Subgroups — #34032913

Order 18688.

286 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#11924040	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#11924040	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₄×#10596833	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11924040	—	2	1	no	—
5	C₂²×#11924040	—	2	1	no	—
6	C₄⋊#11924040	—	2	1	no	—
7	C₄⋊#11924040	C₄	1	2	yes	complemented
8	C₄×#11924040	—	2	1	no	—
9	C₄×#11924040	C₄	1	2	yes	complemented
10	C₂×C₄⋊#10596833	C₄	1	1	yes	characteristic
11	C₂×C₄×#10596833	—	2	1	no	—
12	C₂×C₄×#10596833	C₂²	1	1	yes	characteristic
13	#17989453	—	2	1	no	—
14	C₄⋊C₄×Dic₇₃	C₄	1	1	yes	characteristic
15	C₂×#11924040	—	2	2	no	—
16	C₂×#11924040	—	2	2	no	—
17	C₂×#11924040	—	2	2	no	—
18	C₂×#11924040	—	2	2	no	—
19	C₂×#11924040	—	4	1	no	—
20	C₂×#11924040	—	4	1	no	—
21	C₂²×#10596833	—	2	1	no	—
22	C₂²×#10596833	D₄	1	1	yes	characteristic
23	C₂²×#10596833	D₄	1	1	yes	characteristic
24	C₄⋊#10596833	—	2	1	no	—
25	C₄⋊#10596833	C₈	1	2	yes	—
26	C₄×#10596833	—	4	1	no	—
27	C₄×#10596833	—	4	1	no	—
28	C₄×#10596833	C₂×C₄	1	1	yes	characteristic
29	C₄×#10596833	C₂×C₄	1	1	yes	characteristic
30	C₄×#10596833	D₄	1	1	yes	characteristic
31	C₄×#10596833	Q₈	1	1	yes	characteristic
32	C₂×C₄⋊Dic₇₃	C₈	1	1	yes	characteristic
33	C₂×C₄×Dic₇₃	—	2	1	no	—
34	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
35	#14556498	—	2	1	no	—
36	C₂×C₄⋊C₄×C₇₃	C₈	1	1	yes	characteristic
37	#11924040	—	2	4	no	—
38	#11924040	—	4	2	no	—
39	C₂×#10596833	—	2	1	no	—
40	C₂×#10596833	—	2	1	no	—
41	C₂×#10596833	—	2	2	no	—
42	C₂×#10596833	—	4	1	no	—
43	C₂×#10596833	—	4	1	no	—
44	C₂×#10596833	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×#10596833	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×#10596833	C₄⋊C₄	1	1	yes	characteristic
47	C₂×#10596833	C₄⋊C₄	1	1	yes	characteristic
48	C₂×#10596833	D₈	1	1	yes	characteristic
49	C₂×#10596833	SD₁₆	1	1	yes	characteristic
50	C₂×#10596833	SD₁₆	1	1	yes	characteristic
51	C₂×#10596833	Q₁₆	1	1	yes	characteristic
52	C₂²×Dic₇₃	—	2	1	no	—
53	C₂²×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×Dic₇₃	M₁₆	1	1	yes	characteristic
55	C₄⋊Dic₇₃	—	2	1	no	—
56	C₄⋊Dic₇₃	C₁₆	1	2	yes	complemented
57	C₄×Dic₇₃	—	4	1	no	—
58	C₄×Dic₇₃	—	4	1	no	—
59	C₄×Dic₇₃	C₄²	1	1	yes	characteristic
60	C₄×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
61	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
62	C₄×Dic₇₃	M₁₆	1	1	yes	characteristic
63	C₂²×C₄×C₇₃	—	2	1	no	—
64	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
65	C₄⋊C₄×C₇₃	—	2	1	no	—
66	C₄⋊C₄×C₇₃	C₁₆	1	2	yes	complemented
67	C₂³×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
68	C₂×C₄×C₇₃	—	2	2	no	—
69	C₂×C₄×C₇₃	—	4	1	no	—
70	C₂×C₄×C₇₃	—	4	1	no	—
71	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
72	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
73	C₂×C₄×C₇₃	C₂×C₁₆	1	1	yes	characteristic
74	C₂×C₄×C₇₃	C₂×C₁₆	1	1	yes	characteristic
75	C₂×C₄×C₇₃	M₃₂	1	1	yes	characteristic
76	C₂×C₄×C₇₃	M₃₂	1	1	yes	characteristic
77	C₂×Dic₇₃	—	2	1	no	—
78	C₂×Dic₇₃	—	2	1	no	—
79	C₂×Dic₇₃	—	2	2	no	—
80	C₂×Dic₇₃	—	4	1	no	—
81	C₂×Dic₇₃	—	4	1	no	—
82	C₂×Dic₇₃	#95	1	1	yes	characteristic
83	C₂×Dic₇₃	#97	1	1	yes	characteristic
84	C₂×Dic₇₃	C₂²⋊C₈	1	1	yes	characteristic
85	C₂×Dic₇₃	#102	1	1	yes	characteristic
86	C₂×Dic₇₃	#103	1	1	yes	characteristic
87	C₂×Dic₇₃	#104	1	1	yes	characteristic
88	C₂×Dic₇₃	#104	1	1	yes	characteristic
89	C₂×Dic₇₃	C₄⋊C₈	1	1	yes	characteristic
90	#10596833	—	2	1	no	—
91	#10596833	—	2	1	no	—
92	#10596833	—	4	2	no	—
93	#10596833	#102	1	1	yes	characteristic
94	#10596833	#103	1	1	yes	characteristic
95	#10596833	#106	1	1	yes	characteristic
96	#10596833	#107	1	1	yes	characteristic
97	C₂²×C₇₃	D₄⋊C₈	1	1	yes	characteristic
98	C₂²×C₇₃	Q₈⋊C₈	1	1	yes	characteristic
99	C₂²×C₇₃	#336	1	1	yes	characteristic
100	C₂²×C₇₃	C₂²⋊C₁₆	1	1	yes	characteristic
101	C₂²×C₇₃	C₂²⋊C₁₆	1	1	yes	characteristic
102	C₂²×C₇₃	#350	1	1	yes	characteristic
103	C₂²×C₇₃	#350	1	1	yes	characteristic
104	C₄×C₇₃	—	4	2	no	—
105	C₄×C₇₃	C₄×C₁₆	1	1	yes	characteristic
106	C₄×C₇₃	#346	1	1	yes	characteristic
107	C₄×C₇₃	C₄⋊C₁₆	1	1	yes	characteristic
108	C₄×C₇₃	C₄⋊C₁₆	1	1	yes	characteristic
109	Dic₇₃	—	2	1	no	—
110	Dic₇₃	—	2	1	no	—
111	Dic₇₃	—	4	2	no	—
112	Dic₇₃	D₄⋊C₈	1	1	yes	characteristic
113	Dic₇₃	Q₈⋊C₈	1	1	yes	characteristic
114	Dic₇₃	#339	1	1	yes	characteristic
115	Dic₇₃	#340	1	1	yes	characteristic
116	#7167	—	73	1	no	maximal, Sylow
117	C₂×C₇₃	#1279	1	1	yes	characteristic
118	C₂×C₇₃	D₄⋊C₁₆	1	1	yes	characteristic
119	C₂×C₇₃	D₄⋊C₁₆	1	1	yes	characteristic
120	C₂×C₇₃	Q₈⋊C₁₆	1	1	yes	characteristic
121	C₂×C₇₃	Q₈⋊C₁₆	1	1	yes	characteristic
122	C₂×C₇₃	#1374	1	1	yes	characteristic
123	C₂×C₇₃	#1379	1	1	yes	characteristic
124	C₂×C₄⋊C₁₆	—	73	1	no	—
125	C₂×C₄×C₁₆	—	73	1	no	—
126	C₈×C₄⋊C₄	—	73	1	no	—
127	C₇₃	#7167	1	1	yes	characteristic, complemented, Sylow
128	C₂²×C₁₆	—	146	1	no	—
129	C₂²×C₁₆	—	146	1	no	—
130	C₂×C₄⋊C₈	—	73	1	no	—
131	C₂×C₄×C₈	—	73	1	no	—
132	C₂×C₄×C₈	—	146	1	no	—
133	C₄×C₄⋊C₄	—	73	1	no	—
134	C₄⋊C₁₆	—	73	2	no	—
135	C₄⋊C₁₆	—	146	1	no	—
136	C₄×C₁₆	—	73	2	no	—
137	C₄×C₁₆	—	146	1	no	—
138	#336	—	146	1	no	—
139	C₂²×C₈	—	73	1	no	—
140	C₂²×C₈	—	73	1	no	—
141	C₂²×C₈	—	146	1	no	—
142	C₂×C₄⋊C₄	—	73	1	no	—
143	C₂×C₄⋊C₄	#10596842	1	1	yes	characteristic
144	C₂×C₄²	—	73	1	no	—
145	C₂×C₄²	—	146	1	no	—
146	C₂×C₁₆	—	292	1	no	—
147	C₂×C₁₆	—	292	1	no	—
148	C₂×C₁₆	—	146	2	no	—
149	C₂×C₁₆	—	146	2	no	—
150	C₂×C₁₆	—	146	2	no	—
151	C₂×C₁₆	—	146	2	no	—
152	C₄⋊C₈	—	73	2	no	—
153	C₄⋊C₈	—	146	1	no	—
154	C₄×C₈	—	292	1	no	—
155	C₄×C₈	—	292	1	no	—
156	C₄×C₈	—	73	1	no	—
157	C₄×C₈	—	73	1	no	—
158	C₄×C₈	—	73	1	no	—
159	C₄×C₈	—	73	1	no	—
160	#95	—	146	1	no	—
161	C₂²×C₄	—	2	1	no	—
162	C₂²×C₄	—	73	1	no	—
163	C₂²×C₄	—	73	1	no	—
164	C₂²×C₄	—	146	1	no	—
165	C₂²×C₄	C₂×#10596842	1	1	yes	characteristic
166	C₂×C₈	—	292	1	no	—
167	C₂×C₈	—	292	1	no	—
168	C₂×C₈	—	73	1	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	—	146	2	no	—
177	C₂×C₈	—	146	1	no	—
178	C₂×C₈	—	146	1	no	—
179	C₄⋊C₄	—	2	1	no	—
180	C₄⋊C₄	—	73	2	no	—
181	C₄⋊C₄	—	146	1	no	—
182	C₄⋊C₄	#11924040	1	2	yes	complemented
183	C₄²	—	292	1	no	—
184	C₄²	—	292	1	no	—
185	C₄²	—	73	1	no	—
186	C₄²	—	73	1	no	—
187	C₄²	—	73	1	no	—
188	C₄²	—	73	1	no	—
189	C₁₆	—	292	2	no	—
190	C₁₆	—	146	4	no	—
191	C₂³	C₂²⋊#10596842	1	1	yes	characteristic
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	4	1	no	—
194	C₂×C₄	—	4	1	no	—
195	C₂×C₄	—	292	1	no	—
196	C₂×C₄	—	292	1	no	—
197	C₂×C₄	—	73	1	no	—
198	C₂×C₄	—	73	1	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	1	no	—
205	C₂×C₄	—	146	2	no	—
206	C₂×C₄	—	146	1	no	—
207	C₂×C₄	—	146	1	no	—
208	C₂×C₄	C₄×#10596842	1	1	yes	characteristic
209	C₂×C₄	C₄⋊#10596842	1	1	yes	characteristic
210	C₂×C₄	C₂×#11924040	1	1	yes	characteristic
211	C₂×C₄	#14556686	1	1	yes	characteristic
212	C₂×C₄	#14556687	1	1	yes	characteristic
213	C₂×C₄	#14556688	1	1	yes	characteristic
214	C₈	—	292	2	no	—
215	C₈	—	73	1	no	—
216	C₈	—	73	1	no	—
217	C₈	—	73	1	no	—
218	C₈	—	73	1	no	—
219	C₈	—	146	1	no	—
220	C₈	—	146	1	no	—
221	C₂²	D₄⋊#10596842	1	1	yes	characteristic
222	C₂²	Q₈⋊#10596842	1	1	yes	characteristic
223	C₂²	(C₂×#10596842)⋊C₄	1	1	yes	characteristic
224	C₂²	C₂²⋊#11924040	1	1	yes	characteristic
225	C₂²	#17989684	1	1	yes	characteristic
226	C₂²	#17989687	1	1	yes	characteristic
227	C₂²	#17989688	1	1	yes	characteristic
228	C₄	—	4	2	no	—
229	C₄	—	292	2	no	—
230	C₄	—	73	1	no	—
231	C₄	—	73	1	no	—
232	C₄	—	73	1	no	—
233	C₄	—	73	1	no	—
234	C₄	—	146	1	no	—
235	C₄	—	146	1	no	—
236	C₄	C₄×#11924040	1	1	yes	characteristic
237	C₄	#17989681	1	1	yes	characteristic
238	C₄	C₄⋊#11924040	1	1	yes	characteristic
239	C₄	#17989690	1	1	yes	characteristic
240	C₂	#22626939	1	1	yes	characteristic
241	C₂	D₄⋊#11924040	1	1	yes	characteristic
242	C₂	#22626974	1	1	yes	characteristic
243	C₂	Q₈⋊#11924040	1	1	yes	characteristic
244	C₂	#22626986	1	1	yes	characteristic
245	C₂	#22627001	1	1	yes	characteristic
246	C₂	#22627011	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²⋊#10596842
commutator	C₄×C₇₃	C₄×C₁₆
Frattini	C₂²×C₄	C₂×#10596842
Fitting	C₂×C₄⋊C₄×C₇₃	C₈
socle, prime radical	C₂³×C₇₃	C₂²⋊C₈
cosocle	C₂×C₄×#10596833	C₂²
hypercenter	C₂×C₄⋊C₄	#10596842
hypocenter	C₇₃	#7167
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
7. C₄⋊#11924040	C₄	semidirect
9. C₄×#11924040	C₄	semidirect
56. C₄⋊Dic₇₃	C₁₆	semidirect
66. C₄⋊C₄×C₇₃	C₁₆	semidirect
127. C₇₃	#7167	semidirect
182. C₄⋊C₄	#11924040	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#11924040	2
C₂×C₄×#11924040	2
C₄⋊C₄×#10596833	2
#7167	73