Subgroups — #33238581

Order 18688.

894 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988996	C₂	1	4	yes	maximal
2	#22618329	C₂	1	1	yes	maximal, characteristic
3	#22618294	C₂	1	1	yes	maximal, characteristic
4	#22610099	C₂	1	4	yes	maximal
5	#22610091	C₂	1	2	yes	maximal
6	#22610088	C₂	1	2	yes	maximal
7	C₄×#97×C₇₃	C₂	1	1	yes	maximal, characteristic
8	C₂×#14556591	—	2	2	no	—
9	C₂×#14556591	C₂²	1	2	yes	—
10	C₂×#14556583	C₂²	1	2	yes	—
11	#17988996	—	2	16	no	—
12	C₂×#14556525	C₂²	1	8	yes	—
13	C₂×#14556523	C₂²	1	8	yes	—
14	#17988485	—	2	2	no	—
15	#17988485	—	2	2	no	—
16	#17988457	—	2	2	no	—
17	#17988457	C₂²	1	1	yes	characteristic
18	#17988457	C₂²	1	2	yes	—
19	#17988457	C₂²	1	1	yes	characteristic
20	#17988457	C₂²	1	2	yes	—
21	#17988453	C₂²	1	2	yes	—
22	C₂×#97×C₇₃	C₂²	1	2	yes	—
23	C₂×#97×C₇₃	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₇₃	C₂²	1	2	yes	—
25	C₄³×C₇₃	C₂²	1	1	yes	characteristic
26	C₂²×Dic₁₄₆	—	2	2	no	—
27	#14556591	—	2	8	no	—
28	#14556591	—	4	8	no	—
29	#14556583	—	2	8	no	—
30	C₂×C₄⋊Dic₇₃	—	2	4	no	—
31	C₂×C₄⋊Dic₇₃	—	2	4	no	—
32	C₂×C₄⋊Dic₇₃	—	2	2	no	—
33	C₂×C₄⋊Dic₇₃	—	2	2	no	—
34	C₂×C₄⋊Dic₇₃	—	2	2	no	—
35	C₂×C₄⋊Dic₇₃	C₂³	1	2	yes	—
36	C₂×C₄⋊Dic₇₃	C₂³	1	4	yes	—
37	C₂×C₄⋊Dic₇₃	C₂³	1	2	yes	—
38	C₂×#11924007	—	2	2	no	—
39	C₂×#11924007	—	2	2	no	—
40	#14556525	—	2	16	no	—
41	#14556523	—	2	16	no	—
42	#14556498	—	2	4	no	—
43	C₂²×C₈×C₇₃	C₂³	1	4	yes	—
44	C₂×C₄²×C₇₃	D₄	1	2	yes	—
45	C₂×C₄²×C₇₃	D₄	1	2	yes	—
46	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
49	#97×C₇₃	D₄	1	8	yes	—
50	#97×C₇₃	Q₈	1	8	yes	complemented
51	C₄×C₈×C₇₃	Q₈	1	8	yes	complemented
52	C₂×Dic₁₄₆	—	4	8	no	—
53	C₂×Dic₁₄₆	—	4	4	no	—
54	C₂²×Dic₇₃	—	2	2	no	—
55	C₂²×Dic₇₃	—	2	2	no	—
56	C₂²×Dic₇₃	—	2	4	no	—
57	C₄⋊Dic₇₃	—	2	4	no	—
58	C₄⋊Dic₇₃	—	2	8	no	—
59	C₄⋊Dic₇₃	—	2	4	no	—
60	C₄⋊Dic₇₃	—	4	8	no	—
61	C₄⋊Dic₇₃	—	4	4	no	—
62	C₄⋊Dic₇₃	—	4	4	no	—
63	C₄⋊Dic₇₃	—	4	8	no	—
64	C₄⋊Dic₇₃	—	4	4	no	—
65	#11924007	—	4	4	no	—
66	#11924007	—	4	4	no	—
67	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
72	C₂×C₈×C₇₃	C₂×D₄	1	8	yes	—
73	C₂×C₈×C₇₃	C₂×Q₈	1	8	yes	—
74	C₂×C₈×C₇₃	C₂×Q₈	1	8	yes	—
75	C₄²×C₇₃	—	2	4	no	—
76	C₄²×C₇₃	—	2	4	no	—
77	C₄²×C₇₃	C₂×D₄	1	4	yes	—
78	C₄²×C₇₃	C₂×Q₈	1	4	yes	—
79	C₄²×C₇₃	C₂×Q₈	1	4	yes	—
80	C₂³×C₇₃	C₂²×D₄	1	1	yes	characteristic
81	C₂×C₄×C₇₃	—	2	2	no	—
82	C₂×C₄×C₇₃	—	2	2	no	—
83	C₂×C₄×C₇₃	—	2	2	no	—
84	C₂×C₄×C₇₃	—	2	2	no	—
85	C₂×C₄×C₇₃	#127	1	2	yes	—
86	C₂×C₄×C₇₃	#127	1	2	yes	—
87	C₂×C₄×C₇₃	C₄⋊Q₈	1	4	yes	—
88	C₂×C₄×C₇₃	C₄⋊Q₈	1	4	yes	—
89	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	4	yes	—
90	C₂×C₄×C₇₃	#137	1	4	yes	—
91	C₂×C₄×C₇₃	C₂²×D₄	1	2	yes	—
92	C₂×C₄×C₇₃	C₂²×Q₈	1	2	yes	—
93	C₂×C₄×C₇₃	C₂²×Q₈	1	2	yes	—
94	C₂×Dic₇₃	—	4	2	no	—
95	C₂×Dic₇₃	—	4	2	no	—
96	C₂×Dic₇₃	—	4	4	no	—
97	C₂×Dic₇₃	—	4	8	no	—
98	C₂×Dic₇₃	—	4	4	no	—
99	C₂×Dic₇₃	—	4	4	no	—
100	Dic₁₄₆	—	8	8	no	—
101	C₈×C₇₃	C₄⋊Q₈	1	16	yes	complemented
102	C₂²×C₇₃	C₂×#127	1	1	yes	characteristic
103	C₂²×C₇₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
104	C₂²×C₇₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
105	C₂²×C₇₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
106	C₂²×C₇₃	C₂×#137	1	2	yes	—
107	C₄×C₇₃	—	2	2	no	—
108	C₄×C₇₃	—	2	2	no	—
109	C₄×C₇₃	—	2	4	no	—
110	C₄×C₇₃	#501	1	8	yes	complemented
111	C₄×C₇₃	C₂×C₄⋊Q₈	1	4	yes	—
112	Dic₇₃	—	8	4	no	—
113	Dic₇₃	—	8	2	no	—
114	Dic₇₃	—	8	2	no	—
115	#33316	—	73	1	no	maximal, Sylow
116	C₂×C₇₃	#2868	1	1	yes	characteristic
117	C₂×C₇₃	#3151	1	1	yes	characteristic
118	C₂×C₇₃	#3152	1	1	yes	characteristic
119	C₂×C₇₃	C₂×#501	1	2	yes	—
120	C₂×C₇₃	#3165	1	2	yes	—
121	C₂×#501	—	73	4	no	—
122	#2868	—	73	1	no	—
123	#2859	—	73	1	no	—
124	#1847	—	73	4	no	—
125	#1845	—	73	2	no	—
126	#1844	—	73	2	no	—
127	C₄×#97	D₇₃	1	1	yes	characteristic
128	C₇₃	#33316	1	1	yes	characteristic, complemented, Sylow
129	C₂×C₄⋊Q₈	—	73	2	no	—
130	C₂×C₄⋊Q₈	—	146	2	no	—
131	C₂×#125	—	73	2	no	—
132	#501	—	146	16	no	—
133	C₂×#107	—	73	8	no	—
134	C₂×#106	—	73	8	no	—
135	C₂×#97	C₂×D₇₃	1	2	yes	—
136	C₂×#97	C₂×D₇₃	1	2	yes	—
137	C₂×C₄×C₈	C₂×D₇₃	1	2	yes	—
138	#389	—	146	2	no	—
139	#389	—	146	2	no	—
140	#384	—	73	1	no	—
141	#384	—	73	1	no	—
142	#384	—	73	2	no	—
143	#384	—	73	2	no	—
144	#384	—	146	2	no	—
145	#382	—	73	2	no	—
146	C₄³	C₂×D₇₃	1	1	yes	characteristic
147	C₂²×Q₈	—	146	2	no	—
148	C₂²×C₈	C₂²×D₇₃	1	4	yes	—
149	C₄⋊Q₈	—	292	8	no	—
150	C₄⋊Q₈	—	146	8	no	—
151	#125	—	146	8	no	—
152	C₂×C₄⋊C₄	—	73	2	no	—
153	C₂×C₄⋊C₄	—	73	2	no	—
154	C₂×C₄⋊C₄	—	73	4	no	—
155	C₂×C₄⋊C₄	—	146	2	no	—
156	C₂×C₄⋊C₄	—	146	2	no	—
157	C₂×C₄⋊C₄	—	146	2	no	—
158	C₂×C₄⋊C₄	—	146	4	no	—
159	C₂×C₄⋊C₄	—	146	4	no	—
160	C₂×C₄⋊C₄	—	146	2	no	—
161	C₂×C₄⋊C₄	—	146	2	no	—
162	C₂×C₄²	D₂₉₂	1	2	yes	—
163	C₂×C₄²	D₂₉₂	1	2	yes	—
164	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
165	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
166	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
167	#107	—	146	16	no	—
168	#106	—	146	16	no	—
169	#97	Dic₁₄₆	1	8	yes	complemented
170	#97	D₂₉₂	1	8	yes	—
171	C₄×C₈	Dic₁₄₆	1	8	yes	complemented
172	#95	—	146	4	no	—
173	C₂×Q₈	—	292	8	no	—
174	C₂×Q₈	—	292	4	no	—
175	C₂²×C₄	—	146	2	no	—
176	C₂²×C₄	—	146	2	no	—
177	C₂²×C₄	—	146	4	no	—
178	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
179	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
180	C₂²×C₄	C₂×D₂₉₂	1	2	yes	—
181	C₂²×C₄	C₂×D₂₉₂	1	2	yes	—
182	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
183	C₂×C₈	C₂×D₂₉₂	1	8	yes	—
184	C₂×C₈	C₂×Dic₁₄₆	1	8	yes	—
185	C₂×C₈	C₂×Dic₁₄₆	1	8	yes	—
186	C₄⋊C₄	—	292	8	no	—
187	C₄⋊C₄	—	292	4	no	—
188	C₄⋊C₄	—	292	4	no	—
189	C₄⋊C₄	—	292	4	no	—
190	C₄⋊C₄	—	292	4	no	—
191	C₄⋊C₄	—	292	8	no	—
192	C₄⋊C₄	—	292	4	no	—
193	C₄⋊C₄	—	146	4	no	—
194	C₄⋊C₄	—	146	8	no	—
195	C₄⋊C₄	—	146	4	no	—
196	C₄²	—	2	4	no	—
197	C₄²	—	2	4	no	—
198	C₄²	C₂×D₂₉₂	1	4	yes	—
199	C₄²	C₂×Dic₁₄₆	1	4	yes	—
200	C₄²	C₂×Dic₁₄₆	1	4	yes	—
201	C₂³	C₂²×D₂₉₂	1	1	yes	characteristic
202	Q₈	—	584	8	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	292	8	no	—
208	C₂×C₄	—	292	4	no	—
209	C₂×C₄	—	292	4	no	—
210	C₂×C₄	—	292	2	no	—
211	C₂×C₄	—	292	2	no	—
212	C₂×C₄	—	292	4	no	—
213	C₂×C₄	#14556588	1	2	yes	—
214	C₂×C₄	#14556588	1	2	yes	—
215	C₂×C₄	#14556591	1	4	yes	—
216	C₂×C₄	#14556591	1	4	yes	—
217	C₂×C₄	#14556619	1	4	yes	—
218	C₂×C₄	#14556624	1	4	yes	—
219	C₂×C₄	C₂²×D₂₉₂	1	2	yes	—
220	C₂×C₄	C₂²×Dic₁₄₆	1	2	yes	—
221	C₂×C₄	C₂²×Dic₁₄₆	1	2	yes	—
222	C₈	#14556591	1	16	yes	complemented
223	C₂²	C₂×#14556588	1	1	yes	characteristic
224	C₂²	C₂×#14556591	1	1	yes	characteristic
225	C₂²	C₂×#14556591	1	1	yes	characteristic
226	C₂²	C₂×#14556619	1	2	yes	—
227	C₂²	C₂×#14556624	1	2	yes	—
228	C₄	—	2	4	no	—
229	C₄	—	2	2	no	—
230	C₄	—	2	2	no	—
231	C₄	—	584	4	no	—
232	C₄	—	584	2	no	—
233	C₄	—	584	2	no	—
234	C₄	#17988996	1	8	yes	complemented
235	C₄	C₂×#14556591	1	4	yes	—
236	C₂	#22618329	1	1	yes	characteristic
237	C₂	#22620383	1	1	yes	characteristic
238	C₂	#22620390	1	1	yes	characteristic
239	C₂	C₂×#17988996	1	2	yes	—
240	C₂	#22620477	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₂₉₂
commutator, cosocle	C₂²×C₄×C₇₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₇₃
Fitting	C₄×#97×C₇₃	C₂
socle, prime radical	C₂³×C₇₃	C₂²×D₄
hypercenter	C₄×#97	D₇₃
hypocenter	C₇₃	#33316
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×#97

Complements

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

N	H ≅ G/N	type
50. #97×C₇₃	Q₈	semidirect
51. C₄×C₈×C₇₃	Q₈	semidirect
101. C₈×C₇₃	C₄⋊Q₈	semidirect
110. C₄×C₇₃	#501	semidirect
128. C₇₃	#33316	semidirect
169. #97	Dic₁₄₆	semidirect
171. C₄×C₈	Dic₁₄₆	semidirect
222. C₈	#14556591	semidirect
234. C₄	#17988996	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17988996	2
#22618329	2
#22618294	2
#22610099	2
#22610091	2
#22610088	2
C₄×#97×C₇₃	2
#33316	73