Subgroups — #33999919

Order 18688.

338 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	C₄×C₄⋊#10596833	C₂	1	1	yes	maximal, characteristic
2	#22625645	C₂	1	1	yes	maximal, characteristic
3	#22609786	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊#10596833	—	2	1	no	—
5	C₂×C₄×#10596833	—	2	1	no	—
6	C₂×C₄×#10596833	—	2	1	no	—
7	#17989453	—	2	1	no	—
8	#17989453	C₄	1	2	yes	—
9	#17989453	C₄	1	2	yes	—
10	C₂×C₄⋊#10596831	C₄	1	1	yes	characteristic
11	C₂×#14556501	—	2	1	no	—
12	C₄²×Dic₇₃	C₂²	1	1	yes	characteristic
13	#17988341	—	2	1	no	—
14	C₂×C₄⋊C₈×C₇₃	C₄	1	1	yes	characteristic
15	C₂²×#10596833	—	2	2	no	—
16	C₂²×#10596833	—	2	2	no	—
17	C₄⋊#10596833	—	2	4	no	—
18	C₄×#10596833	—	2	2	no	—
19	C₄×#10596833	—	2	2	no	—
20	C₄×#10596833	—	4	1	no	—
21	C₄×#10596833	—	4	1	no	—
22	C₂²×#10596831	—	2	1	no	—
23	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×Dic₇₃	D₄	1	1	yes	characteristic
26	C₂×C₄×Dic₇₃	D₄	1	1	yes	characteristic
27	C₂×C₄×Dic₇₃	D₄	1	1	yes	characteristic
28	C₂×C₄×Dic₇₃	Q₈	1	1	yes	characteristic
29	C₄⋊#10596831	—	2	1	no	—
30	C₄⋊#10596831	C₈	1	2	yes	complemented
31	#14556501	—	4	1	no	—
32	#14556501	—	4	1	no	—
33	C₂²×C₈×C₇₃	—	2	1	no	—
34	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
35	C₄⋊C₈×C₇₃	—	2	1	no	—
36	C₄⋊C₈×C₇₃	C₈	1	2	yes	complemented
37	C₂×#10596833	—	2	4	no	—
38	C₂×#10596833	—	2	4	no	—
39	C₂×#10596833	—	4	2	no	—
40	C₂×#10596833	—	4	2	no	—
41	C₂×#10596833	—	4	2	no	—
42	C₂×#10596833	—	4	2	no	—
43	C₂²×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
44	C₂²×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
45	C₂²×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
46	C₂²×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
47	C₂×#10596831	—	2	2	no	—
48	C₂×#10596831	—	4	1	no	—
49	C₂×#10596831	—	4	1	no	—
50	C₄×Dic₇₃	—	2	1	no	—
51	C₄×Dic₇₃	—	2	1	no	—
52	C₄×Dic₇₃	—	2	2	no	—
53	C₄×Dic₇₃	—	2	2	no	—
54	C₄×Dic₇₃	—	4	1	no	—
55	C₄×Dic₇₃	—	4	1	no	—
56	C₄×Dic₇₃	D₈	1	1	yes	characteristic
57	C₄×Dic₇₃	SD₁₆	1	1	yes	characteristic
58	C₄×Dic₇₃	SD₁₆	1	1	yes	characteristic
59	C₄×Dic₇₃	Q₁₆	1	1	yes	characteristic
60	C₂²×C₄×C₇₃	C₄²	1	1	yes	characteristic
61	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₈×C₇₃	—	2	2	no	—
64	C₂×C₈×C₇₃	—	4	1	no	—
65	C₂×C₈×C₇₃	—	4	1	no	—
66	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
67	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
68	C₄²×C₇₃	M₁₆	1	1	yes	characteristic
69	C₄²×C₇₃	M₁₆	1	1	yes	characteristic
70	C₂³×C₇₃	#95	1	1	yes	characteristic
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	C₄×C₈	1	1	yes	characteristic
76	C₂×C₄×C₇₃	#97	1	1	yes	characteristic
77	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
78	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
79	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
80	C₂×C₄×C₇₃	#103	1	1	yes	characteristic
81	C₂×C₄×C₇₃	#104	1	1	yes	characteristic
82	C₂×C₄×C₇₃	#104	1	1	yes	characteristic
83	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
84	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
85	C₂×Dic₇₃	—	2	1	no	—
86	C₂×Dic₇₃	—	2	1	no	—
87	C₂×Dic₇₃	—	2	1	no	—
88	C₂×Dic₇₃	—	2	1	no	—
89	C₂×Dic₇₃	—	2	1	no	—
90	C₂×Dic₇₃	—	2	1	no	—
91	C₂×Dic₇₃	—	2	1	no	—
92	C₂×Dic₇₃	—	2	1	no	—
93	C₂×Dic₇₃	#102	1	1	yes	characteristic
94	C₂×Dic₇₃	#103	1	1	yes	characteristic
95	C₂×Dic₇₃	#104	1	2	yes	—
96	C₂×Dic₇₃	#106	1	1	yes	characteristic
97	C₂×Dic₇₃	#107	1	1	yes	characteristic
98	C₂×Dic₇₃	#108	1	2	yes	—
99	#10596833	—	4	4	no	—
100	#10596833	—	4	4	no	—
101	C₈×C₇₃	—	4	2	no	—
102	#10596831	—	4	2	no	—
103	C₂²×C₇₃	#336	1	1	yes	characteristic
104	C₂²×C₇₃	#339	1	1	yes	characteristic
105	C₂²×C₇₃	#339	1	1	yes	characteristic
106	C₂²×C₇₃	#340	1	1	yes	characteristic
107	C₂²×C₇₃	#341	1	1	yes	characteristic
108	C₂²×C₇₃	#344	1	1	yes	characteristic
109	C₂²×C₇₃	#344	1	1	yes	characteristic
110	C₄×C₇₃	—	2	1	no	—
111	C₄×C₇₃	—	2	1	no	—
112	C₄×C₇₃	—	2	1	no	—
113	C₄×C₇₃	—	2	1	no	—
114	C₄×C₇₃	D₄⋊C₈	1	1	yes	characteristic
115	C₄×C₇₃	D₄⋊C₈	1	1	yes	characteristic
116	C₄×C₇₃	Q₈⋊C₈	1	1	yes	characteristic
117	C₄×C₇₃	Q₈⋊C₈	1	1	yes	characteristic
118	Dic₇₃	—	2	2	no	—
119	Dic₇₃	—	2	1	no	—
120	Dic₇₃	—	2	1	no	—
121	Dic₇₃	—	2	2	no	—
122	Dic₇₃	—	2	1	no	—
123	Dic₇₃	—	2	1	no	—
124	#7027	—	73	1	no	maximal, Sylow
125	C₂×C₇₃	#1278	1	1	yes	characteristic
126	C₂×C₇₃	#1279	1	1	yes	characteristic
127	C₂×C₇₃	#1281	1	1	yes	characteristic
128	C₂×C₇₃	#1300	1	1	yes	characteristic
129	C₂×C₇₃	#1301	1	1	yes	characteristic
130	C₂×C₇₃	#1302	1	1	yes	characteristic
131	C₂×C₇₃	#1303	1	1	yes	characteristic
132	#1767	—	73	1	no	—
133	C₄×C₄⋊C₈	—	73	1	no	—
134	#1744	—	73	1	no	—
135	C₇₃	#7027	1	1	yes	characteristic, complemented, Sylow
136	C₂×C₄⋊C₈	—	73	1	no	—
137	C₂×C₄⋊C₈	—	146	1	no	—
138	C₂×C₄⋊C₈	#64408	1	1	yes	characteristic
139	C₂×#97	—	146	1	no	—
140	C₂×C₄×C₈	—	146	1	no	—
141	C₂×C₄×C₈	—	146	1	no	—
142	C₄³	—	73	1	no	—
143	#336	—	73	2	no	—
144	#336	—	73	2	no	—
145	#336	—	146	1	no	—
146	#336	—	146	1	no	—
147	C₂²×C₈	—	2	1	no	—
148	C₂²×C₈	—	146	2	no	—
149	C₂²×C₈	—	146	2	no	—
150	C₂²×C₈	—	146	1	no	—
151	C₂×C₄²	—	73	1	no	—
152	C₂×C₄²	—	73	1	no	—
153	C₂×C₄²	—	73	1	no	—
154	C₂×C₄²	—	73	1	no	—
155	C₂×C₄²	—	73	1	no	—
156	C₂×C₄²	—	73	1	no	—
157	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
158	C₄⋊C₈	—	2	1	no	—
159	C₄⋊C₈	—	73	2	no	—
160	C₄⋊C₈	—	146	4	no	—
161	C₄⋊C₈	—	146	1	no	—
162	C₄⋊C₈	#10596833	1	2	yes	complemented
163	#97	—	292	1	no	—
164	#97	—	292	1	no	—
165	C₄×C₈	—	292	1	no	—
166	C₄×C₈	—	292	1	no	—
167	C₄×C₈	—	146	2	no	—
168	C₄×C₈	—	146	2	no	—
169	C₂²×C₄	—	73	1	no	—
170	C₂²×C₄	—	73	1	no	—
171	C₂²×C₄	—	73	1	no	—
172	C₂²×C₄	—	73	1	no	—
173	C₂²×C₄	C₄×#64408	1	1	yes	characteristic
174	C₂²×C₄	C₂²⋊#64408	1	1	yes	characteristic
175	C₂²×C₄	C₄⋊#64408	1	1	yes	characteristic
176	C₂×C₈	—	2	2	no	—
177	C₂×C₈	—	4	1	no	—
178	C₂×C₈	—	4	1	no	—
179	C₂×C₈	—	292	2	no	—
180	C₂×C₈	—	292	2	no	—
181	C₂×C₈	—	292	2	no	—
182	C₂×C₈	—	292	2	no	—
183	C₂×C₈	—	292	1	no	—
184	C₂×C₈	—	292	1	no	—
185	C₂×C₈	—	146	4	no	—
186	C₂×C₈	—	146	4	no	—
187	C₂×C₈	—	146	2	no	—
188	C₄²	—	292	1	no	—
189	C₄²	—	292	1	no	—
190	C₄²	—	73	1	no	—
191	C₄²	—	73	1	no	—
192	C₄²	—	73	1	no	—
193	C₄²	—	73	1	no	—
194	C₄²	—	146	2	no	—
195	C₄²	—	146	2	no	—
196	C₄²	—	146	1	no	—
197	C₄²	—	146	1	no	—
198	C₄²	C₂×#10596833	1	1	yes	characteristic
199	C₄²	#11924036	1	1	yes	characteristic
200	C₄²	#11924037	1	1	yes	characteristic
201	C₄²	#11924038	1	1	yes	characteristic
202	C₂³	(C₂×#64408)⋊C₄	1	1	yes	characteristic
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	—	73	2	no	—
208	C₂×C₄	—	73	1	no	—
209	C₂×C₄	—	73	1	no	—
210	C₂×C₄	—	73	1	no	—
211	C₂×C₄	—	73	1	no	—
212	C₂×C₄	—	73	2	no	—
213	C₂×C₄	—	146	1	no	—
214	C₂×C₄	—	146	1	no	—
215	C₂×C₄	—	146	1	no	—
216	C₂×C₄	—	146	1	no	—
217	C₂×C₄	—	146	1	no	—
218	C₂×C₄	—	146	1	no	—
219	C₂×C₄	—	146	1	no	—
220	C₂×C₄	—	146	1	no	—
221	C₂×C₄	C₄×#10596833	1	1	yes	characteristic
222	C₂×C₄	#14556646	1	1	yes	characteristic
223	C₂×C₄	C₂²⋊#10596833	1	1	yes	characteristic
224	C₂×C₄	#14556648	1	1	yes	characteristic
225	C₂×C₄	#14556653	1	1	yes	characteristic
226	C₂×C₄	#14556654	1	1	yes	characteristic
227	C₂×C₄	#14556655	1	1	yes	characteristic
228	C₂×C₄	#14556656	1	1	yes	characteristic
229	C₂×C₄	C₄⋊#10596833	1	1	yes	characteristic
230	C₂×C₄	#14556658	1	1	yes	characteristic
231	C₈	—	4	2	no	—
232	C₈	—	292	4	no	—
233	C₈	—	292	4	no	—
234	C₈	—	292	2	no	—
235	C₂²	#17989453	1	1	yes	characteristic
236	C₂²	#17989457	1	1	yes	characteristic
237	C₂²	#17989459	1	1	yes	characteristic
238	C₂²	#17989460	1	1	yes	characteristic
239	C₂²	#17989462	1	1	yes	characteristic
240	C₂²	#17989469	1	1	yes	characteristic
241	C₂²	#17989469	1	1	yes	characteristic
242	C₄	—	2	1	no	—
243	C₄	—	2	1	no	—
244	C₄	—	2	1	no	—
245	C₄	—	2	1	no	—
246	C₄	—	146	2	no	—
247	C₄	—	146	1	no	—
248	C₄	—	146	1	no	—
249	C₄	—	146	2	no	—
250	C₄	—	146	1	no	—
251	C₄	—	146	1	no	—
252	C₄	D₄⋊#10596833	1	1	yes	characteristic
253	C₄	#17989435	1	1	yes	characteristic
254	C₄	Q₈⋊#10596833	1	1	yes	characteristic
255	C₄	#17989437	1	1	yes	characteristic
256	C₂	#22624764	1	1	yes	characteristic
257	C₂	#22624767	1	1	yes	characteristic
258	C₂	#22624771	1	1	yes	characteristic
259	C₂	#22624818	1	1	yes	characteristic
260	C₂	#22624821	1	1	yes	characteristic
261	C₂	#22624824	1	1	yes	characteristic
262	C₂	#22624827	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²⋊#64408
commutator	C₂×C₄×C₇₃	C₄×C₈
Frattini	C₂×C₄²	C₂×#64408
Fitting	C₂×C₄⋊C₈×C₇₃	C₄
socle, prime radical	C₂³×C₇₃	#95
cosocle	C₄²×Dic₇₃	C₂²
hypercenter	C₂×C₄⋊C₈	#64408
hypocenter	C₇₃	#7027
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
30. C₄⋊#10596831	C₈	semidirect
36. C₄⋊C₈×C₇₃	C₈	semidirect
135. C₇₃	#7027	semidirect
162. C₄⋊C₈	#10596833	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊#10596833	2
#22625645	2
#22609786	2
#7027	73