Subgroups — C₄×#17989691 #34034129

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	C₂×#17989691	C₂	1	2	yes	maximal
2	C₄×#14556688	C₂	1	2	yes	maximal
3	#22627006	C₂	1	2	yes	maximal
4	C₄×C₈×#64408	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556688	C₂²	1	4	yes	—
6	#17989691	C₄	1	8	yes	direct product factor
7	#17989681	—	2	2	no	—
8	C₄×#11924040	—	2	2	no	—
9	C₄×#11924036	C₂²	1	1	yes	characteristic
10	C₂×C₈×#64408	C₄	1	1	yes	characteristic
11	C₂×C₈×#64408	C₄	1	1	yes	characteristic
12	C₂×C₈×#64408	C₂²	1	1	yes	characteristic
13	C₂×C₈×#64408	C₂²	1	1	yes	characteristic
14	C₄²×#64408	C₄	1	1	yes	characteristic
15	C₄×C₈×D₇₃	C₄	1	1	yes	characteristic
16	#14556688	—	2	4	no	—
17	#14556688	C₂×C₄	1	8	yes	—
18	C₂×#11924040	—	2	4	no	—
19	C₂×#11924036	C₂×C₄	1	1	yes	characteristic
20	C₂×#11924036	C₂³	1	1	yes	characteristic
21	C₂×C₄×#64408	—	2	1	no	—
22	C₂×C₄×#64408	—	2	1	no	—
23	C₂×C₄×#64408	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×#64408	C₂×C₄	1	1	yes	characteristic
25	C₄×#10596833	D₄	1	1	yes	characteristic
26	C₄×#10596833	D₄	1	1	yes	characteristic
27	C₄×#10596833	Q₈	1	1	yes	characteristic
28	C₄×#10596833	Q₈	1	1	yes	characteristic
29	C₈×#64408	—	2	2	no	—
30	C₈×#64408	—	2	2	no	—
31	C₈×#64408	C₈	1	4	yes	—
32	C₈×#64408	C₂×C₄	1	4	yes	—
33	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
35	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
36	C₄×#10596831	C₈	1	1	yes	characteristic
37	C₈×Dic₇₃	C₈	1	1	yes	characteristic
38	C₈×Dic₇₃	C₈	1	1	yes	characteristic
39	C₄×C₈×C₇₃	C₈	1	1	yes	characteristic
40	#11924040	—	2	8	no	—
41	C₂²×#64408	—	2	1	no	—
42	C₂²×#64408	—	2	1	no	—
43	#11924036	C₂×C₈	1	2	yes	—
44	#11924036	M₁₆	1	2	yes	—
45	#11924036	C₂²×C₄	1	2	yes	—
46	#11924036	#41	1	2	yes	—
47	C₂×#10596833	C₄⋊C₄	1	1	yes	characteristic
48	C₂×#10596833	C₄⋊C₄	1	1	yes	characteristic
49	C₂×#10596833	C₂×D₄	1	1	yes	characteristic
50	C₂×#10596833	C₂×Q₈	1	1	yes	characteristic
51	C₄×#64408	—	2	4	no	—
52	C₄×#64408	—	2	4	no	—
53	C₄×#64408	—	2	1	no	—
54	C₄×#64408	—	2	1	no	—
55	C₄×#64408	C₄²	1	2	yes	—
56	C₄×#64408	C₂×C₈	1	2	yes	—
57	C₂×C₄×D₇₃	M₁₆	1	1	yes	characteristic
58	C₂×C₄×D₇₃	M₁₆	1	1	yes	characteristic
59	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
60	C₂×#10596831	C₂×C₈	1	1	yes	characteristic
61	C₂×#10596831	C₂×C₈	1	1	yes	characteristic
62	C₈×D₇₃	—	2	1	no	—
63	C₈×D₇₃	—	2	1	no	—
64	C₈×D₇₃	C₄²	1	2	yes	—
65	C₈×D₇₃	C₂×C₈	1	2	yes	—
66	C₄×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
67	C₄×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
68	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
69	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
70	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
71	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
72	C₂²×D₇₃	C₂×M₁₆	1	1	yes	characteristic
73	C₂×#64408	—	2	2	no	—
74	C₂×#64408	—	2	2	no	—
75	C₂×#64408	—	2	2	no	—
76	C₂×#64408	—	2	2	no	—
77	C₂×#64408	—	2	2	no	—
78	C₂×#64408	—	2	2	no	—
79	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
80	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
81	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
82	C₂×Dic₇₃	C₄⋊C₈	1	1	yes	characteristic
83	C₂×Dic₇₃	C₄⋊C₈	1	1	yes	characteristic
84	C₂×Dic₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₄×D₇₃	—	2	2	no	—
86	C₄×D₇₃	—	2	2	no	—
87	C₄×D₇₃	C₂×C₄²	1	1	yes	characteristic
88	C₄×D₇₃	#117	1	1	yes	characteristic
89	C₄×D₇₃	C₂²×C₈	1	1	yes	characteristic
90	C₄×D₇₃	C₂×M₁₆	1	1	yes	characteristic
91	#10596833	C₄⋊C₈	1	2	yes	—
92	#10596833	C₄⋊C₈	1	2	yes	—
93	#10596833	C₄×D₄	1	2	yes	—
94	#10596833	C₄×Q₈	1	2	yes	—
95	C₈×C₇₃	—	2	1	no	—
96	C₈×C₇₃	C₄×C₈	1	2	yes	—
97	#10596831	—	2	1	no	—
98	#10596831	C₄×C₈	1	2	yes	—
99	C₂²×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
100	C₂×D₇₃	—	2	1	no	—
101	C₂×D₇₃	—	2	1	no	—
102	C₂×D₇₃	C₄×M₁₆	1	1	yes	characteristic
103	C₂×D₇₃	#431	1	1	yes	characteristic
104	#64408	—	2	4	no	—
105	#64408	—	2	4	no	—
106	C₄×C₇₃	—	2	1	no	—
107	C₄×C₇₃	#364	1	2	yes	complemented
108	C₄×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
109	C₄×C₇₃	#431	1	1	yes	characteristic
110	Dic₇₃	—	2	1	no	—
111	Dic₇₃	#364	1	2	yes	complemented
112	Dic₇₃	C₄×C₄⋊C₄	1	1	yes	characteristic
113	Dic₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
114	C₄×#364	—	73	1	no	maximal, Sylow
115	C₂×C₇₃	C₄×C₄⋊C₈	1	1	yes	characteristic
116	C₂×C₇₃	C₂×#364	1	1	yes	characteristic
117	C₂×C₇₃	C₂×#364	1	1	yes	characteristic
118	D₇₃	—	2	1	no	—
119	D₇₃	—	2	1	no	—
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	8	no	—
134	#346	—	146	2	no	—
135	C₄×C₁₆	—	146	2	no	—
136	C₂²×C₈	—	73	1	no	—
137	C₂²×C₈	—	73	1	no	—
138	C₂²×C₈	—	73	1	no	—
139	C₂²×C₈	—	73	1	no	—
140	C₂×C₄²	—	73	1	no	—
141	C₂×C₄²	—	73	1	no	—
142	C₂×C₄²	—	73	1	no	—
143	C₂×C₄²	—	146	1	no	—
144	C₂×C₄²	—	146	1	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	4	no	—
151	C₄×C₈	—	73	4	no	—
152	C₄×C₈	—	73	1	no	—
153	C₄×C₈	—	73	1	no	—
154	C₄×C₈	—	73	1	no	—
155	C₄×C₈	—	73	1	no	—
156	C₄×C₈	—	73	1	no	—
157	C₄×C₈	—	146	2	no	—
158	C₄×C₈	—	146	2	no	—
159	C₄×C₈	#10596842	1	1	yes	characteristic
160	C₂²×C₄	—	73	1	no	—
161	C₂²×C₄	—	73	1	no	—
162	C₂²×C₄	—	73	1	no	—
163	C₂²×C₄	—	146	1	no	—
164	C₂²×C₄	—	146	1	no	—
165	C₂×C₈	—	73	1	no	—
166	C₂×C₈	—	73	1	no	—
167	C₂×C₈	—	73	1	no	—
168	C₂×C₈	—	73	1	no	—
169	C₂×C₈	—	73	2	no	—
170	C₂×C₈	—	73	2	no	—
171	C₂×C₈	—	73	2	no	—
172	C₂×C₈	—	73	2	no	—
173	C₂×C₈	—	73	1	no	—
174	C₂×C₈	—	73	2	no	—
175	C₂×C₈	—	73	2	no	—
176	C₂×C₈	—	73	1	no	—
177	C₂×C₈	—	146	1	no	—
178	C₂×C₈	—	146	1	no	—
179	C₂×C₈	C₂×#10596842	1	1	yes	characteristic
180	C₂×C₈	C₂×#10596842	1	1	yes	characteristic
181	C₄²	—	73	1	no	—
182	C₄²	—	73	1	no	—
183	C₄²	—	73	1	no	—
184	C₄²	—	73	2	no	—
185	C₄²	—	73	2	no	—
186	C₄²	—	146	1	no	—
187	C₄²	—	146	1	no	—
188	C₄²	—	146	4	no	—
189	C₄²	—	146	4	no	—
190	C₄²	C₂×#10596842	1	1	yes	characteristic
191	C₁₆	—	146	8	no	—
192	C₂³	—	73	1	no	—
193	C₂×C₄	—	73	1	no	—
194	C₂×C₄	—	73	1	no	—
195	C₂×C₄	—	73	1	no	—
196	C₂×C₄	—	73	1	no	—
197	C₂×C₄	—	73	1	no	—
198	C₂×C₄	—	73	1	no	—
199	C₂×C₄	—	73	1	no	—
200	C₂×C₄	—	146	2	no	—
201	C₂×C₄	—	146	2	no	—
202	C₂×C₄	—	146	2	no	—
203	C₂×C₄	—	146	2	no	—
204	C₂×C₄	—	146	2	no	—
205	C₂×C₄	—	146	2	no	—
206	C₂×C₄	—	146	2	no	—
207	C₂×C₄	—	146	2	no	—
208	C₂×C₄	C₄⋊#10596842	1	1	yes	characteristic
209	C₂×C₄	C₄⋊#10596842	1	1	yes	characteristic
210	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
211	C₈	—	2	1	no	—
212	C₈	—	73	2	no	—
213	C₈	—	73	2	no	—
214	C₈	—	73	2	no	—
215	C₈	—	73	2	no	—
216	C₈	—	73	2	no	—
217	C₈	—	146	1	no	—
218	C₈	C₄×#10596842	1	2	yes	—
219	C₂²	—	73	1	no	—
220	C₂²	—	73	1	no	—
221	C₂²	—	146	1	no	—
222	C₂²	—	146	1	no	—
223	C₂²	C₂×C₄⋊#10596842	1	1	yes	characteristic
224	C₄	—	2	1	no	—
225	C₄	—	73	2	no	—
226	C₄	—	73	1	no	—
227	C₄	—	73	1	no	—
228	C₄	—	146	4	no	—
229	C₄	—	146	4	no	—
230	C₄	—	146	1	no	—
231	C₄	#17989691	1	2	yes	direct product factor
232	C₄	C₂×C₄×#10596842	1	1	yes	characteristic
233	C₄	#17989700	1	1	yes	characteristic
234	C₂	—	146	1	no	—
235	C₂	—	146	1	no	—
236	C₂	C₄×C₄⋊#10596842	1	1	yes	characteristic
237	C₂	C₂×#17989691	1	1	yes	characteristic
238	C₂	#22627125	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₄⋊#10596842
commutator	C₄×C₇₃	C₂×C₄×C₈
Frattini	C₂×C₄	C₂²×#10596842
Fitting	C₄×C₈×C₇₃	C₈
socle	C₂²×C₇₃	C₂×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 ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
6. #17989691	C₄	direct (N × H)
107. C₄×C₇₃	#364	semidirect
111. Dic₇₃	#364	semidirect
124. C₇₃	C₄×#364	semidirect
231. C₄	#17989691	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#17989691	2
C₄×#14556688	2
#22627006	2
C₄×C₈×#64408	2
C₄×#364	73

Subgroups — C4×#17989691 #34034129

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#17989691 #34034129