Subgroups — #34030080

Order 18688.

684 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22626900	C₂	1	2	yes	maximal
2	#22626274	C₂	1	2	yes	maximal, complemented
3	#22626273	C₂	1	4	yes	maximal, complemented
4	#22626270	C₂	1	4	yes	maximal, complemented
5	C₂×#17989509	C₂	1	2	yes	maximal, complemented
6	#22618353	C₂	1	1	yes	maximal, characteristic
7	#17989646	—	2	8	no	—
8	#17989643	C₂²	1	8	yes	—
9	C₂×#14556665	C₂²	1	4	yes	—
10	#17989510	C₂²	1	8	yes	—
11	#17989509	C₂²	1	4	yes	—
12	#17989509	C₂²	1	4	yes	—
13	C₂×#14556599	C₂²	1	1	yes	characteristic
14	C₂×#14556597	C₄	1	1	yes	characteristic
15	#17989055	C₄	1	1	yes	characteristic
16	#17988544	C₂²	1	2	yes	—
17	#17988543	C₂²	1	1	yes	characteristic
18	#17988541	C₄	1	2	yes	—
19	#17988539	C₄	1	2	yes	—
20	M₁₆×Dic₇₃	C₄	1	1	yes	characteristic
21	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
22	C₄×C₈×D₇₃	C₂²	1	2	yes	—
23	#405×C₇₃	C₄	1	1	yes	characteristic
24	#14556666	—	2	8	no	—
25	#14556665	—	2	4	no	—
26	#14556665	C₂³	1	8	yes	—
27	#14556664	—	2	8	no	—
28	C₂×#11924031	—	2	4	no	—
29	C₂×#11924028	C₂×C₄	1	1	yes	characteristic
30	#14556599	D₄	1	4	yes	—
31	#14556599	Q₈	1	4	yes	—
32	#14556597	C₈	1	8	yes	—
33	C₂×#11924012	C₂×C₄	1	1	yes	characteristic
34	C₂×#11924011	C₂×C₄	1	2	yes	—
35	C₂×#11924011	C₂×C₄	1	2	yes	—
36	C₂²×#10596831	C₂³	1	1	yes	characteristic
37	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
38	C₂×C₈×D₇₃	C₂³	1	2	yes	—
39	#14556556	—	2	1	no	—
40	#14556553	—	2	1	no	—
41	#14556551	—	2	2	no	—
42	#14556548	C₂×C₄	1	2	yes	—
43	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
44	C₄²×D₇₃	C₂×C₄	1	2	yes	—
45	#14556502	C₂×C₄	1	2	yes	—
46	#14556501	C₂×C₄	1	2	yes	—
47	C₄×#10596831	C₂³	1	2	yes	—
48	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
49	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
50	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
51	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
52	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
53	C₂×M₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
55	#117×C₇₃	C₂×C₄	1	1	yes	characteristic
56	#97×C₇₃	C₂×C₄	1	2	yes	—
57	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
58	#11924031	—	2	8	no	—
59	#11924028	C₄⋊C₄	1	4	yes	—
60	#11924028	C₂×C₈	1	4	yes	—
61	C₂×Dic₁₄₆	M₁₆	1	1	yes	characteristic
62	C₂×D₂₉₂	M₁₆	1	1	yes	characteristic
63	C₂×C₂²⋊D₇₃	M₁₆	1	2	yes	—
64	C₂²×Dic₇₃	—	2	1	no	—
65	C₂×C₄×D₇₃	—	2	2	no	—
66	C₂×C₄×D₇₃	C₂²×C₄	1	2	yes	—
67	#11924012	C₂×C₈	1	4	yes	—
68	#11924011	C₄⋊C₄	1	8	yes	—
69	#11924011	C₂×C₈	1	8	yes	—
70	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
71	C₂×#10596831	C₂²×C₄	1	2	yes	—
72	C₂×#10596831	C₂×D₄	1	2	yes	—
73	C₂×#10596831	C₂×Q₈	1	2	yes	—
74	C₂×#10596831	C₂⁴	1	1	yes	characteristic
75	C₈×D₇₃	C₂×C₈	1	8	yes	—
76	C₈×D₇₃	C₂×D₄	1	4	yes	—
77	C₈×D₇₃	C₂×Q₈	1	4	yes	—
78	#11924007	—	2	2	no	—
79	#11924005	—	2	2	no	—
80	C₄×Dic₇₃	—	2	1	no	—
81	C₄×Dic₇₃	—	2	2	no	—
82	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
83	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
85	M₁₆×C₇₃	C₄⋊C₄	1	4	yes	—
86	C₂×C₈×C₇₃	C₂×C₈	1	4	yes	—
87	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
88	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
90	C₄⋊C₄×C₇₃	—	2	1	no	—
91	C₂²⋊C₄×C₇₃	—	2	1	no	—
92	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
93	C₂³×C₇₃	C₂×M₁₆	1	1	yes	characteristic
94	C₂²×D₇₃	C₂×M₁₆	1	2	yes	—
95	C₂×C₄×C₇₃	—	2	2	no	—
96	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
97	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
98	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
99	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
100	C₂²⋊D₇₃	C₄⋊C₈	1	8	yes	—
101	C₂×Dic₇₃	—	2	1	no	—
102	C₂×Dic₇₃	—	2	1	no	—
103	C₂×Dic₇₃	—	4	2	no	—
104	C₂×Dic₇₃	C₂×M₁₆	1	2	yes	—
105	D₂₉₂	C₄⋊C₈	1	4	yes	—
106	C₄×D₇₃	—	4	4	no	—
107	C₄×D₇₃	C₂×C₄⋊C₄	1	4	yes	—
108	C₄×D₇₃	C₂²×C₈	1	4	yes	—
109	Dic₁₄₆	C₄⋊C₈	1	4	yes	—
110	C₈×C₇₃	C₂×C₄⋊C₄	1	4	yes	—
111	C₈×C₇₃	C₂²×C₈	1	2	yes	—
112	C₈×C₇₃	C₂²×C₈	1	2	yes	—
113	#10596831	C₂×C₄⋊C₄	1	2	yes	—
114	#10596831	C₂²×C₈	1	4	yes	—
115	#10596831	C₂²×D₄	1	1	yes	characteristic
116	#10596831	C₂²×Q₈	1	1	yes	characteristic
117	C₂²×C₇₃	—	4	1	no	—
118	C₂²×C₇₃	C₂×C₄⋊C₈	1	2	yes	—
119	C₂²×C₇₃	C₂²×M₁₆	1	1	yes	characteristic
120	C₂×D₇₃	—	4	2	no	—
121	C₂×D₇₃	C₂×C₄⋊C₈	1	4	yes	—
122	C₄×C₇₃	—	4	2	no	—
123	C₄×C₇₃	C₂×C₄⋊C₈	1	2	yes	—
124	C₄×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
125	C₄×C₇₃	C₂³×C₈	1	1	yes	characteristic
126	Dic₇₃	—	2	2	no	—
127	Dic₇₃	—	4	1	no	—
128	Dic₇₃	C₂×C₄⋊C₈	1	4	yes	—
129	#33401	—	73	1	no	maximal, Sylow
130	C₂×C₇₃	—	2	1	no	—
131	C₂×C₇₃	—	2	2	no	—
132	C₂×C₇₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
133	D₇₃	—	2	4	no	—
134	#3407	—	73	2	no	—
135	C₄×#131	—	73	1	no	—
136	#2153	—	73	4	no	—
137	#2153	—	73	2	no	—
138	C₂×#364	—	73	2	no	—
139	C₂×#364	—	73	4	no	—
140	C₇₃	#33401	1	1	yes	characteristic, complemented, Sylow
141	C₂×#131	—	73	1	no	—
142	C₂×#131	—	73	1	no	—
143	C₄×#41	—	73	1	no	—
144	#504	—	146	8	no	—
145	C₂×M₃₂	—	73	4	no	—
146	C₂×M₃₂	—	73	8	no	—
147	#405	—	73	1	no	—
148	#405	—	73	2	no	—
149	#405	—	73	2	no	—
150	#405	#64408	1	1	yes	characteristic
151	C₄×M₁₆	—	73	1	no	—
152	C₄×M₁₆	—	73	2	no	—
153	C₂×C₄×C₈	—	73	1	no	—
154	C₂×C₄×C₈	—	73	2	no	—
155	#364	—	73	4	no	—
156	#364	—	73	4	no	—
157	#364	—	73	8	no	—
158	C₂×#41	—	73	1	no	—
159	#131	—	73	4	no	—
160	#131	—	73	4	no	—
161	#131	—	73	8	no	—
162	C₂×M₁₆	—	73	1	no	—
163	C₂×M₁₆	—	73	2	no	—
164	C₂×M₁₆	—	73	2	no	—
165	C₂×M₁₆	C₂×#64408	1	1	yes	characteristic
166	C₂²×C₈	—	73	1	no	—
167	C₂²×C₈	—	73	2	no	—
168	C₂²×C₈	—	73	2	no	—
169	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
170	C₄×Q₈	—	146	1	no	—
171	C₄×D₄	—	146	2	no	—
172	C₄×D₄	—	146	1	no	—
173	#117	—	73	2	no	—
174	#117	C₂×#64408	1	1	yes	characteristic
175	C₂×C₄²	—	73	1	no	—
176	C₂×C₄²	—	73	2	no	—
177	M₃₂	—	73	8	no	—
178	M₃₂	—	146	4	no	—
179	M₃₂	—	146	8	no	—
180	C₂×C₁₆	—	146	4	no	—
181	C₂×C₁₆	—	146	8	no	—
182	#97	—	73	2	no	—
183	#97	—	73	2	no	—
184	#97	C₂×#64408	1	2	yes	—
185	C₄×C₈	—	73	1	no	—
186	C₄×C₈	—	73	2	no	—
187	C₄×C₈	—	73	1	no	—
188	C₄×C₈	—	73	1	no	—
189	C₄×C₈	—	73	1	no	—
190	C₄×C₈	—	73	2	no	—
191	C₄×C₈	C₂×#64408	1	2	yes	—
192	#41	—	73	4	no	—
193	#41	—	73	4	no	—
194	C₂×Q₈	—	73	1	no	—
195	C₂×D₄	—	73	2	no	—
196	C₂×D₄	—	73	1	no	—
197	C₂²×C₄	—	73	2	no	—
198	C₂²×C₄	—	146	2	no	—
199	C₂²×C₄	—	146	1	no	—
200	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
201	M₁₆	—	73	4	no	—
202	M₁₆	—	73	8	no	—
203	M₁₆	—	73	8	no	—
204	M₁₆	C₄⋊#64408	1	4	yes	—
205	C₂×C₈	—	73	1	no	—
206	C₂×C₈	—	73	2	no	—
207	C₂×C₈	—	73	1	no	—
208	C₂×C₈	—	73	2	no	—
209	C₂×C₈	—	73	2	no	—
210	C₂×C₈	—	73	8	no	—
211	C₂×C₈	—	73	4	no	—
212	C₂×C₈	—	73	4	no	—
213	C₂×C₈	#11924036	1	4	yes	—
214	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
215	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
216	C₂×C₈	C₂²×#64408	1	2	yes	—
217	C₄⋊C₄	—	2	1	no	—
218	C₄⋊C₄	—	146	2	no	—
219	C₂²⋊C₄	—	2	1	no	—
220	C₂²⋊C₄	—	146	2	no	—
221	C₄²	—	73	1	no	—
222	C₄²	—	73	1	no	—
223	C₄²	—	146	2	no	—
224	C₄²	—	146	1	no	—
225	C₄²	C₂²×#64408	1	2	yes	—
226	C₁₆	—	146	8	no	—
227	C₂³	—	73	2	no	—
228	C₂³	D₇₃⋊M₁₆	1	1	yes	characteristic
229	Q₈	—	73	4	no	—
230	D₄	—	73	8	no	—
231	D₄	—	73	4	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	292	4	no	—
234	C₂×C₄	—	292	2	no	—
235	C₂×C₄	—	73	2	no	—
236	C₂×C₄	—	73	4	no	—
237	C₂×C₄	—	73	4	no	—
238	C₂×C₄	—	146	1	no	—
239	C₂×C₄	—	146	1	no	—
240	C₂×C₄	C₂×C₄⋊#64408	1	2	yes	—
241	C₂×C₄	C₂×#11924036	1	2	yes	—
242	C₂×C₄	D₇₃⋊M₁₆	1	1	yes	characteristic
243	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
244	C₈	—	73	1	no	—
245	C₈	—	73	1	no	—
246	C₈	—	73	2	no	—
247	C₈	—	73	4	no	—
248	C₈	C₂×C₄⋊#64408	1	4	yes	—
249	C₈	C₂×#11924036	1	2	yes	—
250	C₈	C₂×#11924036	1	2	yes	—
251	C₂²	—	4	1	no	—
252	C₂²	—	292	2	no	—
253	C₂²	—	73	4	no	—
254	C₂²	#17989588	1	2	yes	—
255	C₂²	C₂×D₇₃⋊M₁₆	1	1	yes	characteristic
256	C₄	—	4	2	no	—
257	C₄	—	292	1	no	—
258	C₄	—	73	4	no	—
259	C₄	—	146	2	no	—
260	C₄	#17989588	1	2	yes	—
261	C₄	C₂²×C₄⋊#64408	1	1	yes	characteristic
262	C₄	C₂²×#11924036	1	1	yes	characteristic
263	C₂	—	2	1	no	—
264	C₂	—	2	2	no	—
265	C₂	—	146	4	no	—
266	C₂	C₂×#17989588	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×C₄⋊#64408
commutator	C₄×C₇₃	C₂³×C₈
Frattini	C₂×C₄	C₂³×#64408
Fitting	#405×C₇₃	C₄
socle	C₂×C₇₃	C₂²×C₄⋊C₈
prime radical	C₂×#11924028	C₂×C₄
cosocle	C₂×#10596831	C₂⁴
hypercenter	#405	#64408
hypocenter	C₇₃	#33401
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #405

Complements

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

N	H ≅ G/N	type
2. #22626274	C₂	semidirect
3. #22626273	C₂	semidirect
4. #22626270	C₂	semidirect
5. C₂×#17989509	C₂	semidirect
140. C₇₃	#33401	semidirect

Maximal subgroups

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

subgroup	index
#22626900	2
#22626274	2
#22626273	2
#22626270	2
C₂×#17989509	2
#22618353	2
#33401	73