Subgroups — #33239368

Order 18688.

674 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17989019	C₂	1	1	yes	maximal, characteristic
2	#22618805	C₂	1	2	yes	maximal, complemented
3	#22611794	C₂	1	4	yes	maximal, complemented
4	#22611789	C₂	1	2	yes	maximal, complemented
5	#22611777	C₂	1	2	yes	maximal, complemented
6	#22611641	C₂	1	2	yes	maximal, complemented
7	#22611527	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#366×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14556613	C₂²	1	2	yes	—
10	#17989068	—	2	2	no	—
11	#17989019	C₄	1	8	yes	complemented
12	C₂×Dic₅₈₄	C₂²	1	1	yes	characteristic
13	C₂×#14556536	C₂²	1	2	yes	—
14	C₂×D₁₁₆₈	C₂²	1	1	yes	characteristic
15	C₂×#14556531	C₂²	1	1	yes	characteristic
16	#17988686	—	2	4	no	—
17	#17988681	C₂²	1	4	yes	—
18	#17988669	C₂²	1	4	yes	—
19	#17988586	—	2	2	no	—
20	#17988542	C₂²	1	1	yes	characteristic
21	#17988390	C₂²	1	4	yes	—
22	#17988387	C₂²	1	4	yes	—
23	#17988362	C₂²	1	2	yes	—
24	C₁₆×Dic₇₃	C₂²	1	2	yes	—
25	C₂²×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
26	C₂×#107×C₇₃	C₂²	1	2	yes	—
27	#366×C₇₃	C₂²	1	2	yes	—
28	#366×C₇₃	C₂²	1	2	yes	—
29	C₂×#11924028	—	2	2	no	—
30	#14556613	—	2	4	no	—
31	#14556613	C₂×C₄	1	8	yes	complemented
32	C₂×Dic₂₉₂	C₂³	1	2	yes	—
33	C₂×#11924017	—	2	2	no	—
34	C₂×D₅₈₄	C₂³	1	2	yes	—
35	C₂×#11924012	D₄	1	1	yes	characteristic
36	#14556556	—	2	4	no	—
37	#14556553	—	2	4	no	—
38	C₄×C₂²⋊D₇₃	—	4	4	no	—
39	#14556550	D₄	1	1	yes	characteristic
40	#14556549	—	4	2	no	—
41	C₄⋊C₄×D₇₃	—	4	2	no	—
42	Dic₅₈₄	C₂×C₄	1	4	yes	complemented
43	#14556536	C₂×C₄	1	8	yes	complemented
44	D₁₁₆₈	C₂×C₄	1	4	yes	complemented
45	#14556531	—	2	2	no	—
46	C₂×#11924003	C₂³	1	2	yes	—
47	#14556514	—	2	4	no	—
48	#14556511	—	2	4	no	—
49	#14556501	D₄	1	2	yes	—
50	C₈×Dic₇₃	C₂³	1	2	yes	—
51	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
52	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
53	C₂×C₁₆×C₇₃	C₂×C₄	1	4	yes	complemented
54	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
55	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
56	#107×C₇₃	—	2	2	no	—
57	#107×C₇₃	C₂³	1	4	yes	—
58	#11924028	—	2	8	no	—
59	#11924028	—	4	4	no	—
60	C₂×Dic₁₄₆	—	2	2	no	—
61	C₂×D₂₉₂	—	2	2	no	—
62	C₂×C₂²⋊D₇₃	—	4	2	no	—
63	C₂×C₄×D₇₃	—	4	2	no	—
64	C₂×C₄×D₇₃	—	4	4	no	—
65	Dic₂₉₂	—	2	2	no	—
66	Dic₂₉₂	C₂²×C₄	1	4	yes	—
67	#11924017	—	2	8	no	—
68	D₅₈₄	—	2	2	no	—
69	D₅₈₄	C₂²×C₄	1	4	yes	—
70	#11924012	—	2	2	no	—
71	C₂×#10596831	C₂×D₄	1	2	yes	—
72	C₄⋊Dic₇₃	D₈	1	1	yes	characteristic
73	C₄⋊Dic₇₃	D₈	1	1	yes	characteristic
74	#11924007	—	4	4	no	—
75	C₂²⋊Dic₇₃	D₈	1	2	yes	—
76	#11924005	—	4	4	no	—
77	C₄×Dic₇₃	—	4	4	no	—
78	C₄×Dic₇₃	C₂×D₄	1	2	yes	—
79	#11924003	—	2	2	no	—
80	C₂²×C₄×C₇₃	—	4	2	no	—
81	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₇₃	#41	1	2	yes	—
85	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
86	C₄⋊C₄×C₇₃	—	2	4	no	—
87	C₄⋊C₄×C₇₃	—	4	2	no	—
88	C₁₆×C₇₃	C₂²×C₄	1	2	yes	—
89	C₁₆×C₇₃	C₂²×C₄	1	2	yes	—
90	C₂³×C₇₃	C₂×D₈	1	1	yes	characteristic
91	C₂²×D₇₃	—	4	2	no	—
92	C₂×C₄×C₇₃	—	4	4	no	—
93	C₂×C₄×C₇₃	—	16	2	no	—
94	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
95	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
96	C₂×C₄×C₇₃	#135	1	2	yes	—
97	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
98	C₂²⋊D₇₃	—	4	8	no	—
99	C₂×Dic₇₃	—	4	2	no	—
100	C₂×Dic₇₃	C₂×D₈	1	2	yes	—
101	C₂×Dic₇₃	C₂×D₈	1	2	yes	—
102	D₂₉₂	—	2	4	no	—
103	D₂₉₂	—	4	2	no	—
104	C₄×D₇₃	—	4	8	no	—
105	C₄×D₇₃	—	16	4	no	—
106	Dic₁₄₆	—	2	4	no	—
107	Dic₁₄₆	—	4	2	no	—
108	C₈×C₇₃	C₄×D₄	1	2	yes	—
109	C₈×C₇₃	C₂³×C₄	1	1	yes	characteristic
110	C₈×C₇₃	C₂×#41	1	1	yes	characteristic
111	#10596831	—	2	2	no	—
112	C₂²×C₇₃	—	2	1	no	—
113	C₂²×C₇₃	—	2	1	no	—
114	C₂²×C₇₃	C₄×D₈	1	2	yes	—
115	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
116	C₂×D₇₃	—	4	4	no	—
117	C₂×D₇₃	—	8	2	no	—
118	C₂×D₇₃	—	8	2	no	—
119	C₄×C₇₃	—	8	4	no	—
120	C₄×C₇₃	C₄×D₈	1	2	yes	—
121	C₄×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
122	C₄×C₇₃	C₂×#135	1	1	yes	characteristic
123	Dic₇₃	—	2	2	no	—
124	Dic₇₃	—	2	2	no	—
125	Dic₇₃	—	4	4	no	—
126	#33459	—	73	1	no	maximal, Sylow
127	C₂×C₇₃	—	2	2	no	—
128	C₂×C₇₃	C₂×C₄×D₈	1	1	yes	characteristic
129	C₂×C₇₃	#3415	1	1	yes	characteristic
130	C₂×C₇₃	#3417	1	1	yes	characteristic
131	D₇₃	—	8	4	no	—
132	C₂×#508	—	73	1	no	—
133	#2951	—	73	2	no	—
134	#2175	—	73	4	no	—
135	C₄×Q₃₂	—	73	2	no	—
136	C₄×D₁₆	—	73	2	no	—
137	C₂×#366	D₇₃	1	1	yes	characteristic, complemented
138	#2139	—	73	2	no	—
139	#2108	—	73	1	no	—
140	C₇₃	#33459	1	1	yes	characteristic, complemented, Sylow
141	C₂×#135	—	73	2	no	—
142	#520	—	146	2	no	—
143	#508	—	73	8	no	—
144	C₂×Q₃₂	—	73	1	no	—
145	C₂×SD₃₂	—	73	2	no	—
146	C₂×D₁₆	—	73	1	no	—
147	C₂×M₃₂	—	73	1	no	—
148	C₂²×C₁₆	C₂×D₇₃	1	1	yes	characteristic
149	#440	—	146	4	no	—
150	C₄×Q₁₆	—	73	4	no	—
151	C₄×D₈	—	73	4	no	—
152	C₂×#107	C₂×D₇₃	1	2	yes	—
153	#417	—	146	2	no	—
154	#405	—	73	1	no	—
155	#366	C₂×D₇₃	1	2	yes	—
156	#366	C₂×D₇₃	1	2	yes	—
157	#358	—	73	4	no	—
158	#357	—	73	4	no	—
159	#346	—	73	2	no	—
160	C₄×C₁₆	—	73	2	no	—
161	C₂×#41	—	146	2	no	—
162	#135	—	73	8	no	—
163	#135	—	146	4	no	—
164	C₂×Q₁₆	—	73	2	no	—
165	C₂×SD₁₆	—	146	2	no	—
166	C₂×D₈	—	73	2	no	—
167	C₂×M₁₆	—	73	1	no	—
168	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
169	C₄×Q₈	—	146	4	no	—
170	C₄×D₄	—	292	4	no	—
171	C₄×D₄	—	146	4	no	—
172	#117	—	292	2	no	—
173	#117	—	73	1	no	—
174	C₂×C₄⋊C₄	—	2	2	no	—
175	C₂×C₄⋊C₄	—	292	2	no	—
176	Q₃₂	—	73	4	no	—
177	SD₃₂	—	73	8	no	—
178	D₁₆	—	73	4	no	—
179	M₃₂	—	146	2	no	—
180	C₂×C₁₆	—	73	2	no	—
181	C₂×C₁₆	C₄×D₇₃	1	4	yes	complemented
182	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
183	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
184	#107	—	2	2	no	—
185	#107	C₂²×D₇₃	1	4	yes	—
186	#103	—	146	4	no	—
187	#102	—	146	4	no	—
188	#97	—	73	2	no	—
189	C₄×C₈	—	73	2	no	—
190	#41	—	292	4	no	—
191	#41	—	146	8	no	—
192	C₂×Q₈	—	146	2	no	—
193	C₂×D₄	—	292	2	no	—
194	C₂×D₄	—	146	2	no	—
195	C₂²×C₄	—	4	2	no	—
196	C₂²×C₄	—	292	4	no	—
197	C₂²×C₄	—	292	2	no	—
198	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
199	Q₁₆	—	73	4	no	—
200	Q₁₆	—	146	2	no	—
201	SD₁₆	—	146	8	no	—
202	D₈	—	73	4	no	—
203	D₈	—	146	2	no	—
204	M₁₆	—	146	2	no	—
205	C₂×C₈	—	73	2	no	—
206	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
207	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
208	C₂×C₈	#11924029	1	2	yes	—
209	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
210	C₄⋊C₄	—	2	4	no	—
211	C₄⋊C₄	—	4	2	no	—
212	C₄⋊C₄	—	292	4	no	—
213	C₄⋊C₄	—	73	1	no	—
214	C₄⋊C₄	—	73	1	no	—
215	C₂²⋊C₄	—	292	4	no	—
216	C₂²⋊C₄	—	73	2	no	—
217	C₄²	—	292	4	no	—
218	C₄²	—	73	2	no	—
219	C₁₆	—	146	2	no	—
220	C₁₆	C₂×C₄×D₇₃	1	2	yes	—
221	C₁₆	C₂×C₄×D₇₃	1	2	yes	—
222	C₂³	—	292	2	no	—
223	C₂³	D₈×D₇₃	1	1	yes	characteristic
224	Q₈	—	292	2	no	—
225	Q₈	—	146	4	no	—
226	D₄	—	292	8	no	—
227	D₄	—	292	2	no	—
228	D₄	—	146	4	no	—
229	C₂×C₄	—	4	4	no	—
230	C₂×C₄	—	16	2	no	—
231	C₂×C₄	—	292	2	no	—
232	C₂×C₄	—	292	8	no	—
233	C₂×C₄	—	73	2	no	—
234	C₂×C₄	—	73	2	no	—
235	C₂×C₄	—	1168	4	no	—
236	C₂×C₄	#14556553	1	2	yes	—
237	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
238	C₂×C₄	#14556614	1	2	yes	—
239	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
240	C₈	—	146	2	no	—
241	C₈	#14556553	1	2	yes	—
242	C₈	C₂²×C₄×D₇₃	1	1	yes	characteristic
243	C₈	C₂×#11924029	1	1	yes	characteristic
244	C₂²	—	2	1	no	—
245	C₂²	—	2	1	no	—
246	C₂²	—	292	4	no	—
247	C₂²	—	584	2	no	—
248	C₂²	—	584	2	no	—
249	C₂²	#17988669	1	2	yes	—
250	C₂²	C₂×D₈×D₇₃	1	1	yes	characteristic
251	C₄	—	8	4	no	—
252	C₄	—	292	4	no	—
253	C₄	—	146	2	no	—
254	C₄	—	146	2	no	—
255	C₄	#17988669	1	2	yes	—
256	C₄	C₂×#14556553	1	1	yes	characteristic
257	C₄	C₂×#14556614	1	1	yes	characteristic
258	C₂	—	2	2	no	—
259	C₂	—	584	4	no	—
260	C₂	C₂×#17988669	1	1	yes	characteristic
261	C₂	#22623068	1	1	yes	characteristic
262	C₂	#22623080	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₇₃
commutator	C₈×C₇₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₂×#366×C₇₃	C₂
socle	C₂²×C₇₃	C₂²×D₈
prime radical	C₂×#17989019	C₂
cosocle	C₂×C₈×C₇₃	C₂⁴
hypercenter	C₂×#366	D₇₃
hypocenter	C₇₃	#33459
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#366

Complements

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

N	H ≅ G/N	type
2. #22618805	C₂	semidirect
3. #22611794	C₂	semidirect
4. #22611789	C₂	semidirect
5. #22611777	C₂	semidirect
6. #22611641	C₂	semidirect
7. #22611527	C₂	semidirect
8. C₂×#366×C₇₃	C₂	semidirect
11. #17989019	C₄	semidirect
31. #14556613	C₂×C₄	semidirect
42. Dic₅₈₄	C₂×C₄	semidirect
43. #14556536	C₂×C₄	semidirect
44. D₁₁₆₈	C₂×C₄	semidirect
53. C₂×C₁₆×C₇₃	C₂×C₄	semidirect
137. C₂×#366	D₇₃	semidirect
140. C₇₃	#33459	semidirect
181. C₂×C₁₆	C₄×D₇₃	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#17989019	2
#22618805	2
#22611794	2
#22611789	2
#22611777	2
#22611641	2
#22611527	2
C₂×#366×C₇₃	2
#33459	73