Subgroups — #22626840

Order 9344.

330 conjugacy classes of subgroups.

The subgroup lattice has 228 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	#17989638	C₂	1	1	yes	maximal, characteristic, complemented
3	#17989637	C₂	1	1	yes	maximal, characteristic, complemented
4	#17989636	C₂	1	1	yes	maximal, characteristic, complemented
5	#17989636	C₂	1	2	yes	maximal, complemented
6	Q₈×#10596833	C₂	1	1	yes	maximal, characteristic, complemented
7	#17989611	C₂	1	2	yes	maximal
8	#17989611	C₂	1	2	yes	maximal
9	#17989604	C₂	1	1	yes	maximal, characteristic
10	#17989603	C₂	1	1	yes	maximal, characteristic
11	#17989139	C₂	1	1	yes	maximal, characteristic
12	#14556658	C₂²	1	1	yes	characteristic
13	#14556658	C₂²	1	2	yes	—
14	#14556658	C₂²	1	2	yes	—
15	#14556658	C₂²	1	1	yes	characteristic
16	C₄⋊#10596833	C₂²	1	1	yes	characteristic
17	C₄⋊#10596833	C₂²	1	2	yes	—
18	C₄⋊#10596833	C₂²	1	2	yes	—
19	C₄⋊#10596833	C₂²	1	1	yes	characteristic
20	C₂²⋊#10596833	C₂²	1	2	yes	—
21	C₂²⋊#10596833	C₂²	1	1	yes	characteristic
22	C₂²⋊#10596833	C₂²	1	1	yes	characteristic
23	#14556646	C₂²	1	2	yes	—
24	#14556646	C₂²	1	2	yes	—
25	C₄×#10596833	C₂²	1	2	yes	—
26	C₄×#10596833	C₂²	1	1	yes	characteristic
27	C₄×#10596833	C₂²	1	2	yes	—
28	C₄×#10596833	C₂²	1	1	yes	characteristic
29	C₄×#10596833	C₂²	1	2	yes	—
30	#14556590	C₄	1	1	yes	characteristic
31	#14556589	C₄	1	1	yes	characteristic
32	#14556581	C₄	1	2	yes	—
33	#14556570	C₄	1	1	yes	characteristic
34	#14556568	C₄	1	2	yes	—
35	Q₈×Dic₇₃	C₂²	1	1	yes	characteristic
36	#14556556	C₂²	1	1	yes	characteristic
37	#14556556	C₂²	1	2	yes	—
38	#14556547	C₂²	1	2	yes	—
39	C₂×C₄×Dic₇₃	C₂²	1	1	yes	characteristic
40	C₂²⋊Q₈×C₇₃	C₄	1	1	yes	characteristic
41	C₂×#10596833	C₂³	1	1	yes	characteristic
42	C₂×#10596833	C₂³	1	2	yes	—
43	C₂×#10596833	C₂³	1	1	yes	characteristic
44	C₂×#10596833	C₂³	1	1	yes	characteristic
45	C₂×#10596833	C₂³	1	2	yes	—
46	C₂×#10596833	C₂³	1	1	yes	characteristic
47	C₂×Dic₁₄₆	C₂×C₄	1	2	yes	—
48	C₂×Dic₁₄₆	C₂×C₄	1	1	yes	characteristic
49	C₂²×Dic₇₃	C₂×C₄	1	1	yes	characteristic
50	C₂²×Dic₇₃	C₂³	1	1	yes	characteristic
51	C₄⋊Dic₇₃	C₂×C₄	1	2	yes	—
52	C₄⋊Dic₇₃	C₂×C₄	1	1	yes	characteristic
53	#11924007	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	C₂²⋊Dic₇₃	C₂×C₄	1	2	yes	—
59	C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
60	C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
61	C₄×Dic₇₃	C₂³	1	1	yes	characteristic
62	C₄×Dic₇₃	C₂³	1	1	yes	characteristic
63	C₄×Dic₇₃	C₂³	1	2	yes	—
64	C₄×Dic₇₃	C₂³	1	2	yes	—
65	C₂×Q₈×C₇₃	C₂×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
67	C₄⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
68	C₄⋊C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
69	C₂²⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
70	C₂³×C₇₃	C₂²×C₄	1	1	yes	characteristic
71	Q₈×C₇₃	—	2	2	no	—
72	C₂×C₄×C₇₃	—	2	2	no	—
73	C₂×C₄×C₇₃	C₂²×C₄	1	2	yes	—
74	C₂×C₄×C₇₃	C₂²×C₄	1	2	yes	—
75	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×Dic₇₃	—	2	2	no	—
78	C₂×Dic₇₃	—	2	1	no	—
79	C₂×Dic₇₃	—	2	1	no	—
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	C₂×Dic₇₃	C₂⁴	1	1	yes	characteristic
86	Dic₁₄₆	—	2	2	no	—
87	Dic₁₄₆	—	2	4	no	—
88	#10596833	—	2	1	no	—
89	#10596833	—	2	2	no	—
90	#10596833	—	2	1	no	—
91	#10596833	—	2	1	no	—
92	#10596833	—	2	2	no	—
93	#10596833	—	2	1	no	—
94	C₂²×C₇₃	—	2	1	no	—
95	C₂²×C₇₃	—	2	1	no	—
96	C₂²×C₇₃	—	2	1	no	—
97	C₂²×C₇₃	C₂³×C₄	1	1	yes	characteristic
98	C₄×C₇₃	—	2	2	no	—
99	C₄×C₇₃	—	2	1	no	—
100	C₄×C₇₃	—	2	2	no	—
101	C₄×C₇₃	#131	1	2	yes	—
102	Dic₇₃	—	2	1	no	—
103	Dic₇₃	—	2	2	no	—
104	Dic₇₃	—	2	2	no	—
105	Dic₇₃	—	2	1	no	—
106	Dic₇₃	#131	1	2	yes	—
107	Dic₇₃	2₋¹⁺⁴	1	1	yes	characteristic
108	Dic₇₃	2₋¹⁺⁴	1	1	yes	characteristic
109	C₂×C₇₃	—	4	1	no	—
110	C₂×C₇₃	#519	1	1	yes	characteristic
111	C₂×C₇₃	C₂×#131	1	1	yes	characteristic
112	C₂×C₇₃	C₂×#131	1	1	yes	characteristic
113	#2986	—	73	1	no	maximal, Sylow
114	C₇₃	#2986	1	1	yes	characteristic, complemented, Sylow
115	#533	—	73	1	no	—
116	#446	—	73	1	no	—
117	#446	—	73	1	no	—
118	#446	—	73	2	no	—
119	C₈×Q₈	—	73	1	no	—
120	C₈×Q₈	—	73	1	no	—
121	C₈×Q₈	—	73	2	no	—
122	#433	—	73	2	no	—
123	#433	—	73	2	no	—
124	#431	—	73	1	no	—
125	#431	—	73	1	no	—
126	C₄⋊Q₈	—	73	1	no	—
127	C₄⋊Q₈	—	73	1	no	—
128	#125	—	73	2	no	—
129	C₂²⋊Q₈	—	73	1	no	—
130	C₂²⋊Q₈	—	73	2	no	—
131	C₂²⋊Q₈	#64408	1	1	yes	characteristic
132	C₄×Q₈	—	73	1	no	—
133	C₄×Q₈	—	73	1	no	—
134	C₄×Q₈	—	73	2	no	—
135	#117	—	73	2	no	—
136	C₂×C₄²	—	73	1	no	—
137	C₄⋊C₈	—	73	1	no	—
138	C₄⋊C₈	—	73	1	no	—
139	C₄⋊C₈	—	73	1	no	—
140	C₄⋊C₈	—	73	1	no	—
141	C₄⋊C₈	—	73	2	no	—
142	C₄⋊C₈	—	73	2	no	—
143	C₄⋊C₈	—	73	2	no	—
144	C₄⋊C₈	—	73	2	no	—
145	C₂²⋊C₈	—	73	2	no	—
146	C₂²⋊C₈	—	73	1	no	—
147	C₂²⋊C₈	—	73	1	no	—
148	#97	—	73	2	no	—
149	#97	—	73	2	no	—
150	C₄×C₈	—	73	2	no	—
151	C₄×C₈	—	73	1	no	—
152	C₄×C₈	—	73	1	no	—
153	C₄×C₈	—	73	2	no	—
154	C₄×C₈	—	73	2	no	—
155	C₂×Q₈	—	73	2	no	—
156	C₂×Q₈	—	73	1	no	—
157	C₂×Q₈	C₂×#64408	1	1	yes	characteristic
158	C₂²×C₄	—	73	1	no	—
159	C₂²×C₄	—	73	1	no	—
160	C₂²×C₄	C₂×#64408	1	1	yes	characteristic
161	C₂×C₈	—	73	2	no	—
162	C₂×C₈	—	73	1	no	—
163	C₂×C₈	—	73	1	no	—
164	C₂×C₈	—	73	2	no	—
165	C₂×C₈	—	73	1	no	—
166	C₂×C₈	—	73	1	no	—
167	C₄⋊C₄	—	73	2	no	—
168	C₄⋊C₄	—	73	2	no	—
169	C₄⋊C₄	—	73	2	no	—
170	C₄⋊C₄	—	73	2	no	—
171	C₄⋊C₄	—	73	2	no	—
172	C₄⋊C₄	—	73	2	no	—
173	C₄⋊C₄	—	73	1	no	—
174	C₄⋊C₄	C₂×#64408	1	2	yes	—
175	C₄⋊C₄	C₂×#64408	1	1	yes	characteristic
176	C₂²⋊C₄	—	73	2	no	—
177	C₂²⋊C₄	C₂×#64408	1	2	yes	—
178	C₄²	—	73	1	no	—
179	C₄²	—	73	1	no	—
180	C₄²	—	73	1	no	—
181	C₄²	—	73	1	no	—
182	C₄²	—	73	2	no	—
183	C₄²	—	73	2	no	—
184	C₂³	C₂²×#64408	1	1	yes	characteristic
185	Q₈	—	2	2	no	—
186	Q₈	—	146	4	no	—
187	Q₈	—	146	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	73	1	no	—
190	C₂×C₄	—	73	1	no	—
191	C₂×C₄	—	73	1	no	—
192	C₂×C₄	—	73	1	no	—
193	C₂×C₄	—	73	2	no	—
194	C₂×C₄	—	73	2	no	—
195	C₂×C₄	—	146	1	no	—
196	C₂×C₄	—	146	1	no	—
197	C₂×C₄	—	146	2	no	—
198	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
199	C₂×C₄	C₂²×#64408	1	2	yes	—
200	C₂×C₄	C₂²×#64408	1	2	yes	—
201	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
202	C₈	—	146	2	no	—
203	C₈	—	146	1	no	—
204	C₈	—	146	1	no	—
205	C₈	—	146	2	no	—
206	C₈	—	146	1	no	—
207	C₈	—	146	1	no	—
208	C₂²	—	2	1	no	—
209	C₂²	—	2	1	no	—
210	C₂²	—	2	1	no	—
211	C₂²	C₂³×#64408	1	1	yes	characteristic
212	C₄	—	2	1	no	—
213	C₄	—	2	2	no	—
214	C₄	—	2	2	no	—
215	C₄	—	73	1	no	—
216	C₄	—	73	1	no	—
217	C₄	—	73	2	no	—
218	C₄	—	146	1	no	—
219	C₄	—	146	2	no	—
220	C₄	—	146	1	no	—
221	C₄	—	146	2	no	—
222	C₄	#14556678	1	2	yes	—
223	C₂	—	4	1	no	—
224	C₂	#17989655	1	1	yes	characteristic
225	C₂	C₂×#14556678	1	1	yes	characteristic
226	C₂	D₇₃⋊#131	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	#2986
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. #17989639	C₂	semidirect
2. #17989638	C₂	semidirect
3. #17989637	C₂	semidirect
4. #17989636	C₂	semidirect
5. #17989636	C₂	semidirect
6. Q₈×#10596833	C₂	semidirect
114. C₇₃	#2986	semidirect