Subgroups — #22623749

Order 9344.

1060 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×Dic₁₄₆	C₂	1	1	yes	maximal, characteristic, complemented
2	#17989313	C₂	1	2	yes	maximal, complemented
3	#17989298	C₂	1	3	yes	maximal, complemented
4	#17989297	C₂	1	3	yes	maximal, complemented
5	#17989252	C₂	1	6	yes	maximal, complemented
6	#17989214	C₂	1	6	yes	maximal, complemented
7	#17989141	C₂	1	3	yes	maximal, complemented
8	C₄⋊Q₈×D₇₃	C₂	1	3	yes	maximal, complemented
9	#17989069	C₂	1	2	yes	maximal
10	C₄×#11924029	C₂	1	1	yes	maximal, characteristic
11	Q₈²×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#11924029	C₂²	1	1	yes	characteristic
13	C₂×Q₈×D₇₃	C₂²	1	6	yes	—
14	#14556591	C₂²	1	3	yes	—
15	#14556590	C₂²	1	3	yes	—
16	#14556590	C₂²	1	6	yes	—
17	#14556590	C₂²	1	6	yes	—
18	#14556589	C₂²	1	6	yes	—
19	#14556589	C₂²	1	1	yes	characteristic
20	#14556583	C₂²	1	6	yes	—
21	#14556582	C₂²	1	6	yes	—
22	#14556581	C₂²	1	6	yes	—
23	#14556571	C₂²	1	3	yes	—
24	#14556569	C₂²	1	6	yes	—
25	#14556569	C₂²	1	12	yes	—
26	#14556567	C₂²	1	3	yes	—
27	#14556567	C₂²	1	12	yes	—
28	Q₈×Dic₇₃	C₂²	1	1	yes	characteristic
29	Q₈×Dic₇₃	C₂²	1	2	yes	—
30	C₄×Dic₁₄₆	C₂²	1	6	yes	—
31	C₄×Dic₁₄₆	C₂²	1	3	yes	—
32	C₄×D₂₉₂	C₂²	1	6	yes	—
33	C₄×D₂₉₂	C₂²	1	3	yes	—
34	#14556553	C₂²	1	3	yes	—
35	#14556553	C₂²	1	6	yes	—
36	#14556549	C₂²	1	3	yes	—
37	#14556548	C₂²	1	6	yes	—
38	C₄⋊C₄×D₇₃	C₂²	1	6	yes	—
39	C₄⋊C₄×D₇₃	C₂²	1	6	yes	—
40	C₄²×D₇₃	C₂²	1	3	yes	—
41	C₄⋊Q₈×C₇₃	C₂²	1	3	yes	—
42	C₄⋊Q₈×C₇₃	C₂²	1	6	yes	—
43	C₄×Q₈×C₇₃	C₂²	1	3	yes	—
44	C₄×Q₈×C₇₃	C₂²	1	1	yes	characteristic
45	C₄×Q₈×C₇₃	C₂²	1	2	yes	—
46	#11924029	Q₈	1	8	yes	complemented
47	C₂×Dic₁₄₆	C₂³	1	1	yes	characteristic
48	C₂×Dic₁₄₆	C₂³	1	6	yes	—
49	C₂×Dic₁₄₆	C₂³	1	6	yes	—
50	Q₈×D₇₃	—	4	12	no	—
51	C₂×D₂₉₂	C₂³	1	3	yes	—
52	C₂×C₄×D₇₃	C₂³	1	3	yes	—
53	C₂×C₄×D₇₃	C₂³	1	3	yes	—
54	C₂×C₄×D₇₃	C₂³	1	12	yes	—
55	C₂×C₄×D₇₃	C₂³	1	3	yes	—
56	C₄⋊Dic₇₃	C₂³	1	3	yes	—
57	C₄⋊Dic₇₃	C₂³	1	6	yes	—
58	C₄⋊Dic₇₃	C₂³	1	3	yes	—
59	C₄⋊Dic₇₃	C₂³	1	6	yes	—
60	#11924007	C₂³	1	6	yes	—
61	#11924007	C₂³	1	12	yes	—
62	#11924007	C₂³	1	2	yes	—
63	#11924007	C₂³	1	2	yes	—
64	#11924007	C₂³	1	6	yes	—
65	#11924007	C₂³	1	6	yes	—
66	#11924005	C₂³	1	6	yes	—
67	#11924005	C₂³	1	6	yes	—
68	#11924005	C₂³	1	6	yes	—
69	C₄×Dic₇₃	C₂³	1	1	yes	characteristic
70	C₄×Dic₇₃	C₂³	1	3	yes	—
71	C₄×Dic₇₃	C₂³	1	3	yes	—
72	C₄×Dic₇₃	C₂³	1	6	yes	—
73	C₂×Q₈×C₇₃	C₂³	1	1	yes	characteristic
74	C₂×Q₈×C₇₃	C₂³	1	6	yes	—
75	C₂×Q₈×C₇₃	C₂³	1	1	yes	characteristic
76	C₄⋊C₄×C₇₃	C₂³	1	3	yes	—
77	C₄⋊C₄×C₇₃	C₂³	1	6	yes	—
78	C₄⋊C₄×C₇₃	C₂³	1	3	yes	—
79	C₄⋊C₄×C₇₃	C₂³	1	6	yes	—
80	C₄²×C₇₃	C₂³	1	3	yes	—
81	C₄²×C₇₃	C₂³	1	6	yes	—
82	C₂²×D₇₃	C₂⁴	1	3	yes	—
83	Q₈×C₇₃	—	2	2	no	—
84	Q₈×C₇₃	—	4	6	no	—
85	Q₈×C₇₃	C₂×Q₈	1	4	yes	complemented
86	C₂×C₄×C₇₃	C₂⁴	1	2	yes	—
87	C₂×C₄×C₇₃	C₂⁴	1	3	yes	—
88	C₂×C₄×C₇₃	C₂⁴	1	6	yes	—
89	C₂×C₄×C₇₃	C₂⁴	1	3	yes	—
90	C₂×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
91	C₂×Dic₇₃	C₂⁴	1	2	yes	—
92	C₂×Dic₇₃	C₂⁴	1	3	yes	—
93	C₂×Dic₇₃	C₂⁴	1	6	yes	—
94	C₂×Dic₇₃	C₂⁴	1	1	yes	characteristic
95	C₂×Dic₇₃	C₂⁴	1	1	yes	characteristic
96	D₂₉₂	C₂×Q₈	1	12	yes	complemented
97	C₄×D₇₃	—	2	6	no	—
98	C₄×D₇₃	—	2	6	no	—
99	C₄×D₇₃	—	4	12	no	—
100	C₄×D₇₃	C₂×Q₈	1	12	yes	complemented
101	Dic₁₄₆	—	2	2	no	—
102	Dic₁₄₆	—	2	12	no	—
103	Dic₁₄₆	—	4	6	no	—
104	C₂²×C₇₃	C₂⁵	1	1	yes	characteristic
105	C₂×D₇₃	—	2	3	no	—
106	C₂×D₇₃	—	2	3	no	—
107	C₂×D₇₃	C₂²×Q₈	1	6	yes	—
108	C₄×C₇₃	—	2	6	no	—
109	C₄×C₇₃	—	2	3	no	—
110	C₄×C₇₃	—	2	2	no	—
111	C₄×C₇₃	C₂²×Q₈	1	6	yes	—
112	C₄×C₇₃	2₋¹⁺⁴	1	2	yes	—
113	Dic₇₃	—	2	6	no	—
114	Dic₇₃	—	2	2	no	—
115	Dic₇₃	—	2	3	no	—
116	Dic₇₃	C₂²×Q₈	1	2	yes	—
117	Dic₇₃	H(5,𝔽₂)	1	2	yes	—
118	C₂×C₇₃	C₂³×Q₈	1	1	yes	characteristic
119	C₂×C₇₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
120	C₂×C₇₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
121	D₇₃	—	2	6	no	—
122	#3503	—	73	1	no	maximal, Sylow
123	C₇₃	#3503	1	1	yes	characteristic, complemented, Sylow
124	Q₈²	—	73	1	no	—
125	Q₈²	D₇₃	1	1	yes	characteristic, complemented
126	#557	—	73	2	no	—
127	#554	—	73	3	no	—
128	#554	—	73	3	no	—
129	D₄×Q₈	—	73	6	no	—
130	#544	—	73	6	no	—
131	#533	—	73	3	no	—
132	C₂×C₄⋊Q₈	—	73	3	no	—
133	#520	—	73	2	no	—
134	C₄×#41	—	73	1	no	—
135	C₂×#41	—	73	1	no	—
136	C₂²×Q₈	—	73	6	no	—
137	C₄⋊Q₈	—	73	6	no	—
138	C₄⋊Q₈	—	73	1	no	—
139	C₄⋊Q₈	—	73	3	no	—
140	C₄⋊Q₈	—	73	6	no	—
141	C₄⋊Q₈	—	73	6	no	—
142	C₄⋊Q₈	—	73	3	no	—
143	C₄⋊Q₈	C₂×D₇₃	1	3	yes	—
144	C₄⋊Q₈	C₂×D₇₃	1	6	yes	—
145	#125	—	73	6	no	—
146	#125	—	73	6	no	—
147	#125	—	73	6	no	—
148	C₂²⋊Q₈	—	73	3	no	—
149	C₂²⋊Q₈	—	73	12	no	—
150	C₂²⋊Q₈	—	73	6	no	—
151	C₂²⋊Q₈	—	73	3	no	—
152	C₂²⋊Q₈	—	73	12	no	—
153	C₄×Q₈	—	73	1	no	—
154	C₄×Q₈	—	73	2	no	—
155	C₄×Q₈	—	73	6	no	—
156	C₄×Q₈	—	73	3	no	—
157	C₄×Q₈	C₂×D₇₃	1	1	yes	characteristic
158	C₄×Q₈	C₂×D₇₃	1	2	yes	—
159	C₄×Q₈	C₂×D₇₃	1	3	yes	—
160	C₄×D₄	—	73	3	no	—
161	C₄×D₄	—	73	6	no	—
162	C₄×D₄	—	73	3	no	—
163	C₄×D₄	—	73	6	no	—
164	#117	—	73	3	no	—
165	#117	—	73	6	no	—
166	C₂×C₄⋊C₄	—	73	6	no	—
167	C₂×C₄⋊C₄	—	73	6	no	—
168	C₂×C₄²	—	73	3	no	—
169	#41	—	73	8	no	—
170	C₂×Q₈	—	292	12	no	—
171	C₂×Q₈	—	73	6	no	—
172	C₂×Q₈	—	73	1	no	—
173	C₂×Q₈	—	73	6	no	—
174	C₂×Q₈	C₂²×D₇₃	1	1	yes	characteristic
175	C₂×Q₈	C₂²×D₇₃	1	6	yes	—
176	C₂×Q₈	C₂²×D₇₃	1	1	yes	characteristic
177	C₂×D₄	—	73	3	no	—
178	C₂²×C₄	—	73	12	no	—
179	C₂²×C₄	—	73	3	no	—
180	C₂²×C₄	—	73	3	no	—
181	C₂²×C₄	—	73	3	no	—
182	C₄⋊C₄	—	73	6	no	—
183	C₄⋊C₄	—	73	12	no	—
184	C₄⋊C₄	—	73	2	no	—
185	C₄⋊C₄	—	73	2	no	—
186	C₄⋊C₄	—	73	6	no	—
187	C₄⋊C₄	—	73	6	no	—
188	C₄⋊C₄	—	73	3	no	—
189	C₄⋊C₄	—	73	6	no	—
190	C₄⋊C₄	—	73	3	no	—
191	C₄⋊C₄	—	73	6	no	—
192	C₄⋊C₄	C₂²×D₇₃	1	3	yes	—
193	C₄⋊C₄	C₂²×D₇₃	1	6	yes	—
194	C₄⋊C₄	C₂²×D₇₃	1	3	yes	—
195	C₄⋊C₄	C₂²×D₇₃	1	6	yes	—
196	C₂²⋊C₄	—	73	6	no	—
197	C₂²⋊C₄	—	73	6	no	—
198	C₂²⋊C₄	—	73	6	no	—
199	C₄²	—	73	1	no	—
200	C₄²	—	73	3	no	—
201	C₄²	—	73	3	no	—
202	C₄²	—	73	6	no	—
203	C₄²	C₂²×D₇₃	1	3	yes	—
204	C₄²	C₂²×D₇₃	1	6	yes	—
205	C₂³	—	73	3	no	—
206	Q₈	—	2	2	no	—
207	Q₈	—	4	6	no	—
208	Q₈	—	292	6	no	—
209	Q₈	—	146	12	no	—
210	Q₈	—	146	2	no	—
211	Q₈	Q₈×D₇₃	1	4	yes	complemented
212	D₄	—	73	12	no	—
213	C₂×C₄	—	292	12	no	—
214	C₂×C₄	—	73	1	no	—
215	C₂×C₄	—	73	1	no	—
216	C₂×C₄	—	73	12	no	—
217	C₂×C₄	—	73	2	no	—
218	C₂×C₄	—	73	3	no	—
219	C₂×C₄	—	73	6	no	—
220	C₂×C₄	—	146	6	no	—
221	C₂×C₄	—	146	6	no	—
222	C₂×C₄	C₂³×D₇₃	1	3	yes	—
223	C₂×C₄	C₂³×D₇₃	1	2	yes	—
224	C₂×C₄	C₂³×D₇₃	1	6	yes	—
225	C₂×C₄	C₂³×D₇₃	1	3	yes	—
226	C₂×C₄	C₂³×D₇₃	1	1	yes	characteristic
227	C₂²	—	73	6	no	—
228	C₂²	—	146	3	no	—
229	C₂²	—	146	3	no	—
230	C₂²	C₂⁴×D₇₃	1	1	yes	characteristic
231	C₄	—	2	6	no	—
232	C₄	—	2	3	no	—
233	C₄	—	2	2	no	—
234	C₄	—	73	2	no	—
235	C₄	—	73	2	no	—
236	C₄	—	146	6	no	—
237	C₄	—	146	3	no	—
238	C₄	—	146	2	no	—
239	C₄	C₂×Q₈×D₇₃	1	6	yes	—
240	C₄	#14556639	1	2	yes	—
241	C₂	—	146	6	no	—
242	C₂	C₂²×Q₈×D₇₃	1	1	yes	characteristic
243	C₂	H(5,𝔽₂)×D₇₃	1	1	yes	characteristic
244	C₂	C₂×#14556639	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₇₃
commutator, socle	C₂²×C₇₃	C₂⁵
Fitting	Q₈²×C₇₃	C₂
hypercenter	Q₈²	D₇₃
hypocenter	C₇₃	#3503
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ Q₈²

Complements

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

N	H ≅ G/N	type
1. Q₈×Dic₁₄₆	C₂	semidirect
2. #17989313	C₂	semidirect
3. #17989298	C₂	semidirect
4. #17989297	C₂	semidirect
5. #17989252	C₂	semidirect
6. #17989214	C₂	semidirect
7. #17989141	C₂	semidirect
8. C₄⋊Q₈×D₇₃	C₂	semidirect
11. Q₈²×C₇₃	C₂	semidirect
46. #11924029	Q₈	semidirect
85. Q₈×C₇₃	C₂×Q₈	semidirect
96. D₂₉₂	C₂×Q₈	semidirect
100. C₄×D₇₃	C₂×Q₈	semidirect
123. C₇₃	#3503	semidirect
125. Q₈²	D₇₃	semidirect
211. Q₈	Q₈×D₇₃	semidirect