Subgroups — #22626771

Order 9344.

406 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17989639	C₂	1	2	yes	maximal, complemented
2	#17989619	C₂	1	2	yes	maximal, complemented
3	#17989618	C₂	1	2	yes	maximal, complemented
4	#17989614	C₂	1	2	yes	maximal, complemented
5	#17989610	C₂	1	1	yes	maximal, characteristic, complemented
6	#17989608	C₂	1	1	yes	maximal, characteristic, complemented
7	#17989607	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#11924037	C₂	1	1	yes	maximal, characteristic, complemented
9	#17989546	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#14556646	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#11924027	C₂	1	1	yes	maximal, characteristic
12	C₂×#11924037	C₂²	1	2	yes	—
13	C₂×#11924036	C₂²	1	2	yes	—
14	C₂²×#10596833	C₂²	1	2	yes	—
15	#14556658	C₂²	1	2	yes	—
16	#14556658	C₂²	1	2	yes	—
17	C₄⋊#10596833	C₂²	1	2	yes	—
18	#14556648	C₂²	1	2	yes	—
19	C₂²⋊#10596833	C₂²	1	2	yes	—
20	C₂²⋊#10596833	C₂²	1	2	yes	—
21	#14556646	C₂²	1	2	yes	—
22	#14556646	C₂²	1	2	yes	—
23	#14556646	C₂²	1	1	yes	characteristic
24	#14556646	C₂²	1	1	yes	characteristic
25	#14556646	C₂²	1	1	yes	characteristic
26	#14556646	C₂²	1	1	yes	characteristic
27	C₄×#10596833	C₂²	1	2	yes	—
28	C₂×#11924027	C₄	1	1	yes	characteristic
29	C₄×Dic₁₄₆	C₄	1	1	yes	characteristic
30	#14556556	C₂²	1	1	yes	characteristic
31	D₄×Dic₇₃	C₂²	1	1	yes	characteristic
32	C₄×C₂²⋊D₇₃	C₄	1	1	yes	characteristic
33	C₄×C₂²⋊D₇₃	C₄	1	1	yes	characteristic
34	#14556551	C₂²	1	1	yes	characteristic
35	#14556551	C₂²	1	1	yes	characteristic
36	#14556549	C₄	1	1	yes	characteristic
37	#14556547	C₄	1	1	yes	characteristic
38	#14556547	C₄	1	1	yes	characteristic
39	C₂×C₄×Dic₇₃	C₂²	1	1	yes	characteristic
40	C₂×C₄×Dic₇₃	C₂²	1	1	yes	characteristic
41	C₄²×D₇₃	C₂²	1	1	yes	characteristic
42	C₄×D₄×C₇₃	C₄	1	1	yes	characteristic
43	#11924037	—	2	4	no	—
44	#11924036	—	2	4	no	—
45	C₂×#10596833	—	2	4	no	—
46	C₂×#10596833	C₂³	1	2	yes	—
47	C₂×#10596833	C₂³	1	2	yes	—
48	C₂×#10596833	C₂³	1	2	yes	—
49	C₂×#10596833	C₂³	1	2	yes	—
50	#11924027	—	2	4	no	—
51	C₂×Dic₁₄₆	C₂×C₄	1	1	yes	characteristic
52	C₂×C₂²⋊D₇₃	C₂×C₄	1	1	yes	characteristic
53	C₂×C₂²⋊D₇₃	C₂×C₄	1	1	yes	characteristic
54	C₂²×Dic₇₃	C₂×C₄	1	1	yes	characteristic
55	C₂²×Dic₇₃	C₂×C₄	1	1	yes	characteristic
56	C₂²×Dic₇₃	C₂³	1	1	yes	characteristic
57	C₂²×Dic₇₃	C₂³	1	1	yes	characteristic
58	C₂×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄×D₇₃	C₂³	1	1	yes	characteristic
61	C₄⋊Dic₇₃	C₂×C₄	1	1	yes	characteristic
62	#11924007	C₂×C₄	1	1	yes	characteristic
63	#11924007	C₂×C₄	1	1	yes	characteristic
64	#11924007	C₂×C₄	1	1	yes	characteristic
65	#11924007	C₂×C₄	1	1	yes	characteristic
66	C₂²⋊Dic₇₃	C₂×C₄	1	1	yes	characteristic
67	C₂²⋊Dic₇₃	C₂×C₄	1	1	yes	characteristic
68	#11924005	C₂×C₄	1	1	yes	characteristic
69	#11924005	C₂×C₄	1	1	yes	characteristic
70	C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
71	C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
72	C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
73	C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
74	C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
75	C₄×Dic₇₃	C₂³	1	1	yes	characteristic
76	C₄×Dic₇₃	C₂³	1	1	yes	characteristic
77	C₄×Dic₇₃	C₂³	1	1	yes	characteristic
78	C₄×Dic₇₃	C₂³	1	1	yes	characteristic
79	C₂×D₄×C₇₃	C₂×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
82	C₄⋊C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
83	C₂²⋊C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
84	C₂²⋊C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
85	C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
86	C₂³×C₇₃	C₂²×C₄	1	1	yes	characteristic
87	C₂³×C₇₃	C₂²×C₄	1	1	yes	characteristic
88	C₂²×D₇₃	C₂²×C₄	1	1	yes	characteristic
89	D₄×C₇₃	—	2	2	no	—
90	C₂×C₄×C₇₃	—	4	1	no	—
91	C₂×C₄×C₇₃	—	4	1	no	—
92	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
97	C₂²⋊D₇₃	—	2	2	no	—
98	C₂²⋊D₇₃	M₁₆	1	4	yes	complemented
99	C₂×Dic₇₃	—	2	2	no	—
100	C₂×Dic₇₃	—	2	2	no	—
101	C₂×Dic₇₃	—	2	2	no	—
102	C₂×Dic₇₃	—	2	2	no	—
103	C₂×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
104	C₂×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
105	C₂×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
106	C₂×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
107	C₂×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
108	C₂×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
109	C₂×Dic₇₃	C₂⁴	1	1	yes	characteristic
110	C₄×D₇₃	—	2	2	no	—
111	C₄×D₇₃	—	2	2	no	—
112	C₄×D₇₃	—	4	1	no	—
113	Dic₁₄₆	—	2	2	no	—
114	#10596833	—	2	2	no	—
115	#10596833	—	2	2	no	—
116	#10596833	—	2	2	no	—
117	#10596833	#41	1	4	yes	complemented
118	C₂²×C₇₃	—	2	1	no	—
119	C₂²×C₇₃	—	2	1	no	—
120	C₂²×C₇₃	—	2	1	no	—
121	C₂²×C₇₃	—	2	1	no	—
122	C₂²×C₇₃	—	2	1	no	—
123	C₂²×C₇₃	C₂×M₁₆	1	2	yes	—
124	C₂²×C₇₃	C₂³×C₄	1	1	yes	characteristic
125	C₂×D₇₃	—	2	1	no	—
126	C₂×D₇₃	—	2	1	no	—
127	C₂×D₇₃	C₂×M₁₆	1	2	yes	—
128	C₄×C₇₃	—	2	1	no	—
129	C₄×C₇₃	—	2	1	no	—
130	C₄×C₇₃	—	2	1	no	—
131	C₄×C₇₃	—	2	1	no	—
132	C₄×C₇₃	—	2	1	no	—
133	Dic₇₃	—	2	1	no	—
134	Dic₇₃	—	2	1	no	—
135	Dic₇₃	—	2	1	no	—
136	Dic₇₃	—	2	1	no	—
137	Dic₇₃	—	2	1	no	—
138	Dic₇₃	C₂×M₁₆	1	2	yes	—
139	Dic₇₃	C₂×#41	1	2	yes	—
140	C₂×C₇₃	—	2	2	no	—
141	C₂×C₇₃	—	4	1	no	—
142	C₂×C₇₃	C₄×#41	1	1	yes	characteristic
143	C₂×C₇₃	C₂²×M₁₆	1	1	yes	characteristic
144	C₂×C₇₃	#568	1	1	yes	characteristic
145	D₇₃	—	2	2	no	—
146	#2970	—	73	1	no	maximal, Sylow
147	C₇₃	#2970	1	1	yes	characteristic, complemented, Sylow
148	C₄×#41	—	73	1	no	—
149	#446	—	73	2	no	—
150	#435	—	73	2	no	—
151	#435	—	73	2	no	—
152	C₈×D₄	—	73	2	no	—
153	#432	—	73	1	no	—
154	#432	—	73	1	no	—
155	#432	—	73	1	no	—
156	C₄×M₁₆	—	73	1	no	—
157	C₂×#97	—	73	1	no	—
158	C₂×#97	—	73	1	no	—
159	C₂×#41	—	73	1	no	—
160	C₂×M₁₆	—	73	2	no	—
161	C₂²×C₈	—	73	2	no	—
162	C₂²×C₈	—	73	2	no	—
163	C₄×Q₈	—	73	1	no	—
164	C₄×Q₈	—	73	1	no	—
165	C₄×D₄	—	73	1	no	—
166	C₄×D₄	—	73	1	no	—
167	C₄×D₄	—	73	1	no	—
168	C₄×D₄	—	73	1	no	—
169	C₄×D₄	—	73	1	no	—
170	C₄×D₄	#64408	1	1	yes	characteristic
171	#117	—	73	1	no	—
172	#117	—	73	1	no	—
173	#117	—	73	1	no	—
174	C₂×C₄²	—	73	1	no	—
175	C₂×C₄²	—	73	1	no	—
176	C₂×C₄²	—	73	1	no	—
177	C₄⋊C₈	—	73	2	no	—
178	C₄⋊C₈	—	73	2	no	—
179	C₄⋊C₈	—	73	2	no	—
180	C₂²⋊C₈	—	73	2	no	—
181	C₂²⋊C₈	—	73	2	no	—
182	C₂²⋊C₈	—	73	2	no	—
183	#97	—	73	1	no	—
184	#97	—	73	1	no	—
185	#97	—	73	1	no	—
186	#97	—	73	2	no	—
187	#97	—	73	2	no	—
188	#97	—	73	1	no	—
189	C₄×C₈	—	73	2	no	—
190	#41	—	146	4	no	—
191	C₂×Q₈	—	73	1	no	—
192	C₂×D₄	—	73	1	no	—
193	C₂×D₄	—	73	1	no	—
194	C₂×D₄	C₂×#64408	1	1	yes	characteristic
195	C₂²×C₄	—	73	1	no	—
196	C₂²×C₄	—	73	1	no	—
197	C₂²×C₄	—	73	1	no	—
198	C₂²×C₄	—	73	1	no	—
199	C₂²×C₄	—	73	1	no	—
200	C₂²×C₄	—	73	1	no	—
201	C₂²×C₄	—	73	1	no	—
202	C₂²×C₄	C₂×#64408	1	1	yes	characteristic
203	C₂²×C₄	C₂×#64408	1	1	yes	characteristic
204	M₁₆	—	146	4	no	—
205	C₂×C₈	—	73	2	no	—
206	C₂×C₈	—	73	2	no	—
207	C₂×C₈	—	73	2	no	—
208	C₂×C₈	—	73	2	no	—
209	C₂×C₈	—	146	4	no	—
210	C₂×C₈	—	146	4	no	—
211	C₄⋊C₄	—	73	1	no	—
212	C₄⋊C₄	—	73	1	no	—
213	C₄⋊C₄	—	73	1	no	—
214	C₄⋊C₄	—	73	1	no	—
215	C₄⋊C₄	—	73	1	no	—
216	C₄⋊C₄	C₂×#64408	1	1	yes	characteristic
217	C₂²⋊C₄	—	73	1	no	—
218	C₂²⋊C₄	—	73	1	no	—
219	C₂²⋊C₄	—	73	1	no	—
220	C₂²⋊C₄	—	73	1	no	—
221	C₂²⋊C₄	C₂×#64408	1	1	yes	characteristic
222	C₂²⋊C₄	C₂×#64408	1	1	yes	characteristic
223	C₄²	—	73	1	no	—
224	C₄²	—	73	1	no	—
225	C₄²	—	73	1	no	—
226	C₄²	—	73	1	no	—
227	C₄²	—	73	1	no	—
228	C₄²	—	73	1	no	—
229	C₄²	—	73	1	no	—
230	C₄²	—	73	1	no	—
231	C₄²	—	73	1	no	—
232	C₄²	C₂×#64408	1	1	yes	characteristic
233	C₂³	—	73	1	no	—
234	C₂³	C₂²×#64408	1	1	yes	characteristic
235	C₂³	C₂²×#64408	1	1	yes	characteristic
236	Q₈	—	146	2	no	—
237	D₄	—	2	2	no	—
238	D₄	—	73	4	no	—
239	D₄	—	146	2	no	—
240	C₂×C₄	—	4	1	no	—
241	C₂×C₄	—	4	1	no	—
242	C₂×C₄	—	292	1	no	—
243	C₂×C₄	—	73	1	no	—
244	C₂×C₄	—	73	1	no	—
245	C₂×C₄	—	73	1	no	—
246	C₂×C₄	—	73	1	no	—
247	C₂×C₄	—	73	1	no	—
248	C₂×C₄	—	73	1	no	—
249	C₂×C₄	—	73	1	no	—
250	C₂×C₄	—	146	2	no	—
251	C₂×C₄	—	146	2	no	—
252	C₂×C₄	—	146	2	no	—
253	C₂×C₄	—	146	2	no	—
254	C₂×C₄	—	146	2	no	—
255	C₂×C₄	—	146	2	no	—
256	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
257	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
258	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
259	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
260	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
261	C₈	—	73	4	no	—
262	C₈	—	146	2	no	—
263	C₈	—	146	2	no	—
264	C₈	—	146	2	no	—
265	C₂²	—	2	1	no	—
266	C₂²	—	2	1	no	—
267	C₂²	—	2	1	no	—
268	C₂²	—	2	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	73	2	no	—
271	C₂²	—	146	1	no	—
272	C₂²	—	146	1	no	—
273	C₂²	D₇₃⋊M₁₆	1	2	yes	—
274	C₂²	C₂³×#64408	1	1	yes	characteristic
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	73	2	no	—
281	C₄	—	73	2	no	—
282	C₄	—	146	1	no	—
283	C₄	—	146	1	no	—
284	C₄	—	146	1	no	—
285	C₄	—	146	1	no	—
286	C₄	—	146	1	no	—
287	C₂	—	2	2	no	—
288	C₂	—	4	1	no	—
289	C₂	—	146	2	no	—
290	C₂	#41×#64408	1	1	yes	characteristic
291	C₂	C₂×D₇₃⋊M₁₆	1	1	yes	characteristic
292	C₂	#17989665	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#64408
commutator, socle	C₂²×C₇₃	C₂³×C₄
Fitting	C₄×D₄×C₇₃	C₄
hypercenter	C₄×D₄	#64408
hypocenter	C₇₃	#2970
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
1. #17989639	C₂	semidirect
2. #17989619	C₂	semidirect
3. #17989618	C₂	semidirect
4. #17989614	C₂	semidirect
5. #17989610	C₂	semidirect
6. #17989608	C₂	semidirect
7. #17989607	C₂	semidirect
8. C₄×#11924037	C₂	semidirect
9. #17989546	C₂	semidirect
10. C₂×#14556646	C₂	semidirect
98. C₂²⋊D₇₃	M₁₆	semidirect
117. #10596833	#41	semidirect
147. C₇₃	#2970	semidirect