Subgroups — #32984850

Order 18688.

486 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₂₉₂	C₂	1	1	yes	maximal, characteristic
2	#22610866	C₂	1	2	yes	maximal
3	C₂×#17988317	C₂	1	2	yes	maximal
4	C₂×#17988308	C₂	1	1	yes	maximal, characteristic
5	C₂×#334×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₉₂	C₄	1	1	yes	characteristic
7	C₂×C₄×Dic₁₄₆	C₂²	1	2	yes	—
8	C₄×Dic₂₉₂	—	2	2	no	—
9	C₄×Dic₂₉₂	—	2	2	no	—
10	C₂×#14556525	C₄	1	1	yes	characteristic
11	C₂×#14556516	C₄	1	2	yes	—
12	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
13	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
14	#17988341	—	2	4	no	—
15	#17988317	—	2	4	no	—
16	#17988308	—	2	2	no	—
17	C₂×C₄⋊C₈×C₇₃	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
19	#334×C₇₃	—	2	2	no	—
20	C₂²×Dic₁₄₆	C₂×C₄	1	2	yes	—
21	C₂×Dic₂₉₂	—	2	2	no	—
22	C₂×Dic₂₉₂	—	2	2	no	—
23	C₂×Dic₂₉₂	C₈	1	4	yes	complemented
24	C₂²×#10596831	—	2	1	no	—
25	C₂²×#10596831	D₄	1	2	yes	—
26	C₄×Dic₁₄₆	—	2	2	no	—
27	C₄×Dic₁₄₆	—	2	2	no	—
28	C₄×Dic₁₄₆	—	2	2	no	—
29	C₄×Dic₁₄₆	—	2	2	no	—
30	C₂×C₄⋊Dic₇₃	C₂×C₄	1	2	yes	—
31	C₂×#11924007	—	2	2	no	—
32	C₂×C₄×Dic₇₃	—	2	2	no	—
33	#14556525	—	2	2	no	—
34	#14556516	—	2	4	no	—
35	C₄×#10596831	—	2	1	no	—
36	C₄×#10596831	—	2	1	no	—
37	C₄×#10596831	—	2	1	no	—
38	C₄×#10596831	D₄	1	2	yes	—
39	C₂²×C₈×C₇₃	—	2	2	no	—
40	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
43	C₄⋊C₈×C₇₃	—	2	2	no	—
44	C₄⋊C₈×C₇₃	—	2	2	no	—
45	C₄×C₈×C₇₃	D₄	1	2	yes	—
46	C₄×C₈×C₇₃	D₄	1	2	yes	—
47	C₂×Dic₁₄₆	—	2	2	no	—
48	C₂×Dic₁₄₆	—	2	2	no	—
49	C₂×Dic₁₄₆	—	2	2	no	—
50	C₂×Dic₁₄₆	—	2	2	no	—
51	C₂×Dic₁₄₆	—	2	2	no	—
52	C₂×Dic₁₄₆	C₂×C₈	1	4	yes	—
53	C₂²×Dic₇₃	—	2	2	no	—
54	C₂²×Dic₇₃	—	2	2	no	—
55	Dic₂₉₂	—	2	8	no	—
56	C₂×#10596831	—	2	2	no	—
57	C₂×#10596831	—	2	2	no	—
58	C₂×#10596831	—	2	4	no	—
59	C₂×#10596831	—	4	2	no	—
60	C₄⋊Dic₇₃	—	2	2	no	—
61	C₄⋊Dic₇₃	—	2	2	no	—
62	#11924007	—	4	2	no	—
63	#11924007	—	4	2	no	—
64	C₄×Dic₇₃	—	4	2	no	—
65	C₄×Dic₇₃	—	4	2	no	—
66	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
69	C₂×C₈×C₇₃	—	2	1	no	—
70	C₂×C₈×C₇₃	—	2	1	no	—
71	C₂×C₈×C₇₃	—	4	2	no	—
72	C₂×C₈×C₇₃	—	4	4	no	—
73	C₂×C₈×C₇₃	C₂²⋊C₄	1	2	yes	—
74	C₂×C₈×C₇₃	C₂²⋊C₄	1	2	yes	—
75	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
76	C₂×C₈×C₇₃	M₁₆	1	2	yes	—
77	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₇₃	#41	1	1	yes	characteristic
81	C₂³×C₇₃	C₄×D₄	1	1	yes	characteristic
82	C₂×C₄×C₇₃	—	2	2	no	—
83	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
85	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
86	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
87	C₂×C₄×C₇₃	#124	1	1	yes	characteristic
88	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
89	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
90	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
91	C₂×C₄×C₇₃	#131	1	1	yes	characteristic
92	C₂×C₄×C₇₃	#131	1	1	yes	characteristic
93	C₂×C₄×C₇₃	#137	1	4	yes	—
94	C₂×Dic₇₃	—	2	4	no	—
95	C₂×Dic₇₃	—	4	2	no	—
96	C₂×Dic₇₃	—	4	2	no	—
97	C₂×Dic₇₃	—	4	2	no	—
98	C₂×Dic₇₃	—	4	4	no	—
99	Dic₁₄₆	—	2	4	no	—
100	Dic₁₄₆	—	2	4	no	—
101	Dic₁₄₆	—	4	2	no	—
102	Dic₁₄₆	—	4	2	no	—
103	C₈×C₇₃	—	2	2	no	—
104	C₈×C₇₃	—	8	2	no	—
105	C₈×C₇₃	C₂²⋊C₈	1	4	yes	—
106	#10596831	—	4	2	no	—
107	#10596831	—	4	2	no	—
108	C₂²×C₇₃	#390	1	1	yes	characteristic
109	C₂²×C₇₃	C₈×D₄	1	1	yes	characteristic
110	C₂²×C₇₃	#436	1	1	yes	characteristic
111	C₂²×C₇₃	#441	1	2	yes	—
112	C₂²×C₇₃	#444	1	2	yes	—
113	C₄×C₇₃	—	2	1	no	—
114	C₄×C₇₃	—	2	1	no	—
115	C₄×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
116	C₄×C₇₃	#408	1	1	yes	characteristic
117	C₄×C₇₃	C₈×D₄	1	1	yes	characteristic
118	C₄×C₇₃	#436	1	1	yes	characteristic
119	C₄×C₇₃	#470	1	2	yes	—
120	C₄×C₇₃	#497	1	2	yes	—
121	Dic₇₃	—	4	4	no	—
122	Dic₇₃	—	8	2	no	—
123	#9658	—	73	1	no	maximal, Sylow
124	C₂×C₇₃	#1579	1	2	yes	—
125	C₂×C₇₃	#1593	1	2	yes	—
126	C₂×C₇₃	#1954	1	1	yes	characteristic
127	C₂×C₇₃	#1971	1	1	yes	characteristic
128	C₂×C₇₃	#1975	1	1	yes	characteristic
129	C₂×C₄×Q₁₆	—	73	1	no	—
130	#1984	—	73	2	no	—
131	C₂×#334	D₇₃	1	1	yes	characteristic
132	C₂×Q₈⋊C₈	—	73	2	no	—
133	C₂×#322	—	73	1	no	—
134	C₇₃	#9658	1	1	yes	characteristic, complemented, Sylow
135	C₂²×Q₁₆	—	73	1	no	—
136	C₂×C₄×Q₈	—	73	2	no	—
137	C₄×Q₁₆	—	146	2	no	—
138	C₄×Q₁₆	—	146	2	no	—
139	C₂×#107	—	73	1	no	—
140	C₂×C₄⋊C₈	C₂×D₇₃	1	2	yes	—
141	C₂×#103	—	73	2	no	—
142	C₂×C₄×C₈	—	73	1	no	—
143	C₂×C₄×C₈	—	73	1	no	—
144	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
145	#336	—	146	4	no	—
146	#334	—	2	2	no	—
147	Q₈⋊C₈	—	146	4	no	—
148	#322	—	146	2	no	—
149	C₂²×Q₈	—	73	2	no	—
150	C₂×Q₁₆	—	73	4	no	—
151	C₂×Q₁₆	—	146	2	no	—
152	C₂×Q₁₆	—	146	2	no	—
153	C₂²×C₈	—	2	2	no	—
154	C₂²×C₈	—	73	2	no	—
155	C₂²×C₈	—	146	1	no	—
156	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
157	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
158	C₄×Q₈	—	146	2	no	—
159	C₄×Q₈	—	146	2	no	—
160	C₄×Q₈	—	146	2	no	—
161	C₄×Q₈	—	146	2	no	—
162	C₂×C₄⋊C₄	—	73	2	no	—
163	C₂×C₄⋊C₄	—	146	2	no	—
164	C₂×C₄²	—	146	2	no	—
165	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
166	#107	—	146	2	no	—
167	C₄⋊C₈	—	2	2	no	—
168	C₄⋊C₈	—	2	2	no	—
169	#103	—	146	4	no	—
170	C₄×C₈	—	73	2	no	—
171	C₄×C₈	—	146	1	no	—
172	C₄×C₈	—	146	1	no	—
173	C₄×C₈	—	146	1	no	—
174	C₄×C₈	D₂₉₂	1	2	yes	—
175	C₄×C₈	C₂²⋊D₇₃	1	2	yes	—
176	C₂×Q₈	—	73	4	no	—
177	C₂×Q₈	—	146	2	no	—
178	C₂×Q₈	—	146	2	no	—
179	C₂×Q₈	—	146	2	no	—
180	C₂×Q₈	—	146	2	no	—
181	C₂×Q₈	—	146	2	no	—
182	C₂²×C₄	—	146	2	no	—
183	C₂²×C₄	—	146	2	no	—
184	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
185	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
186	C₂²×C₄	#11924029	1	1	yes	characteristic
187	Q₁₆	—	146	8	no	—
188	C₂×C₈	—	2	1	no	—
189	C₂×C₈	—	2	1	no	—
190	C₂×C₈	—	4	2	no	—
191	C₂×C₈	—	4	4	no	—
192	C₂×C₈	—	292	2	no	—
193	C₂×C₈	—	146	2	no	—
194	C₂×C₈	—	146	2	no	—
195	C₂×C₈	—	146	4	no	—
196	C₂×C₈	#11924005	1	2	yes	—
197	C₂×C₈	#11924005	1	2	yes	—
198	C₂×C₈	C₈×D₇₃	1	2	yes	—
199	C₂×C₈	#11924011	1	2	yes	—
200	C₄⋊C₄	—	292	2	no	—
201	C₄⋊C₄	—	292	2	no	—
202	C₄⋊C₄	—	146	2	no	—
203	C₄⋊C₄	—	146	2	no	—
204	C₄²	—	292	2	no	—
205	C₄²	—	292	2	no	—
206	C₄²	D₄×D₇₃	1	1	yes	characteristic
207	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
208	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
209	C₄²	#11924029	1	1	yes	characteristic
210	C₂³	#14556553	1	1	yes	characteristic
211	Q₈	—	292	2	no	—
212	Q₈	—	292	2	no	—
213	Q₈	—	146	4	no	—
214	Q₈	—	146	4	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	292	2	no	—
217	C₂×C₄	—	292	2	no	—
218	C₂×C₄	—	292	4	no	—
219	C₂×C₄	—	292	2	no	—
220	C₂×C₄	—	146	4	no	—
221	C₂×C₄	C₂×#11924005	1	1	yes	characteristic
222	C₂×C₄	#14556553	1	1	yes	characteristic
223	C₂×C₄	#14556566	1	1	yes	characteristic
224	C₂×C₄	#14556566	1	1	yes	characteristic
225	C₂×C₄	#14556580	1	1	yes	characteristic
226	C₂×C₄	#14556587	1	1	yes	characteristic
227	C₂×C₄	C₂×C₈×D₇₃	1	1	yes	characteristic
228	C₂×C₄	C₂×#11924011	1	1	yes	characteristic
229	C₂×C₄	#14556599	1	1	yes	characteristic
230	C₂×C₄	#14556599	1	1	yes	characteristic
231	C₂×C₄	#14556620	1	4	yes	—
232	C₈	—	2	2	no	—
233	C₈	—	8	2	no	—
234	C₈	—	292	2	no	—
235	C₈	—	292	2	no	—
236	C₈	#14556503	1	4	yes	—
237	C₂²	#17988488	1	1	yes	characteristic
238	C₂²	#17988652	1	1	yes	characteristic
239	C₂²	#17988664	1	1	yes	characteristic
240	C₂²	#17988693	1	2	yes	—
241	C₂²	#17988708	1	2	yes	—
242	C₄	—	2	1	no	—
243	C₄	—	2	1	no	—
244	C₄	—	292	4	no	—
245	C₄	—	584	2	no	—
246	C₄	C₂×#14556503	1	1	yes	characteristic
247	C₄	#17988555	1	1	yes	characteristic
248	C₄	#17988652	1	1	yes	characteristic
249	C₄	#17988664	1	1	yes	characteristic
250	C₄	#17988846	1	2	yes	—
251	C₄	#17988978	1	2	yes	—
252	C₂	#22608749	1	2	yes	—
253	C₂	#22608839	1	2	yes	—
254	C₂	#22610724	1	1	yes	characteristic
255	C₂	#22610800	1	1	yes	characteristic
256	C₂	#22610822	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₇₃
commutator	C₂×C₄×C₇₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₂×#334×C₇₃	C₂
socle, prime radical	C₂³×C₇₃	C₄×D₄
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	C₂×#334	D₇₃
hypocenter	C₇₃	#9658
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
23. C₂×Dic₂₉₂	C₈	semidirect
134. C₇₃	#9658	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₂₉₂	2
#22610866	2
C₂×#17988317	2
C₂×#17988308	2
C₂×#334×C₇₃	2
#9658	73