Subgroups — #34034980

Order 18688.

444 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22627158	C₂	1	4	yes	maximal, complemented
2	#22627157	C₂	1	4	yes	maximal, complemented
3	#22627137	C₂	1	4	yes	maximal
4	#22627135	C₂	1	1	yes	maximal, characteristic
5	#22627135	C₂	1	1	yes	maximal, characteristic
6	#22626720	C₂	1	1	yes	maximal, characteristic
7	#17989690	C₂²	1	2	yes	—
8	#17989690	C₂²	1	2	yes	—
9	#17989690	C₂²	1	4	yes	—
10	C₄⋊#11924040	C₂²	1	4	yes	—
11	#17989684	C₂²	1	2	yes	—
12	#17989684	C₂²	1	2	yes	—
13	#17989681	C₂²	1	2	yes	—
14	#17989681	C₂²	1	2	yes	—
15	#17989681	C₂²	1	4	yes	—
16	C₄×#11924040	C₂²	1	4	yes	—
17	#17989638	C₄	1	2	yes	—
18	#17989637	C₄	1	2	yes	—
19	#17989636	C₂²	1	2	yes	—
20	Q₈×#10596833	C₂²	1	2	yes	—
21	#17989594	C₄	1	2	yes	—
22	#17989591	C₄	1	1	yes	characteristic
23	#17989552	C₂²	1	2	yes	—
24	C₄×#11924036	C₂²	1	1	yes	characteristic
25	#17989142	C₄	1	1	yes	characteristic
26	C₂×#11924040	C₂³	1	4	yes	—
27	C₂×#11924040	C₂³	1	4	yes	—
28	C₂×#11924038	C₂×C₄	1	2	yes	—
29	C₂×#11924036	C₂×C₄	1	1	yes	characteristic
30	C₂×#11924036	C₂³	1	1	yes	characteristic
31	#14556658	C₂×C₄	1	4	yes	—
32	C₄⋊#10596833	C₂×C₄	1	4	yes	—
33	C₄⋊#10596833	C₂×C₄	1	2	yes	—
34	C₄⋊#10596833	C₂×C₄	1	2	yes	—
35	#14556646	C₂×C₄	1	4	yes	—
36	C₄×#10596833	C₂×C₄	1	1	yes	characteristic
37	C₄×#10596833	C₂×C₄	1	1	yes	characteristic
38	C₄×#10596833	C₂³	1	1	yes	characteristic
39	C₄×#10596833	C₂³	1	4	yes	—
40	C₄×#10596833	C₂³	1	1	yes	characteristic
41	#14556591	C₈	1	1	yes	characteristic
42	#14556582	C₈	1	2	yes	—
43	#14556571	C₈	1	2	yes	—
44	#14556567	C₈	1	2	yes	—
45	Q₈×Dic₇₃	C₂×C₄	1	2	yes	—
46	#14556556	C₂×C₄	1	2	yes	—
47	#14556549	C₂×C₄	1	2	yes	—
48	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
49	C₄⋊Q₈×C₇₃	C₈	1	1	yes	characteristic
50	#11924040	—	2	4	no	—
51	#11924040	—	2	4	no	—
52	#11924038	—	4	2	no	—
53	#11924036	—	2	2	no	—
54	#11924036	—	4	1	no	—
55	C₂×#10596833	C₂²×C₄	1	1	yes	characteristic
56	C₂×#10596833	C₂²×C₄	1	1	yes	characteristic
57	C₂×#10596833	C₂²×C₄	1	1	yes	characteristic
58	C₂×#10596833	C₂²×C₄	1	4	yes	—
59	C₂×#10596833	C₂⁴	1	1	yes	characteristic
60	C₂×Dic₁₄₆	C₂×C₈	1	2	yes	—
61	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
62	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
63	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
64	C₄⋊Dic₇₃	C₂×C₈	1	2	yes	—
65	C₄⋊Dic₇₃	C₂×C₈	1	2	yes	—
66	C₄⋊Dic₇₃	C₂×C₈	1	4	yes	—
67	#11924007	C₂×C₈	1	4	yes	—
68	#11924005	C₂×C₈	1	4	yes	—
69	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
70	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
71	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
72	C₄×Dic₇₃	C₂²×C₄	1	4	yes	—
73	C₂×Q₈×C₇₃	C₂×C₈	1	2	yes	—
74	C₄⋊C₄×C₇₃	C₂×C₈	1	2	yes	—
75	C₄⋊C₄×C₇₃	C₂×C₈	1	2	yes	—
76	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
77	C₂²×D₇₃	C₂²×C₈	1	1	yes	characteristic
78	Q₈×C₇₃	—	4	2	no	—
79	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
80	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
81	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
82	C₂×C₄×C₇₃	C₂²×C₈	1	4	yes	—
83	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
84	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
85	C₂×Dic₇₃	C₂²×C₈	1	4	yes	—
86	C₂×Dic₇₃	C₂³×C₄	1	1	yes	characteristic
87	C₄×D₇₃	—	2	1	no	—
88	C₄×D₇₃	—	2	1	no	—
89	C₄×D₇₃	—	4	1	no	—
90	C₄×D₇₃	—	4	1	no	—
91	Dic₁₄₆	—	4	2	no	—
92	#10596833	—	2	1	no	—
93	#10596833	—	2	1	no	—
94	#10596833	—	2	1	no	—
95	#10596833	—	2	4	no	—
96	#10596833	2₋¹⁺⁴	1	2	yes	—
97	C₂²×C₇₃	C₂³×C₈	1	1	yes	characteristic
98	C₂×D₇₃	—	2	1	no	—
99	C₂×D₇₃	—	2	1	no	—
100	C₂×D₇₃	—	2	1	no	—
101	C₄×C₇₃	—	2	1	no	—
102	C₄×C₇₃	—	2	1	no	—
103	C₄×C₇₃	—	2	4	no	—
104	C₄×C₇₃	—	2	1	no	—
105	Dic₇₃	—	2	4	no	—
106	Dic₇₃	—	2	1	no	—
107	Dic₇₃	—	2	1	no	—
108	Dic₇₃	#519	1	1	yes	characteristic
109	Dic₇₃	#568	1	1	yes	characteristic
110	#33501	—	73	1	no	maximal, Sylow
111	C₂×C₇₃	#2982	1	1	yes	characteristic
112	C₂×C₇₃	#3407	1	1	yes	characteristic
113	C₂×C₇₃	#3407	1	1	yes	characteristic
114	D₇₃	—	4	1	no	—
115	#2960	—	73	1	no	—
116	#2183	—	73	4	no	—
117	#2183	—	73	4	no	—
118	#2164	—	73	4	no	—
119	#2163	—	73	1	no	—
120	#2163	—	73	1	no	—
121	C₇₃	#33501	1	1	yes	characteristic, complemented, Sylow
122	#533	—	73	1	no	—
123	#446	—	73	2	no	—
124	#446	—	73	2	no	—
125	C₈×Q₈	—	73	2	no	—
126	C₈×Q₈	—	73	2	no	—
127	#424	—	73	2	no	—
128	#423	—	73	1	no	—
129	#405	—	73	2	no	—
130	C₂×C₄×C₈	—	73	1	no	—
131	C₄⋊C₁₆	—	73	4	no	—
132	C₄⋊C₁₆	—	73	2	no	—
133	C₄⋊C₁₆	—	73	2	no	—
134	C₄⋊C₁₆	—	73	4	no	—
135	C₂²⋊C₁₆	—	73	2	no	—
136	C₂²⋊C₁₆	—	73	2	no	—
137	#346	—	73	2	no	—
138	#346	—	73	2	no	—
139	#346	—	73	4	no	—
140	C₄×C₁₆	—	73	4	no	—
141	C₂×M₁₆	—	73	2	no	—
142	C₂²×C₈	—	73	1	no	—
143	C₂²×C₈	—	73	1	no	—
144	C₄⋊Q₈	—	73	1	no	—
145	C₄⋊Q₈	#10596842	1	1	yes	characteristic
146	#125	—	73	2	no	—
147	C₂²⋊Q₈	—	73	2	no	—
148	C₂²⋊Q₈	—	73	2	no	—
149	C₄×Q₈	—	73	2	no	—
150	C₄×Q₈	—	73	2	no	—
151	#117	—	73	2	no	—
152	C₂×C₄²	—	73	1	no	—
153	C₂×C₁₆	—	73	4	no	—
154	C₂×C₁₆	—	73	4	no	—
155	C₄⋊C₈	—	73	2	no	—
156	C₄⋊C₈	—	73	2	no	—
157	C₄⋊C₈	—	73	4	no	—
158	C₄⋊C₈	—	73	4	no	—
159	#97	—	73	4	no	—
160	C₄×C₈	—	73	4	no	—
161	C₄×C₈	—	73	1	no	—
162	C₄×C₈	—	73	1	no	—
163	C₄×C₈	—	73	1	no	—
164	C₄×C₈	—	73	1	no	—
165	C₂×Q₈	—	73	2	no	—
166	C₂×Q₈	C₂×#10596842	1	2	yes	—
167	C₂²×C₄	—	73	1	no	—
168	C₂²×C₄	—	73	1	no	—
169	C₂²×C₄	—	73	1	no	—
170	M₁₆	—	292	2	no	—
171	C₂×C₈	—	292	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	—	73	4	no	—
177	C₂×C₈	—	146	2	no	—
178	C₄⋊C₄	—	73	4	no	—
179	C₄⋊C₄	—	73	2	no	—
180	C₄⋊C₄	—	73	4	no	—
181	C₄⋊C₄	—	73	2	no	—
182	C₄⋊C₄	C₂×#10596842	1	2	yes	—
183	C₄⋊C₄	C₂×#10596842	1	2	yes	—
184	C₂²⋊C₄	—	73	4	no	—
185	C₄²	—	73	1	no	—
186	C₄²	—	73	1	no	—
187	C₄²	—	73	1	no	—
188	C₄²	—	73	4	no	—
189	C₄²	C₂×#10596842	1	1	yes	characteristic
190	C₁₆	—	146	4	no	—
191	C₁₆	—	146	4	no	—
192	C₂³	—	73	1	no	—
193	Q₈	—	4	2	no	—
194	Q₈	—	292	2	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	4	no	—
201	C₂×C₄	—	146	1	no	—
202	C₂×C₄	—	146	1	no	—
203	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
204	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
205	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
206	C₂×C₄	C₂²×#10596842	1	4	yes	—
207	C₈	—	73	2	no	—
208	C₈	—	146	1	no	—
209	C₈	—	146	1	no	—
210	C₈	—	146	1	no	—
211	C₈	—	146	4	no	—
212	C₂²	—	146	1	no	—
213	C₂²	—	146	1	no	—
214	C₂²	—	146	1	no	—
215	C₂²	C₂³×#10596842	1	1	yes	characteristic
216	C₄	—	2	1	no	—
217	C₄	—	2	1	no	—
218	C₄	—	2	4	no	—
219	C₄	—	2	1	no	—
220	C₄	—	73	1	no	—
221	C₄	—	73	1	no	—
222	C₄	—	146	1	no	—
223	C₄	—	146	1	no	—
224	C₄	—	146	4	no	—
225	C₂	—	292	1	no	—
226	C₂	#22627174	1	1	yes	characteristic
227	C₂	#22627184	1	1	yes	characteristic
228	C₂	#22627186	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#10596842
commutator, socle	C₂²×C₇₃	C₂³×C₈
Fitting	C₄⋊Q₈×C₇₃	C₈
prime radical	C₂²×D₇₃	C₂²×C₈
cosocle	C₂×#10596833	C₂⁴
hypercenter	C₄⋊Q₈	#10596842
hypocenter	C₇₃	#33501
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. #22627158	C₂	semidirect
2. #22627157	C₂	semidirect
121. C₇₃	#33501	semidirect

Maximal subgroups

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

subgroup	index
#22627158	2
#22627157	2
#22627137	2
#22627135	2
#22626720	2
#33501	73