Subgroups — #107×#10596842 #34033293

Order 18688.

446 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊C₄×#10596842	C₂	1	2	yes	maximal
2	C₂×#17989677	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×#10596842	C₂	1	1	yes	maximal, characteristic
4	#22626939	C₂	1	2	yes	maximal
5	#107×#64408	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#10596842	C₂²	1	2	yes	—
7	C₂×C₄×#10596842	—	2	2	no	—
8	C₂×C₄×#10596842	C₂²	1	1	yes	characteristic
9	C₂×C₄×#10596842	C₂²	1	1	yes	characteristic
10	(C₂×#10596842)⋊C₄	—	2	2	no	—
11	#17989677	C₄	1	4	yes	complemented
12	C₈×#10596842	C₄	1	4	yes	complemented
13	C₂×#14556660	C₄	1	1	yes	characteristic
14	C₂×C₈×#64408	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×#64408	C₂²	1	2	yes	—
16	#17989460	C₄	1	2	yes	—
17	#107×D₇₃	C₄	1	1	yes	characteristic
18	C₂²×#10596842	—	2	2	no	—
19	C₂²×#10596842	—	2	1	no	—
20	C₂²×#10596842	D₄	1	1	yes	characteristic
21	C₂²×#10596842	D₄	1	1	yes	characteristic
22	C₄⋊#10596842	—	2	2	no	—
23	C₄⋊#10596842	C₂×C₄	1	4	yes	—
24	C₄×#10596842	—	2	1	no	—
25	C₄×#10596842	—	4	4	no	—
26	C₄×#10596842	C₂×C₄	1	2	yes	—
27	C₄×#10596842	C₂×C₄	1	2	yes	—
28	C₄×#10596842	Q₈	1	2	yes	—
29	C₂×#11924036	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄⋊#64408	C₂×C₄	1	2	yes	—
31	C₂×C₄×#64408	—	2	2	no	—
32	C₂×C₄×#64408	C₂³	1	1	yes	characteristic
33	#14556660	C₈	1	4	yes	complemented
34	C₈×#64408	C₂×C₄	1	2	yes	—
35	C₈×#64408	C₂×C₄	1	2	yes	—
36	(C₂×#64408)⋊C₄	—	2	2	no	—
37	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
38	C₄⋊C₄×D₇₃	C₂×C₄	1	2	yes	—
39	#14556525	C₈	1	1	yes	characteristic, complemented
40	#14556524	C₈	1	2	yes	complemented
41	#107×C₇₃	C₈	1	1	yes	characteristic, complemented
42	C₂×#10596842	—	2	2	no	—
43	C₂×#10596842	—	2	2	no	—
44	C₂×#10596842	—	2	2	no	—
45	C₂×#10596842	—	2	1	no	—
46	C₂×#10596842	—	2	1	no	—
47	C₂×#10596842	—	2	4	no	—
48	C₂×#10596842	—	4	2	no	—
49	C₂×#10596842	—	4	2	no	—
50	C₂×#10596842	C₄⋊C₄	1	2	yes	—
51	C₂×#10596842	C₄⋊C₄	1	2	yes	—
52	C₂×#10596842	D₈	1	2	yes	—
53	C₂×#10596842	Q₁₆	1	2	yes	—
54	C₂²×#64408	—	2	2	no	—
55	C₂²×#64408	C₂×D₄	1	1	yes	characteristic
56	C₂²×#64408	#41	1	1	yes	characteristic
57	#11924036	—	2	1	no	—
58	#11924036	C₄²	1	2	yes	—
59	C₂×#10596833	—	2	1	no	—
60	C₄⋊#64408	—	2	2	no	—
61	C₄⋊#64408	C₂×C₈	1	4	yes	—
62	C₄×#64408	—	4	2	no	—
63	C₄×#64408	—	4	2	no	—
64	C₄×#64408	C₂²×C₄	1	1	yes	characteristic
65	C₄×#64408	C₂²×C₄	1	1	yes	characteristic
66	C₄×#64408	C₂×Q₈	1	1	yes	characteristic
67	C₄×#64408	#41	1	1	yes	characteristic
68	C₂×C₄×D₇₃	—	2	2	no	—
69	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
70	C₂×#10596831	C₂×C₈	1	1	yes	characteristic
71	C₈×D₇₃	C₄²	1	2	yes	—
72	C₈×D₇₃	C₂×C₈	1	2	yes	—
73	C₄⋊Dic₇₃	C₂×C₈	1	2	yes	—
74	#11924007	—	2	2	no	—
75	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
76	C₄⋊C₄×C₇₃	C₂×C₈	1	2	yes	—
77	C₂²×D₇₃	C₄×D₄	1	1	yes	characteristic
78	C₂×#64408	—	2	1	no	—
79	C₂×#64408	—	2	1	no	—
80	C₂×#64408	—	2	4	no	—
81	C₂×#64408	—	4	2	no	—
82	C₂×#64408	—	4	2	no	—
83	C₂×#64408	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂×#64408	#117	1	1	yes	characteristic
85	C₂×#64408	C₄×D₄	1	1	yes	characteristic
86	C₂×#64408	C₄×Q₈	1	1	yes	characteristic
87	C₂×#64408	C₂×D₈	1	1	yes	characteristic
88	C₂×#64408	C₂×Q₁₆	1	1	yes	characteristic
89	C₂×#64408	#135	1	1	yes	characteristic
90	C₂×#64408	#135	1	1	yes	characteristic
91	#10596842	—	2	4	no	—
92	#10596842	—	2	2	no	—
93	#10596842	—	4	4	no	—
94	#10596842	#107	1	4	yes	direct product factor
95	C₂×C₄×C₇₃	—	2	2	no	—
96	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
97	C₂×Dic₇₃	—	2	2	no	—
98	C₂×Dic₇₃	#131	1	1	yes	characteristic
99	C₄×D₇₃	—	4	2	no	—
100	C₄×D₇₃	—	4	2	no	—
101	C₄×D₇₃	C₂×C₄²	1	1	yes	characteristic
102	C₄×D₇₃	C₄×Q₈	1	1	yes	characteristic
103	C₄×D₇₃	C₂²×C₈	1	1	yes	characteristic
104	C₄×D₇₃	#131	1	1	yes	characteristic
105	#10596833	—	2	2	no	—
106	C₈×C₇₃	C₄×C₈	1	2	yes	complemented
107	#10596831	C₄×C₈	1	2	yes	complemented
108	C₂²×C₇₃	C₈×D₄	1	1	yes	characteristic
109	C₂×D₇₃	C₄×C₄⋊C₄	1	1	yes	characteristic
110	C₂×D₇₃	C₈×D₄	1	1	yes	characteristic
111	C₂×D₇₃	C₄×D₈	1	1	yes	characteristic
112	C₂×D₇₃	C₄×Q₁₆	1	1	yes	characteristic
113	C₂×D₇₃	#443	1	1	yes	characteristic
114	C₂×D₇₃	#443	1	1	yes	characteristic
115	#64408	—	2	1	no	—
116	#64408	—	2	1	no	—
117	#64408	—	4	4	no	—
118	#64408	C₂×#107	1	1	yes	characteristic
119	#64408	#427	1	1	yes	characteristic
120	#64408	C₄×D₈	1	1	yes	characteristic
121	#64408	C₄×Q₁₆	1	1	yes	characteristic
122	C₄×C₇₃	—	4	2	no	—
123	C₄×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
124	C₄×C₇₃	C₈×Q₈	1	1	yes	characteristic
125	Dic₇₃	—	4	2	no	—
126	Dic₇₃	#405	1	1	yes	characteristic
127	Dic₇₃	C₈×Q₈	1	1	yes	characteristic
128	C₈×#107	—	73	1	no	maximal, Sylow
129	C₂×C₇₃	C₈×D₈	1	1	yes	characteristic
130	C₂×C₇₃	C₈×Q₁₆	1	1	yes	characteristic
131	C₂×C₇₃	C₈×C₄⋊C₄	1	1	yes	characteristic
132	D₇₃	C₈×D₈	1	1	yes	characteristic
133	D₇₃	C₈×Q₁₆	1	1	yes	characteristic
134	D₇₃	#1772	1	1	yes	characteristic
135	D₇₃	C₄×#107	1	1	yes	characteristic
136	C₄×#107	—	73	1	no	—
137	C₈×C₄⋊C₄	—	73	2	no	—
138	C₂×#335	—	73	1	no	—
139	C₂×C₈²	—	73	1	no	—
140	#1279	—	73	2	no	—
141	C₇₃	C₈×#107	1	1	yes	characteristic, complemented, Sylow
142	C₂×#107	—	73	1	no	—
143	C₂×#107	—	73	1	no	—
144	C₂×C₄⋊C₈	—	73	2	no	—
145	C₂×C₄×C₈	—	73	1	no	—
146	C₂×C₄×C₈	—	73	1	no	—
147	C₂×C₄×C₈	—	73	1	no	—
148	C₂×C₄×C₈	—	146	2	no	—
149	C₄×C₄⋊C₄	—	73	2	no	—
150	#340	—	73	2	no	—
151	#336	—	146	2	no	—
152	#335	—	73	4	no	—
153	C₈²	—	73	4	no	—
154	C₂²×C₈	—	73	1	no	—
155	C₂²×C₈	—	73	1	no	—
156	C₂²×C₈	—	73	1	no	—
157	C₂²×C₈	—	73	1	no	—
158	C₂²×C₈	—	146	2	no	—
159	C₂²×C₈	—	146	1	no	—
160	C₂×C₄⋊C₄	—	73	2	no	—
161	C₂×C₄⋊C₄	—	73	2	no	—
162	C₂×C₄²	—	73	1	no	—
163	C₂×C₄²	—	146	2	no	—
164	#107	—	73	4	no	—
165	#107	—	73	1	no	—
166	#107	—	73	2	no	—
167	#107	#10596842	1	1	yes	characteristic, direct product factor
168	C₄⋊C₈	—	73	4	no	—
169	C₄⋊C₈	—	146	2	no	—
170	C₄×C₈	—	292	4	no	—
171	C₄×C₈	—	73	2	no	—
172	C₄×C₈	—	73	2	no	—
173	C₄×C₈	—	73	2	no	—
174	C₄×C₈	—	73	2	no	—
175	C₄×C₈	—	73	2	no	—
176	C₄×C₈	—	146	1	no	—
177	#95	—	146	2	no	—
178	C₂²×C₄	—	73	1	no	—
179	C₂²×C₄	—	73	1	no	—
180	C₂²×C₄	—	73	1	no	—
181	C₂²×C₄	—	146	2	no	—
182	C₂²×C₄	—	146	2	no	—
183	C₂×C₈	—	292	2	no	—
184	C₂×C₈	—	292	2	no	—
185	C₂×C₈	—	73	2	no	—
186	C₂×C₈	—	73	2	no	—
187	C₂×C₈	—	73	2	no	—
188	C₂×C₈	—	73	1	no	—
189	C₂×C₈	—	73	2	no	—
190	C₂×C₈	—	73	2	no	—
191	C₂×C₈	—	73	2	no	—
192	C₂×C₈	—	73	2	no	—
193	C₂×C₈	—	146	4	no	—
194	C₂×C₈	—	146	1	no	—
195	C₂×C₈	—	146	1	no	—
196	C₂×C₈	—	146	2	no	—
197	C₂×C₈	—	146	1	no	—
198	C₂×C₈	—	146	1	no	—
199	C₂×C₈	—	146	2	no	—
200	C₂×C₈	—	146	2	no	—
201	C₂×C₈	C₂×#10596842	1	1	yes	characteristic
202	C₄⋊C₄	—	73	4	no	—
203	C₄⋊C₄	—	73	2	no	—
204	C₄⋊C₄	—	146	2	no	—
205	C₄⋊C₄	—	146	2	no	—
206	C₄⋊C₄	C₂×#10596842	1	2	yes	—
207	C₄²	—	292	2	no	—
208	C₄²	—	292	2	no	—
209	C₄²	—	73	1	no	—
210	C₄²	—	73	1	no	—
211	C₄²	—	73	1	no	—
212	C₄²	—	73	1	no	—
213	C₂³	—	73	1	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	292	2	no	—
216	C₂×C₄	—	292	2	no	—
217	C₂×C₄	—	292	2	no	—
218	C₂×C₄	—	292	2	no	—
219	C₂×C₄	—	73	1	no	—
220	C₂×C₄	—	73	1	no	—
221	C₂×C₄	—	73	1	no	—
222	C₂×C₄	—	73	1	no	—
223	C₂×C₄	—	73	1	no	—
224	C₂×C₄	—	73	1	no	—
225	C₂×C₄	—	73	1	no	—
226	C₂×C₄	—	73	1	no	—
227	C₂×C₄	—	73	1	no	—
228	C₂×C₄	—	73	1	no	—
229	C₂×C₄	—	73	1	no	—
230	C₂×C₄	—	73	1	no	—
231	C₂×C₄	—	73	1	no	—
232	C₂×C₄	—	146	4	no	—
233	C₂×C₄	—	146	2	no	—
234	C₂×C₄	—	146	1	no	—
235	C₂×C₄	—	146	1	no	—
236	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
237	C₈	—	292	4	no	—
238	C₈	—	73	4	no	—
239	C₈	—	73	2	no	—
240	C₈	—	146	2	no	—
241	C₈	—	146	2	no	—
242	C₈	—	146	4	no	—
243	C₈	C₄×#10596842	1	2	yes	complemented
244	C₂²	—	73	1	no	—
245	C₂²	—	73	1	no	—
246	C₂²	—	73	1	no	—
247	C₂²	—	73	1	no	—
248	C₂²	—	73	1	no	—
249	C₂²	—	73	1	no	—
250	C₂²	D₄×#10596842	1	1	yes	characteristic
251	C₄	—	4	2	no	—
252	C₄	—	292	4	no	—
253	C₄	—	292	2	no	—
254	C₄	—	73	1	no	—
255	C₄	—	73	1	no	—
256	C₄	—	73	1	no	—
257	C₄	—	73	1	no	—
258	C₄	—	73	1	no	—
259	C₄	—	73	1	no	—
260	C₄	—	146	1	no	—
261	C₄	—	146	1	no	—
262	C₄	C₂×C₄×#10596842	1	1	yes	characteristic
263	C₄	Q₈×#10596842	1	1	yes	characteristic
264	C₂	—	73	1	no	—
265	C₂	—	73	1	no	—
266	C₂	—	73	1	no	—
267	C₂	—	73	1	no	—
268	C₂	D₈×#10596842	1	1	yes	characteristic
269	C₂	Q₁₆×#10596842	1	1	yes	characteristic
270	C₂	C₄⋊C₄×#10596842	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×#10596842
commutator	C₄×C₇₃	C₂×C₄×C₈
Frattini	C₂×C₄	C₂²×#10596842
Fitting	#107×C₇₃	C₈
socle	C₂²×C₇₃	C₈×D₄
prime radical	C₂²×D₇₃	C₄×D₄
cosocle	C₂×C₄×#64408	C₂³
hypercenter	#107	#10596842
hypocenter	C₇₃	C₈×#107
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

Complements

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

N	H ≅ G/N	type
11. #17989677	C₄	semidirect
12. C₈×#10596842	C₄	semidirect
33. #14556660	C₈	semidirect
39. #14556525	C₈	semidirect
40. #14556524	C₈	semidirect
41. #107×C₇₃	C₈	semidirect
94. #10596842	#107	direct (N × H)
106. C₈×C₇₃	C₄×C₈	semidirect
107. #10596831	C₄×C₈	semidirect
141. C₇₃	C₈×#107	semidirect
167. #107	#10596842	direct (N × H)
243. C₈	C₄×#10596842	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊C₄×#10596842	2
C₂×#17989677	2
C₂×C₈×#10596842	2
#22626939	2
#107×#64408	2
C₈×#107	73