
164	C₄×C₈	—	73	4	no	—
165	C₄×C₈	—	73	4	no	—
166	C₄×C₈	—	73	1	no	—
167	C₄×C₈	—	73	1	no	—
168	C₄×C₈	—	73	1	no	—
169	C₄×C₈	—	73	1	no	—
170	C₄×C₈	—	73	1	no	—
171	C₄×C₈	—	146	1	no	—
172	C₄×C₈	—	146	1	no	—
173	C₄×C₈	—	146	2	no	—
174	C₄×C₈	C₂×#64408	1	1	yes	characteristic
175	C₂²×C₄	—	73	1	no	—
176	C₂²×C₄	—	73	2	no	—
177	C₂²×C₄	—	146	1	no	—
178	C₂²×C₄	—	146	1	no	—
179	C₂×C₈	—	73	2	no	—
180	C₂×C₈	—	73	1	no	—
181	C₂×C₈	—	73	1	no	—
182	C₂×C₈	—	73	2	no	—
183	C₂×C₈	—	73	1	no	—
184	C₂×C₈	—	73	1	no	—
185	C₂×C₈	—	73	1	no	—
186	C₂×C₈	—	73	1	no	—
187	C₂×C₈	—	73	2	no	—
188	C₂×C₈	—	73	2	no	—
189	C₂×C₈	—	73	1	no	—
190	C₂×C₈	—	73	1	no	—
191	C₂×C₈	—	73	2	no	—
192	C₂×C₈	—	146	1	no	—
193	C₂×C₈	—	146	1	no	—
194	C₂×C₈	C₄×#64408	1	1	yes	characteristic
195	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
196	C₄²	—	73	1	no	—
197	C₄²	—	73	1	no	—
198	C₄²	—	73	2	no	—
199	C₄²	—	73	2	no	—
200	C₄²	—	73	1	no	—
201	C₄²	—	146	1	no	—
202	C₄²	—	146	1	no	—
203	C₄²	—	146	8	no	—
204	C₄²	C₄×#64408	1	1	yes	characteristic
205	C₁₆	—	2	2	no	—
206	C₁₆	—	146	4	no	—
207	C₁₆	—	146	2	no	—
208	C₂³	—	73	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	73	2	no	—
211	C₂×C₄	—	73	1	no	—
212	C₂×C₄	—	73	1	no	—
213	C₂×C₄	—	73	1	no	—
214	C₂×C₄	—	73	1	no	—
215	C₂×C₄	—	73	1	no	—
216	C₂×C₄	—	73	2	no	—
217	C₂×C₄	—	146	4	no	—
218	C₂×C₄	—	146	4	no	—
219	C₂×C₄	—	146	2	no	—
220	C₂×C₄	—	146	4	no	—
221	C₂×C₄	—	146	1	no	—
222	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
223	C₈	—	73	2	no	—
224	C₈	—	73	2	no	—
225	C₈	—	73	1	no	—
226	C₈	—	73	1	no	—
227	C₈	—	73	2	no	—
228	C₈	—	146	2	no	—
229	C₈	C₈×#64408	1	2	yes	—
230	C₈	D₄×#64408	1	1	yes	characteristic
231	C₈	Q₈×#64408	1	1	yes	characteristic
232	C₂²	—	73	1	no	—
233	C₂²	—	73	1	no	—
234	C₂²	—	146	2	no	—
235	C₂²	M₁₆×#64408	1	1	yes	characteristic
236	C₄	—	2	2	no	—
237	C₄	—	73	4	no	—
238	C₄	—	73	1	no	—
239	C₄	—	73	1	no	—
240	C₄	—	146	2	no	—
241	C₄	—	146	2	no	—
242	C₄	—	146	4	no	—
243	C₄	C₄⋊C₄×#64408	1	1	yes	characteristic
244	C₄	C₂×C₈×#64408	1	1	yes	characteristic
245	C₂	—	2	1	no	—
246	C₂	—	73	2	no	—
247	C₂	—	146	1	no	—
248	C₂	C₄⋊C₈×#64408	1	1	yes	characteristic

	subgroup	G/H
center	C₈	D₄×#64408
commutator	C₄×C₇₃	C₂×C₄×C₈
Frattini	C₂×C₈	C₂²×#64408
Fitting	#364×C₇₃	C₄
socle	C₂×C₇₃	C₄×C₄⋊C₈
prime radical	C₂²×D₇₃	C₂×M₁₆
cosocle	C₂×C₈×D₇₃	C₂³
hypercenter	#364	#64408
hypocenter	C₇₃	C₄×#364
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ #364

Complements

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

N	H ≅ G/N	type
13. #17989510	C₄	semidirect
14. #17989509	C₄	semidirect
19. #17988408	C₄	semidirect
20. #17988407	C₄	semidirect
21. #364×C₇₃	C₄	semidirect
113. #64408	#364	direct (N × H)
130. C₇₃	C₄×#364	semidirect
144. #364	#64408	direct (N × H)

Maximal subgroups

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

subgroup	index
#22626270	2
M₃₂×#64408	2
C₄×C₈×#64408	2
#22624974	2
#364×D₇₃	2
C₄×#364	73