Subgroups — #22626882

Order 9344.

364 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17989638	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×#10596833	C₂	1	1	yes	maximal, characteristic, complemented
3	#17989621	C₂	1	1	yes	maximal, characteristic, complemented
4	#17989616	C₂	1	2	yes	maximal, complemented
5	#17989615	C₂	1	1	yes	maximal, characteristic, complemented
6	#17989614	C₂	1	2	yes	maximal, complemented
7	#17989612	C₂	1	2	yes	maximal, complemented
8	#17989604	C₂	1	1	yes	maximal, characteristic, complemented
9	#17989594	C₂	1	2	yes	maximal, complemented
10	C₂×C₄⋊#10596833	C₂	1	1	yes	maximal, characteristic, complemented
11	#17989123	C₂	1	1	yes	maximal, characteristic
12	C₂×#11924038	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	1	yes	characteristic
17	#14556658	C₂²	1	1	yes	characteristic
18	C₄⋊#10596833	C₂²	1	1	yes	characteristic
19	C₄⋊#10596833	C₂²	1	2	yes	—
20	C₄⋊#10596833	C₂²	1	1	yes	characteristic
21	C₄⋊#10596833	C₂²	1	2	yes	—
22	#14556648	C₂²	1	2	yes	—
23	#14556648	C₂²	1	2	yes	—
24	C₂²⋊#10596833	C₂²	1	1	yes	characteristic
25	C₂²⋊#10596833	C₂²	1	1	yes	characteristic
26	#14556646	C₂²	1	2	yes	—
27	C₄×#10596833	C₂²	1	1	yes	characteristic
28	C₄×#10596833	C₂²	1	2	yes	—
29	C₄×#10596833	C₂²	1	1	yes	characteristic
30	#14556586	C₄	1	2	yes	—
31	#14556582	C₄	1	1	yes	characteristic
32	#14556580	C₄	1	1	yes	characteristic
33	#14556575	C₄	1	2	yes	—
34	#14556565	C₄	1	1	yes	characteristic
35	Q₈×Dic₇₃	C₂²	1	1	yes	characteristic
36	#14556553	C₂²	1	1	yes	characteristic
37	#14556551	C₂²	1	2	yes	—
38	#14556549	C₂²	1	2	yes	—
39	C₂×C₄×Dic₇₃	C₂²	1	1	yes	characteristic
40	C₂²⋊Q₈×C₇₃	C₄	1	1	yes	characteristic
41	#11924038	—	4	2	no	—
42	#11924036	—	4	2	no	—
43	C₂×#10596833	—	4	2	no	—
44	C₂×#10596833	C₂³	1	2	yes	—
45	C₂×#10596833	C₂³	1	2	yes	—
46	C₂×#10596833	C₂³	1	1	yes	characteristic
47	C₂×#10596833	C₂³	1	1	yes	characteristic
48	C₂×#10596833	C₂³	1	1	yes	characteristic
49	C₂×#10596833	C₂³	1	1	yes	characteristic
50	C₂×D₂₉₂	C₂×C₄	1	1	yes	characteristic
51	C₂×C₂²⋊D₇₃	C₂×C₄	1	2	yes	—
52	C₂²×Dic₇₃	C₂×C₄	1	1	yes	characteristic
53	C₂²×Dic₇₃	C₂³	1	1	yes	characteristic
54	C₂×C₄×D₇₃	C₂³	1	2	yes	—
55	C₄⋊Dic₇₃	C₂×C₄	1	2	yes	—
56	C₄⋊Dic₇₃	C₂×C₄	1	1	yes	characteristic
57	C₄⋊Dic₇₃	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	#11924005	C₂×C₄	1	2	yes	—
62	#11924005	C₂×C₄	1	2	yes	—
63	C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
64	C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
65	C₄×Dic₇₃	C₂³	1	1	yes	characteristic
66	C₄×Dic₇₃	C₂³	1	1	yes	characteristic
67	C₄×Dic₇₃	C₂³	1	2	yes	—
68	C₂×Q₈×C₇₃	C₂×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
70	C₄⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
71	C₄⋊C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
72	C₂²⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
73	C₂³×C₇₃	C₂²×C₄	1	1	yes	characteristic
74	C₂²×D₇₃	C₂²×C₄	1	2	yes	—
75	Q₈×C₇₃	—	2	2	no	—
76	C₂×C₄×C₇₃	—	2	2	no	—
77	C₂×C₄×C₇₃	C₂²×C₄	1	2	yes	—
78	C₂×C₄×C₇₃	C₂²×C₄	1	2	yes	—
79	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
81	C₂²⋊D₇₃	—	2	4	no	—
82	C₂×Dic₇₃	—	2	2	no	—
83	C₂×Dic₇₃	—	2	1	no	—
84	C₂×Dic₇₃	—	2	1	no	—
85	C₂×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
88	C₂×Dic₇₃	C₂²×C₄	1	2	yes	—
89	C₂×Dic₇₃	C₂⁴	1	1	yes	characteristic
90	D₂₉₂	—	2	2	no	—
91	C₄×D₇₃	—	2	2	no	—
92	C₄×D₇₃	—	2	2	no	—
93	#10596833	—	2	2	no	—
94	#10596833	—	2	2	no	—
95	#10596833	—	2	1	no	—
96	#10596833	—	2	1	no	—
97	#10596833	—	2	1	no	—
98	#10596833	—	2	1	no	—
99	C₂²×C₇₃	—	2	1	no	—
100	C₂²×C₇₃	—	2	1	no	—
101	C₂²×C₇₃	#131	1	2	yes	—
102	C₂²×C₇₃	C₂³×C₄	1	1	yes	characteristic
103	C₂×D₇₃	—	2	2	no	—
104	C₂×D₇₃	—	2	2	no	—
105	C₂×D₇₃	—	2	2	no	—
106	C₄×C₇₃	—	2	2	no	—
107	C₄×C₇₃	—	2	1	no	—
108	C₄×C₇₃	—	2	2	no	—
109	C₄×C₇₃	#131	1	2	yes	—
110	Dic₇₃	—	2	1	no	—
111	Dic₇₃	—	2	1	no	—
112	Dic₇₃	—	2	1	no	—
113	Dic₇₃	—	2	2	no	—
114	Dic₇₃	H(5,𝔽₂)	1	1	yes	characteristic
115	Dic₇₃	2₋¹⁺⁴	1	1	yes	characteristic
116	C₂×C₇₃	—	2	2	no	—
117	C₂×C₇₃	#520	1	1	yes	characteristic
118	C₂×C₇₃	C₂×#131	1	1	yes	characteristic
119	C₂×C₇₃	C₂×#131	1	1	yes	characteristic
120	D₇₃	—	4	2	no	—
121	#2994	—	73	1	no	maximal, Sylow
122	C₇₃	#2994	1	1	yes	characteristic, complemented, Sylow
123	#529	—	73	1	no	—
124	#446	—	73	1	no	—
125	C₈×Q₈	—	73	1	no	—
126	#436	—	73	1	no	—
127	#435	—	73	2	no	—
128	C₈×D₄	—	73	2	no	—
129	C₈×D₄	—	73	1	no	—
130	#433	—	73	2	no	—
131	#431	—	73	1	no	—
132	#424	—	73	2	no	—
133	C₂×C₄⋊C₈	—	73	1	no	—
134	C₂×M₁₆	—	73	2	no	—
135	C₂²×C₈	—	73	2	no	—
136	C₂²×C₈	—	73	2	no	—
137	#126	—	73	2	no	—
138	#125	—	73	1	no	—
139	#124	—	73	1	no	—
140	#123	—	73	2	no	—
141	C₂²⋊Q₈	#64408	1	1	yes	characteristic
142	#121	—	73	1	no	—
143	C₄×Q₈	—	73	1	no	—
144	C₄×D₄	—	73	1	no	—
145	C₄×D₄	—	73	2	no	—
146	#117	—	73	2	no	—
147	C₂×C₄²	—	73	1	no	—
148	C₄⋊C₈	—	73	2	no	—
149	C₄⋊C₈	—	73	1	no	—
150	C₄⋊C₈	—	73	1	no	—
151	C₄⋊C₈	—	73	1	no	—
152	C₄⋊C₈	—	73	2	no	—
153	C₄⋊C₈	—	73	2	no	—
154	C₄⋊C₈	—	73	1	no	—
155	C₂²⋊C₈	—	73	1	no	—
156	C₂²⋊C₈	—	73	1	no	—
157	C₂²⋊C₈	—	73	2	no	—
158	C₂²⋊C₈	—	73	2	no	—
159	#97	—	73	2	no	—
160	C₄×C₈	—	73	1	no	—
161	C₄×C₈	—	73	1	no	—
162	C₄×C₈	—	73	2	no	—
163	C₂×Q₈	C₂×#64408	1	1	yes	characteristic
164	C₂×D₄	—	73	2	no	—
165	C₂×D₄	—	73	1	no	—
166	C₂²×C₄	—	73	1	no	—
167	C₂²×C₄	—	73	1	no	—
168	C₂²×C₄	—	73	2	no	—
169	C₂²×C₄	C₂×#64408	1	1	yes	characteristic
170	M₁₆	—	292	2	no	—
171	C₂×C₈	—	292	2	no	—
172	C₂×C₈	—	292	2	no	—
173	C₂×C₈	—	73	1	no	—
174	C₂×C₈	—	73	1	no	—
175	C₂×C₈	—	73	2	no	—
176	C₂×C₈	—	73	1	no	—
177	C₂×C₈	—	73	1	no	—
178	C₂×C₈	—	73	2	no	—
179	C₄⋊C₄	—	73	2	no	—
180	C₄⋊C₄	—	73	2	no	—
181	C₄⋊C₄	—	73	1	no	—
182	C₄⋊C₄	—	73	2	no	—
183	C₄⋊C₄	C₂×#64408	1	2	yes	—
184	C₄⋊C₄	C₂×#64408	1	1	yes	characteristic
185	C₂²⋊C₄	—	73	2	no	—
186	C₂²⋊C₄	—	73	2	no	—
187	C₂²⋊C₄	—	73	2	no	—
188	C₂²⋊C₄	—	73	2	no	—
189	C₂²⋊C₄	C₂×#64408	1	2	yes	—
190	C₄²	—	73	1	no	—
191	C₄²	—	73	1	no	—
192	C₄²	—	73	1	no	—
193	C₄²	—	73	2	no	—
194	C₄²	—	73	1	no	—
195	C₂³	—	73	2	no	—
196	C₂³	C₂²×#64408	1	1	yes	characteristic
197	Q₈	—	2	2	no	—
198	D₄	—	146	4	no	—
199	D₄	—	146	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	73	1	no	—
202	C₂×C₄	—	73	1	no	—
203	C₂×C₄	—	73	1	no	—
204	C₂×C₄	—	73	1	no	—
205	C₂×C₄	—	73	2	no	—
206	C₂×C₄	—	146	2	no	—
207	C₂×C₄	—	146	2	no	—
208	C₂×C₄	—	146	1	no	—
209	C₂×C₄	—	146	1	no	—
210	C₂×C₄	—	146	2	no	—
211	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
212	C₂×C₄	C₂²×#64408	1	2	yes	—
213	C₂×C₄	C₂²×#64408	1	2	yes	—
214	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
215	C₈	—	146	2	no	—
216	C₈	—	146	1	no	—
217	C₈	—	146	2	no	—
218	C₈	—	146	1	no	—
219	C₈	—	146	1	no	—
220	C₈	—	146	1	no	—
221	C₂²	—	2	1	no	—
222	C₂²	—	2	1	no	—
223	C₂²	—	146	2	no	—
224	C₂²	—	146	2	no	—
225	C₂²	—	146	2	no	—
226	C₂²	#14556679	1	2	yes	—
227	C₂²	C₂³×#64408	1	1	yes	characteristic
228	C₄	—	2	2	no	—
229	C₄	—	2	1	no	—
230	C₄	—	2	2	no	—
231	C₄	—	73	1	no	—
232	C₄	—	73	1	no	—
233	C₄	—	146	1	no	—
234	C₄	—	146	1	no	—
235	C₄	—	146	1	no	—
236	C₄	—	146	2	no	—
237	C₄	#14556678	1	2	yes	—
238	C₂	—	2	2	no	—
239	C₂	—	292	2	no	—
240	C₂	#17989656	1	1	yes	characteristic
241	C₂	C₂×#14556678	1	1	yes	characteristic
242	C₂	C₂×#14556679	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#64408
commutator, socle	C₂²×C₇₃	C₂³×C₄
Fitting	C₂²⋊Q₈×C₇₃	C₄
hypercenter	C₂²⋊Q₈	#64408
hypocenter	C₇₃	#2994
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
1. #17989638	C₂	semidirect
2. Q₈×#10596833	C₂	semidirect
3. #17989621	C₂	semidirect
4. #17989616	C₂	semidirect
5. #17989615	C₂	semidirect
6. #17989614	C₂	semidirect
7. #17989612	C₂	semidirect
8. #17989604	C₂	semidirect
9. #17989594	C₂	semidirect
10. C₂×C₄⋊#10596833	C₂	semidirect
122. C₇₃	#2994	semidirect