Subgroups — #34030251

Order 18688.

648 conjugacy classes of subgroups.

The subgroup lattice has 308 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, complemented
2	#22626323	C₂	1	4	yes	maximal, complemented
3	#22626320	C₂	1	4	yes	maximal, complemented
4	C₂×#17989509	C₂	1	1	yes	maximal, characteristic, complemented
5	#22626175	C₂	1	2	yes	maximal, complemented
6	#22626137	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#17988703	C₂	1	1	yes	maximal, characteristic
8	#17989646	—	2	4	no	—
9	#17989646	C₂²	1	8	yes	complemented
10	#17989643	—	2	4	no	—
11	C₂×#14556665	C₂²	1	2	yes	—
12	C₂×#14556665	C₂²	1	1	yes	characteristic
13	C₂×#14556665	C₂²	1	1	yes	characteristic
14	#17989509	C₂²	1	2	yes	complemented
15	#17989509	C₂²	1	2	yes	complemented
16	#17989486	C₂²	1	8	yes	complemented
17	#17989476	C₂²	1	2	yes	complemented
18	#17989473	C₂²	1	2	yes	complemented
19	C₂×#14556613	C₄	1	1	yes	characteristic
20	C₂×#14556599	C₂²	1	2	yes	—
21	#17988703	C₄	1	2	yes	—
22	#17988703	C₄	1	2	yes	—
23	#17988703	C₂²	1	2	yes	—
24	#17988703	C₂²	1	2	yes	—
25	C₂×#14556517	C₄	1	2	yes	—
26	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
27	C₂×#108×C₇₃	C₄	1	1	yes	characteristic
28	#14556666	—	2	8	no	—
29	#14556665	—	2	2	no	—
30	#14556665	—	2	1	no	—
31	#14556665	—	2	1	no	—
32	#14556665	D₄	1	2	yes	—
33	#14556665	D₄	1	2	yes	—
34	#14556665	C₂³	1	4	yes	—
35	#14556664	—	2	8	no	—
36	C₂×#11924031	—	2	2	no	—
37	C₂×#11924031	C₂³	1	2	yes	—
38	C₂×#11924031	C₂³	1	2	yes	—
39	C₂×#11924028	C₂×C₄	1	2	yes	—
40	#14556613	C₈	1	2	yes	—
41	#14556613	C₈	1	2	yes	—
42	#14556613	C₂×C₄	1	2	yes	—
43	#14556613	C₂×C₄	1	2	yes	—
44	C₂×Dic₂₉₂	C₈	1	1	yes	characteristic
45	C₂×#11924017	C₈	1	2	yes	—
46	C₂×D₅₈₄	C₈	1	1	yes	characteristic
47	#14556599	—	2	2	no	—
48	#14556599	—	2	2	no	—
49	#14556599	C₂×C₄	1	4	yes	—
50	#14556599	C₂³	1	4	yes	—
51	C₂×#11924011	—	2	2	no	—
52	C₂²×#10596831	C₂³	1	1	yes	characteristic
53	C₂×C₈×D₇₃	—	2	2	no	—
54	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
55	#14556517	C₂×C₄	1	4	yes	—
56	#14556517	C₂×C₄	1	4	yes	—
57	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
58	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
59	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
60	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
61	C₂×M₁₆×C₇₃	C₂×C₄	1	2	yes	—
62	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
63	#108×C₇₃	C₂×C₄	1	2	yes	—
64	#108×C₇₃	C₂×C₄	1	2	yes	—
65	#11924031	—	2	4	no	—
66	#11924031	—	2	2	no	—
67	#11924031	C₂×D₄	1	4	yes	—
68	#11924028	—	2	2	no	—
69	#11924028	—	2	2	no	—
70	#11924028	C₂×C₈	1	4	yes	—
71	#11924028	C₂²×C₄	1	4	yes	—
72	C₂×Dic₁₄₆	C₂×C₈	1	2	yes	—
73	C₂×D₂₉₂	C₂×C₈	1	2	yes	—
74	C₂×C₂²⋊D₇₃	—	2	2	no	—
75	C₂²×Dic₇₃	—	2	1	no	—
76	C₂×C₄×D₇₃	—	2	2	no	—
77	Dic₂₉₂	C₂×C₈	1	2	yes	—
78	Dic₂₉₂	C₂×C₈	1	2	yes	—
79	#11924017	C₂×C₈	1	4	yes	—
80	#11924017	C₂×C₈	1	4	yes	—
81	D₅₈₄	C₂×C₈	1	2	yes	—
82	D₅₈₄	C₂×C₈	1	2	yes	—
83	#11924011	—	2	4	no	—
84	#11924011	—	2	4	no	—
85	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
86	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
87	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
88	C₂×#10596831	#41	1	1	yes	characteristic
89	C₂×#10596831	#41	1	1	yes	characteristic
90	C₂×#10596831	C₂⁴	1	1	yes	characteristic
91	C₈×D₇₃	—	2	4	no	—
92	C₈×D₇₃	—	2	4	no	—
93	C₄×Dic₇₃	—	2	1	no	—
94	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
95	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
96	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
97	M₁₆×C₇₃	—	2	2	no	—
98	M₁₆×C₇₃	C₂²×C₄	1	4	yes	—
99	C₂×C₈×C₇₃	—	2	2	no	—
100	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
101	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
102	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
103	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
104	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
106	C₂³×C₇₃	#131	1	1	yes	characteristic
107	C₂²×D₇₃	—	2	2	no	—
108	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
111	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
112	C₂×C₄×C₇₃	#131	1	1	yes	characteristic
113	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
114	C₂²⋊D₇₃	—	2	4	no	—
115	C₂²⋊D₇₃	—	2	4	no	—
116	C₂×Dic₇₃	—	2	1	no	—
117	C₂×Dic₇₃	—	2	2	no	—
118	C₂×Dic₇₃	—	2	1	no	—
119	C₂×Dic₇₃	—	4	2	no	—
120	D₂₉₂	—	2	2	no	—
121	D₂₉₂	C₂²×C₈	1	4	yes	—
122	C₄×D₇₃	—	2	4	no	—
123	C₄×D₇₃	—	2	4	no	—
124	Dic₁₄₆	—	2	2	no	—
125	Dic₁₄₆	C₂²×C₈	1	4	yes	—
126	C₈×C₇₃	—	2	4	no	—
127	C₈×C₇₃	C₂²×C₈	1	1	yes	characteristic
128	C₈×C₇₃	C₂²×C₈	1	1	yes	characteristic
129	C₈×C₇₃	C₂²×C₈	1	1	yes	characteristic
130	C₈×C₇₃	C₂²×C₈	1	1	yes	characteristic
131	#10596831	C₄×D₄	1	1	yes	characteristic
132	#10596831	C₄×D₄	1	1	yes	characteristic
133	#10596831	C₂²×D₄	1	1	yes	characteristic
134	#10596831	C₂×#41	1	1	yes	characteristic
135	C₂²×C₇₃	—	4	1	no	—
136	C₂²×C₇₃	C₈×D₄	1	1	yes	characteristic
137	C₂²×C₇₃	C₈×D₄	1	1	yes	characteristic
138	C₂²×C₇₃	C₂×#131	1	1	yes	characteristic
139	C₂×D₇₃	—	2	4	no	—
140	C₂×D₇₃	—	8	2	no	—
141	C₄×C₇₃	C₈×D₄	1	1	yes	characteristic
142	C₄×C₇₃	C₈×D₄	1	1	yes	characteristic
143	C₄×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
144	C₄×C₇₃	C₂³×C₈	1	1	yes	characteristic
145	Dic₇₃	—	2	2	no	—
146	Dic₇₃	—	2	4	no	—
147	Dic₇₃	—	4	1	no	—
148	#33443	—	73	1	no	maximal, Sylow
149	C₂×C₇₃	—	2	1	no	—
150	C₂×C₇₃	—	2	1	no	—
151	C₂×C₇₃	—	2	1	no	—
152	C₂×C₇₃	C₂×C₈×D₄	1	1	yes	characteristic
153	D₇₃	—	4	4	no	—
154	#3407	—	73	2	no	—
155	C₂×#443	—	73	1	no	—
156	#2171	—	73	4	no	—
157	#2170	—	73	4	no	—
158	C₂×#364	—	73	1	no	—
159	#2117	—	73	2	no	—
160	C₄×M₃₂	—	73	1	no	—
161	C₇₃	#33443	1	1	yes	characteristic, complemented, Sylow
162	C₂×#135	—	73	1	no	—
163	C₂×#131	—	73	2	no	—
164	#504	—	73	8	no	—
165	#504	—	146	4	no	—
166	C₂×M₃₂	—	73	1	no	—
167	C₂×M₃₂	—	73	1	no	—
168	C₂×M₃₂	—	73	2	no	—
169	C₂×M₃₂	—	146	4	no	—
170	#443	—	73	2	no	—
171	#443	—	73	2	no	—
172	#443	—	73	2	no	—
173	#443	—	73	2	no	—
174	C₂×#108	#64408	1	1	yes	characteristic
175	C₂×#104	—	73	2	no	—
176	C₂×C₄×C₈	—	73	1	no	—
177	#364	—	73	2	no	—
178	#364	—	73	2	no	—
179	#350	—	73	8	no	—
180	#346	—	73	2	no	—
181	C₄×C₁₆	—	73	2	no	—
182	C₂×#41	—	73	2	no	—
183	#135	—	73	2	no	—
184	#135	—	73	2	no	—
185	#135	—	73	2	no	—
186	#135	—	73	2	no	—
187	C₂×Q₁₆	—	73	1	no	—
188	C₂×SD₁₆	—	73	2	no	—
189	C₂×D₈	—	73	1	no	—
190	#131	—	73	4	no	—
191	#131	—	73	4	no	—
192	#131	—	146	2	no	—
193	#131	—	146	2	no	—
194	C₂×M₁₆	—	146	2	no	—
195	C₂×M₁₆	C₂×#64408	1	2	yes	—
196	C₂²×C₈	—	73	1	no	—
197	C₂²×C₈	—	146	2	no	—
198	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
199	C₂×C₄²	—	73	1	no	—
200	M₃₂	—	73	2	no	—
201	M₃₂	—	73	2	no	—
202	M₃₂	—	73	4	no	—
203	M₃₂	—	146	8	no	—
204	M₃₂	—	146	2	no	—
205	M₃₂	—	146	1	no	—
206	M₃₂	—	146	1	no	—
207	C₂×C₁₆	—	73	2	no	—
208	C₂×C₁₆	—	73	2	no	—
209	C₂×C₁₆	—	146	8	no	—
210	C₂×C₁₆	—	146	2	no	—
211	#108	C₂×#64408	1	2	yes	—
212	#108	C₂×#64408	1	2	yes	—
213	#104	—	73	4	no	—
214	#104	—	73	4	no	—
215	C₄×C₈	—	73	1	no	—
216	C₄×C₈	—	73	1	no	—
217	C₄×C₈	—	73	1	no	—
218	C₄×C₈	—	73	1	no	—
219	#41	—	73	4	no	—
220	#41	—	73	4	no	—
221	#41	—	146	2	no	—
222	#41	—	146	2	no	—
223	C₂×Q₈	—	73	2	no	—
224	C₂×D₄	—	73	2	no	—
225	C₂×D₄	—	146	2	no	—
226	C₂²×C₄	—	146	1	no	—
227	C₂²×C₄	—	146	2	no	—
228	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
229	Q₁₆	—	73	2	no	—
230	Q₁₆	—	73	2	no	—
231	SD₁₆	—	73	4	no	—
232	SD₁₆	—	73	4	no	—
233	D₈	—	73	2	no	—
234	D₈	—	73	2	no	—
235	M₁₆	—	2	2	no	—
236	M₁₆	—	146	4	no	—
237	M₁₆	—	146	4	no	—
238	M₁₆	C₂²×#64408	1	4	yes	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	73	1	no	—
241	C₂×C₈	—	73	1	no	—
242	C₂×C₈	—	73	1	no	—
243	C₂×C₈	—	73	1	no	—
244	C₂×C₈	—	73	1	no	—
245	C₂×C₈	—	73	1	no	—
246	C₂×C₈	—	146	4	no	—
247	C₂×C₈	—	146	4	no	—
248	C₂×C₈	#11924036	1	1	yes	characteristic
249	C₂×C₈	#11924036	1	1	yes	characteristic
250	C₂×C₈	#11924036	1	1	yes	characteristic
251	C₂×C₈	#11924036	1	1	yes	characteristic
252	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
253	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
254	C₄²	—	73	1	no	—
255	C₄²	—	73	1	no	—
256	C₄²	—	146	1	no	—
257	C₁₆	—	73	4	no	—
258	C₁₆	—	146	2	no	—
259	C₁₆	—	146	4	no	—
260	C₂³	—	146	2	no	—
261	C₂³	#14556679	1	1	yes	characteristic
262	Q₈	—	73	4	no	—
263	Q₈	—	146	2	no	—
264	D₄	—	73	4	no	—
265	D₄	—	146	4	no	—
266	D₄	—	146	4	no	—
267	D₄	—	146	2	no	—
268	C₂×C₄	—	292	2	no	—
269	C₂×C₄	—	146	1	no	—
270	C₂×C₄	—	146	1	no	—
271	C₂×C₄	—	146	2	no	—
272	C₂×C₄	—	146	4	no	—
273	C₂×C₄	—	146	4	no	—
274	C₂×C₄	D₄×#64408	1	1	yes	characteristic
275	C₂×C₄	D₄×#64408	1	1	yes	characteristic
276	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
277	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
278	C₂×C₄	#14556679	1	1	yes	characteristic
279	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
280	C₈	—	2	4	no	—
281	C₈	—	73	1	no	—
282	C₈	—	73	1	no	—
283	C₈	—	73	1	no	—
284	C₈	—	73	1	no	—
285	C₈	C₂×#11924036	1	1	yes	characteristic
286	C₈	C₂×#11924036	1	1	yes	characteristic
287	C₈	C₂×#11924036	1	1	yes	characteristic
288	C₈	C₂×#11924036	1	1	yes	characteristic
289	C₂²	—	4	1	no	—
290	C₂²	—	584	2	no	—
291	C₂²	—	146	4	no	—
292	C₂²	#17989615	1	1	yes	characteristic
293	C₂²	#17989615	1	1	yes	characteristic
294	C₂²	C₂×#14556679	1	1	yes	characteristic
295	C₄	—	292	1	no	—
296	C₄	—	146	2	no	—
297	C₄	—	146	4	no	—
298	C₄	#17989615	1	1	yes	characteristic
299	C₄	#17989615	1	1	yes	characteristic
300	C₄	C₂×D₄×#64408	1	1	yes	characteristic
301	C₄	C₂²×#11924036	1	1	yes	characteristic
302	C₂	—	2	1	no	—
303	C₂	—	2	1	no	—
304	C₂	—	2	1	no	—
305	C₂	—	292	4	no	—
306	C₂	C₂×#17989615	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄×#64408
commutator	C₄×C₇₃	C₂³×C₈
Frattini	C₂×C₄	C₂³×#64408
Fitting	C₂×#108×C₇₃	C₄
socle	C₂×C₇₃	C₂×C₈×D₄
prime radical	C₂×#14556613	C₄
cosocle	C₂×#10596831	C₂⁴
hypercenter	C₂×#108	#64408
hypocenter	C₇₃	#33443
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#108

Complements

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

N	H ≅ G/N	type
1. #22626900	C₂	semidirect
2. #22626323	C₂	semidirect
3. #22626320	C₂	semidirect
4. C₂×#17989509	C₂	semidirect
5. #22626175	C₂	semidirect
6. #22626137	C₂	semidirect
9. #17989646	C₂²	semidirect
14. #17989509	C₂²	semidirect
15. #17989509	C₂²	semidirect
16. #17989486	C₂²	semidirect
17. #17989476	C₂²	semidirect
18. #17989473	C₂²	semidirect
161. C₇₃	#33443	semidirect

Maximal subgroups

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

subgroup	index
#22626900	2
#22626323	2
#22626320	2
C₂×#17989509	2
#22626175	2
#22626137	2
C₂×#17988703	2
#33443	73