Subgroups — #32972553

Order 18688.

352 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22610083	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#10596831	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊C₈×C₇₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊#10596831	—	2	1	no	—
5	C₂×C₄⋊#10596831	—	2	1	no	—
6	C₂×C₄⋊#10596831	C₄	1	2	yes	—
7	C₂×C₄⋊#10596831	C₄	1	2	yes	—
8	C₂×C₄×#10596831	—	2	1	no	—
9	#17988342	—	2	1	no	—
10	C₂×C₄⋊C₈×C₇₃	—	2	1	no	—
11	C₂×C₄×C₈×C₇₃	—	2	1	no	—
12	C₄³×C₇₃	C₂²	1	1	yes	characteristic
13	#336×C₇₃	C₄	1	2	yes	—
14	C₂²×#10596831	—	2	2	no	—
15	C₂²×#10596831	—	2	2	no	—
16	C₄⋊#10596831	—	2	2	no	—
17	C₄⋊#10596831	—	4	2	no	—
18	C₄⋊#10596831	—	4	1	no	—
19	C₄⋊#10596831	—	4	1	no	—
20	C₄⋊#10596831	—	4	1	no	—
21	C₄⋊#10596831	—	4	1	no	—
22	C₄⋊#10596831	C₈	1	4	yes	complemented
23	C₄×#10596831	—	4	1	no	—
24	C₄×#10596831	—	4	1	no	—
25	C₂²×C₈×C₇₃	—	2	2	no	—
26	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₇₃	Q₈	1	1	yes	characteristic
33	C₄⋊C₈×C₇₃	—	2	4	no	—
34	C₄×C₈×C₇₃	—	4	1	no	—
35	C₄×C₈×C₇₃	—	4	1	no	—
36	C₂×#10596831	—	2	4	no	—
37	C₂×#10596831	—	4	2	no	—
38	C₂×#10596831	—	4	2	no	—
39	C₂×#10596831	—	4	2	no	—
40	C₂×#10596831	—	4	2	no	—
41	C₂×#10596831	—	4	2	no	—
42	C₂²×C₄×C₇₃	C₄²	1	1	yes	characteristic
43	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₇₃	—	2	4	no	—
50	C₂×C₈×C₇₃	—	4	2	no	—
51	C₂×C₈×C₇₃	—	4	2	no	—
52	C₄²×C₇₃	—	2	1	no	—
53	C₄²×C₇₃	—	2	1	no	—
54	C₄²×C₇₃	—	2	2	no	—
55	C₄²×C₇₃	—	2	2	no	—
56	C₄²×C₇₃	—	4	1	no	—
57	C₄²×C₇₃	C₂×C₈	1	2	yes	—
58	C₄²×C₇₃	M₁₆	1	2	yes	—
59	C₄²×C₇₃	D₈	1	1	yes	characteristic
60	C₄²×C₇₃	D₈	1	1	yes	characteristic
61	C₄²×C₇₃	SD₁₆	1	1	yes	characteristic
62	C₄²×C₇₃	SD₁₆	1	1	yes	characteristic
63	C₄²×C₇₃	SD₁₆	1	1	yes	characteristic
64	C₄²×C₇₃	SD₁₆	1	1	yes	characteristic
65	C₄²×C₇₃	Q₁₆	1	1	yes	characteristic
66	C₄²×C₇₃	Q₁₆	1	1	yes	characteristic
67	C₂³×C₇₃	#95	1	1	yes	characteristic
68	C₂×C₄×C₇₃	—	2	1	no	—
69	C₂×C₄×C₇₃	—	2	1	no	—
70	C₂×C₄×C₇₃	—	2	1	no	—
71	C₂×C₄×C₇₃	—	2	1	no	—
72	C₂×C₄×C₇₃	—	2	1	no	—
73	C₂×C₄×C₇₃	—	2	1	no	—
74	C₂×C₄×C₇₃	—	2	1	no	—
75	C₂×C₄×C₇₃	—	2	1	no	—
76	C₂×C₄×C₇₃	—	2	1	no	—
77	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
78	C₂×C₄×C₇₃	#97	1	1	yes	characteristic
79	C₂×C₄×C₇₃	C₂²⋊C₈	1	2	yes	—
80	C₂×C₄×C₇₃	#100	1	1	yes	characteristic
81	C₂×C₄×C₇₃	#101	1	1	yes	characteristic
82	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
83	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
84	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
85	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
86	C₂×C₄×C₇₃	#103	1	1	yes	characteristic
87	C₂×C₄×C₇₃	#103	1	1	yes	characteristic
88	C₂×C₄×C₇₃	#103	1	1	yes	characteristic
89	C₂×C₄×C₇₃	#103	1	1	yes	characteristic
90	C₂×C₄×C₇₃	#104	1	2	yes	—
91	C₂×C₄×C₇₃	C₄⋊C₈	1	2	yes	—
92	C₂×C₄×C₇₃	#106	1	1	yes	characteristic
93	C₂×C₄×C₇₃	#106	1	1	yes	characteristic
94	C₂×C₄×C₇₃	#107	1	1	yes	characteristic
95	C₂×C₄×C₇₃	#107	1	1	yes	characteristic
96	C₂×C₄×C₇₃	#108	1	2	yes	—
97	C₈×C₇₃	—	4	4	no	—
98	#10596831	—	4	4	no	—
99	#10596831	—	8	2	no	—
100	C₂²×C₇₃	#336	1	1	yes	characteristic
101	C₂²×C₇₃	#339	1	1	yes	characteristic
102	C₂²×C₇₃	#340	1	1	yes	characteristic
103	C₂²×C₇₃	#340	1	1	yes	characteristic
104	C₂²×C₇₃	#341	1	1	yes	characteristic
105	C₂²×C₇₃	#343	1	1	yes	characteristic
106	C₂²×C₇₃	#344	1	1	yes	characteristic
107	C₄×C₇₃	—	2	1	no	—
108	C₄×C₇₃	—	2	1	no	—
109	C₄×C₇₃	—	2	1	no	—
110	C₄×C₇₃	—	2	1	no	—
111	C₄×C₇₃	—	2	1	no	—
112	C₄×C₇₃	—	2	1	no	—
113	C₄×C₇₃	—	4	1	no	—
114	C₄×C₇₃	D₄⋊C₈	1	2	yes	—
115	C₄×C₇₃	Q₈⋊C₈	1	2	yes	—
116	C₄×C₇₃	#331	1	1	yes	characteristic
117	C₄×C₇₃	#332	1	1	yes	characteristic
118	C₄×C₇₃	#332	1	1	yes	characteristic
119	C₄×C₇₃	#333	1	1	yes	characteristic
120	C₄×C₇₃	#334	1	2	yes	—
121	C₄×C₇₃	#335	1	2	yes	—
122	#7031	—	73	1	no	maximal, Sylow
123	C₂×C₇₃	#1277	1	1	yes	characteristic
124	C₂×C₇₃	#1279	1	1	yes	characteristic
125	C₂×C₇₃	#1283	1	1	yes	characteristic
126	C₂×C₇₃	#1296	1	1	yes	characteristic
127	C₂×C₇₃	#1299	1	1	yes	characteristic
128	C₂×C₇₃	#1301	1	1	yes	characteristic
129	C₂×C₇₃	#1303	1	1	yes	characteristic
130	#1842	—	73	1	no	—
131	C₄×C₄⋊C₈	—	73	1	no	—
132	C₄×C₄⋊C₈	D₇₃	1	1	yes	characteristic
133	C₇₃	#7031	1	1	yes	characteristic, complemented, Sylow
134	C₂×C₄⋊C₈	—	2	1	no	—
135	C₂×C₄⋊C₈	—	73	2	no	—
136	C₂×C₄⋊C₈	—	73	2	no	—
137	C₂×C₄⋊C₈	—	146	1	no	—
138	C₂×C₄⋊C₈	—	146	1	no	—
139	C₂×C₄×C₈	—	2	1	no	—
140	C₂×C₄×C₈	—	146	1	no	—
141	C₄³	C₂×D₇₃	1	1	yes	characteristic
142	#336	—	146	1	no	—
143	#336	Dic₇₃	1	2	yes	—
144	C₂²×C₈	—	2	2	no	—
145	C₂²×C₈	—	146	2	no	—
146	C₂²×C₈	—	146	2	no	—
147	C₂×C₄²	Dic₁₄₆	1	1	yes	characteristic
148	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
149	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
150	C₂×C₄²	D₂₉₂	1	1	yes	characteristic
151	C₂×C₄²	C₂×Dic₇₃	1	1	yes	characteristic
152	C₂×C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
153	C₂×C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
154	C₄⋊C₈	—	2	4	no	—
155	C₄⋊C₈	—	292	2	no	—
156	C₄⋊C₈	—	292	1	no	—
157	C₄⋊C₈	—	292	1	no	—
158	C₄⋊C₈	—	292	1	no	—
159	C₄⋊C₈	—	292	1	no	—
160	C₄⋊C₈	—	73	4	no	—
161	C₄⋊C₈	—	146	2	no	—
162	C₄×C₈	—	4	1	no	—
163	C₄×C₈	—	4	1	no	—
164	C₄×C₈	—	292	1	no	—
165	C₄×C₈	—	292	1	no	—
166	C₂²×C₄	C₄×Dic₇₃	1	1	yes	characteristic
167	C₂²×C₄	#11924005	1	1	yes	characteristic
168	C₂²×C₄	#11924005	1	1	yes	characteristic
169	C₂²×C₄	C₂²⋊Dic₇₃	1	1	yes	characteristic
170	C₂²×C₄	#11924007	1	1	yes	characteristic
171	C₂²×C₄	#11924007	1	1	yes	characteristic
172	C₂²×C₄	C₄⋊Dic₇₃	1	1	yes	characteristic
173	C₂×C₈	—	2	4	no	—
174	C₂×C₈	—	4	2	no	—
175	C₂×C₈	—	4	2	no	—
176	C₂×C₈	—	292	2	no	—
177	C₂×C₈	—	292	2	no	—
178	C₂×C₈	—	292	2	no	—
179	C₂×C₈	—	292	2	no	—
180	C₂×C₈	—	292	2	no	—
181	C₂×C₈	—	146	4	no	—
182	C₄²	—	2	1	no	—
183	C₄²	—	2	1	no	—
184	C₄²	—	2	2	no	—
185	C₄²	—	2	2	no	—
186	C₄²	—	4	1	no	—
187	C₄²	C₈×D₇₃	1	2	yes	—
188	C₄²	#11924011	1	2	yes	—
189	C₄²	#11924013	1	1	yes	characteristic
190	C₄²	D₅₈₄	1	1	yes	characteristic
191	C₄²	#11924015	1	1	yes	characteristic
192	C₄²	#11924016	1	1	yes	characteristic
193	C₄²	#11924017	1	1	yes	characteristic
194	C₄²	#11924017	1	1	yes	characteristic
195	C₄²	#11924018	1	1	yes	characteristic
196	C₄²	Dic₂₉₂	1	1	yes	characteristic
197	C₂³	#14556498	1	1	yes	characteristic
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	C₈×Dic₇₃	1	1	yes	characteristic
208	C₂×C₄	#14556501	1	1	yes	characteristic
209	C₂×C₄	#14556503	1	2	yes	—
210	C₂×C₄	#14556507	1	1	yes	characteristic
211	C₂×C₄	#14556509	1	1	yes	characteristic
212	C₂×C₄	#14556511	1	1	yes	characteristic
213	C₂×C₄	#14556512	1	1	yes	characteristic
214	C₂×C₄	#14556513	1	1	yes	characteristic
215	C₂×C₄	#14556513	1	1	yes	characteristic
216	C₂×C₄	#14556514	1	1	yes	characteristic
217	C₂×C₄	#14556515	1	1	yes	characteristic
218	C₂×C₄	#14556516	1	1	yes	characteristic
219	C₂×C₄	#14556516	1	1	yes	characteristic
220	C₂×C₄	#14556519	1	2	yes	—
221	C₂×C₄	#14556520	1	2	yes	—
222	C₂×C₄	#14556522	1	1	yes	characteristic
223	C₂×C₄	#14556523	1	1	yes	characteristic
224	C₂×C₄	#14556524	1	1	yes	characteristic
225	C₂×C₄	#14556525	1	1	yes	characteristic
226	C₂×C₄	#14556526	1	2	yes	—
227	C₈	—	4	4	no	—
228	C₈	—	292	4	no	—
229	C₈	—	584	2	no	—
230	C₂²	#17988341	1	1	yes	characteristic
231	C₂²	#17988347	1	1	yes	characteristic
232	C₂²	#17988349	1	1	yes	characteristic
233	C₂²	#17988350	1	1	yes	characteristic
234	C₂²	#17988351	1	1	yes	characteristic
235	C₂²	#17988355	1	1	yes	characteristic
236	C₂²	#17988357	1	1	yes	characteristic
237	C₄	—	2	1	no	—
238	C₄	—	2	1	no	—
239	C₄	—	2	1	no	—
240	C₄	—	2	1	no	—
241	C₄	—	2	1	no	—
242	C₄	—	2	1	no	—
243	C₄	—	4	1	no	—
244	C₄	#17988316	1	2	yes	—
245	C₄	#17988319	1	2	yes	—
246	C₄	#17988330	1	1	yes	characteristic
247	C₄	#17988332	1	1	yes	characteristic
248	C₄	#17988334	1	1	yes	characteristic
249	C₄	#17988335	1	1	yes	characteristic
250	C₄	#17988337	1	2	yes	—
251	C₄	#17988339	1	2	yes	—
252	C₂	#22607765	1	1	yes	characteristic
253	C₂	#22607771	1	1	yes	characteristic
254	C₂	#22607780	1	1	yes	characteristic
255	C₂	#22607815	1	1	yes	characteristic
256	C₂	#22607821	1	1	yes	characteristic
257	C₂	#22607827	1	1	yes	characteristic
258	C₂	#22607834	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#11924005
commutator	C₂×C₄×C₇₃	C₄×C₈
Frattini	C₄³	C₂×D₇₃
Fitting	C₄×C₄⋊C₈×C₇₃	C₂
socle, prime radical	C₂³×C₇₃	#95
cosocle	C₄³×C₇₃	C₂²
hypercenter	C₄×C₄⋊C₈	D₇₃
hypocenter	C₇₃	#7031
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
22. C₄⋊#10596831	C₈	semidirect
133. C₇₃	#7031	semidirect

Maximal subgroups

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

subgroup	index
#22610083	2
C₄×C₄⋊#10596831	2
C₄×C₄⋊C₈×C₇₃	2
#7031	73