Subgroups — #34030174

Order 18688.

524 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22626306	C₂	1	4	yes	maximal, complemented
2	#22626302	C₂	1	4	yes	maximal, complemented
3	#22626272	C₂	1	2	yes	maximal, complemented
4	C₂×#17989484	C₂	1	2	yes	maximal
5	#22626152	C₂	1	2	yes	maximal, complemented
6	#22618347	C₂	1	1	yes	maximal, characteristic
7	C₂×#14556665	C₂²	1	4	yes	—
8	#17989509	C₂²	1	8	yes	complemented
9	#17989484	C₂²	1	4	yes	—
10	#17989484	C₂²	1	4	yes	—
11	#17989479	C₂²	1	8	yes	complemented
12	C₂²×#11924011	C₂²	1	1	yes	characteristic
13	C₂×M₁₆×D₇₃	C₄	1	1	yes	characteristic
14	#117×D₇₃	C₄	1	1	yes	characteristic
15	#17988545	C₂²	1	1	yes	characteristic
16	#17988544	C₄	1	2	yes	—
17	#17988543	C₂²	1	1	yes	characteristic
18	#17988542	C₄	1	1	yes	characteristic
19	#17988541	C₂²	1	2	yes	—
20	#17988539	C₂²	1	2	yes	—
21	C₄×#11924011	C₄	1	2	yes	—
22	#405×C₇₃	C₄	1	1	yes	characteristic
23	#14556665	—	2	4	no	—
24	#14556665	C₂³	1	8	yes	—
25	C₂×#11924031	—	2	4	no	—
26	C₂²×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
27	C₂×#11924012	C₂×C₄	1	1	yes	characteristic
28	C₂×#11924011	—	2	4	no	—
29	C₂×#11924011	C₂×C₄	1	1	yes	characteristic
30	C₂×#11924011	C₂×C₄	1	2	yes	—
31	C₂×#11924011	C₂×C₄	1	1	yes	characteristic
32	C₂×#11924011	C₂³	1	1	yes	characteristic
33	C₂×#11924011	C₂³	1	1	yes	characteristic
34	M₁₆×D₇₃	—	2	4	no	—
35	C₂²×#10596831	C₂³	1	1	yes	characteristic
36	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
37	#14556550	C₂×C₄	1	1	yes	characteristic
38	#14556549	—	2	1	no	—
39	#14556548	C₂×C₄	1	2	yes	—
40	#14556547	—	2	1	no	—
41	C₄⋊C₄×D₇₃	—	2	1	no	—
42	C₂²⋊C₄×D₇₃	—	2	1	no	—
43	C₄²×D₇₃	C₂×C₄	1	2	yes	—
44	#14556502	C₂³	1	2	yes	—
45	#14556501	C₂×C₄	1	2	yes	—
46	#14556501	C₂×C₄	1	2	yes	—
47	C₄×#10596831	C₂×C₄	1	2	yes	—
48	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
49	C₈×Dic₇₃	C₂³	1	2	yes	—
50	C₂×M₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
51	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
52	#117×C₇₃	C₂×C₄	1	1	yes	characteristic
53	#97×C₇₃	C₂×C₄	1	2	yes	—
54	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
55	#11924031	—	2	8	no	—
56	C₂³×D₇₃	M₁₆	1	1	yes	characteristic
57	C₂²×Dic₇₃	M₁₆	1	1	yes	characteristic
58	C₂×C₄×D₇₃	—	2	2	no	—
59	C₂×C₄×D₇₃	—	2	2	no	—
60	C₂×C₄×D₇₃	—	2	2	no	—
61	C₂×C₄×D₇₃	M₁₆	1	1	yes	characteristic
62	C₂×C₄×D₇₃	M₁₆	1	1	yes	characteristic
63	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
65	#11924012	—	2	2	no	—
66	#11924011	—	2	4	no	—
67	#11924011	—	2	2	no	—
68	#11924011	—	2	2	no	—
69	#11924011	—	2	2	no	—
70	#11924011	—	2	2	no	—
71	C₂×#10596831	C₂²×C₄	1	2	yes	—
72	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
73	C₂×#10596831	#41	1	4	yes	—
74	C₂×#10596831	C₂⁴	1	1	yes	characteristic
75	C₈×D₇₃	—	2	4	no	—
76	C₄⋊Dic₇₃	—	2	1	no	—
77	#11924007	—	2	2	no	—
78	C₂²⋊Dic₇₃	—	2	1	no	—
79	#11924005	—	2	2	no	—
80	C₄×Dic₇₃	—	2	2	no	—
81	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
82	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
83	M₁₆×C₇₃	M₁₆	1	4	yes	—
84	C₂×C₈×C₇₃	—	2	2	no	—
85	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
88	C₄⋊C₄×C₇₃	—	2	1	no	—
89	C₂²⋊C₄×C₇₃	—	2	1	no	—
90	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
91	C₂³×C₇₃	C₂×M₁₆	1	1	yes	characteristic
92	C₂²×D₇₃	—	2	2	no	—
93	C₂²×D₇₃	—	8	1	no	—
94	C₂²×D₇₃	C₂×M₁₆	1	1	yes	characteristic
95	C₂²×D₇₃	C₂×M₁₆	1	1	yes	characteristic
96	C₂×C₄×C₇₃	—	2	2	no	—
97	C₂×C₄×C₇₃	#117	1	2	yes	—
98	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
99	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
100	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
101	C₂×Dic₇₃	—	2	2	no	—
102	C₂×Dic₇₃	—	2	2	no	—
103	C₂×Dic₇₃	C₂×M₁₆	1	1	yes	characteristic
104	C₂×Dic₇₃	C₂×M₁₆	1	1	yes	characteristic
105	C₄×D₇₃	—	2	1	no	—
106	C₄×D₇₃	—	2	1	no	—
107	C₄×D₇₃	—	2	1	no	—
108	C₄×D₇₃	—	2	1	no	—
109	C₄×D₇₃	—	2	2	no	—
110	C₄×D₇₃	—	2	2	no	—
111	C₄×D₇₃	—	8	2	no	—
112	C₈×C₇₃	—	2	1	no	—
113	C₈×C₇₃	—	2	1	no	—
114	C₈×C₇₃	C₂×M₁₆	1	4	yes	—
115	#10596831	—	2	2	no	—
116	#10596831	#117	1	2	yes	—
117	#10596831	C₂×#41	1	2	yes	—
118	C₂²×C₇₃	—	4	1	no	—
119	C₂²×C₇₃	#432	1	2	yes	—
120	C₂²×C₇₃	C₂²×M₁₆	1	1	yes	characteristic
121	C₂×D₇₃	—	2	1	no	—
122	C₂×D₇₃	—	2	1	no	—
123	C₂×D₇₃	—	4	1	no	—
124	C₂×D₇₃	—	4	1	no	—
125	C₂×D₇₃	—	8	2	no	—
126	C₄×C₇₃	—	4	2	no	—
127	C₄×C₇₃	#432	1	2	yes	—
128	C₄×C₇₃	C₂×#117	1	1	yes	characteristic
129	C₄×C₇₃	C₂²×M₁₆	1	1	yes	characteristic
130	Dic₇₃	—	2	1	no	—
131	Dic₇₃	—	2	1	no	—
132	Dic₇₃	—	4	2	no	—
133	#33427	—	73	1	no	maximal, Sylow
134	C₂×C₇₃	—	2	1	no	—
135	C₂×C₇₃	—	2	2	no	—
136	C₂×C₇₃	C₂×#432	1	1	yes	characteristic
137	D₇₃	—	4	1	no	—
138	D₇₃	—	4	1	no	—
139	#2873	—	73	1	no	—
140	#2166	—	73	4	no	—
141	#2165	—	73	4	no	—
142	#2153	—	73	2	no	—
143	C₂×#349	—	73	2	no	—
144	#2110	—	73	2	no	—
145	C₇₃	#33427	1	1	yes	characteristic, complemented, Sylow
146	C₂²×M₁₆	—	73	1	no	—
147	C₂²×M₁₆	—	73	1	no	—
148	C₂×#117	—	73	1	no	—
149	C₂×M₃₂	—	73	4	no	—
150	#405	—	73	1	no	—
151	#405	—	73	1	no	—
152	#405	—	73	1	no	—
153	#405	—	73	2	no	—
154	#405	—	73	2	no	—
155	#405	#64408	1	1	yes	characteristic
156	C₄×M₁₆	—	73	2	no	—
157	C₄×M₁₆	—	73	2	no	—
158	#364	—	73	8	no	—
159	#349	—	73	4	no	—
160	#349	—	73	4	no	—
161	#347	—	73	8	no	—
162	C₂³×C₄	—	73	1	no	—
163	C₂×M₁₆	—	73	1	no	—
164	C₂×M₁₆	—	73	1	no	—
165	C₂×M₁₆	—	73	1	no	—
166	C₂×M₁₆	—	73	1	no	—
167	C₂×M₁₆	—	73	2	no	—
168	C₂×M₁₆	—	73	1	no	—
169	C₂×M₁₆	—	146	4	no	—
170	C₂×M₁₆	—	146	4	no	—
171	C₂×M₁₆	C₂×#64408	1	1	yes	characteristic
172	C₂²×C₈	—	73	1	no	—
173	C₂²×C₈	—	73	2	no	—
174	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
175	#117	—	73	1	no	—
176	#117	—	73	2	no	—
177	#117	—	146	1	no	—
178	#117	—	146	1	no	—
179	#117	C₂×#64408	1	1	yes	characteristic
180	C₂×C₄⋊C₄	—	146	1	no	—
181	C₂×C₂²⋊C₄	—	146	1	no	—
182	C₂×C₄²	—	73	2	no	—
183	M₃₂	—	73	8	no	—
184	M₃₂	—	146	4	no	—
185	C₂×C₁₆	—	146	4	no	—
186	#97	—	73	2	no	—
187	#97	—	73	2	no	—
188	#97	—	73	2	no	—
189	#97	C₂×#64408	1	2	yes	—
190	C₄×C₈	—	73	2	no	—
191	C₄×C₈	—	73	2	no	—
192	C₄×C₈	—	73	2	no	—
193	C₄×C₈	C₂×#64408	1	2	yes	—
194	C₂⁴	—	73	1	no	—
195	C₂²×C₄	—	73	1	no	—
196	C₂²×C₄	—	73	1	no	—
197	C₂²×C₄	—	73	1	no	—
198	C₂²×C₄	—	73	1	no	—
199	C₂²×C₄	—	73	1	no	—
200	C₂²×C₄	—	146	2	no	—
201	C₂²×C₄	—	146	2	no	—
202	C₂²×C₄	—	146	2	no	—
203	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
204	M₁₆	—	146	2	no	—
205	M₁₆	—	146	2	no	—
206	M₁₆	—	146	2	no	—
207	M₁₆	—	146	2	no	—
208	M₁₆	—	146	2	no	—
209	M₁₆	—	146	4	no	—
210	M₁₆	#11924038	1	4	yes	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	73	1	no	—
213	C₂×C₈	—	73	1	no	—
214	C₂×C₈	—	73	4	no	—
215	C₂×C₈	—	73	2	no	—
216	C₂×C₈	—	146	4	no	—
217	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
218	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
219	C₂×C₈	C₂²×#64408	1	2	yes	—
220	C₄⋊C₄	—	2	1	no	—
221	C₄⋊C₄	—	146	2	no	—
222	C₄⋊C₄	—	146	1	no	—
223	C₂²⋊C₄	—	2	1	no	—
224	C₂²⋊C₄	—	146	2	no	—
225	C₂²⋊C₄	—	146	1	no	—
226	C₄²	—	73	2	no	—
227	C₄²	—	146	2	no	—
228	C₄²	C₂²×#64408	1	2	yes	—
229	C₁₆	—	146	8	no	—
230	C₂³	—	584	1	no	—
231	C₂³	—	73	1	no	—
232	C₂³	—	73	1	no	—
233	C₂³	—	146	2	no	—
234	C₂³	D₇₃⋊M₁₆	1	1	yes	characteristic
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	584	2	no	—
237	C₂×C₄	—	73	1	no	—
238	C₂×C₄	—	73	1	no	—
239	C₂×C₄	—	146	2	no	—
240	C₂×C₄	—	146	2	no	—
241	C₂×C₄	—	146	1	no	—
242	C₂×C₄	—	146	1	no	—
243	C₂×C₄	—	146	1	no	—
244	C₂×C₄	—	146	1	no	—
245	C₂×C₄	—	146	2	no	—
246	C₂×C₄	—	146	2	no	—
247	C₂×C₄	#14556670	1	2	yes	—
248	C₂×C₄	D₇₃⋊M₁₆	1	1	yes	characteristic
249	C₂×C₄	C₂×#11924038	1	2	yes	—
250	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
251	C₈	—	2	1	no	—
252	C₈	—	2	1	no	—
253	C₈	—	73	2	no	—
254	C₈	—	73	2	no	—
255	C₈	—	146	2	no	—
256	C₈	C₂×#11924038	1	4	yes	—
257	C₂²	—	4	1	no	—
258	C₂²	—	292	1	no	—
259	C₂²	—	292	1	no	—
260	C₂²	—	584	2	no	—
261	C₂²	—	146	1	no	—
262	C₂²	—	146	1	no	—
263	C₂²	#17989609	1	2	yes	—
264	C₂²	C₂×D₇₃⋊M₁₆	1	1	yes	characteristic
265	C₄	—	4	2	no	—
266	C₄	—	292	2	no	—
267	C₄	—	146	1	no	—
268	C₄	—	146	1	no	—
269	C₄	#17989609	1	2	yes	—
270	C₄	C₂×#14556670	1	1	yes	characteristic
271	C₄	C₂²×#11924038	1	1	yes	characteristic
272	C₂	—	2	1	no	—
273	C₂	—	2	2	no	—
274	C₂	—	292	1	no	—
275	C₂	—	292	1	no	—
276	C₂	C₂×#17989609	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#14556670
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#64408
Fitting	#405×C₇₃	C₄
socle	C₂×C₇₃	C₂×#432
prime radical	C₂³×D₇₃	M₁₆
cosocle	C₂×#10596831	C₂⁴
hypercenter	#405	#64408
hypocenter	C₇₃	#33427
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ #117 ◁ #405

Complements

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

N	H ≅ G/N	type
1. #22626306	C₂	semidirect
2. #22626302	C₂	semidirect
3. #22626272	C₂	semidirect
5. #22626152	C₂	semidirect
8. #17989509	C₂²	semidirect
11. #17989479	C₂²	semidirect
145. C₇₃	#33427	semidirect

Maximal subgroups

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

subgroup	index
#22626306	2
#22626302	2
#22626272	2
C₂×#17989484	2
#22626152	2
#22618347	2
#33427	73