Subgroups — #34000280

Order 18688.

274 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17989476	C₂	1	1	yes	maximal, characteristic
2	C₂×#17989448	C₂	1	1	yes	maximal, characteristic
3	#107×Dic₇₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11924031	—	2	1	no	—
5	C₂×C₄⋊#10596833	—	2	1	no	—
6	#17989476	—	2	1	no	—
7	#17989476	C₄	1	2	yes	complemented
8	#17989448	—	2	1	no	—
9	#17989448	C₄	1	2	yes	complemented
10	C₂×#14556525	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
12	C₄⋊C₄×Dic₇₃	—	2	1	no	—
13	#17988349	—	2	1	no	—
14	C₂×#107×C₇₃	C₄	1	1	yes	characteristic
15	C₂²×#10596833	—	4	1	no	—
16	C₂×#11924031	—	2	2	no	—
17	C₂×#11924031	—	2	2	no	—
18	C₂×#11924031	—	2	2	no	—
19	C₄⋊#10596833	—	2	2	no	—
20	C₄⋊#10596833	—	4	1	no	—
21	C₂²×#10596831	D₄	1	1	yes	characteristic
22	C₂×C₄⋊Dic₇₃	—	2	1	no	—
23	C₂×C₄×Dic₇₃	—	4	1	no	—
24	C₂×C₄×Dic₇₃	D₄	1	1	yes	characteristic
25	#14556525	—	2	1	no	—
26	#14556525	C₈	1	2	yes	complemented
27	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
28	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
29	C₈×Dic₇₃	D₄	1	1	yes	characteristic
30	C₈×Dic₇₃	Q₈	1	1	yes	characteristic
31	#14556498	—	4	1	no	—
32	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
34	#107×C₇₃	—	2	1	no	—
35	#107×C₇₃	C₈	1	2	yes	complemented
36	C₂×#10596833	—	4	2	no	—
37	C₂×#10596833	—	8	1	no	—
38	C₂×#10596833	—	8	1	no	—
39	#11924031	—	2	4	no	—
40	C₂²×Dic₇₃	—	4	1	no	—
41	C₂²×Dic₇₃	D₈	1	1	yes	characteristic
42	C₂²×Dic₇₃	SD₁₆	1	1	yes	characteristic
43	C₂×#10596831	—	2	1	no	—
44	C₂×#10596831	—	2	1	no	—
45	C₂×#10596831	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
47	C₄⋊Dic₇₃	—	2	2	no	—
48	C₄⋊Dic₇₃	—	4	1	no	—
49	C₄×Dic₇₃	—	8	2	no	—
50	C₄×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
51	C₄×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
52	C₄×Dic₇₃	SD₁₆	1	1	yes	characteristic
53	C₄×Dic₇₃	Q₁₆	1	1	yes	characteristic
54	C₂²×C₄×C₇₃	—	4	1	no	—
55	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
57	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
59	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
60	C₂×C₈×C₇₃	M₁₆	1	1	yes	characteristic
61	C₂×C₈×C₇₃	M₁₆	1	1	yes	characteristic
62	C₄⋊C₄×C₇₃	—	2	2	no	—
63	C₄⋊C₄×C₇₃	—	4	1	no	—
64	C₂³×C₇₃	#102	1	1	yes	characteristic
65	C₂×C₄×C₇₃	—	4	2	no	—
66	C₂×C₄×C₇₃	—	8	1	no	—
67	C₂×C₄×C₇₃	—	8	1	no	—
68	C₂×C₄×C₇₃	#95	1	1	yes	characteristic
69	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
70	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
71	C₂×C₄×C₇₃	#103	1	1	yes	characteristic
72	C₂×C₄×C₇₃	#104	1	1	yes	characteristic
73	C₂×C₄×C₇₃	#104	1	1	yes	characteristic
74	C₂×Dic₇₃	—	2	1	no	—
75	C₂×Dic₇₃	—	2	1	no	—
76	C₂×Dic₇₃	—	2	1	no	—
77	C₂×Dic₇₃	—	2	1	no	—
78	C₂×Dic₇₃	—	4	2	no	—
79	C₂×Dic₇₃	—	8	1	no	—
80	C₂×Dic₇₃	—	8	1	no	—
81	C₂×Dic₇₃	#102	1	1	yes	characteristic
82	C₂×Dic₇₃	#103	1	1	yes	characteristic
83	C₂×Dic₇₃	#106	1	1	yes	characteristic
84	C₂×Dic₇₃	#107	1	1	yes	characteristic
85	#10596833	—	8	2	no	—
86	C₈×C₇₃	C₄×C₈	1	1	yes	characteristic
87	C₈×C₇₃	#97	1	1	yes	characteristic
88	C₈×C₇₃	C₄⋊C₈	1	1	yes	characteristic
89	C₈×C₇₃	C₄⋊C₈	1	1	yes	characteristic
90	#10596831	—	2	1	no	—
91	#10596831	—	2	1	no	—
92	C₂²×C₇₃	D₄⋊C₈	1	1	yes	characteristic
93	C₂²×C₇₃	D₄⋊C₈	1	1	yes	characteristic
94	C₂²×C₇₃	#340	1	1	yes	characteristic
95	C₂²×C₇₃	#360	1	1	yes	characteristic
96	C₂²×C₇₃	#360	1	1	yes	characteristic
97	C₂²×C₇₃	#361	1	1	yes	characteristic
98	C₂²×C₇₃	#362	1	1	yes	characteristic
99	C₄×C₇₃	—	8	2	no	—
100	C₄×C₇₃	Q₈⋊C₈	1	1	yes	characteristic
101	C₄×C₇₃	Q₈⋊C₈	1	1	yes	characteristic
102	C₄×C₇₃	#336	1	1	yes	characteristic
103	C₄×C₇₃	#339	1	1	yes	characteristic
104	Dic₇₃	—	2	1	no	—
105	Dic₇₃	—	2	1	no	—
106	Dic₇₃	—	2	1	no	—
107	Dic₇₃	—	2	1	no	—
108	Dic₇₃	—	8	2	no	—
109	#7178	—	73	1	no	maximal, Sylow
110	C₂×C₇₃	#1279	1	1	yes	characteristic
111	C₂×C₇₃	#1333	1	1	yes	characteristic
112	C₂×C₇₃	#1333	1	1	yes	characteristic
113	C₂×C₇₃	#1334	1	1	yes	characteristic
114	C₂×C₇₃	#1334	1	1	yes	characteristic
115	C₂×C₇₃	#1386	1	1	yes	characteristic
116	C₂×C₇₃	#1387	1	1	yes	characteristic
117	C₂×#346	—	73	1	no	—
118	C₄×#107	—	73	1	no	—
119	C₂×#334	—	73	1	no	—
120	C₇₃	#7178	1	1	yes	characteristic, complemented, Sylow
121	C₂²×C₁₆	—	146	1	no	—
122	C₂×#107	—	73	1	no	—
123	C₂×#107	#64408	1	1	yes	characteristic
124	C₂×C₄⋊C₈	—	146	1	no	—
125	C₂×C₄×C₈	—	73	1	no	—
126	C₄×C₄⋊C₄	—	146	1	no	—
127	#346	—	73	2	no	—
128	#346	—	146	1	no	—
129	#340	—	146	1	no	—
130	#334	—	73	2	no	—
131	#334	—	146	1	no	—
132	C₂²×C₈	—	292	1	no	—
133	C₂²×C₈	—	73	1	no	—
134	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
135	C₂×C₄⋊C₄	—	2	1	no	—
136	C₂×C₄⋊C₄	—	146	1	no	—
137	C₂×C₄²	—	292	1	no	—
138	C₂×C₄²	—	73	1	no	—
139	C₂×C₁₆	—	146	2	no	—
140	C₂×C₁₆	—	146	2	no	—
141	C₂×C₁₆	—	146	2	no	—
142	#107	—	2	1	no	—
143	#107	—	73	2	no	—
144	#107	—	146	1	no	—
145	#107	#10596833	1	2	yes	complemented
146	C₄⋊C₈	—	292	1	no	—
147	C₄⋊C₈	—	146	2	no	—
148	C₄×C₈	—	73	1	no	—
149	C₄×C₈	—	73	1	no	—
150	C₄×C₈	—	73	1	no	—
151	C₄×C₈	—	73	1	no	—
152	#95	—	292	1	no	—
153	C₂²×C₄	—	4	1	no	—
154	C₂²×C₄	—	292	1	no	—
155	C₂²×C₄	—	73	1	no	—
156	C₂²×C₄	—	73	1	no	—
157	C₂²×C₄	C₂²⋊#64408	1	1	yes	characteristic
158	C₂×C₈	—	292	2	no	—
159	C₂×C₈	—	584	1	no	—
160	C₂×C₈	—	584	1	no	—
161	C₂×C₈	—	73	1	no	—
162	C₂×C₈	—	73	1	no	—
163	C₂×C₈	—	146	1	no	—
164	C₂×C₈	—	146	1	no	—
165	C₂×C₈	C₄×#64408	1	1	yes	characteristic
166	C₂×C₈	C₄⋊#64408	1	1	yes	characteristic
167	C₂×C₈	C₂×#10596833	1	1	yes	characteristic
168	C₂×C₈	#11924036	1	1	yes	characteristic
169	C₂×C₈	#11924037	1	1	yes	characteristic
170	C₂×C₈	#11924038	1	1	yes	characteristic
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	4	1	no	—
173	C₄⋊C₄	—	292	1	no	—
174	C₄⋊C₄	—	146	2	no	—
175	C₄²	—	584	2	no	—
176	C₄²	—	73	1	no	—
177	C₄²	—	73	1	no	—
178	C₄²	—	73	1	no	—
179	C₄²	—	73	1	no	—
180	C₁₆	—	146	4	no	—
181	C₂³	#14556653	1	1	yes	characteristic
182	C₂×C₄	—	4	2	no	—
183	C₂×C₄	—	8	1	no	—
184	C₂×C₄	—	8	1	no	—
185	C₂×C₄	—	292	2	no	—
186	C₂×C₄	—	584	1	no	—
187	C₂×C₄	—	584	1	no	—
188	C₂×C₄	—	73	1	no	—
189	C₂×C₄	—	73	1	no	—
190	C₂×C₄	—	73	1	no	—
191	C₂×C₄	—	73	1	no	—
192	C₂×C₄	—	146	1	no	—
193	C₂×C₄	—	146	1	no	—
194	C₂×C₄	—	146	1	no	—
195	C₂×C₄	—	146	1	no	—
196	C₂×C₄	(C₂×#64408)⋊C₄	1	1	yes	characteristic
197	C₂×C₄	C₂²⋊#10596833	1	1	yes	characteristic
198	C₂×C₄	#14556648	1	1	yes	characteristic
199	C₂×C₄	#14556654	1	1	yes	characteristic
200	C₂×C₄	#14556655	1	1	yes	characteristic
201	C₂×C₄	#14556656	1	1	yes	characteristic
202	C₈	—	584	2	no	—
203	C₈	—	146	1	no	—
204	C₈	—	146	1	no	—
205	C₈	C₄×#10596833	1	1	yes	characteristic
206	C₈	#14556646	1	1	yes	characteristic
207	C₈	C₄⋊#10596833	1	1	yes	characteristic
208	C₈	#14556658	1	1	yes	characteristic
209	C₂²	D₄⋊#10596833	1	1	yes	characteristic
210	C₂²	#17989435	1	1	yes	characteristic
211	C₂²	#17989460	1	1	yes	characteristic
212	C₂²	#17989503	1	1	yes	characteristic
213	C₂²	#17989504	1	1	yes	characteristic
214	C₂²	#17989505	1	1	yes	characteristic
215	C₂²	#17989506	1	1	yes	characteristic
216	C₄	—	8	2	no	—
217	C₄	—	584	2	no	—
218	C₄	—	146	1	no	—
219	C₄	—	146	1	no	—
220	C₄	—	146	1	no	—
221	C₄	—	146	1	no	—
222	C₄	Q₈⋊#10596833	1	1	yes	characteristic
223	C₄	#17989437	1	1	yes	characteristic
224	C₄	#17989453	1	1	yes	characteristic
225	C₄	#17989457	1	1	yes	characteristic
226	C₂	#22624767	1	1	yes	characteristic
227	C₂	#22624889	1	1	yes	characteristic
228	C₂	#22624890	1	1	yes	characteristic
229	C₂	#22624891	1	1	yes	characteristic
230	C₂	#22624892	1	1	yes	characteristic
231	C₂	#22625002	1	1	yes	characteristic
232	C₂	#22625005	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14556653
commutator	C₈×C₇₃	C₄×C₈
Frattini	C₂²×C₈	C₂×#64408
Fitting	C₂×#107×C₇₃	C₄
socle, prime radical	C₂³×C₇₃	#102
cosocle	C₂×C₈×Dic₇₃	C₂²
hypercenter	C₂×#107	#64408
hypocenter	C₇₃	#7178
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#107

Complements

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

N	H ≅ G/N	type
7. #17989476	C₄	semidirect
9. #17989448	C₄	semidirect
26. #14556525	C₈	semidirect
35. #107×C₇₃	C₈	semidirect
120. C₇₃	#7178	semidirect
145. #107	#10596833	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17989476	2
C₂×#17989448	2
#107×Dic₇₃	2
#7178	73