Subgroups — #33004451

Order 18688.

598 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	C₂×#17988964	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#11924003	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#17988387	C₂	1	4	yes	maximal, complemented
4	C₄×#107×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₅₈₄	—	2	1	no	—
6	C₂²×D₅₈₄	C₂²	1	2	yes	—
7	C₂×#14556588	—	2	1	no	—
8	C₂²×#11924003	C₂²	1	2	yes	—
9	#17988964	—	2	2	no	—
10	#17988964	C₄	1	4	yes	complemented
11	C₄⋊#11924003	—	2	2	no	—
12	#17988387	—	2	8	no	—
13	C₂×#107×C₇₃	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
15	C₄×C₄⋊C₄×C₇₃	—	2	1	no	—
16	#340×C₇₃	—	2	1	no	—
17	C₂²×D₂₉₂	—	2	2	no	—
18	C₂²×D₂₉₂	—	4	1	no	—
19	C₂²×D₂₉₂	—	4	1	no	—
20	C₂×D₅₈₄	—	2	4	no	—
21	C₂×D₅₈₄	—	2	4	no	—
22	C₂×D₅₈₄	—	2	2	no	—
23	C₂×D₅₈₄	—	2	4	no	—
24	C₂×D₅₈₄	—	4	4	no	—
25	C₂×D₅₈₄	C₂×C₄	1	4	yes	—
26	#14556588	—	2	4	no	—
27	#14556588	—	4	2	no	—
28	C₂×#11924003	—	2	4	no	—
29	C₂×#11924003	—	2	2	no	—
30	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
32	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
33	C₂×C₄²×C₇₃	—	4	1	no	—
34	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
35	#107×C₇₃	—	2	2	no	—
36	#107×C₇₃	D₄	1	4	yes	complemented
37	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₇₃	D₄	1	2	yes	—
39	#95×C₇₃	—	4	1	no	—
40	C₂³×D₇₃	—	4	2	no	—
41	C₂×D₂₉₂	—	2	4	no	—
42	C₂×D₂₉₂	—	4	4	no	—
43	C₂×D₂₉₂	—	4	4	no	—
44	C₂×D₂₉₂	—	4	4	no	—
45	C₂×D₂₉₂	—	4	4	no	—
46	C₂×D₂₉₂	—	4	2	no	—
47	C₂×D₂₉₂	—	8	4	no	—
48	C₂×D₂₉₂	—	8	2	no	—
49	C₂×D₂₉₂	—	8	2	no	—
50	D₅₈₄	—	2	8	no	—
51	D₅₈₄	—	4	8	no	—
52	D₅₈₄	—	4	2	no	—
53	D₅₈₄	—	4	2	no	—
54	#11924003	—	4	2	no	—
55	C₂²×C₄×C₇₃	—	4	2	no	—
56	C₂²×C₄×C₇₃	D₈	1	1	yes	characteristic
57	C₂²×C₄×C₇₃	D₈	1	1	yes	characteristic
58	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₇₃	—	2	2	no	—
60	C₂×C₈×C₇₃	C₂²⋊C₄	1	2	yes	—
61	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
63	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
64	C₂×C₈×C₇₃	#41	1	2	yes	—
65	C₄⋊C₄×C₇₃	—	2	4	no	—
66	C₄⋊C₄×C₇₃	—	4	2	no	—
67	C₄²×C₇₃	—	8	2	no	—
68	C₄²×C₇₃	C₂²⋊C₄	1	2	yes	—
69	C₄²×C₇₃	SD₁₆	1	2	yes	—
70	C₂³×C₇₃	C₂×D₈	1	1	yes	characteristic
71	C₂²×D₇₃	—	4	4	no	—
72	C₂²×D₇₃	—	8	4	no	—
73	C₂²×D₇₃	—	8	2	no	—
74	C₂²×D₇₃	—	8	2	no	—
75	C₂²×D₇₃	—	8	4	no	—
76	C₂×C₄×C₇₃	—	2	2	no	—
77	C₂×C₄×C₇₃	—	4	4	no	—
78	C₂×C₄×C₇₃	—	8	2	no	—
79	C₂×C₄×C₇₃	—	8	2	no	—
80	C₂×C₄×C₇₃	#102	1	2	yes	—
81	C₂×C₄×C₇₃	#102	1	2	yes	—
82	C₂×C₄×C₇₃	D₁₆	1	2	yes	—
83	C₂×C₄×C₇₃	SD₃₂	1	2	yes	—
84	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₇₃	#121	1	2	yes	—
86	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
87	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
88	D₂₉₂	—	4	4	no	—
89	D₂₉₂	—	4	4	no	—
90	D₂₉₂	—	8	8	no	—
91	D₂₉₂	—	8	8	no	—
92	D₂₉₂	—	8	4	no	—
93	C₈×C₇₃	—	2	2	no	—
94	C₈×C₇₃	C₄×D₄	1	2	yes	—
95	C₈×C₇₃	#124	1	1	yes	characteristic
96	C₈×C₇₃	#127	1	1	yes	characteristic
97	C₂²×C₇₃	C₂×#102	1	1	yes	characteristic
98	C₂²×C₇₃	#459	1	2	yes	—
99	C₂²×C₇₃	C₂×D₁₆	1	1	yes	characteristic
100	C₂²×C₇₃	C₂×SD₃₂	1	1	yes	characteristic
101	C₂²×C₇₃	#509	1	2	yes	—
102	C₂×D₇₃	—	8	4	no	—
103	C₂×D₇₃	—	8	4	no	—
104	C₂×D₇₃	—	8	4	no	—
105	C₂×D₇₃	—	16	4	no	—
106	C₂×D₇₃	—	16	4	no	—
107	C₄×C₇₃	—	2	2	no	—
108	C₄×C₇₃	—	8	4	no	—
109	C₄×C₇₃	#357	1	4	yes	complemented
110	C₄×C₇₃	#390	1	1	yes	characteristic
111	C₄×C₇₃	#418	1	1	yes	characteristic
112	C₄×C₇₃	#460	1	2	yes	—
113	#13252	—	73	1	no	maximal, Sylow
114	C₂×C₇₃	#1982	1	1	yes	characteristic
115	C₂×C₇₃	C₂×#357	1	1	yes	characteristic
116	C₂×C₇₃	#2140	1	1	yes	characteristic
117	C₂×C₇₃	#2206	1	2	yes	—
118	C₂×C₇₃	#2207	1	2	yes	—
119	D₇₃	—	16	4	no	—
120	C₂×#493	—	73	1	no	—
121	C₂×C₄⋊C₁₆	—	73	1	no	—
122	C₂×#357	—	73	4	no	—
123	C₄×#107	D₇₃	1	1	yes	characteristic, complemented
124	C₇₃	#13252	1	1	yes	characteristic, complemented, Sylow
125	C₂²×D₈	—	73	2	no	—
126	C₂²×D₈	—	146	1	no	—
127	C₂×#127	—	146	1	no	—
128	C₂²×C₁₆	—	73	2	no	—
129	#493	—	73	4	no	—
130	#493	—	146	2	no	—
131	C₂×#107	C₂×D₇₃	1	2	yes	—
132	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
133	C₄×C₄⋊C₄	—	2	1	no	—
134	C₄⋊C₁₆	—	146	2	no	—
135	#357	—	146	8	no	—
136	#340	—	2	1	no	—
137	C₂²×D₄	—	292	1	no	—
138	C₂²×D₄	—	292	1	no	—
139	C₂²×D₄	—	146	2	no	—
140	C₂×D₈	—	292	4	no	—
141	C₂×D₈	—	73	4	no	—
142	C₂×D₈	—	146	4	no	—
143	C₂×D₈	—	146	2	no	—
144	C₂×D₈	—	146	4	no	—
145	C₂×D₈	—	146	4	no	—
146	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
147	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
148	#127	—	292	2	no	—
149	#127	—	146	4	no	—
150	C₂×C₄⋊C₄	—	2	2	no	—
151	C₂×C₄²	—	4	1	no	—
152	C₂×C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
153	C₂×C₁₆	—	146	4	no	—
154	C₂×C₁₆	—	146	2	no	—
155	#107	—	2	2	no	—
156	#107	D₂₉₂	1	4	yes	complemented
157	C₄×C₈	C₄×D₇₃	1	2	yes	—
158	C₄×C₈	D₂₉₂	1	2	yes	—
159	#95	—	4	1	no	—
160	C₂⁴	—	292	2	no	—
161	C₂×D₄	—	292	4	no	—
162	C₂×D₄	—	292	4	no	—
163	C₂×D₄	—	292	4	no	—
164	C₂×D₄	—	292	2	no	—
165	C₂×D₄	—	292	4	no	—
166	C₂×D₄	—	584	4	no	—
167	C₂×D₄	—	584	2	no	—
168	C₂×D₄	—	584	2	no	—
169	C₂×D₄	—	146	4	no	—
170	C₂²×C₄	—	4	2	no	—
171	C₂²×C₄	#11924013	1	1	yes	characteristic
172	C₂²×C₄	#11924013	1	1	yes	characteristic
173	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
174	D₈	—	292	8	no	—
175	D₈	—	292	2	no	—
176	D₈	—	292	2	no	—
177	D₈	—	146	8	no	—
178	C₂×C₈	—	2	2	no	—
179	C₂×C₈	#11924005	1	2	yes	—
180	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
181	C₂×C₈	C₂×D₂₉₂	1	2	yes	—
182	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
183	C₂×C₈	#11924028	1	2	yes	—
184	C₄⋊C₄	—	2	4	no	—
185	C₄⋊C₄	—	4	2	no	—
186	C₄²	—	8	2	no	—
187	C₄²	#11924005	1	2	yes	—
188	C₄²	#11924015	1	2	yes	—
189	C₁₆	—	292	2	no	—
190	C₂³	—	292	4	no	—
191	C₂³	—	584	2	no	—
192	C₂³	—	584	2	no	—
193	C₂³	—	584	4	no	—
194	C₂³	—	584	4	no	—
195	C₂³	C₂×#11924013	1	1	yes	characteristic
196	D₄	—	292	4	no	—
197	D₄	—	292	4	no	—
198	D₄	—	584	8	no	—
199	D₄	—	584	8	no	—
200	D₄	—	584	4	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	4	4	no	—
203	C₂×C₄	—	8	2	no	—
204	C₂×C₄	—	8	2	no	—
205	C₂×C₄	#14556511	1	2	yes	—
206	C₂×C₄	#14556511	1	2	yes	—
207	C₂×C₄	#14556532	1	2	yes	—
208	C₂×C₄	#14556534	1	2	yes	—
209	C₂×C₄	C₂×#11924005	1	1	yes	characteristic
210	C₂×C₄	#14556565	1	2	yes	—
211	C₂×C₄	C₂×#11924015	1	1	yes	characteristic
212	C₂×C₄	#14556616	1	2	yes	—
213	C₈	—	2	2	no	—
214	C₈	C₄×D₂₉₂	1	2	yes	—
215	C₈	#14556579	1	1	yes	characteristic
216	C₈	#14556588	1	1	yes	characteristic
217	C₂²	—	584	4	no	—
218	C₂²	—	584	4	no	—
219	C₂²	—	584	4	no	—
220	C₂²	—	1168	4	no	—
221	C₂²	—	1168	4	no	—
222	C₂²	C₂×#14556511	1	1	yes	characteristic
223	C₂²	#17988782	1	2	yes	—
224	C₂²	C₂×#14556532	1	1	yes	characteristic
225	C₂²	C₂×#14556534	1	1	yes	characteristic
226	C₂²	#17989022	1	2	yes	—
227	C₄	—	2	2	no	—
228	C₄	—	8	4	no	—
229	C₄	#17988387	1	4	yes	complemented
230	C₄	#17988489	1	1	yes	characteristic
231	C₄	#17988591	1	1	yes	characteristic
232	C₄	#17988787	1	2	yes	—
233	C₂	—	1168	4	no	—
234	C₂	#22610858	1	1	yes	characteristic
235	C₂	C₂×#17988387	1	1	yes	characteristic
236	C₂	#22611646	1	1	yes	characteristic
237	C₂	#22611966	1	2	yes	—
238	C₂	#22611971	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#11924013
commutator	C₂×C₈×C₇₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₄×#107×C₇₃	C₂
socle	C₂³×C₇₃	C₂×D₈
prime radical	C₂×#17988964	C₂
cosocle	C₂²×C₈×C₇₃	C₂³
hypercenter	C₄×#107	D₇₃
hypocenter	C₇₃	#13252
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#107

Complements

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

N	H ≅ G/N	type
2. C₂×C₄⋊#11924003	C₂	semidirect
3. C₂×#17988387	C₂	semidirect
4. C₄×#107×C₇₃	C₂	semidirect
10. #17988964	C₄	semidirect
36. #107×C₇₃	D₄	semidirect
109. C₄×C₇₃	#357	semidirect
123. C₄×#107	D₇₃	semidirect
124. C₇₃	#13252	semidirect
156. #107	D₂₉₂	semidirect
229. C₄	#17988387	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17988964	2
C₂×C₄⋊#11924003	2
C₂×#17988387	2
C₄×#107×C₇₃	2
#13252	73