Subgroups — #34007476

Order 18688.

398 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22626029	C₂	1	2	yes	maximal
2	C₂×#17989450	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×#10596833	C₂	1	1	yes	maximal, characteristic
4	#22624767	C₂	1	2	yes	maximal
5	#107×Dic₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556658	—	2	2	no	—
7	C₂×C₄⋊#10596833	C₂²	1	2	yes	—
8	C₂×C₄×#10596833	C₂²	1	1	yes	characteristic
9	C₂×C₄×#10596833	C₂²	1	1	yes	characteristic
10	#17989453	—	2	2	no	—
11	#17989450	—	2	2	no	—
12	C₈×#10596833	—	2	2	no	—
13	C₂×#14556525	C₄	1	1	yes	characteristic
14	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×Dic₇₃	C₂²	1	2	yes	—
16	#17988349	C₄	1	2	yes	—
17	C₂×#107×C₇₃	C₄	1	1	yes	characteristic
18	C₂²×#10596833	—	2	2	no	—
19	C₂²×#10596833	—	2	1	no	—
20	C₂²×#10596833	D₄	1	1	yes	characteristic
21	C₂²×#10596833	D₄	1	1	yes	characteristic
22	#14556658	—	4	4	no	—
23	C₄⋊#10596833	—	2	2	no	—
24	C₄⋊#10596833	—	2	2	no	—
25	C₄×#10596833	—	2	1	no	—
26	C₄×#10596833	—	2	1	no	—
27	C₄×#10596833	—	2	1	no	—
28	C₄×#10596833	Q₈	1	2	yes	—
29	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄⋊Dic₇₃	C₂×C₄	1	2	yes	—
31	C₂×C₄×Dic₇₃	—	2	2	no	—
32	C₂×C₄×Dic₇₃	C₂³	1	1	yes	characteristic
33	#14556525	C₈	1	4	yes	complemented
34	C₈×Dic₇₃	D₄	1	2	yes	—
35	C₈×Dic₇₃	Q₈	1	2	yes	—
36	#14556498	—	2	2	no	—
37	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
39	#107×C₇₃	—	2	2	no	—
40	C₂×#10596833	—	2	2	no	—
41	C₂×#10596833	—	2	2	no	—
42	C₂×#10596833	—	2	1	no	—
43	C₂×#10596833	—	2	1	no	—
44	C₂×#10596833	—	2	2	no	—
45	C₂×#10596833	—	2	1	no	—
46	C₂×#10596833	—	2	1	no	—
47	C₂×#10596833	—	4	2	no	—
48	C₂×#10596833	—	4	2	no	—
49	C₂×#10596833	—	4	2	no	—
50	C₂×#10596833	D₈	1	2	yes	—
51	C₂×#10596833	Q₁₆	1	2	yes	—
52	C₂²×Dic₇₃	—	2	2	no	—
53	C₂²×Dic₇₃	C₂×D₄	1	1	yes	characteristic
54	C₂²×Dic₇₃	#41	1	1	yes	characteristic
55	C₂×#10596831	—	2	1	no	—
56	C₂×#10596831	—	2	1	no	—
57	C₂×#10596831	C₄⋊C₄	1	2	yes	—
58	C₄⋊Dic₇₃	—	2	2	no	—
59	C₄⋊Dic₇₃	C₂×C₈	1	4	yes	—
60	C₄×Dic₇₃	—	4	2	no	—
61	C₄×Dic₇₃	—	4	2	no	—
62	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
63	C₄×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
64	C₄×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
65	C₄×Dic₇₃	#41	1	1	yes	characteristic
66	C₂²×C₄×C₇₃	—	2	2	no	—
67	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
69	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
70	C₂×C₈×C₇₃	M₁₆	1	2	yes	—
71	C₄⋊C₄×C₇₃	—	2	2	no	—
72	C₄⋊C₄×C₇₃	—	2	2	no	—
73	C₂³×C₇₃	C₄×D₄	1	1	yes	characteristic
74	C₂×C₄×C₇₃	—	4	2	no	—
75	C₂×C₄×C₇₃	—	4	2	no	—
76	C₂×C₄×C₇₃	—	4	2	no	—
77	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₇₃	C₄×Q₈	1	1	yes	characteristic
79	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
80	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
81	C₂×C₄×C₇₃	#131	1	1	yes	characteristic
82	C₂×C₄×C₇₃	#131	1	1	yes	characteristic
83	C₂×Dic₇₃	—	2	1	no	—
84	C₂×Dic₇₃	—	2	1	no	—
85	C₂×Dic₇₃	—	2	4	no	—
86	C₂×Dic₇₃	—	4	2	no	—
87	C₂×Dic₇₃	—	4	2	no	—
88	C₂×Dic₇₃	#121	1	1	yes	characteristic
89	C₂×Dic₇₃	C₂²⋊Q₈	1	1	yes	characteristic
90	C₂×Dic₇₃	#125	1	1	yes	characteristic
91	C₂×Dic₇₃	C₄⋊Q₈	1	1	yes	characteristic
92	C₂×Dic₇₃	C₂×D₈	1	1	yes	characteristic
93	C₂×Dic₇₃	C₂×Q₁₆	1	1	yes	characteristic
94	C₂×Dic₇₃	#135	1	1	yes	characteristic
95	C₂×Dic₇₃	#135	1	1	yes	characteristic
96	#10596833	—	2	4	no	—
97	#10596833	—	2	2	no	—
98	#10596833	—	2	2	no	—
99	#10596833	—	8	2	no	—
100	C₈×C₇₃	C₄⋊C₈	1	4	yes	complemented
101	#10596831	—	2	2	no	—
102	C₂²×C₇₃	#389	1	1	yes	characteristic
103	C₂²×C₇₃	C₈×D₄	1	1	yes	characteristic
104	C₂²×C₇₃	#436	1	1	yes	characteristic
105	C₂²×C₇₃	C₄×D₈	1	1	yes	characteristic
106	C₂²×C₇₃	C₄×Q₁₆	1	1	yes	characteristic
107	C₂²×C₇₃	#443	1	1	yes	characteristic
108	C₂²×C₇₃	#443	1	1	yes	characteristic
109	C₄×C₇₃	—	8	2	no	—
110	C₄×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
111	C₄×C₇₃	#424	1	1	yes	characteristic
112	C₄×C₇₃	C₈×Q₈	1	1	yes	characteristic
113	C₄×C₇₃	#446	1	1	yes	characteristic
114	Dic₇₃	—	2	1	no	—
115	Dic₇₃	—	2	1	no	—
116	Dic₇₃	—	4	4	no	—
117	Dic₇₃	#466	1	1	yes	characteristic
118	Dic₇₃	#467	1	1	yes	characteristic
119	Dic₇₃	#499	1	1	yes	characteristic
120	Dic₇₃	#500	1	1	yes	characteristic
121	#9594	—	73	1	no	maximal, Sylow
122	C₂×C₇₃	C₈×D₈	1	1	yes	characteristic
123	C₂×C₇₃	C₈×Q₁₆	1	1	yes	characteristic
124	C₂×C₇₃	#1589	1	1	yes	characteristic
125	C₂×C₇₃	#1591	1	1	yes	characteristic
126	C₂×C₇₃	#1939	1	1	yes	characteristic
127	C₂×C₇₃	#1942	1	1	yes	characteristic
128	C₂×C₇₃	#1945	1	1	yes	characteristic
129	#1939	—	73	2	no	—
130	C₄×#107	—	73	1	no	—
131	C₂×#335	—	73	1	no	—
132	C₂×C₈²	—	73	1	no	—
133	#1279	—	73	2	no	—
134	C₇₃	#9594	1	1	yes	characteristic, complemented, Sylow
135	C₂×#107	—	73	1	no	—
136	C₂×#107	#64408	1	1	yes	characteristic
137	C₂×C₄⋊C₈	—	73	2	no	—
138	C₂×C₄⋊C₈	—	146	2	no	—
139	C₂×C₄×C₈	—	73	1	no	—
140	C₂×C₄×C₈	—	73	1	no	—
141	C₂×C₄×C₈	—	73	1	no	—
142	C₄×C₄⋊C₄	—	73	2	no	—
143	#340	—	73	2	no	—
144	#336	—	146	2	no	—
145	#335	—	146	2	no	—
146	C₈²	—	146	2	no	—
147	C₂²×C₈	—	73	1	no	—
148	C₂²×C₈	—	73	1	no	—
149	C₂²×C₈	—	73	1	no	—
150	C₂²×C₈	—	146	2	no	—
151	C₂²×C₈	—	146	1	no	—
152	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
153	C₂×C₄⋊C₄	—	73	2	no	—
154	C₂×C₄⋊C₄	C₂×#64408	1	2	yes	—
155	C₂×C₄²	—	73	1	no	—
156	C₂×C₄²	—	146	2	no	—
157	#107	—	2	2	no	—
158	#107	—	73	4	no	—
159	C₄⋊C₈	—	292	4	no	—
160	C₄⋊C₈	—	146	2	no	—
161	C₄⋊C₈	—	146	2	no	—
162	C₄×C₈	—	73	2	no	—
163	C₄×C₈	—	73	2	no	—
164	C₄×C₈	—	73	2	no	—
165	C₄×C₈	—	146	1	no	—
166	C₄×C₈	—	146	1	no	—
167	C₄×C₈	—	146	1	no	—
168	#95	—	146	2	no	—
169	C₂²×C₄	—	2	2	no	—
170	C₂²×C₄	—	73	1	no	—
171	C₂²×C₄	—	73	1	no	—
172	C₂²×C₄	—	146	2	no	—
173	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
174	C₂×C₈	—	292	2	no	—
175	C₂×C₈	—	292	2	no	—
176	C₂×C₈	—	292	2	no	—
177	C₂×C₈	—	73	2	no	—
178	C₂×C₈	—	73	2	no	—
179	C₂×C₈	—	73	2	no	—
180	C₂×C₈	—	146	2	no	—
181	C₂×C₈	—	146	1	no	—
182	C₂×C₈	—	146	1	no	—
183	C₂×C₈	—	146	1	no	—
184	C₂×C₈	—	146	1	no	—
185	C₂×C₈	—	146	2	no	—
186	C₂×C₈	—	146	2	no	—
187	C₂×C₈	—	146	1	no	—
188	C₂×C₈	—	146	1	no	—
189	C₂×C₈	C₄⋊#64408	1	2	yes	—
190	C₂×C₈	#11924036	1	2	yes	—
191	C₂×C₈	#11924037	1	2	yes	—
192	C₄⋊C₄	—	2	2	no	—
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	—	73	4	no	—
195	C₄⋊C₄	—	146	2	no	—
196	C₄²	—	292	2	no	—
197	C₄²	—	292	2	no	—
198	C₄²	—	73	1	no	—
199	C₄²	—	73	1	no	—
200	C₄²	—	73	1	no	—
201	C₄²	—	73	1	no	—
202	C₂³	D₄×#64408	1	1	yes	characteristic
203	C₂×C₄	—	4	2	no	—
204	C₂×C₄	—	4	2	no	—
205	C₂×C₄	—	4	2	no	—
206	C₂×C₄	—	292	2	no	—
207	C₂×C₄	—	292	2	no	—
208	C₂×C₄	—	73	1	no	—
209	C₂×C₄	—	73	1	no	—
210	C₂×C₄	—	73	1	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	—	146	4	no	—
217	C₂×C₄	—	146	1	no	—
218	C₂×C₄	—	146	1	no	—
219	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
220	C₂×C₄	Q₈×#64408	1	1	yes	characteristic
221	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
222	C₂×C₄	C₂×#11924037	1	1	yes	characteristic
223	C₂×C₄	#14556678	1	1	yes	characteristic
224	C₂×C₄	#14556679	1	1	yes	characteristic
225	C₈	—	584	2	no	—
226	C₈	—	146	2	no	—
227	C₈	—	146	2	no	—
228	C₈	—	146	4	no	—
229	C₈	—	146	2	no	—
230	C₈	#14556658	1	4	yes	complemented
231	C₂²	#17989538	1	1	yes	characteristic
232	C₂²	#17989615	1	1	yes	characteristic
233	C₂²	#17989622	1	1	yes	characteristic
234	C₂²	D₈×#64408	1	1	yes	characteristic
235	C₂²	Q₁₆×#64408	1	1	yes	characteristic
236	C₂²	#17989629	1	1	yes	characteristic
237	C₂²	#17989631	1	1	yes	characteristic
238	C₄	—	8	2	no	—
239	C₄	—	292	4	no	—
240	C₄	—	73	1	no	—
241	C₄	—	73	1	no	—
242	C₄	—	73	1	no	—
243	C₄	—	73	1	no	—
244	C₄	—	146	1	no	—
245	C₄	—	146	1	no	—
246	C₄	C₂×#14556658	1	1	yes	characteristic
247	C₄	#17989594	1	1	yes	characteristic
248	C₄	#17989636	1	1	yes	characteristic
249	C₄	#17989638	1	1	yes	characteristic
250	C₂	#22625512	1	1	yes	characteristic
251	C₂	#22625519	1	1	yes	characteristic
252	C₂	#22625564	1	1	yes	characteristic
253	C₂	#22625571	1	1	yes	characteristic
254	C₂	#22626029	1	1	yes	characteristic
255	C₂	#22626035	1	1	yes	characteristic
256	C₂	#22626039	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₄×#64408
commutator	C₂×C₄×C₇₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#64408
Fitting	C₂×#107×C₇₃	C₄
socle, prime radical	C₂³×C₇₃	C₄×D₄
cosocle	C₂×C₄×Dic₇₃	C₂³
hypercenter	C₂×#107	#64408
hypocenter	C₇₃	#9594
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

Complements

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

N	H ≅ G/N	type
33. #14556525	C₈	semidirect
100. C₈×C₇₃	C₄⋊C₈	semidirect
134. C₇₃	#9594	semidirect
230. C₈	#14556658	semidirect

Maximal subgroups

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

subgroup	index
#22626029	2
C₂×#17989450	2
C₂×C₈×#10596833	2
#22624767	2
#107×Dic₇₃	2
#9594	73