Subgroups — #64408⋊#364 #34034127

Order 18688.

398 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22627125	C₂	1	2	yes	maximal
2	#64408⋊M₃₂	C₂	1	2	yes	maximal
3	#22627005	C₂	1	2	yes	maximal
4	C₄×C₈×#64408	C₂	1	1	yes	maximal, characteristic
5	#17989706	C₂²	1	4	yes	—
6	#17989692	C₄	1	4	yes	complemented
7	#17989691	C₄	1	4	yes	complemented
8	#17989682	—	2	2	no	—
9	#64408⋊C₁₆	—	2	2	no	—
10	C₄×#11924036	C₂²	1	1	yes	characteristic
11	C₂×C₈×#64408	C₄	1	1	yes	characteristic
12	C₂×C₈×#64408	C₄	1	1	yes	characteristic
13	C₂×C₈×#64408	C₂²	1	1	yes	characteristic
14	C₂×C₈×#64408	C₂²	1	1	yes	characteristic
15	C₄²×#64408	C₄	1	1	yes	characteristic
16	C₄×C₈×D₇₃	C₄	1	1	yes	characteristic
17	#14556688	C₂×C₄	1	8	yes	—
18	#14556687	—	2	4	no	—
19	#14556686	—	2	4	no	—
20	C₂×#11924036	C₂×C₄	1	1	yes	characteristic
21	C₂×#11924036	C₂³	1	1	yes	characteristic
22	C₂×C₄×#64408	—	2	2	no	—
23	C₂×C₄×#64408	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×#64408	C₂×C₄	1	1	yes	characteristic
25	C₄×#10596833	—	2	2	no	—
26	C₈×#64408	—	2	1	no	—
27	C₈×#64408	—	2	1	no	—
28	C₈×#64408	C₈	1	2	yes	—
29	C₈×#64408	C₈	1	2	yes	—
30	C₈×#64408	C₂×C₄	1	2	yes	—
31	C₈×#64408	C₂×C₄	1	2	yes	—
32	C₈×#64408	D₄	1	1	yes	characteristic
33	C₈×#64408	D₄	1	1	yes	characteristic
34	C₈×#64408	Q₈	1	1	yes	characteristic
35	C₈×#64408	Q₈	1	1	yes	characteristic
36	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
38	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
39	C₄×#10596831	C₈	1	1	yes	characteristic
40	C₈×Dic₇₃	C₈	1	1	yes	characteristic
41	C₈×Dic₇₃	C₈	1	1	yes	characteristic
42	C₄×C₈×C₇₃	C₈	1	1	yes	characteristic
43	#11924040	—	2	8	no	—
44	C₂²×#64408	—	2	1	no	—
45	C₂²×#64408	M₁₆	1	2	yes	—
46	#11924036	—	2	1	no	—
47	#11924036	C₂×C₈	1	2	yes	—
48	#11924036	C₂²×C₄	1	2	yes	—
49	#11924036	C₂×D₄	1	1	yes	characteristic
50	#11924036	C₂×Q₈	1	1	yes	characteristic
51	C₂×#10596833	—	2	1	no	—
52	C₂×#10596833	#41	1	2	yes	—
53	C₄×#64408	—	2	8	no	—
54	C₄×#64408	—	2	1	no	—
55	C₄×#64408	C₄²	1	2	yes	—
56	C₄×#64408	C₄⋊C₄	1	1	yes	characteristic
57	C₄×#64408	C₄⋊C₄	1	1	yes	characteristic
58	C₄×#64408	C₂×C₈	1	2	yes	—
59	C₂×C₄×D₇₃	—	2	1	no	—
60	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
61	C₂×#10596831	C₂×C₈	1	1	yes	characteristic
62	C₂×#10596831	C₂×C₈	1	1	yes	characteristic
63	C₈×D₇₃	C₄²	1	1	yes	characteristic
64	C₈×D₇₃	C₄²	1	1	yes	characteristic
65	C₈×D₇₃	C₄⋊C₄	1	1	yes	characteristic
66	C₈×D₇₃	C₄⋊C₄	1	1	yes	characteristic
67	C₈×D₇₃	C₂×C₈	1	1	yes	characteristic
68	C₈×D₇₃	C₂×C₈	1	1	yes	characteristic
69	C₈×D₇₃	M₁₆	1	1	yes	characteristic
70	C₈×D₇₃	M₁₆	1	1	yes	characteristic
71	C₄×Dic₇₃	—	2	1	no	—
72	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
73	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
74	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
75	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
76	C₂²×D₇₃	C₂×M₁₆	1	1	yes	characteristic
77	C₂×#64408	—	2	4	no	—
78	C₂×#64408	—	2	2	no	—
79	C₂×#64408	—	2	4	no	—
80	C₂×#64408	C₄⋊C₈	1	2	yes	—
81	C₂×#64408	C₄⋊C₈	1	2	yes	—
82	C₂×C₄×C₇₃	—	2	1	no	—
83	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
84	C₂×Dic₇₃	—	2	1	no	—
85	C₂×Dic₇₃	#117	1	1	yes	characteristic
86	C₄×D₇₃	—	2	4	no	—
87	C₄×D₇₃	C₂×C₄²	1	1	yes	characteristic
88	C₄×D₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₄×D₇₃	C₂²×C₈	1	1	yes	characteristic
90	C₄×D₇₃	C₂×M₁₆	1	1	yes	characteristic
91	#10596833	—	2	2	no	—
92	#10596833	C₄×D₄	1	2	yes	—
93	#10596833	C₄×Q₈	1	2	yes	—
94	C₈×C₇₃	C₄×C₈	1	1	yes	characteristic
95	C₈×C₇₃	C₄×C₈	1	1	yes	characteristic
96	C₈×C₇₃	C₄⋊C₈	1	1	yes	characteristic
97	C₈×C₇₃	C₄⋊C₈	1	1	yes	characteristic
98	#10596831	C₄×C₈	1	1	yes	characteristic
99	#10596831	C₄×C₈	1	1	yes	characteristic
100	#10596831	C₄⋊C₈	1	1	yes	characteristic
101	#10596831	C₄⋊C₈	1	1	yes	characteristic
102	C₂²×C₇₃	#431	1	1	yes	characteristic
103	C₂×D₇₃	—	2	2	no	—
104	C₂×D₇₃	C₄×M₁₆	1	1	yes	characteristic
105	C₂×D₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
106	#64408	—	2	4	no	—
107	#64408	—	2	2	no	—
108	#64408	#364	1	4	yes	complemented
109	C₄×C₇₃	—	2	2	no	—
110	C₄×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
111	C₄×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
112	Dic₇₃	—	2	2	no	—
113	Dic₇₃	C₄×C₄⋊C₄	1	1	yes	characteristic
114	Dic₇₃	#431	1	1	yes	characteristic
115	C₄×#364	—	73	1	no	maximal, Sylow
116	C₂×C₇₃	—	2	1	no	—
117	C₂×C₇₃	C₄×C₄⋊C₈	1	1	yes	characteristic
118	D₇₃	—	2	1	no	—
119	D₇₃	C₂×#364	1	2	yes	—
120	C₂×#364	—	73	2	no	—
121	C₄×M₃₂	—	73	2	no	—
122	C₄²×C₈	—	73	1	no	—
123	#1376	—	73	2	no	—
124	C₇₃	C₄×#364	1	1	yes	characteristic, complemented, Sylow
125	C₂×M₃₂	—	73	4	no	—
126	C₂×C₄×C₈	—	73	1	no	—
127	C₂×C₄×C₈	—	73	1	no	—
128	C₂×C₄×C₈	—	73	1	no	—
129	C₂×C₄×C₈	—	73	1	no	—
130	C₂×C₄×C₈	—	73	1	no	—
131	C₂×C₄×C₈	—	73	1	no	—
132	C₄³	—	73	1	no	—
133	#364	—	73	4	no	—
134	#364	—	73	4	no	—
135	#346	—	146	2	no	—
136	C₄×C₁₆	—	146	2	no	—
137	C₂²×C₈	—	73	1	no	—
138	C₂²×C₈	—	73	1	no	—
139	C₂²×C₈	—	73	1	no	—
140	C₂²×C₈	—	73	1	no	—
141	C₂×C₄²	—	73	1	no	—
142	C₂×C₄²	—	73	1	no	—
143	C₂×C₄²	—	73	1	no	—
144	C₂×C₄²	—	146	2	no	—
145	M₃₂	—	73	8	no	—
146	M₃₂	—	146	4	no	—
147	C₂×C₁₆	—	146	4	no	—
148	C₄×C₈	—	73	1	no	—
149	C₄×C₈	—	73	1	no	—
150	C₄×C₈	—	73	2	no	—
151	C₄×C₈	—	73	2	no	—
152	C₄×C₈	—	73	2	no	—
153	C₄×C₈	—	73	2	no	—
154	C₄×C₈	—	73	1	no	—
155	C₄×C₈	—	73	1	no	—
156	C₄×C₈	—	73	1	no	—
157	C₄×C₈	—	73	1	no	—
158	C₄×C₈	—	73	1	no	—
159	C₄×C₈	—	146	1	no	—
160	C₄×C₈	—	146	1	no	—
161	C₄×C₈	—	146	2	no	—
162	C₄×C₈	#10596842	1	1	yes	characteristic
163	C₂²×C₄	—	73	1	no	—
164	C₂²×C₄	—	73	2	no	—
165	C₂²×C₄	—	146	1	no	—
166	C₂²×C₄	—	146	1	no	—
167	C₂×C₈	—	73	2	no	—
168	C₂×C₈	—	73	2	no	—
169	C₂×C₈	—	73	1	no	—
170	C₂×C₈	—	73	1	no	—
171	C₂×C₈	—	73	1	no	—
172	C₂×C₈	—	73	1	no	—
173	C₂×C₈	—	73	1	no	—
174	C₂×C₈	—	73	1	no	—
175	C₂×C₈	—	73	1	no	—
176	C₂×C₈	—	73	1	no	—
177	C₂×C₈	—	73	1	no	—
178	C₂×C₈	—	73	1	no	—
179	C₂×C₈	—	73	1	no	—
180	C₂×C₈	—	73	1	no	—
181	C₂×C₈	—	73	2	no	—
182	C₂×C₈	—	146	1	no	—
183	C₂×C₈	—	146	1	no	—
184	C₂×C₈	C₂×#10596842	1	1	yes	characteristic
185	C₂×C₈	C₂×#10596842	1	1	yes	characteristic
186	C₄²	—	73	1	no	—
187	C₄²	—	73	1	no	—
188	C₄²	—	73	2	no	—
189	C₄²	—	73	2	no	—
190	C₄²	—	73	1	no	—
191	C₄²	—	146	1	no	—
192	C₄²	—	146	1	no	—
193	C₄²	—	146	8	no	—
194	C₄²	C₂×#10596842	1	1	yes	characteristic
195	C₁₆	—	146	8	no	—
196	C₂³	—	73	1	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	73	2	no	—
199	C₂×C₄	—	73	1	no	—
200	C₂×C₄	—	73	1	no	—
201	C₂×C₄	—	73	1	no	—
202	C₂×C₄	—	73	1	no	—
203	C₂×C₄	—	73	1	no	—
204	C₂×C₄	—	73	2	no	—
205	C₂×C₄	—	146	4	no	—
206	C₂×C₄	—	146	4	no	—
207	C₂×C₄	—	146	2	no	—
208	C₂×C₄	—	146	4	no	—
209	C₂×C₄	—	146	1	no	—
210	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
211	C₈	—	73	2	no	—
212	C₈	—	73	2	no	—
213	C₈	—	73	1	no	—
214	C₈	—	73	1	no	—
215	C₈	—	73	1	no	—
216	C₈	—	73	1	no	—
217	C₈	—	146	2	no	—
218	C₈	C₄×#10596842	1	1	yes	characteristic
219	C₈	C₄×#10596842	1	1	yes	characteristic
220	C₈	C₄⋊#10596842	1	1	yes	characteristic
221	C₈	C₄⋊#10596842	1	1	yes	characteristic
222	C₂²	—	73	1	no	—
223	C₂²	—	73	1	no	—
224	C₂²	—	146	2	no	—
225	C₂²	#17989700	1	1	yes	characteristic
226	C₄	—	2	2	no	—
227	C₄	—	73	4	no	—
228	C₄	—	73	1	no	—
229	C₄	—	73	1	no	—
230	C₄	—	146	2	no	—
231	C₄	—	146	2	no	—
232	C₄	—	146	4	no	—
233	C₄	C₂×C₄×#10596842	1	1	yes	characteristic
234	C₄	C₂×C₄⋊#10596842	1	1	yes	characteristic
235	C₂	—	2	1	no	—
236	C₂	—	73	2	no	—
237	C₂	—	146	1	no	—
238	C₂	C₄×C₄⋊#10596842	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₄⋊#10596842
commutator	C₄×C₇₃	C₂×C₄×C₈
Frattini	C₂×C₄	C₂²×#10596842
Fitting	C₄×C₈×C₇₃	C₈
socle	C₂×C₇₃	C₄×C₄⋊C₈
prime radical	C₂²×D₇₃	C₂×M₁₆
cosocle	C₂×#11924036	C₂³
hypercenter	C₄×C₈	#10596842
hypocenter	C₇₃	C₄×#364
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
6. #17989692	C₄	semidirect
7. #17989691	C₄	semidirect
108. #64408	#364	semidirect
124. C₇₃	C₄×#364	semidirect

Maximal subgroups

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

subgroup	index
#22627125	2
#64408⋊M₃₂	2
#22627005	2
C₄×C₈×#64408	2
C₄×#364	73