Subgroups — #22618879

Order 9344.

356 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17989141	C₂	1	1	yes	maximal, characteristic
2	#17988715	C₂	1	2	yes	maximal, complemented
3	#17988714	C₂	1	2	yes	maximal, complemented
4	#17988714	C₂	1	1	yes	maximal, characteristic, complemented
5	Q₈×#10596831	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×Dic₁₄₆	C₂	1	1	yes	maximal, characteristic, complemented
7	#17988612	C₂	1	2	yes	maximal
8	C₄⋊C₈×D₇₃	C₂	1	1	yes	maximal, characteristic
9	#17988544	C₂	1	2	yes	maximal
10	C₄×#11924011	C₂	1	1	yes	maximal, characteristic
11	#446×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#11924011	C₂²	1	1	yes	characteristic
13	C₂×#11924011	C₂²	1	1	yes	characteristic
14	C₂×C₈×D₇₃	C₂²	1	2	yes	—
15	#14556590	C₄	1	1	yes	characteristic
16	#14556589	C₄	1	1	yes	characteristic
17	#14556581	C₄	1	2	yes	—
18	#14556571	C₄	1	1	yes	characteristic
19	#14556569	C₄	1	2	yes	—
20	#14556567	C₄	1	1	yes	characteristic
21	C₄×Dic₁₄₆	C₂²	1	1	yes	characteristic
22	C₄×Dic₁₄₆	C₂²	1	2	yes	—
23	#14556548	C₂²	1	2	yes	—
24	C₄²×D₇₃	C₂²	1	1	yes	characteristic
25	C₄⋊#10596831	C₂²	1	2	yes	—
26	C₄⋊#10596831	C₂²	1	1	yes	characteristic
27	#14556520	C₂²	1	2	yes	—
28	#14556520	C₂²	1	2	yes	—
29	#14556520	C₂²	1	2	yes	—
30	#14556501	C₂²	1	1	yes	characteristic
31	#14556501	C₂²	1	2	yes	—
32	#14556501	C₂²	1	1	yes	characteristic
33	C₄×#10596831	C₂²	1	1	yes	characteristic
34	C₄×#10596831	C₂²	1	2	yes	—
35	C₈×Dic₇₃	C₂²	1	2	yes	—
36	C₄×Q₈×C₇₃	C₂²	1	1	yes	characteristic
37	C₄⋊C₈×C₇₃	C₂²	1	2	yes	—
38	C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
39	#97×C₇₃	C₂²	1	2	yes	—
40	C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
41	C₂×Dic₁₄₆	C₂×C₄	1	1	yes	characteristic
42	C₂×Dic₁₄₆	C₂×C₄	1	2	yes	—
43	C₂×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×D₇₃	C₂³	1	1	yes	characteristic
46	#11924011	—	2	2	no	—
47	#11924011	Q₈	1	4	yes	complemented
48	C₂×#10596831	C₂³	1	2	yes	—
49	C₂×#10596831	C₂³	1	1	yes	characteristic
50	C₂×#10596831	C₂³	1	1	yes	characteristic
51	C₈×D₇₃	—	2	4	no	—
52	C₄⋊Dic₇₃	C₂×C₄	1	1	yes	characteristic
53	C₄⋊Dic₇₃	C₂×C₄	1	2	yes	—
54	#11924007	C₂×C₄	1	2	yes	—
55	#11924007	C₂×C₄	1	2	yes	—
56	#11924007	C₂×C₄	1	2	yes	—
57	#11924007	C₂×C₄	1	2	yes	—
58	#11924007	C₂×C₄	1	2	yes	—
59	#11924005	C₂×C₄	1	2	yes	—
60	#11924005	C₂×C₄	1	2	yes	—
61	C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
62	C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
63	C₄×Dic₇₃	C₂³	1	2	yes	—
64	C₄×Dic₇₃	C₂³	1	1	yes	characteristic
65	C₂×Q₈×C₇₃	C₂×C₄	1	1	yes	characteristic
66	C₂×C₈×C₇₃	C₂³	1	1	yes	characteristic
67	C₂×C₈×C₇₃	C₂³	1	2	yes	—
68	C₂×C₈×C₇₃	C₂³	1	1	yes	characteristic
69	C₄⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
70	C₄⋊C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
71	C₄²×C₇₃	C₂³	1	1	yes	characteristic
72	C₄²×C₇₃	C₂³	1	2	yes	—
73	C₂²×D₇₃	C₂²×C₄	1	1	yes	characteristic
74	Q₈×C₇₃	—	2	2	no	—
75	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×C₇₃	C₂²×C₄	1	2	yes	—
77	C₂×C₄×C₇₃	C₂²×C₄	1	2	yes	—
78	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
80	C₂×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
82	C₂×Dic₇₃	C₂²×C₄	1	2	yes	—
83	C₂×Dic₇₃	C₂²×C₄	1	2	yes	—
84	C₂×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
85	C₄×D₇₃	—	2	2	no	—
86	C₄×D₇₃	—	4	1	no	—
87	C₄×D₇₃	C₂×Q₈	1	2	yes	—
88	C₄×D₇₃	#41	1	2	yes	—
89	Dic₁₄₆	—	2	2	no	—
90	Dic₁₄₆	—	4	2	no	—
91	C₈×C₇₃	—	2	2	no	—
92	C₈×C₇₃	—	2	1	no	—
93	C₈×C₇₃	C₂×Q₈	1	2	yes	—
94	#10596831	—	2	2	no	—
95	#10596831	—	2	1	no	—
96	#10596831	C₂×Q₈	1	2	yes	—
97	C₂²×C₇₃	C₂³×C₄	1	1	yes	characteristic
98	C₂×D₇₃	—	2	1	no	—
99	C₂×D₇₃	—	2	1	no	—
100	C₂×D₇₃	C₄×Q₈	1	2	yes	—
101	C₄×C₇₃	—	2	2	no	—
102	C₄×C₇₃	—	2	2	no	—
103	C₄×C₇₃	—	2	1	no	—
104	C₄×C₇₃	#131	1	2	yes	—
105	C₄×C₇₃	C₂²×Q₈	1	1	yes	characteristic
106	C₄×C₇₃	C₂×#41	1	1	yes	characteristic
107	Dic₇₃	—	2	2	no	—
108	Dic₇₃	—	2	1	no	—
109	Dic₇₃	—	2	1	no	—
110	Dic₇₃	—	2	2	no	—
111	Dic₇₃	C₄×Q₈	1	2	yes	—
112	C₂×C₇₃	C₂×C₄×Q₈	1	1	yes	characteristic
113	C₂×C₇₃	C₂×#131	1	1	yes	characteristic
114	C₂×C₇₃	#568	1	1	yes	characteristic
115	D₇₃	—	2	2	no	—
116	#2962	—	73	1	no	maximal, Sylow
117	C₇₃	#2962	1	1	yes	characteristic, complemented, Sylow
118	#533	—	73	1	no	—
119	#446	—	73	1	no	—
120	#446	—	73	2	no	—
121	#446	—	73	2	no	—
122	#446	D₇₃	1	1	yes	characteristic, complemented
123	C₈×Q₈	—	73	1	no	—
124	C₈×Q₈	—	73	1	no	—
125	#424	—	73	2	no	—
126	C₂×C₄⋊C₈	—	73	1	no	—
127	#405	—	73	2	no	—
128	C₄×M₁₆	—	73	1	no	—
129	C₂×M₁₆	—	73	1	no	—
130	C₂×M₁₆	—	73	1	no	—
131	C₂²×C₈	—	73	2	no	—
132	C₄⋊Q₈	—	73	1	no	—
133	C₄⋊Q₈	—	73	1	no	—
134	#125	—	73	2	no	—
135	C₂²⋊Q₈	—	73	1	no	—
136	C₂²⋊Q₈	—	73	2	no	—
137	C₂²⋊Q₈	—	73	1	no	—
138	C₄×Q₈	—	73	1	no	—
139	C₄×Q₈	—	73	2	no	—
140	C₄×Q₈	C₂×D₇₃	1	1	yes	characteristic
141	#117	—	73	2	no	—
142	C₂×C₄²	—	73	1	no	—
143	C₄⋊C₈	—	73	2	no	—
144	C₄⋊C₈	—	73	2	no	—
145	C₄⋊C₈	—	73	2	no	—
146	C₄⋊C₈	—	73	1	no	—
147	C₄⋊C₈	—	73	2	no	—
148	C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
149	C₄⋊C₈	C₂×D₇₃	1	2	yes	—
150	#97	—	73	1	no	—
151	#97	—	73	1	no	—
152	#97	—	73	2	no	—
153	#97	C₂×D₇₃	1	2	yes	—
154	C₄×C₈	—	73	2	no	—
155	C₄×C₈	—	73	1	no	—
156	C₄×C₈	—	73	2	no	—
157	C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
158	C₂×Q₈	—	73	2	no	—
159	C₂×Q₈	—	73	1	no	—
160	C₂×Q₈	C₄×D₇₃	1	1	yes	characteristic
161	C₂²×C₄	—	73	1	no	—
162	C₂²×C₄	—	73	1	no	—
163	C₂²×C₄	—	73	1	no	—
164	M₁₆	—	73	4	no	—
165	M₁₆	—	146	2	no	—
166	C₂×C₈	—	73	1	no	—
167	C₂×C₈	—	73	2	no	—
168	C₂×C₈	—	73	1	no	—
169	C₂×C₈	—	146	4	no	—
170	C₂×C₈	C₂²×D₇₃	1	1	yes	characteristic
171	C₂×C₈	C₂²×D₇₃	1	2	yes	—
172	C₂×C₈	C₂²×D₇₃	1	1	yes	characteristic
173	C₄⋊C₄	—	73	2	no	—
174	C₄⋊C₄	—	73	2	no	—
175	C₄⋊C₄	—	73	2	no	—
176	C₄⋊C₄	—	73	2	no	—
177	C₄⋊C₄	—	73	2	no	—
178	C₄⋊C₄	—	73	2	no	—
179	C₄⋊C₄	—	73	1	no	—
180	C₄⋊C₄	C₄×D₇₃	1	2	yes	—
181	C₄⋊C₄	C₄×D₇₃	1	1	yes	characteristic
182	C₂²⋊C₄	—	73	2	no	—
183	C₂²⋊C₄	—	73	2	no	—
184	C₄²	—	73	1	no	—
185	C₄²	—	73	1	no	—
186	C₄²	—	73	2	no	—
187	C₄²	—	73	1	no	—
188	C₄²	C₂²×D₇₃	1	1	yes	characteristic
189	C₄²	C₂²×D₇₃	1	2	yes	—
190	C₂³	—	73	1	no	—
191	Q₈	—	2	2	no	—
192	Q₈	—	292	2	no	—
193	Q₈	—	146	2	no	—
194	C₂×C₄	—	292	1	no	—
195	C₂×C₄	—	73	1	no	—
196	C₂×C₄	—	73	2	no	—
197	C₂×C₄	—	73	2	no	—
198	C₂×C₄	—	73	1	no	—
199	C₂×C₄	—	73	1	no	—
200	C₂×C₄	—	73	2	no	—
201	C₂×C₄	—	73	2	no	—
202	C₂×C₄	—	146	2	no	—
203	C₂×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
204	C₂×C₄	C₂×C₄×D₇₃	1	2	yes	—
205	C₂×C₄	C₂×C₄×D₇₃	1	2	yes	—
206	C₂×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
207	C₂×C₄	C₂³×D₇₃	1	1	yes	characteristic
208	C₈	—	2	2	no	—
209	C₈	—	2	1	no	—
210	C₈	—	73	2	no	—
211	C₈	—	146	2	no	—
212	C₈	—	146	1	no	—
213	C₈	Q₈×D₇₃	1	2	yes	—
214	C₂²	—	73	2	no	—
215	C₂²	—	146	1	no	—
216	C₂²	—	146	1	no	—
217	C₂²	C₂²×C₄×D₇₃	1	1	yes	characteristic
218	C₄	—	2	2	no	—
219	C₄	—	2	1	no	—
220	C₄	—	2	2	no	—
221	C₄	—	73	2	no	—
222	C₄	—	146	1	no	—
223	C₄	—	146	2	no	—
224	C₄	—	146	1	no	—
225	C₄	—	146	2	no	—
226	C₄	#14556599	1	2	yes	—
227	C₄	C₂×Q₈×D₇₃	1	1	yes	characteristic
228	C₄	#41×D₇₃	1	1	yes	characteristic
229	C₂	—	146	2	no	—
230	C₂	C₄×Q₈×D₇₃	1	1	yes	characteristic
231	C₂	C₂×#14556599	1	1	yes	characteristic
232	C₂	#17989352	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₇₃
commutator, socle	C₂²×C₇₃	C₂³×C₄
Fitting	#446×C₇₃	C₂
hypercenter	#446	D₇₃
hypocenter	C₇₃	#2962
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ #446

Complements

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

N	H ≅ G/N	type
2. #17988715	C₂	semidirect
3. #17988714	C₂	semidirect
4. #17988714	C₂	semidirect
5. Q₈×#10596831	C₂	semidirect
6. C₈×Dic₁₄₆	C₂	semidirect
11. #446×C₇₃	C₂	semidirect
47. #11924011	Q₈	semidirect
117. C₇₃	#2962	semidirect
122. #446	D₇₃	semidirect