Subgroups — #22625639

Order 9344.

388 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₇₃⋊M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×D₇₃⋊M₁₆	C₂	1	1	yes	maximal, characteristic
3	#17989468	C₂	1	4	yes	maximal, complemented
4	#117×D₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#11924038	—	2	1	no	—
6	C₂×#11924038	C₄	1	2	yes	complemented
7	D₇₃⋊M₁₆	—	2	1	no	—
8	D₇₃⋊M₁₆	—	2	1	no	—
9	D₇₃⋊M₁₆	—	2	1	no	—
10	D₇₃⋊M₁₆	—	2	1	no	—
11	D₇₃⋊M₁₆	—	2	1	no	—
12	D₇₃⋊M₁₆	—	2	1	no	—
13	D₇₃⋊M₁₆	C₄	1	2	yes	complemented
14	D₇₃⋊M₁₆	C₄	1	2	yes	complemented
15	C₂×#11924037	C₂²	1	2	yes	—
16	C₂×#11924037	C₂²	1	2	yes	—
17	C₂×#11924036	—	2	1	no	—
18	C₂×#11924036	C₄	1	2	yes	complemented
19	C₂²×C₄×D₇₃	C₂²	1	1	yes	characteristic
20	#14556550	C₄	1	1	yes	characteristic
21	#14556549	—	2	1	no	—
22	#14556548	—	2	1	no	—
23	#14556547	C₂²	1	2	yes	—
24	C₄⋊C₄×D₇₃	—	2	1	no	—
25	C₂²⋊C₄×D₇₃	C₄	1	2	yes	—
26	C₄²×D₇₃	—	2	1	no	—
27	#117×C₇₃	C₄	1	1	yes	characteristic
28	#11924038	—	2	2	no	—
29	#11924038	—	2	2	no	—
30	#11924038	—	2	2	no	—
31	#11924038	—	2	2	no	—
32	#11924037	—	2	2	no	—
33	#11924037	—	2	2	no	—
34	#11924037	—	2	2	no	—
35	#11924037	C₂×C₄	1	4	yes	complemented
36	#11924036	—	2	2	no	—
37	#11924036	—	2	2	no	—
38	#11924036	—	2	2	no	—
39	#11924036	—	2	2	no	—
40	C₂×#10596833	—	2	2	no	—
41	C₂×#10596833	C₂×C₄	1	4	yes	complemented
42	C₂³×D₇₃	C₂×C₄	1	1	yes	characteristic
43	C₂²×Dic₇₃	C₂³	1	1	yes	characteristic
44	C₂×C₄×D₇₃	—	2	2	no	—
45	C₂×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×D₇₃	D₄	1	1	yes	characteristic
50	C₂×C₄×D₇₃	D₄	1	1	yes	characteristic
51	C₂×C₄×D₇₃	D₄	1	1	yes	characteristic
52	C₂×C₄×D₇₃	D₄	1	1	yes	characteristic
53	C₂×C₄×D₇₃	D₄	1	1	yes	characteristic
54	C₂×C₄×D₇₃	D₄	1	1	yes	characteristic
55	C₂×C₄×D₇₃	Q₈	1	1	yes	characteristic
56	C₂×C₄×D₇₃	Q₈	1	1	yes	characteristic
57	C₄⋊Dic₇₃	—	2	1	no	—
58	#11924007	—	2	2	no	—
59	C₂²⋊Dic₇₃	C₂×C₄	1	2	yes	—
60	#11924005	—	2	2	no	—
61	C₄×Dic₇₃	—	2	1	no	—
62	C₄×Dic₇₃	—	2	2	no	—
63	C₂²×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
64	C₄⋊C₄×C₇₃	—	2	1	no	—
65	C₂²⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
66	C₄²×C₇₃	—	2	1	no	—
67	C₂³×C₇₃	C₂²×C₄	1	1	yes	characteristic
68	C₂²×D₇₃	—	4	2	no	—
69	C₂²×D₇₃	C₄²	1	1	yes	characteristic
70	C₂²×D₇₃	C₄²	1	1	yes	characteristic
71	C₂²×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×D₇₃	C₄⋊C₄	1	1	yes	characteristic
74	C₂²×D₇₃	C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₇₃	—	2	2	no	—
76	C₂×C₄×C₇₃	C₄²	1	1	yes	characteristic
77	C₂×C₄×C₇₃	C₄²	1	1	yes	characteristic
78	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
82	C₂×Dic₇₃	—	2	2	no	—
83	C₂×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
84	C₂×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×Dic₇₃	C₂×D₄	1	1	yes	characteristic
86	C₂×Dic₇₃	C₂×D₄	1	1	yes	characteristic
87	C₂×Dic₇₃	C₂×D₄	1	1	yes	characteristic
88	C₂×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
89	C₄×D₇₃	—	4	4	no	—
90	C₄×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
91	C₄×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
92	C₄×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
93	C₄×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
94	C₄×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
95	C₄×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
96	C₄×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
97	C₄×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
98	C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
99	C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
100	C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
101	C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
102	C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
103	C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
104	C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
105	C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
106	#10596833	—	2	4	no	—
107	#10596833	—	2	4	no	—
108	C₂²×C₇₃	—	4	1	no	—
109	C₂²×C₇₃	C₂×C₄²	1	1	yes	characteristic
110	C₂²×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂²×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×D₇₃	—	2	2	no	—
113	C₂×D₇₃	—	2	2	no	—
114	C₂×D₇₃	—	2	2	no	—
115	C₂×D₇₃	—	4	1	no	—
116	C₂×D₇₃	—	4	1	no	—
117	C₂×D₇₃	—	4	1	no	—
118	C₂×D₇₃	#95	1	1	yes	characteristic
119	C₂×D₇₃	#95	1	1	yes	characteristic
120	C₂×D₇₃	#95	1	1	yes	characteristic
121	C₂×D₇₃	#95	1	1	yes	characteristic
122	C₄×C₇₃	—	4	2	no	—
123	C₄×C₇₃	#95	1	1	yes	characteristic
124	C₄×C₇₃	#95	1	1	yes	characteristic
125	C₄×C₇₃	#95	1	1	yes	characteristic
126	C₄×C₇₃	#95	1	1	yes	characteristic
127	Dic₇₃	—	4	2	no	—
128	Dic₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
129	Dic₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	Dic₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
131	Dic₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
132	C₂×C₇₃	—	2	1	no	—
133	C₂×C₇₃	—	2	1	no	—
134	C₂×C₇₃	—	2	1	no	—
135	C₂×C₇₃	C₂×#95	1	1	yes	characteristic
136	D₇₃	—	2	1	no	—
137	D₇₃	—	2	1	no	—
138	D₇₃	—	2	1	no	—
139	D₇₃	#344	1	2	yes	complemented
140	C₂×#344	—	73	1	no	maximal, Sylow
141	C₇₃	C₂×#344	1	1	yes	characteristic, complemented, Sylow
142	C₂²×M₁₆	—	73	1	no	—
143	C₂²×M₁₆	—	73	1	no	—
144	C₂×#117	—	73	1	no	—
145	#344	—	73	4	no	—
146	C₂³×C₄	—	73	1	no	—
147	C₂×M₁₆	—	73	2	no	—
148	C₂×M₁₆	—	73	2	no	—
149	C₂×M₁₆	—	73	2	no	—
150	C₂×M₁₆	—	73	2	no	—
151	C₂×M₁₆	—	73	2	no	—
152	C₂×M₁₆	—	146	1	no	—
153	C₂×M₁₆	—	146	1	no	—
154	C₂×M₁₆	—	146	1	no	—
155	C₂×M₁₆	—	146	1	no	—
156	C₂×M₁₆	—	146	1	no	—
157	C₂×M₁₆	—	146	1	no	—
158	C₂×M₁₆	—	146	1	no	—
159	C₂²×C₈	—	73	2	no	—
160	C₂²×C₈	—	146	1	no	—
161	#117	—	73	1	no	—
162	#117	—	73	2	no	—
163	#117	—	146	1	no	—
164	#117	—	146	1	no	—
165	#117	#64408	1	1	yes	characteristic
166	C₂×C₄⋊C₄	—	146	1	no	—
167	C₂×C₂²⋊C₄	—	73	2	no	—
168	C₂×C₄²	—	146	1	no	—
169	C₂⁴	—	73	1	no	—
170	C₂²×C₄	—	73	1	no	—
171	C₂²×C₄	—	73	1	no	—
172	C₂²×C₄	—	73	1	no	—
173	C₂²×C₄	—	73	1	no	—
174	C₂²×C₄	—	73	1	no	—
175	C₂²×C₄	—	73	1	no	—
176	C₂²×C₄	—	73	1	no	—
177	C₂²×C₄	—	73	1	no	—
178	C₂²×C₄	—	73	1	no	—
179	C₂²×C₄	—	73	1	no	—
180	C₂²×C₄	—	73	1	no	—
181	C₂²×C₄	—	73	1	no	—
182	C₂²×C₄	—	73	1	no	—
183	C₂²×C₄	—	146	2	no	—
184	C₂²×C₄	C₂×#64408	1	1	yes	characteristic
185	M₁₆	—	73	4	no	—
186	M₁₆	—	146	2	no	—
187	M₁₆	—	146	2	no	—
188	M₁₆	—	146	2	no	—
189	M₁₆	—	146	2	no	—
190	M₁₆	—	146	2	no	—
191	M₁₆	—	146	2	no	—
192	M₁₆	—	146	2	no	—
193	C₂×C₈	—	73	4	no	—
194	C₂×C₈	—	146	2	no	—
195	C₂×C₈	—	146	2	no	—
196	C₂×C₈	—	146	2	no	—
197	C₂×C₈	—	146	2	no	—
198	C₂×C₈	—	146	2	no	—
199	C₄⋊C₄	—	2	1	no	—
200	C₄⋊C₄	—	146	2	no	—
201	C₄⋊C₄	—	146	1	no	—
202	C₂²⋊C₄	—	73	2	no	—
203	C₂²⋊C₄	—	146	2	no	—
204	C₂²⋊C₄	C₂×#64408	1	2	yes	—
205	C₄²	—	2	1	no	—
206	C₄²	—	146	2	no	—
207	C₄²	—	146	1	no	—
208	C₂³	—	292	2	no	—
209	C₂³	—	73	1	no	—
210	C₂³	—	73	1	no	—
211	C₂³	—	73	1	no	—
212	C₂³	—	73	1	no	—
213	C₂³	—	73	1	no	—
214	C₂³	—	73	1	no	—
215	C₂³	C₂²×#64408	1	1	yes	characteristic
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	292	4	no	—
218	C₂×C₄	—	73	1	no	—
219	C₂×C₄	—	73	1	no	—
220	C₂×C₄	—	73	1	no	—
221	C₂×C₄	—	73	1	no	—
222	C₂×C₄	—	73	1	no	—
223	C₂×C₄	—	73	1	no	—
224	C₂×C₄	—	73	1	no	—
225	C₂×C₄	—	73	1	no	—
226	C₂×C₄	—	73	1	no	—
227	C₂×C₄	—	73	1	no	—
228	C₂×C₄	—	73	1	no	—
229	C₂×C₄	—	73	1	no	—
230	C₂×C₄	—	73	1	no	—
231	C₂×C₄	—	73	1	no	—
232	C₂×C₄	—	73	1	no	—
233	C₂×C₄	—	73	1	no	—
234	C₂×C₄	—	73	1	no	—
235	C₂×C₄	—	73	1	no	—
236	C₂×C₄	—	73	1	no	—
237	C₂×C₄	—	73	1	no	—
238	C₂×C₄	—	73	1	no	—
239	C₂×C₄	—	73	1	no	—
240	C₂×C₄	—	146	2	no	—
241	C₂×C₄	C₄×#64408	1	1	yes	characteristic
242	C₂×C₄	C₄×#64408	1	1	yes	characteristic
243	C₂×C₄	C₂²⋊#64408	1	1	yes	characteristic
244	C₂×C₄	C₂²⋊#64408	1	1	yes	characteristic
245	C₂×C₄	C₄⋊#64408	1	1	yes	characteristic
246	C₂×C₄	C₄⋊#64408	1	1	yes	characteristic
247	C₈	—	146	4	no	—
248	C₈	—	146	4	no	—
249	C₂²	—	4	1	no	—
250	C₂²	—	292	1	no	—
251	C₂²	—	292	1	no	—
252	C₂²	—	292	1	no	—
253	C₂²	—	73	1	no	—
254	C₂²	—	73	1	no	—
255	C₂²	—	73	1	no	—
256	C₂²	—	73	1	no	—
257	C₂²	—	146	2	no	—
258	C₂²	—	146	2	no	—
259	C₂²	—	146	2	no	—
260	C₂²	C₂×C₄×#64408	1	1	yes	characteristic
261	C₂²	C₂×C₂²⋊#64408	1	1	yes	characteristic
262	C₂²	C₂×C₄⋊#64408	1	1	yes	characteristic
263	C₄	—	4	2	no	—
264	C₄	—	292	2	no	—
265	C₄	—	73	1	no	—
266	C₄	—	73	1	no	—
267	C₄	—	73	1	no	—
268	C₄	—	73	1	no	—
269	C₄	(C₂×#64408)⋊C₄	1	1	yes	characteristic
270	C₄	(C₂×#64408)⋊C₄	1	1	yes	characteristic
271	C₄	(C₂×#64408)⋊C₄	1	1	yes	characteristic
272	C₄	(C₂×#64408)⋊C₄	1	1	yes	characteristic
273	C₂	—	2	1	no	—
274	C₂	—	2	1	no	—
275	C₂	—	2	1	no	—
276	C₂	—	73	2	no	—
277	C₂	—	146	1	no	—
278	C₂	—	146	1	no	—
279	C₂	—	146	1	no	—
280	C₂	C₂×(C₂×#64408)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	(C₂×#64408)⋊C₄
commutator	C₂²×C₇₃	C₂×C₄²
Frattini	C₂³	C₂²×#64408
Fitting	#117×C₇₃	C₄
socle	C₂×C₇₃	C₂×#95
hypercenter	#117	#64408
hypocenter	C₇₃	C₂×#344
LCS	G ▷ C₂²×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
3. #17989468	C₂	semidirect
6. C₂×#11924038	C₄	semidirect
13. D₇₃⋊M₁₆	C₄	semidirect
14. D₇₃⋊M₁₆	C₄	semidirect
18. C₂×#11924036	C₄	semidirect
35. #11924037	C₂×C₄	semidirect
41. C₂×#10596833	C₂×C₄	semidirect
139. D₇₃	#344	semidirect
141. C₇₃	C₂×#344	semidirect