Subgroups — #22626726

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	#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	#17989637	C₂	1	2	yes	maximal, complemented
5	#17989636	C₂	1	1	yes	maximal, characteristic, complemented
6	Q₈×#10596833	C₂	1	1	yes	maximal, characteristic, complemented
7	#17989593	C₂	1	2	yes	maximal
8	C₂×C₄⋊#10596833	C₂	1	1	yes	maximal, characteristic
9	#17989551	C₂	1	2	yes	maximal
10	C₄×#11924037	C₂	1	1	yes	maximal, characteristic
11	#17989139	C₂	1	1	yes	maximal, characteristic
12	C₂×#11924037	C₂²	1	1	yes	characteristic
13	C₂×#11924037	C₂²	1	1	yes	characteristic
14	C₂²×#10596833	C₂²	1	2	yes	—
15	#14556658	C₂²	1	2	yes	—
16	#14556658	C₂²	1	2	yes	—
17	C₄⋊#10596833	C₂²	1	1	yes	characteristic
18	C₄⋊#10596833	C₂²	1	2	yes	—
19	C₄⋊#10596833	C₂²	1	2	yes	—
20	C₄⋊#10596833	C₂²	1	1	yes	characteristic
21	C₄⋊#10596833	C₂²	1	2	yes	—
22	#14556646	C₂²	1	1	yes	characteristic
23	#14556646	C₂²	1	2	yes	—
24	#14556646	C₂²	1	2	yes	—
25	#14556646	C₂²	1	1	yes	characteristic
26	C₄×#10596833	C₂²	1	1	yes	characteristic
27	C₄×#10596833	C₂²	1	2	yes	—
28	C₄×#10596833	C₂²	1	2	yes	—
29	C₄×#10596833	C₂²	1	1	yes	characteristic
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	#11924037	—	2	2	no	—
42	#11924037	Q₈	1	4	yes	complemented
43	C₂×#10596833	—	2	4	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×Dic₁₄₆	C₂×C₄	1	1	yes	characteristic
51	C₂×Dic₁₄₆	C₂×C₄	1	2	yes	—
52	C₂²×Dic₇₃	C₂×C₄	1	1	yes	characteristic
53	C₂²×Dic₇₃	C₂³	1	1	yes	characteristic
54	C₄⋊Dic₇₃	C₂×C₄	1	2	yes	—
55	C₄⋊Dic₇₃	C₂×C₄	1	1	yes	characteristic
56	#11924007	C₂×C₄	1	2	yes	—
57	#11924007	C₂×C₄	1	2	yes	—
58	#11924007	C₂×C₄	1	2	yes	—
59	#11924007	C₂×C₄	1	2	yes	—
60	#11924007	C₂×C₄	1	2	yes	—
61	C₂²⋊Dic₇₃	C₂×C₄	1	2	yes	—
62	C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
63	C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
64	C₄×Dic₇₃	C₂³	1	1	yes	characteristic
65	C₄×Dic₇₃	C₂³	1	1	yes	characteristic
66	C₄×Dic₇₃	C₂³	1	2	yes	—
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	Q₈×C₇₃	—	2	2	no	—
75	C₂×C₄×C₇₃	—	2	2	no	—
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	C₂²×C₄	1	1	yes	characteristic
80	C₂×Dic₇₃	—	4	1	no	—
81	C₂×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
82	C₂×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
83	C₂×Dic₇₃	C₂²×C₄	1	2	yes	—
84	C₂×Dic₇₃	C₂²×C₄	1	2	yes	—
85	C₂×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×Dic₇₃	C₂×Q₈	1	2	yes	—
87	C₂×Dic₇₃	#41	1	2	yes	—
88	C₂×Dic₇₃	C₂⁴	1	1	yes	characteristic
89	Dic₁₄₆	—	2	2	no	—
90	Dic₁₄₆	—	4	2	no	—
91	#10596833	—	2	2	no	—
92	#10596833	—	2	2	no	—
93	#10596833	—	2	1	no	—
94	#10596833	—	2	1	no	—
95	#10596833	C₂×Q₈	1	2	yes	—
96	#10596833	C₂×Q₈	1	2	yes	—
97	C₂²×C₇₃	—	2	1	no	—
98	C₂²×C₇₃	—	2	1	no	—
99	C₂²×C₇₃	C₄×Q₈	1	2	yes	—
100	C₂²×C₇₃	C₂³×C₄	1	1	yes	characteristic
101	C₄×C₇₃	—	2	1	no	—
102	C₄×C₇₃	—	2	2	no	—
103	C₄×C₇₃	—	2	2	no	—
104	C₄×C₇₃	#131	1	2	yes	—
105	Dic₇₃	—	2	2	no	—
106	Dic₇₃	—	2	1	no	—
107	Dic₇₃	—	2	2	no	—
108	Dic₇₃	—	2	1	no	—
109	Dic₇₃	C₄×Q₈	1	2	yes	—
110	Dic₇₃	C₂²×Q₈	1	1	yes	characteristic
111	Dic₇₃	C₂×#41	1	1	yes	characteristic
112	C₂×C₇₃	—	2	2	no	—
113	C₂×C₇₃	C₂×C₄×Q₈	1	1	yes	characteristic
114	C₂×C₇₃	C₂×#131	1	1	yes	characteristic
115	C₂×C₇₃	#568	1	1	yes	characteristic
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	1	no	—
122	#446	—	73	2	no	—
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	#64408	1	1	yes	characteristic
138	C₄×Q₈	—	73	1	no	—
139	C₄×Q₈	—	73	1	no	—
140	C₄×Q₈	—	73	2	no	—
141	#117	—	73	2	no	—
142	C₂×C₄²	—	73	1	no	—
143	C₄⋊C₈	—	73	1	no	—
144	C₄⋊C₈	—	73	1	no	—
145	C₄⋊C₈	—	73	2	no	—
146	C₄⋊C₈	—	73	2	no	—
147	C₄⋊C₈	—	73	2	no	—
148	C₄⋊C₈	—	73	2	no	—
149	C₄⋊C₈	—	73	2	no	—
150	#97	—	73	1	no	—
151	#97	—	73	1	no	—
152	#97	—	73	2	no	—
153	#97	—	73	2	no	—
154	C₄×C₈	—	73	1	no	—
155	C₄×C₈	—	73	1	no	—
156	C₄×C₈	—	73	2	no	—
157	C₄×C₈	—	73	2	no	—
158	C₂×Q₈	—	73	2	no	—
159	C₂×Q₈	—	73	1	no	—
160	C₂×Q₈	C₂×#64408	1	1	yes	characteristic
161	C₂²×C₄	—	73	1	no	—
162	C₂²×C₄	—	73	1	no	—
163	C₂²×C₄	C₂×#64408	1	1	yes	characteristic
164	M₁₆	—	73	4	no	—
165	M₁₆	—	146	2	no	—
166	C₂×C₈	—	73	1	no	—
167	C₂×C₈	—	73	1	no	—
168	C₂×C₈	—	73	2	no	—
169	C₂×C₈	—	73	1	no	—
170	C₂×C₈	—	73	1	no	—
171	C₂×C₈	—	73	2	no	—
172	C₂×C₈	—	146	4	no	—
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	C₂×#64408	1	2	yes	—
181	C₄⋊C₄	C₂×#64408	1	1	yes	characteristic
182	C₂²⋊C₄	—	73	2	no	—
183	C₂²⋊C₄	C₂×#64408	1	2	yes	—
184	C₄²	—	73	1	no	—
185	C₄²	—	73	1	no	—
186	C₄²	—	73	1	no	—
187	C₄²	—	73	2	no	—
188	C₄²	—	73	2	no	—
189	C₄²	—	73	1	no	—
190	C₂³	C₂²×#64408	1	1	yes	characteristic
191	Q₈	—	2	2	no	—
192	Q₈	—	292	2	no	—
193	Q₈	—	146	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	292	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	1	no	—
201	C₂×C₄	—	73	1	no	—
202	C₂×C₄	—	73	2	no	—
203	C₂×C₄	—	73	2	no	—
204	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
205	C₂×C₄	C₂²×#64408	1	2	yes	—
206	C₂×C₄	C₂²×#64408	1	2	yes	—
207	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
208	C₈	—	73	2	no	—
209	C₈	—	73	2	no	—
210	C₈	—	146	2	no	—
211	C₈	—	146	2	no	—
212	C₈	—	146	1	no	—
213	C₈	—	146	1	no	—
214	C₂²	—	2	1	no	—
215	C₂²	—	2	1	no	—
216	C₂²	Q₈×#64408	1	2	yes	—
217	C₂²	C₂³×#64408	1	1	yes	characteristic
218	C₄	—	2	1	no	—
219	C₄	—	2	2	no	—
220	C₄	—	2	2	no	—
221	C₄	—	73	1	no	—
222	C₄	—	73	1	no	—
223	C₄	—	73	2	no	—
224	C₄	—	146	2	no	—
225	C₄	—	146	1	no	—
226	C₄	—	146	1	no	—
227	C₄	—	146	2	no	—
228	C₄	#14556678	1	2	yes	—
229	C₂	—	2	2	no	—
230	C₂	C₂×Q₈×#64408	1	1	yes	characteristic
231	C₂	C₂×#14556678	1	1	yes	characteristic
232	C₂	#17989666	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	#2962
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. #17989637	C₂	semidirect
5. #17989636	C₂	semidirect
6. Q₈×#10596833	C₂	semidirect
42. #11924037	Q₈	semidirect
117. C₇₃	#2962	semidirect