Subgroups — #32984876

Order 18688.

482 conjugacy classes of subgroups.

The subgroup lattice has 248 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	#22610867	C₂	1	2	yes	maximal
3	C₂×#17988340	C₂	1	1	yes	maximal, characteristic
4	C₂×#17988319	C₂	1	2	yes	maximal
5	C₂×#322×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×C₄⋊#10596831	C₂²	1	2	yes	—
12	C₂×#14556516	C₄	1	2	yes	—
13	#17988341	—	2	4	no	—
14	#17988340	—	2	2	no	—
15	#17988319	—	2	4	no	—
16	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
19	#322×C₇₃	—	2	2	no	—
20	C₂²×Dic₁₄₆	C₂×C₄	1	2	yes	—
21	C₂×Dic₂₉₂	—	2	4	no	—
22	C₂×Dic₂₉₂	—	2	2	no	—
23	C₂²×#10596831	—	2	2	no	—
24	C₄×Dic₁₄₆	—	2	2	no	—
25	C₄×Dic₁₄₆	—	2	2	no	—
26	C₄×Dic₁₄₆	—	2	2	no	—
27	C₄×Dic₁₄₆	—	2	2	no	—
28	C₂×C₄⋊Dic₇₃	C₂×C₄	1	2	yes	—
29	C₂×#11924007	—	2	2	no	—
30	C₂×C₄×Dic₇₃	—	2	2	no	—
31	#14556525	—	2	2	no	—
32	C₄⋊#10596831	—	2	2	no	—
33	C₄⋊#10596831	—	2	2	no	—
34	#14556516	—	2	4	no	—
35	C₂²×C₈×C₇₃	—	2	1	no	—
36	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₇₃	D₄	1	2	yes	—
39	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
40	C₄×C₈×C₇₃	—	2	1	no	—
41	C₄×C₈×C₇₃	—	2	1	no	—
42	C₄×C₈×C₇₃	—	2	1	no	—
43	C₄×C₈×C₇₃	D₄	1	2	yes	—
44	C₄×C₈×C₇₃	D₄	1	2	yes	—
45	C₄×C₈×C₇₃	D₄	1	2	yes	—
46	C₂×Dic₁₄₆	—	2	2	no	—
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	M₁₆	1	4	yes	—
52	C₂²×Dic₇₃	—	2	2	no	—
53	C₂²×Dic₇₃	—	2	2	no	—
54	Dic₂₉₂	—	2	8	no	—
55	C₂×#10596831	—	4	2	no	—
56	C₂×#10596831	—	4	4	no	—
57	C₄⋊Dic₇₃	—	2	2	no	—
58	C₄⋊Dic₇₃	—	2	2	no	—
59	#11924007	—	4	2	no	—
60	#11924007	—	4	2	no	—
61	C₄×Dic₇₃	—	4	2	no	—
62	C₄×Dic₇₃	—	4	2	no	—
63	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
66	C₂×C₈×C₇₃	—	2	2	no	—
67	C₂×C₈×C₇₃	—	2	2	no	—
68	C₂×C₈×C₇₃	—	2	2	no	—
69	C₂×C₈×C₇₃	—	2	2	no	—
70	C₂×C₈×C₇₃	—	2	2	no	—
71	C₂×C₈×C₇₃	—	4	2	no	—
72	C₂×C₈×C₇₃	C₂²⋊C₄	1	2	yes	—
73	C₂×C₈×C₇₃	C₂²⋊C₄	1	2	yes	—
74	C₂×C₈×C₇₃	Q₁₆	1	4	yes	complemented
75	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
76	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₇₃	#41	1	1	yes	characteristic
79	C₂³×C₇₃	C₄×D₄	1	1	yes	characteristic
80	C₂×C₄×C₇₃	—	2	2	no	—
81	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
83	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
84	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
85	C₂×C₄×C₇₃	#124	1	1	yes	characteristic
86	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
87	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
88	C₂×C₄×C₇₃	#131	1	2	yes	—
89	C₂×C₄×C₇₃	C₂×Q₁₆	1	2	yes	—
90	C₂×C₄×C₇₃	#135	1	2	yes	—
91	C₂×Dic₇₃	—	2	4	no	—
92	C₂×Dic₇₃	—	4	2	no	—
93	C₂×Dic₇₃	—	4	2	no	—
94	C₂×Dic₇₃	—	4	2	no	—
95	C₂×Dic₇₃	—	4	4	no	—
96	Dic₁₄₆	—	2	4	no	—
97	Dic₁₄₆	—	2	4	no	—
98	Dic₁₄₆	—	4	2	no	—
99	Dic₁₄₆	—	4	2	no	—
100	C₈×C₇₃	—	2	2	no	—
101	C₈×C₇₃	—	2	2	no	—
102	C₈×C₇₃	—	2	4	no	—
103	C₈×C₇₃	—	4	2	no	—
104	#10596831	—	8	2	no	—
105	C₂²×C₇₃	#390	1	1	yes	characteristic
106	C₂²×C₇₃	#435	1	2	yes	—
107	C₂²×C₇₃	C₄×Q₁₆	1	2	yes	—
108	C₂²×C₇₃	#444	1	2	yes	—
109	C₄×C₇₃	—	2	1	no	—
110	C₄×C₇₃	—	2	1	no	—
111	C₄×C₇₃	#407	1	1	yes	characteristic
112	C₄×C₇₃	#408	1	1	yes	characteristic
113	C₄×C₇₃	#435	1	2	yes	—
114	C₄×C₇₃	#467	1	2	yes	—
115	C₄×C₇₃	#494	1	1	yes	characteristic
116	C₄×C₇₃	#495	1	1	yes	characteristic
117	Dic₇₃	—	4	4	no	—
118	Dic₇₃	—	8	2	no	—
119	#9662	—	73	1	no	maximal, Sylow
120	C₂×C₇₃	#1584	1	4	yes	—
121	C₂×C₇₃	#1955	1	1	yes	characteristic
122	C₂×C₇₃	#1969	1	1	yes	characteristic
123	C₂×C₇₃	#1975	1	1	yes	characteristic
124	C₂×C₄×Q₁₆	—	73	1	no	—
125	#1984	—	73	2	no	—
126	C₂×#335	—	73	1	no	—
127	C₂×Q₈⋊C₈	—	73	2	no	—
128	C₂×#322	D₇₃	1	1	yes	characteristic
129	C₇₃	#9662	1	1	yes	characteristic, complemented, Sylow
130	C₂²×Q₁₆	—	73	1	no	—
131	C₂×C₄×Q₈	—	73	2	no	—
132	C₄×Q₁₆	—	146	2	no	—
133	C₄×Q₁₆	—	146	2	no	—
134	C₂×#107	—	73	1	no	—
135	C₂×C₄⋊C₈	—	73	2	no	—
136	C₂×#103	—	73	2	no	—
137	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
138	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
139	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
140	#336	—	146	4	no	—
141	#335	—	146	2	no	—
142	Q₈⋊C₈	—	146	4	no	—
143	#322	—	2	2	no	—
144	C₂²×Q₈	—	73	2	no	—
145	C₂×Q₁₆	—	146	4	no	—
146	C₂×Q₁₆	—	146	2	no	—
147	C₂²×C₈	—	2	1	no	—
148	C₂²×C₈	—	146	2	no	—
149	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
150	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
151	C₂²×C₈	D₂₉₂	1	2	yes	—
152	C₄×Q₈	—	146	2	no	—
153	C₄×Q₈	—	146	2	no	—
154	C₄×Q₈	—	146	2	no	—
155	C₄×Q₈	—	146	2	no	—
156	C₂×C₄⋊C₄	—	73	2	no	—
157	C₂×C₄⋊C₄	—	146	2	no	—
158	C₂×C₄²	—	146	2	no	—
159	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
160	#107	—	146	2	no	—
161	C₄⋊C₈	—	146	2	no	—
162	C₄⋊C₈	—	146	2	no	—
163	#103	—	146	4	no	—
164	C₄×C₈	—	2	1	no	—
165	C₄×C₈	—	2	1	no	—
166	C₄×C₈	—	2	1	no	—
167	C₄×C₈	D₂₉₂	1	2	yes	—
168	C₄×C₈	D₂₉₂	1	2	yes	—
169	C₄×C₈	C₂²⋊D₇₃	1	2	yes	—
170	C₂×Q₈	—	73	4	no	—
171	C₂×Q₈	—	146	2	no	—
172	C₂×Q₈	—	146	2	no	—
173	C₂×Q₈	—	146	2	no	—
174	C₂×Q₈	—	146	2	no	—
175	C₂×Q₈	—	146	2	no	—
176	C₂²×C₄	—	146	2	no	—
177	C₂²×C₄	—	146	2	no	—
178	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
179	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
180	C₂²×C₄	#11924028	1	1	yes	characteristic
181	Q₁₆	—	146	8	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	4	2	no	—
188	C₂×C₈	—	292	2	no	—
189	C₂×C₈	—	292	4	no	—
190	C₂×C₈	#11924005	1	2	yes	—
191	C₂×C₈	#11924005	1	2	yes	—
192	C₂×C₈	Dic₂₉₂	1	4	yes	complemented
193	C₄⋊C₄	—	292	2	no	—
194	C₄⋊C₄	—	292	2	no	—
195	C₄⋊C₄	—	146	2	no	—
196	C₄⋊C₄	—	146	2	no	—
197	C₄²	—	292	2	no	—
198	C₄²	—	292	2	no	—
199	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
200	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
201	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
202	C₄²	#11924028	1	1	yes	characteristic
203	C₂³	C₄×D₂₉₂	1	1	yes	characteristic
204	Q₈	—	292	2	no	—
205	Q₈	—	292	2	no	—
206	Q₈	—	146	4	no	—
207	Q₈	—	146	4	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	292	2	no	—
210	C₂×C₄	—	292	2	no	—
211	C₂×C₄	—	292	4	no	—
212	C₂×C₄	—	292	2	no	—
213	C₂×C₄	—	146	4	no	—
214	C₂×C₄	C₂×#11924005	1	1	yes	characteristic
215	C₂×C₄	C₄×D₂₉₂	1	1	yes	characteristic
216	C₂×C₄	#14556565	1	1	yes	characteristic
217	C₂×C₄	#14556565	1	1	yes	characteristic
218	C₂×C₄	#14556579	1	1	yes	characteristic
219	C₂×C₄	#14556588	1	1	yes	characteristic
220	C₂×C₄	M₁₆×D₇₃	1	2	yes	—
221	C₂×C₄	#14556599	1	2	yes	—
222	C₂×C₄	C₂×Dic₂₉₂	1	2	yes	—
223	C₂×C₄	#14556613	1	2	yes	—
224	C₈	—	2	2	no	—
225	C₈	—	2	2	no	—
226	C₈	—	2	4	no	—
227	C₈	—	4	2	no	—
228	C₈	—	584	2	no	—
229	C₂²	#17988489	1	1	yes	characteristic
230	C₂²	#17988660	1	2	yes	—
231	C₂²	C₄×Dic₂₉₂	1	2	yes	—
232	C₂²	#17988710	1	2	yes	—
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	292	4	no	—
236	C₄	—	584	2	no	—
237	C₄	#17988550	1	1	yes	characteristic
238	C₄	#17988555	1	1	yes	characteristic
239	C₄	#17988660	1	2	yes	—
240	C₄	#17988828	1	2	yes	—
241	C₄	#17988967	1	1	yes	characteristic
242	C₄	#17988971	1	1	yes	characteristic
243	C₂	#22608783	1	4	yes	—
244	C₂	#22610729	1	1	yes	characteristic
245	C₂	#22610792	1	1	yes	characteristic
246	C₂	#22610824	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₉₂
commutator	C₂²×C₄×C₇₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₂×#322×C₇₃	C₂
socle, prime radical	C₂³×C₇₃	C₄×D₄
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	C₂×#322	D₇₃
hypocenter	C₇₃	#9662
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
74. C₂×C₈×C₇₃	Q₁₆	semidirect
129. C₇₃	#9662	semidirect
192. C₂×C₈	Dic₂₉₂	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
#22610867	2
C₂×#17988340	2
C₂×#17988319	2
C₂×#322×C₇₃	2
#9662	73