Subgroups — #22620007

Order 9344.

390 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₁₄₆	C₂	1	1	yes	maximal, characteristic, complemented
2	#17988915	C₂	1	1	yes	maximal, characteristic, complemented
3	#17988898	C₂	1	1	yes	maximal, characteristic
4	#17988891	C₂	1	2	yes	maximal, complemented
5	#17988806	C₂	1	2	yes	maximal, complemented
6	#17988742	C₂	1	1	yes	maximal, characteristic, complemented
7	#17988725	C₂	1	1	yes	maximal, characteristic
8	C₄×#11924018	C₂	1	2	yes	maximal, complemented
9	C₄×#11924016	C₂	1	2	yes	maximal, complemented
10	#17988640	C₂	1	1	yes	maximal, characteristic, complemented
11	#529×C₇₃	C₂	1	1	yes	maximal, characteristic
12	C₂²×Dic₁₄₆	C₂²	1	1	yes	characteristic
13	C₂×#11924018	C₂²	1	2	yes	—
14	C₂×#11924016	C₂²	1	2	yes	—
15	C₄×Dic₁₄₆	—	2	1	no	—
16	C₄×Dic₁₄₆	—	2	1	no	—
17	C₄×Dic₁₄₆	C₂²	1	2	yes	—
18	C₄×Dic₁₄₆	C₂²	1	2	yes	—
19	C₂×C₄⋊Dic₇₃	C₂²	1	1	yes	characteristic
20	C₂×#11924007	—	2	1	no	—
21	C₂×C₄×Dic₇₃	—	2	1	no	—
22	#14556524	C₂²	1	2	yes	—
23	#14556522	C₂²	1	2	yes	—
24	C₄⋊#10596831	C₂²	1	2	yes	—
25	#14556515	C₂²	1	2	yes	—
26	#14556514	C₂²	1	2	yes	—
27	#14556514	C₂²	1	2	yes	—
28	#14556512	C₂²	1	2	yes	—
29	C₂²⋊#10596831	C₂²	1	1	yes	characteristic
30	C₂²⋊#10596831	C₂²	1	1	yes	characteristic
31	C₄×#10596831	C₂²	1	2	yes	—
32	#126×C₇₃	—	2	1	no	—
33	#125×C₇₃	C₂²	1	1	yes	characteristic
34	#124×C₇₃	C₂²	1	1	yes	characteristic
35	#123×C₇₃	—	2	1	no	—
36	C₂²⋊Q₈×C₇₃	C₂²	1	1	yes	characteristic
37	#121×C₇₃	C₂²	1	1	yes	characteristic
38	C₄×Q₈×C₇₃	C₂²	1	1	yes	characteristic
39	C₄×D₄×C₇₃	—	2	1	no	—
40	C₄×D₄×C₇₃	C₂²	1	1	yes	characteristic
41	#117×C₇₃	—	2	1	no	—
42	C₂×C₄²×C₇₃	C₂²	1	1	yes	characteristic
43	C₂×Dic₁₄₆	—	2	1	no	—
44	C₂×Dic₁₄₆	—	2	4	no	—
45	C₂×Dic₁₄₆	C₂³	1	2	yes	—
46	C₂²×Dic₇₃	—	2	1	no	—
47	C₂²×Dic₇₃	—	2	1	no	—
48	#11924018	—	2	4	no	—
49	#11924016	—	2	4	no	—
50	C₂×#10596831	C₂³	1	2	yes	—
51	C₂×#10596831	C₂³	1	2	yes	—
52	C₄⋊Dic₇₃	—	2	1	no	—
53	C₄⋊Dic₇₃	C₂³	1	2	yes	—
54	#11924007	—	2	2	no	—
55	#11924007	—	2	2	no	—
56	C₄×Dic₇₃	—	2	2	no	—
57	C₄×Dic₇₃	—	2	2	no	—
58	C₂×Q₈×C₇₃	C₂³	1	1	yes	characteristic
59	C₂×D₄×C₇₃	—	2	1	no	—
60	C₂×D₄×C₇₃	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₇₃	—	2	1	no	—
62	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
63	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
64	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
65	C₄⋊C₄×C₇₃	—	2	1	no	—
66	C₄⋊C₄×C₇₃	—	2	1	no	—
67	C₄⋊C₄×C₇₃	—	2	1	no	—
68	C₄⋊C₄×C₇₃	—	2	1	no	—
69	C₄⋊C₄×C₇₃	C₂³	1	1	yes	characteristic
70	C₄⋊C₄×C₇₃	C₂³	1	1	yes	characteristic
71	C₂²⋊C₄×C₇₃	—	2	1	no	—
72	C₂²⋊C₄×C₇₃	—	2	1	no	—
73	C₂²⋊C₄×C₇₃	—	2	1	no	—
74	C₂²⋊C₄×C₇₃	—	2	1	no	—
75	C₂²⋊C₄×C₇₃	—	2	1	no	—
76	C₄²×C₇₃	—	2	1	no	—
77	C₄²×C₇₃	D₄	1	1	yes	characteristic
78	C₄²×C₇₃	D₄	1	1	yes	characteristic
79	C₄²×C₇₃	C₂³	1	1	yes	characteristic
80	C₄²×C₇₃	C₂³	1	1	yes	characteristic
81	C₂³×C₇₃	—	2	1	no	—
82	C₂³×C₇₃	C₂×D₄	1	1	yes	characteristic
83	Q₈×C₇₃	—	2	2	no	—
84	D₄×C₇₃	—	2	2	no	—
85	D₄×C₇₃	—	4	2	no	—
86	C₂×C₄×C₇₃	—	2	2	no	—
87	C₂×C₄×C₇₃	—	2	2	no	—
88	C₂×C₄×C₇₃	—	2	2	no	—
89	C₂×C₄×C₇₃	—	2	1	no	—
90	C₂×C₄×C₇₃	—	2	1	no	—
91	C₂×C₄×C₇₃	—	2	1	no	—
92	C₂×C₄×C₇₃	—	4	2	no	—
93	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
99	C₂×Dic₇₃	—	2	4	no	—
100	C₂×Dic₇₃	—	2	2	no	—
101	C₂×Dic₇₃	—	2	2	no	—
102	C₂×Dic₇₃	—	4	2	no	—
103	Dic₁₄₆	—	2	4	no	—
104	Dic₁₄₆	—	2	2	no	—
105	Dic₁₄₆	#41	1	4	yes	—
106	#10596831	—	2	2	no	—
107	#10596831	—	2	2	no	—
108	C₂²×C₇₃	—	2	1	no	—
109	C₂²×C₇₃	—	2	1	no	—
110	C₂²×C₇₃	—	4	1	no	—
111	C₂²×C₇₃	—	4	1	no	—
112	C₂²×C₇₃	—	4	1	no	—
113	C₂²×C₇₃	#135	1	2	yes	—
114	C₂²×C₇₃	C₂²×D₄	1	1	yes	characteristic
115	C₄×C₇₃	—	2	1	no	—
116	C₄×C₇₃	—	2	1	no	—
117	C₄×C₇₃	—	2	1	no	—
118	C₄×C₇₃	—	2	1	no	—
119	C₄×C₇₃	—	4	1	no	—
120	C₄×C₇₃	—	4	1	no	—
121	C₄×C₇₃	—	4	1	no	—
122	C₄×C₇₃	#137	1	2	yes	—
123	C₄×C₇₃	C₂×#41	1	2	yes	—
124	Dic₇₃	—	2	4	no	—
125	Dic₇₃	—	4	2	no	—
126	C₂×C₇₃	—	2	2	no	—
127	C₂×C₇₃	—	8	1	no	—
128	C₂×C₇₃	#525	1	1	yes	characteristic
129	C₂×C₇₃	C₂×#135	1	1	yes	characteristic
130	C₂×C₇₃	C₂×#137	1	1	yes	characteristic
131	#3108	—	73	1	no	maximal, Sylow
132	C₇₃	#3108	1	1	yes	characteristic, complemented, Sylow
133	#529	D₇₃	1	1	yes	characteristic
134	C₂×C₄×Q₈	—	73	1	no	—
135	#483	—	73	1	no	—
136	#480	—	73	1	no	—
137	#479	—	73	2	no	—
138	#464	—	73	2	no	—
139	#451	—	73	1	no	—
140	#448	—	73	1	no	—
141	C₄×Q₁₆	—	73	2	no	—
142	C₄×SD₁₆	—	73	2	no	—
143	#431	—	73	1	no	—
144	C₂²×Q₈	—	73	1	no	—
145	C₂×Q₁₆	—	73	2	no	—
146	C₂×SD₁₆	—	73	2	no	—
147	#126	—	2	1	no	—
148	#125	C₂×D₇₃	1	1	yes	characteristic
149	#124	C₂×D₇₃	1	1	yes	characteristic
150	#123	—	2	1	no	—
151	C₂²⋊Q₈	C₂×D₇₃	1	1	yes	characteristic
152	#121	C₂×D₇₃	1	1	yes	characteristic
153	C₄×Q₈	—	73	2	no	—
154	C₄×Q₈	—	73	2	no	—
155	C₄×Q₈	—	146	1	no	—
156	C₄×Q₈	—	146	1	no	—
157	C₄×Q₈	C₂×D₇₃	1	1	yes	characteristic
158	C₄×D₄	—	2	1	no	—
159	C₄×D₄	C₂×D₇₃	1	1	yes	characteristic
160	#117	—	2	1	no	—
161	C₂×C₄⋊C₄	—	73	1	no	—
162	C₂×C₄⋊C₄	—	146	1	no	—
163	C₂×C₄²	—	146	1	no	—
164	C₂×C₄²	C₂×D₇₃	1	1	yes	characteristic
165	#107	—	73	2	no	—
166	#106	—	73	2	no	—
167	C₄⋊C₈	—	73	2	no	—
168	#103	—	73	2	no	—
169	#103	—	73	2	no	—
170	#103	—	73	2	no	—
171	#102	—	73	2	no	—
172	C₂²⋊C₈	—	73	1	no	—
173	C₂²⋊C₈	—	73	1	no	—
174	C₄×C₈	—	73	2	no	—
175	C₂×Q₈	—	73	2	no	—
176	C₂×Q₈	—	146	4	no	—
177	C₂×Q₈	—	146	1	no	—
178	C₂×Q₈	C₂²×D₇₃	1	1	yes	characteristic
179	C₂×D₄	—	2	1	no	—
180	C₂×D₄	C₂²×D₇₃	1	1	yes	characteristic
181	C₂²×C₄	—	2	1	no	—
182	C₂²×C₄	—	146	1	no	—
183	C₂²×C₄	—	146	1	no	—
184	C₂²×C₄	C₂²⋊D₇₃	1	1	yes	characteristic
185	C₂²×C₄	C₂²⋊D₇₃	1	1	yes	characteristic
186	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
187	Q₁₆	—	146	4	no	—
188	SD₁₆	—	146	4	no	—
189	C₂×C₈	—	73	2	no	—
190	C₂×C₈	—	73	2	no	—
191	C₄⋊C₄	—	2	1	no	—
192	C₄⋊C₄	—	2	1	no	—
193	C₄⋊C₄	—	2	1	no	—
194	C₄⋊C₄	—	2	1	no	—
195	C₄⋊C₄	—	73	2	no	—
196	C₄⋊C₄	—	146	2	no	—
197	C₄⋊C₄	—	146	2	no	—
198	C₄⋊C₄	—	146	1	no	—
199	C₄⋊C₄	C₂²×D₇₃	1	1	yes	characteristic
200	C₄⋊C₄	C₂²×D₇₃	1	1	yes	characteristic
201	C₂²⋊C₄	—	2	1	no	—
202	C₂²⋊C₄	—	2	1	no	—
203	C₂²⋊C₄	—	2	1	no	—
204	C₂²⋊C₄	—	2	1	no	—
205	C₂²⋊C₄	—	2	1	no	—
206	C₄²	—	2	1	no	—
207	C₄²	—	146	2	no	—
208	C₄²	—	146	2	no	—
209	C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
210	C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
211	C₄²	C₂²×D₇₃	1	1	yes	characteristic
212	C₄²	C₂²×D₇₃	1	1	yes	characteristic
213	C₂³	—	2	1	no	—
214	C₂³	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
215	Q₈	—	2	2	no	—
216	Q₈	—	73	4	no	—
217	Q₈	—	146	4	no	—
218	Q₈	—	146	2	no	—
219	D₄	—	2	2	no	—
220	D₄	—	4	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	292	2	no	—
229	C₂×C₄	—	146	2	no	—
230	C₂×C₄	—	146	4	no	—
231	C₂×C₄	—	146	2	no	—
232	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
233	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
234	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
235	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
236	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
237	C₂×C₄	C₂³×D₇₃	1	1	yes	characteristic
238	C₈	—	146	2	no	—
239	C₈	—	146	2	no	—
240	C₂²	—	2	1	no	—
241	C₂²	—	2	1	no	—
242	C₂²	—	4	1	no	—
243	C₂²	—	4	1	no	—
244	C₂²	—	4	1	no	—
245	C₂²	#14556611	1	2	yes	—
246	C₂²	C₂²×C₂²⋊D₇₃	1	1	yes	characteristic
247	C₄	—	2	1	no	—
248	C₄	—	2	1	no	—
249	C₄	—	2	1	no	—
250	C₄	—	2	1	no	—
251	C₄	—	4	1	no	—
252	C₄	—	4	1	no	—
253	C₄	—	4	1	no	—
254	C₄	—	292	2	no	—
255	C₄	—	146	4	no	—
256	C₄	#14556621	1	2	yes	—
257	C₄	#41×D₇₃	1	2	yes	—
258	C₂	—	2	2	no	—
259	C₂	—	8	1	no	—
260	C₂	#17989094	1	1	yes	characteristic
261	C₂	C₂×#14556611	1	1	yes	characteristic
262	C₂	C₂×#14556621	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₇₃
commutator	C₂×C₄×C₇₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₃
Fitting	#529×C₇₃	C₂
socle	C₂²×C₇₃	C₂²×D₄
hypercenter	#529	D₇₃
hypocenter	C₇₃	#3108
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #529

Complements

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

N	H ≅ G/N	type
1. C₂×C₄×Dic₁₄₆	C₂	semidirect
2. #17988915	C₂	semidirect
4. #17988891	C₂	semidirect
5. #17988806	C₂	semidirect
6. #17988742	C₂	semidirect
8. C₄×#11924018	C₂	semidirect
9. C₄×#11924016	C₂	semidirect
10. #17988640	C₂	semidirect
132. C₇₃	#3108	semidirect