Subgroups — #34034745

Order 18688.

684 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22627185	C₂	1	2	yes	maximal
2	#22627127	C₂	1	6	yes	maximal, complemented
3	C₂×#17989692	C₂	1	3	yes	maximal, complemented
4	#22627125	C₂	1	3	yes	maximal, complemented
5	#131×#64408	C₂	1	1	yes	maximal, characteristic
6	#17989709	—	2	2	no	—
7	#17989708	—	2	6	no	—
8	C₂×#14556688	C₂²	1	6	yes	—
9	#17989706	C₂²	1	6	yes	—
10	#17989692	C₂²	1	1	yes	characteristic
11	#17989692	C₂²	1	3	yes	—
12	#17989692	C₂²	1	6	yes	—
13	#17989691	C₂²	1	6	yes	—
14	D₇₃⋊#131	C₂²	1	1	yes	characteristic
15	#41×#64408	C₄	1	1	yes	characteristic
16	#17989553	C₂²	1	3	yes	—
17	#17989550	C₄	1	3	yes	—
18	M₁₆×#64408	C₄	1	3	yes	—
19	C₂×C₈×#64408	C₂²	1	3	yes	—
20	#131×D₇₃	C₄	1	1	yes	characteristic
21	#14556688	—	2	6	no	—
22	#14556688	C₂³	1	2	yes	—
23	#14556688	C₂³	1	6	yes	—
24	#14556687	—	2	6	no	—
25	#14556686	—	2	6	no	—
26	C₂×#11924040	—	2	6	no	—
27	#14556679	D₄	1	1	yes	characteristic
28	#14556679	Q₈	1	1	yes	characteristic
29	#14556678	D₄	1	3	yes	—
30	#14556678	Q₈	1	3	yes	—
31	D₇₃⋊M₁₆	C₂×C₄	1	3	yes	—
32	C₂×#11924036	C₂³	1	3	yes	—
33	Q₈×#64408	—	2	1	no	—
34	D₄×#64408	—	2	3	no	—
35	#14556670	C₂×C₄	1	3	yes	—
36	C₂×C₄×#64408	C₂×C₄	1	3	yes	—
37	#14556645	C₂×C₄	1	6	yes	—
38	C₈×#64408	C₂×C₄	1	3	yes	—
39	C₈×#64408	C₂×C₄	1	3	yes	—
40	C₈×#64408	C₂³	1	3	yes	—
41	C₈×#64408	C₂³	1	1	yes	characteristic
42	#41×D₇₃	C₂×C₄	1	1	yes	characteristic
43	#14556599	C₈	1	3	yes	—
44	#14556598	C₈	1	1	yes	characteristic
45	#14556597	C₈	1	3	yes	—
46	M₁₆×D₇₃	C₂×C₄	1	3	yes	—
47	C₂×C₈×D₇₃	C₂×C₄	1	3	yes	—
48	#131×C₇₃	C₈	1	1	yes	characteristic
49	#11924040	—	2	6	no	—
50	#11924040	—	2	2	no	—
51	C₂²×#64408	—	2	3	no	—
52	#11924038	C₄⋊C₄	1	3	yes	—
53	#11924038	C₄⋊C₄	1	3	yes	—
54	#11924037	C₄⋊C₄	1	6	yes	—
55	#11924036	C₂²×C₄	1	3	yes	—
56	#11924036	C₂×D₄	1	3	yes	—
57	#11924036	C₂×Q₈	1	3	yes	—
58	#11924036	C₂⁴	1	1	yes	characteristic
59	C₂×#10596833	C₂×D₄	1	3	yes	—
60	C₂×#10596833	C₂×Q₈	1	3	yes	—
61	C₄⋊#64408	—	2	3	no	—
62	C₂²⋊#64408	—	2	3	no	—
63	C₄×#64408	—	2	3	no	—
64	C₄×#64408	C₂²×C₄	1	1	yes	characteristic
65	C₄×#64408	C₂²×C₄	1	3	yes	—
66	#11924029	C₄⋊C₄	1	1	yes	characteristic
67	#11924028	C₂×C₈	1	3	yes	—
68	#11924027	C₄⋊C₄	1	3	yes	—
69	Q₈×D₇₃	M₁₆	1	1	yes	characteristic
70	D₄×D₇₃	M₁₆	1	3	yes	—
71	C₂×C₄×D₇₃	C₂²×C₄	1	3	yes	—
72	#11924012	C₂×C₈	1	3	yes	—
73	#11924011	C₂×C₈	1	6	yes	—
74	C₂×#10596831	C₂×C₈	1	3	yes	—
75	C₈×D₇₃	C₂×C₈	1	3	yes	—
76	C₈×D₇₃	C₂×C₈	1	3	yes	—
77	C₈×D₇₃	C₂²×C₄	1	1	yes	characteristic
78	C₈×D₇₃	C₂²×C₄	1	3	yes	—
79	#41×C₇₃	C₂×C₈	1	1	yes	characteristic
80	M₁₆×C₇₃	C₂×C₈	1	3	yes	—
81	C₂×C₈×C₇₃	C₂×C₈	1	3	yes	—
82	C₂²×D₇₃	C₂×M₁₆	1	3	yes	—
83	C₂×#64408	—	2	1	no	—
84	C₂×#64408	—	2	3	no	—
85	C₂×#64408	—	4	6	no	—
86	Q₈×C₇₃	C₄⋊C₈	1	1	yes	characteristic
87	D₄×C₇₃	C₄⋊C₈	1	3	yes	—
88	C₂×C₄×C₇₃	C₂²×C₈	1	3	yes	—
89	C₂²⋊D₇₃	C₄⋊C₈	1	6	yes	—
90	C₂×Dic₇₃	C₂×C₄⋊C₄	1	3	yes	—
91	D₂₉₂	C₄⋊C₈	1	3	yes	—
92	C₄×D₇₃	C₂×C₄⋊C₄	1	3	yes	—
93	C₄×D₇₃	C₂²×C₈	1	3	yes	—
94	C₄×D₇₃	C₂×M₁₆	1	3	yes	—
95	C₄×D₇₃	C₂³×C₄	1	1	yes	characteristic
96	Dic₁₄₆	C₄⋊C₈	1	3	yes	—
97	#10596833	C₂×C₄⋊C₄	1	3	yes	—
98	#10596833	C₂×C₄⋊C₄	1	3	yes	—
99	#10596833	C₂²×D₄	1	1	yes	characteristic
100	#10596833	C₂²×Q₈	1	1	yes	characteristic
101	C₈×C₇₃	C₂²×C₈	1	1	yes	characteristic
102	C₈×C₇₃	C₂²×C₈	1	3	yes	—
103	#10596831	C₂²×C₈	1	1	yes	characteristic
104	#10596831	C₂²×C₈	1	3	yes	—
105	C₂²×C₇₃	C₂×C₄⋊C₈	1	3	yes	—
106	C₂×D₇₃	—	4	3	no	—
107	C₂×D₇₃	C₂×C₄⋊C₈	1	3	yes	—
108	C₂×D₇₃	C₂²×M₁₆	1	1	yes	characteristic
109	#64408	—	2	2	no	—
110	#64408	—	4	3	no	—
111	C₄×C₇₃	C₂×C₄⋊C₈	1	3	yes	—
112	C₄×C₇₃	C₂³×C₈	1	1	yes	characteristic
113	Dic₇₃	C₂×C₄⋊C₈	1	3	yes	—
114	Dic₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
115	#33401	—	73	1	no	maximal, Sylow
116	C₂×C₇₃	—	2	3	no	—
117	C₂×C₇₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
118	D₇₃	—	2	1	no	—
119	D₇₃	—	2	3	no	—
120	#3407	—	73	2	no	—
121	C₄×#131	—	73	1	no	—
122	#2153	—	73	6	no	—
123	C₂×#364	—	73	3	no	—
124	C₂×#364	—	73	3	no	—
125	C₇₃	#33401	1	1	yes	characteristic, complemented, Sylow
126	C₂×#131	—	73	1	no	—
127	C₂×#131	—	73	1	no	—
128	C₄×#41	—	73	1	no	—
129	#504	—	146	6	no	—
130	#504	—	146	2	no	—
131	C₂×M₃₂	—	73	6	no	—
132	C₂×M₃₂	—	73	6	no	—
133	#405	—	73	3	no	—
134	#405	—	73	3	no	—
135	C₄×M₁₆	—	73	3	no	—
136	C₂×C₄×C₈	—	73	3	no	—
137	#364	—	73	6	no	—
138	#364	—	73	1	no	—
139	#364	—	73	3	no	—
140	#364	—	73	6	no	—
141	C₂×#41	—	73	1	no	—
142	#131	—	73	3	no	—
143	#131	—	73	3	no	—
144	#131	—	73	1	no	—
145	#131	—	73	1	no	—
146	#131	—	73	1	no	—
147	#131	—	73	3	no	—
148	#131	—	73	3	no	—
149	#131	#10596842	1	1	yes	characteristic
150	C₂×M₁₆	—	73	3	no	—
151	C₂×M₁₆	—	73	3	no	—
152	C₂²×C₈	—	73	3	no	—
153	C₂²×C₈	—	73	3	no	—
154	C₄×Q₈	—	146	1	no	—
155	C₄×D₄	—	146	3	no	—
156	#117	—	73	3	no	—
157	C₂×C₄²	—	73	3	no	—
158	M₃₂	—	73	2	no	—
159	M₃₂	—	73	6	no	—
160	M₃₂	—	146	6	no	—
161	M₃₂	—	146	6	no	—
162	C₂×C₁₆	—	146	6	no	—
163	C₂×C₁₆	—	146	6	no	—
164	#97	—	73	6	no	—
165	C₄×C₈	—	73	1	no	—
166	C₄×C₈	—	73	3	no	—
167	C₄×C₈	—	73	3	no	—
168	C₄×C₈	—	73	3	no	—
169	#41	—	73	1	no	—
170	#41	—	73	3	no	—
171	#41	—	73	3	no	—
172	#41	C₂×#10596842	1	1	yes	characteristic
173	C₂×Q₈	—	73	1	no	—
174	C₂×D₄	—	73	3	no	—
175	C₂²×C₄	—	73	3	no	—
176	C₂²×C₄	—	146	3	no	—
177	M₁₆	—	73	6	no	—
178	M₁₆	—	73	3	no	—
179	M₁₆	—	73	3	no	—
180	M₁₆	—	73	3	no	—
181	M₁₆	—	73	6	no	—
182	M₁₆	C₂×#10596842	1	3	yes	—
183	C₂×C₈	—	73	3	no	—
184	C₂×C₈	—	73	3	no	—
185	C₂×C₈	—	73	3	no	—
186	C₂×C₈	—	73	3	no	—
187	C₂×C₈	—	73	3	no	—
188	C₂×C₈	—	73	1	no	—
189	C₂×C₈	—	73	3	no	—
190	C₂×C₈	—	73	3	no	—
191	C₂×C₈	—	73	3	no	—
192	C₂×C₈	—	73	1	no	—
193	C₂×C₈	—	73	3	no	—
194	C₂×C₈	C₂×#10596842	1	3	yes	—
195	C₄⋊C₄	—	146	3	no	—
196	C₂²⋊C₄	—	146	3	no	—
197	C₄²	—	73	1	no	—
198	C₄²	—	73	3	no	—
199	C₄²	—	146	3	no	—
200	C₁₆	—	146	6	no	—
201	C₁₆	—	146	2	no	—
202	C₂³	—	73	3	no	—
203	Q₈	—	73	3	no	—
204	Q₈	C₄⋊#10596842	1	1	yes	characteristic
205	D₄	—	73	6	no	—
206	D₄	—	73	3	no	—
207	D₄	C₄⋊#10596842	1	3	yes	—
208	C₂×C₄	—	292	6	no	—
209	C₂×C₄	—	73	3	no	—
210	C₂×C₄	—	73	3	no	—
211	C₂×C₄	—	73	3	no	—
212	C₂×C₄	—	73	1	no	—
213	C₂×C₄	—	73	3	no	—
214	C₂×C₄	—	146	3	no	—
215	C₂×C₄	—	146	1	no	—
216	C₂×C₄	C₂²×#10596842	1	3	yes	—
217	C₈	—	73	1	no	—
218	C₈	—	73	1	no	—
219	C₈	—	73	3	no	—
220	C₈	—	73	3	no	—
221	C₈	—	73	1	no	—
222	C₈	—	73	3	no	—
223	C₈	C₂²×#10596842	1	1	yes	characteristic
224	C₈	C₂²×#10596842	1	3	yes	—
225	C₂²	—	292	3	no	—
226	C₂²	—	73	1	no	—
227	C₂²	—	73	3	no	—
228	C₂²	C₂×C₄⋊#10596842	1	3	yes	—
229	C₄	—	292	3	no	—
230	C₄	—	73	1	no	—
231	C₄	—	73	3	no	—
232	C₄	—	146	2	no	—
233	C₄	C₂×C₄⋊#10596842	1	3	yes	—
234	C₄	C₂³×#10596842	1	1	yes	characteristic
235	C₂	—	2	3	no	—
236	C₂	—	146	1	no	—
237	C₂	—	146	3	no	—
238	C₂	C₂²×C₄⋊#10596842	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×C₄⋊#10596842
commutator	C₄×C₇₃	C₂³×C₈
Frattini	C₄	C₂³×#10596842
Fitting	#131×C₇₃	C₈
socle	C₂×C₇₃	C₂²×C₄⋊C₈
prime radical	#41×D₇₃	C₂×C₄
cosocle	#11924036	C₂⁴
hypercenter	#131	#10596842
hypocenter	C₇₃	#33401
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131

Complements

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

N	H ≅ G/N	type
2. #22627127	C₂	semidirect
3. C₂×#17989692	C₂	semidirect
4. #22627125	C₂	semidirect
125. C₇₃	#33401	semidirect

Maximal subgroups

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

subgroup	index
#22627185	2
#22627127	2
C₂×#17989692	2
#22627125	2
#131×#64408	2
#33401	73