Subgroups — #34034974

Order 18688.

452 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22627157	C₂	1	4	yes	maximal, complemented
2	#22627156	C₂	1	2	yes	maximal, complemented
3	#22627155	C₂	1	2	yes	maximal, complemented
4	#22627137	C₂	1	4	yes	maximal
5	#22627134	C₂	1	1	yes	maximal, characteristic
6	#22627130	C₂	1	1	yes	maximal, characteristic
7	#22626720	C₂	1	1	yes	maximal, characteristic
8	#17989690	C₂²	1	2	yes	—
9	#17989690	C₂²	1	4	yes	—
10	C₄⋊#11924040	C₂²	1	2	yes	—
11	C₄⋊#11924040	C₂²	1	4	yes	—
12	#17989684	C₂²	1	2	yes	—
13	#17989684	C₂²	1	2	yes	—
14	#17989681	C₂²	1	2	yes	—
15	#17989681	C₂²	1	4	yes	—
16	C₄×#11924040	C₂²	1	1	yes	characteristic
17	C₄×#11924040	C₂²	1	1	yes	characteristic
18	C₄×#11924040	C₂²	1	4	yes	—
19	#17989638	C₄	1	2	yes	—
20	#17989637	C₄	1	2	yes	—
21	#17989636	C₂²	1	2	yes	—
22	Q₈×#10596833	C₂²	1	2	yes	—
23	#17989594	C₄	1	2	yes	—
24	#17989591	C₄	1	1	yes	characteristic
25	#17989552	C₂²	1	2	yes	—
26	C₄×#11924036	C₂²	1	1	yes	characteristic
27	#17989142	C₄	1	1	yes	characteristic
28	C₂×#11924040	C₂³	1	2	yes	—
29	C₂×#11924040	C₂³	1	2	yes	—
30	C₂×#11924040	C₂³	1	4	yes	—
31	C₂×#11924038	C₂×C₄	1	2	yes	—
32	C₂×#11924036	C₂×C₄	1	1	yes	characteristic
33	C₂×#11924036	C₂³	1	1	yes	characteristic
34	#14556658	C₂×C₄	1	4	yes	—
35	C₄⋊#10596833	C₂×C₄	1	4	yes	—
36	C₄⋊#10596833	C₂×C₄	1	2	yes	—
37	C₄⋊#10596833	C₂×C₄	1	2	yes	—
38	#14556646	C₂×C₄	1	4	yes	—
39	C₄×#10596833	C₂×C₄	1	1	yes	characteristic
40	C₄×#10596833	C₂×C₄	1	1	yes	characteristic
41	C₄×#10596833	C₂³	1	1	yes	characteristic
42	C₄×#10596833	C₂³	1	4	yes	—
43	C₄×#10596833	C₂³	1	1	yes	characteristic
44	#14556591	C₈	1	1	yes	characteristic
45	#14556582	C₈	1	2	yes	—
46	#14556571	C₈	1	2	yes	—
47	#14556567	C₈	1	2	yes	—
48	Q₈×Dic₇₃	C₂×C₄	1	2	yes	—
49	#14556556	C₂×C₄	1	2	yes	—
50	#14556549	C₂×C₄	1	2	yes	—
51	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
52	C₄⋊Q₈×C₇₃	C₈	1	1	yes	characteristic
53	#11924040	—	2	2	no	—
54	#11924040	—	2	2	no	—
55	#11924040	—	2	4	no	—
56	#11924038	—	4	2	no	—
57	#11924036	—	2	2	no	—
58	#11924036	—	4	1	no	—
59	C₂×#10596833	C₂²×C₄	1	1	yes	characteristic
60	C₂×#10596833	C₂²×C₄	1	1	yes	characteristic
61	C₂×#10596833	C₂²×C₄	1	1	yes	characteristic
62	C₂×#10596833	C₂²×C₄	1	4	yes	—
63	C₂×#10596833	C₂⁴	1	1	yes	characteristic
64	C₂×Dic₁₄₆	C₂×C₈	1	2	yes	—
65	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
66	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
67	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
68	C₄⋊Dic₇₃	C₂×C₈	1	2	yes	—
69	C₄⋊Dic₇₃	C₂×C₈	1	2	yes	—
70	C₄⋊Dic₇₃	C₂×C₈	1	4	yes	—
71	#11924007	C₂×C₈	1	4	yes	—
72	#11924005	C₂×C₈	1	4	yes	—
73	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
74	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
75	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
76	C₄×Dic₇₃	C₂²×C₄	1	4	yes	—
77	C₂×Q₈×C₇₃	C₂×C₈	1	2	yes	—
78	C₄⋊C₄×C₇₃	C₂×C₈	1	2	yes	—
79	C₄⋊C₄×C₇₃	C₂×C₈	1	2	yes	—
80	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
81	C₂²×D₇₃	C₂²×C₈	1	1	yes	characteristic
82	Q₈×C₇₃	—	4	2	no	—
83	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
84	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
85	C₂×C₄×C₇₃	C₂²×C₈	1	4	yes	—
86	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
87	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
88	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
89	C₂×Dic₇₃	C₂²×C₈	1	4	yes	—
90	C₂×Dic₇₃	C₂³×C₄	1	1	yes	characteristic
91	C₄×D₇₃	—	2	2	no	—
92	C₄×D₇₃	—	2	1	no	—
93	C₄×D₇₃	—	2	1	no	—
94	C₄×D₇₃	—	4	1	no	—
95	Dic₁₄₆	—	2	4	no	—
96	#10596833	—	2	1	no	—
97	#10596833	—	2	1	no	—
98	#10596833	—	2	1	no	—
99	#10596833	—	2	4	no	—
100	#10596833	2₋¹⁺⁴	1	2	yes	—
101	C₂²×C₇₃	C₂³×C₈	1	1	yes	characteristic
102	C₂×D₇₃	—	2	1	no	—
103	C₂×D₇₃	—	2	1	no	—
104	C₂×D₇₃	—	2	1	no	—
105	C₄×C₇₃	—	2	1	no	—
106	C₄×C₇₃	—	2	4	no	—
107	C₄×C₇₃	—	2	1	no	—
108	C₄×C₇₃	#504	1	2	yes	—
109	Dic₇₃	—	2	4	no	—
110	Dic₇₃	—	2	1	no	—
111	Dic₇₃	—	2	1	no	—
112	Dic₇₃	#519	1	1	yes	characteristic
113	Dic₇₃	#568	1	1	yes	characteristic
114	#33499	—	73	1	no	maximal, Sylow
115	C₂×C₇₃	#2982	1	1	yes	characteristic
116	C₂×C₇₃	C₂×#504	1	1	yes	characteristic
117	C₂×C₇₃	#3407	1	1	yes	characteristic
118	D₇₃	—	4	1	no	—
119	#2960	—	73	1	no	—
120	#2183	—	73	2	no	—
121	#2183	—	73	4	no	—
122	Q₈×C₁₆	—	73	2	no	—
123	#2164	—	73	4	no	—
124	#2163	—	73	1	no	—
125	#2162	—	73	1	no	—
126	C₇₃	#33499	1	1	yes	characteristic, complemented, Sylow
127	#533	—	73	1	no	—
128	#446	—	73	2	no	—
129	#446	—	73	2	no	—
130	C₈×Q₈	—	73	2	no	—
131	C₈×Q₈	—	73	2	no	—
132	#424	—	73	2	no	—
133	#423	—	73	1	no	—
134	#405	—	73	2	no	—
135	C₂×C₄×C₈	—	73	1	no	—
136	C₄⋊C₁₆	—	73	2	no	—
137	C₄⋊C₁₆	—	73	4	no	—
138	C₄⋊C₁₆	—	73	2	no	—
139	C₄⋊C₁₆	—	73	4	no	—
140	C₂²⋊C₁₆	—	73	2	no	—
141	C₂²⋊C₁₆	—	73	2	no	—
142	#346	—	73	2	no	—
143	#346	—	73	4	no	—
144	C₄×C₁₆	—	73	1	no	—
145	C₄×C₁₆	—	73	1	no	—
146	C₄×C₁₆	—	73	4	no	—
147	C₂×M₁₆	—	73	2	no	—
148	C₂²×C₈	—	73	1	no	—
149	C₂²×C₈	—	73	1	no	—
150	C₄⋊Q₈	—	73	1	no	—
151	C₄⋊Q₈	#10596842	1	1	yes	characteristic
152	#125	—	73	2	no	—
153	C₂²⋊Q₈	—	73	2	no	—
154	C₂²⋊Q₈	—	73	2	no	—
155	C₄×Q₈	—	73	2	no	—
156	C₄×Q₈	—	73	2	no	—
157	#117	—	73	2	no	—
158	C₂×C₄²	—	73	1	no	—
159	C₂×C₁₆	—	73	2	no	—
160	C₂×C₁₆	—	73	2	no	—
161	C₂×C₁₆	—	73	4	no	—
162	C₄⋊C₈	—	73	2	no	—
163	C₄⋊C₈	—	73	2	no	—
164	C₄⋊C₈	—	73	4	no	—
165	C₄⋊C₈	—	73	4	no	—
166	#97	—	73	4	no	—
167	C₄×C₈	—	73	1	no	—
168	C₄×C₈	—	73	4	no	—
169	C₄×C₈	—	73	1	no	—
170	C₄×C₈	—	73	1	no	—
171	C₄×C₈	—	73	1	no	—
172	C₂×Q₈	—	73	2	no	—
173	C₂×Q₈	C₂×#10596842	1	2	yes	—
174	C₂²×C₄	—	73	1	no	—
175	C₂²×C₄	—	73	1	no	—
176	C₂²×C₄	—	73	1	no	—
177	M₁₆	—	292	2	no	—
178	C₂×C₈	—	292	1	no	—
179	C₂×C₈	—	73	1	no	—
180	C₂×C₈	—	73	1	no	—
181	C₂×C₈	—	73	1	no	—
182	C₂×C₈	—	73	1	no	—
183	C₂×C₈	—	73	4	no	—
184	C₂×C₈	—	146	2	no	—
185	C₄⋊C₄	—	73	4	no	—
186	C₄⋊C₄	—	73	2	no	—
187	C₄⋊C₄	—	73	4	no	—
188	C₄⋊C₄	—	73	2	no	—
189	C₄⋊C₄	C₂×#10596842	1	2	yes	—
190	C₄⋊C₄	C₂×#10596842	1	2	yes	—
191	C₂²⋊C₄	—	73	4	no	—
192	C₄²	—	73	1	no	—
193	C₄²	—	73	1	no	—
194	C₄²	—	73	1	no	—
195	C₄²	—	73	4	no	—
196	C₄²	C₂×#10596842	1	1	yes	characteristic
197	C₁₆	—	146	2	no	—
198	C₁₆	—	146	2	no	—
199	C₁₆	—	146	4	no	—
200	C₂³	—	73	1	no	—
201	Q₈	—	4	2	no	—
202	Q₈	—	146	4	no	—
203	C₂×C₄	—	292	1	no	—
204	C₂×C₄	—	73	1	no	—
205	C₂×C₄	—	73	1	no	—
206	C₂×C₄	—	73	1	no	—
207	C₂×C₄	—	73	4	no	—
208	C₂×C₄	—	146	1	no	—
209	C₂×C₄	—	146	1	no	—
210	C₂×C₄	—	146	2	no	—
211	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
212	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
213	C₂×C₄	C₂²×#10596842	1	4	yes	—
214	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
215	C₈	—	73	2	no	—
216	C₈	—	146	1	no	—
217	C₈	—	146	1	no	—
218	C₈	—	146	1	no	—
219	C₈	—	146	4	no	—
220	C₂²	—	146	1	no	—
221	C₂²	—	146	1	no	—
222	C₂²	—	146	1	no	—
223	C₂²	C₂³×#10596842	1	1	yes	characteristic
224	C₄	—	2	1	no	—
225	C₄	—	2	1	no	—
226	C₄	—	2	4	no	—
227	C₄	—	73	1	no	—
228	C₄	—	73	1	no	—
229	C₄	—	146	1	no	—
230	C₄	—	146	4	no	—
231	C₄	—	146	1	no	—
232	C₄	#17989709	1	2	yes	—
233	C₂	—	292	1	no	—
234	C₂	#22627174	1	1	yes	characteristic
235	C₂	C₂×#17989709	1	1	yes	characteristic
236	C₂	#22627184	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	#33499
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. #22627157	C₂	semidirect
2. #22627156	C₂	semidirect
3. #22627155	C₂	semidirect
126. C₇₃	#33499	semidirect

Maximal subgroups

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

subgroup	index
#22627157	2
#22627156	2
#22627155	2
#22627137	2
#22627134	2
#22627130	2
#22626720	2
#33499	73