Subgroups — #34030095

Order 18688.

674 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17989597	C₂	1	2	yes	maximal
2	#22626283	C₂	1	8	yes	maximal, complemented
3	#22626282	C₂	1	4	yes	maximal, complemented
4	C₂×M₃₂×D₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556670	—	2	2	no	—
6	#17989597	—	2	2	no	—
7	#17989597	—	2	2	no	—
8	#17989597	C₂²	1	4	yes	—
9	#17989597	C₂²	1	4	yes	—
10	C₂×#14556660	C₂²	1	4	yes	—
11	C₂×#14556659	—	2	2	no	—
12	#17989514	C₂²	1	8	yes	complemented
13	#17989513	C₂²	1	8	yes	complemented
14	C₂²×C₈×D₇₃	C₂²	1	1	yes	characteristic
15	C₂×#14556531	C₄	1	1	yes	characteristic, complemented
16	C₂×#14556530	C₄	1	2	yes	complemented
17	M₃₂×D₇₃	C₄	1	4	yes	complemented
18	M₃₂×D₇₃	C₂²	1	4	yes	—
19	C₂×C₁₆×D₇₃	C₂²	1	2	yes	—
20	C₂×M₃₂×C₇₃	C₄	1	1	yes	characteristic, complemented
21	#14556670	—	2	4	no	—
22	#14556670	—	2	4	no	—
23	#14556670	—	4	2	no	—
24	#14556670	—	4	2	no	—
25	C₂×C₄⋊#64408	—	2	4	no	—
26	C₂×C₂²⋊#64408	—	4	2	no	—
27	C₂×C₄×#64408	—	4	2	no	—
28	#14556660	—	2	4	no	—
29	#14556660	C₂³	1	8	yes	—
30	#14556659	—	2	4	no	—
31	#14556659	—	2	4	no	—
32	C₂²×C₄×D₇₃	D₄	1	1	yes	characteristic
33	C₂²×#10596831	Q₈	1	1	yes	characteristic
34	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
35	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₈×D₇₃	D₄	1	2	yes	—
37	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
38	C₂×C₈×D₇₃	Q₈	1	2	yes	—
39	C₂×C₈×D₇₃	Q₈	1	1	yes	characteristic
40	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
41	C₂×C₈×D₇₃	C₂³	1	2	yes	—
42	#14556531	C₂×C₄	1	4	yes	complemented
43	#14556530	C₂×C₄	1	8	yes	complemented
44	C₂×#11924003	C₂×C₄	1	2	yes	complemented
45	C₁₆×D₇₃	C₂×C₄	1	4	yes	complemented
46	C₁₆×D₇₃	C₂³	1	4	yes	—
47	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
48	M₃₂×C₇₃	C₂×C₄	1	4	yes	complemented
49	C₂×C₁₆×C₇₃	C₂×C₄	1	2	yes	complemented
50	C₂²×#64408	—	4	4	no	—
51	C₄⋊#64408	—	2	8	no	—
52	C₄⋊#64408	—	4	4	no	—
53	C₂²⋊#64408	—	4	4	no	—
54	C₂²⋊#64408	—	4	4	no	—
55	C₄×#64408	—	4	4	no	—
56	C₄×#64408	—	4	4	no	—
57	C₂³×D₇₃	D₈	1	1	yes	characteristic
58	C₂²×Dic₇₃	Q₁₆	1	1	yes	characteristic
59	C₂×C₄×D₇₃	C₄⋊C₄	1	2	yes	—
60	C₂×C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×D₇₃	D₈	1	2	yes	—
62	C₂×C₄×D₇₃	D₈	1	1	yes	characteristic
63	C₂×C₄×D₇₃	Q₁₆	1	1	yes	characteristic
64	C₂×C₄×D₇₃	Q₁₆	1	2	yes	—
65	C₂×C₄×D₇₃	C₂×D₄	1	2	yes	—
66	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
67	C₂×#10596831	C₄⋊C₄	1	2	yes	—
68	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
69	C₂×#10596831	C₂×Q₈	1	1	yes	characteristic
70	C₂×#10596831	C₂×Q₈	1	2	yes	—
71	C₈×D₇₃	C₄⋊C₄	1	2	yes	—
72	C₈×D₇₃	C₄⋊C₄	1	2	yes	—
73	C₈×D₇₃	C₄⋊C₄	1	2	yes	—
74	C₈×D₇₃	C₂²×C₄	1	1	yes	characteristic
75	C₈×D₇₃	C₂²×C₄	1	2	yes	—
76	C₈×D₇₃	C₂×D₄	1	1	yes	characteristic
77	C₈×D₇₃	C₂×D₄	1	2	yes	—
78	C₈×D₇₃	C₂×Q₈	1	1	yes	characteristic
79	C₈×D₇₃	C₂×Q₈	1	2	yes	—
80	C₈×D₇₃	C₂⁴	1	1	yes	characteristic
81	#11924003	C₂²×C₄	1	4	yes	—
82	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
83	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
84	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
85	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
87	C₁₆×C₇₃	C₂²×C₄	1	4	yes	—
88	C₂³×C₇₃	#107	1	1	yes	characteristic
89	C₂²×D₇₃	—	4	2	no	—
90	C₂²×D₇₃	#107	1	2	yes	—
91	C₂²×D₇₃	#107	1	1	yes	characteristic
92	C₂²×D₇₃	C₂×D₈	1	1	yes	characteristic
93	C₂²×D₇₃	C₂×D₈	1	2	yes	—
94	C₂×#64408	—	4	8	no	—
95	C₂×#64408	—	8	8	no	—
96	C₂×C₄×C₇₃	#107	1	1	yes	characteristic
97	C₂×C₄×C₇₃	#107	1	2	yes	—
98	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
99	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂×Dic₇₃	#107	1	2	yes	—
101	C₂×Dic₇₃	#107	1	1	yes	characteristic
102	C₂×Dic₇₃	C₂×Q₁₆	1	2	yes	—
103	C₂×Dic₇₃	C₂×Q₁₆	1	1	yes	characteristic
104	C₄×D₇₃	#107	1	2	yes	—
105	C₄×D₇₃	#107	1	2	yes	—
106	C₄×D₇₃	#107	1	2	yes	—
107	C₄×D₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₄×D₇₃	C₂×C₄⋊C₄	1	2	yes	—
109	C₄×D₇₃	C₂×D₈	1	1	yes	characteristic
110	C₄×D₇₃	C₂×D₈	1	2	yes	—
111	C₄×D₇₃	C₂×Q₁₆	1	2	yes	—
112	C₄×D₇₃	C₂×Q₁₆	1	1	yes	characteristic
113	C₄×D₇₃	C₂²×D₄	1	1	yes	characteristic
114	C₈×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₈×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
116	C₈×C₇₃	C₂³×C₄	1	1	yes	characteristic
117	#10596831	C₂×C₄⋊C₄	1	1	yes	characteristic
118	#10596831	C₂×C₄⋊C₄	1	2	yes	—
119	#10596831	C₂²×Q₈	1	1	yes	characteristic
120	C₂²×C₇₃	—	4	1	no	—
121	C₂²×C₇₃	C₂×#107	1	1	yes	characteristic
122	C₂²×C₇₃	C₂×#107	1	2	yes	—
123	C₂×D₇₃	—	2	4	no	—
124	C₂×D₇₃	—	2	2	no	—
125	C₂×D₇₃	—	4	1	no	—
126	C₂×D₇₃	—	4	2	no	—
127	C₂×D₇₃	C₂×#107	1	1	yes	characteristic
128	C₂×D₇₃	C₂×#107	1	2	yes	—
129	C₂×D₇₃	C₂²×D₈	1	1	yes	characteristic
130	#64408	—	8	8	no	—
131	C₄×C₇₃	C₂×#107	1	2	yes	—
132	C₄×C₇₃	C₂×#107	1	1	yes	characteristic
133	C₄×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
134	Dic₇₃	C₂×#107	1	1	yes	characteristic
135	Dic₇₃	C₂×#107	1	2	yes	—
136	Dic₇₃	C₂²×Q₁₆	1	1	yes	characteristic
137	#33409	—	73	1	no	maximal, Sylow
138	C₂×C₇₃	—	2	2	no	—
139	C₂×C₇₃	—	2	1	no	—
140	C₂×C₇₃	C₂²×#107	1	1	yes	characteristic
141	D₇₃	—	2	2	no	—
142	D₇₃	—	2	1	no	—
143	D₇₃	#3414	1	2	yes	—
144	C₂²×M₃₂	—	73	1	no	—
145	C₂×#427	—	73	2	no	—
146	#2159	—	73	8	no	—
147	#2158	—	73	4	no	—
148	C₇₃	#33409	1	1	yes	characteristic, complemented, Sylow
149	C₂³×C₈	—	73	1	no	—
150	C₂×#117	—	146	2	no	—
151	C₂×M₃₂	—	73	1	no	—
152	C₂×M₃₂	—	73	4	no	—
153	C₂×M₃₂	—	73	4	no	—
154	C₂×M₃₂	—	73	2	no	—
155	C₂×M₃₂	#64408	1	1	yes	characteristic, complemented
156	C₂²×C₁₆	—	73	2	no	—
157	#427	—	73	4	no	—
158	#427	—	73	4	no	—
159	#427	—	146	2	no	—
160	#427	—	146	2	no	—
161	C₂×#107	—	73	4	no	—
162	C₂×#106	—	146	2	no	—
163	#367	—	73	8	no	—
164	#366	—	73	8	no	—
165	C₂³×C₄	—	73	1	no	—
166	C₂²×C₈	—	73	1	no	—
167	C₂²×C₈	—	73	2	no	—
168	C₂²×C₈	—	73	1	no	—
169	C₂²×C₈	—	73	2	no	—
170	C₂²×C₈	—	73	1	no	—
171	C₂²×C₈	—	73	2	no	—
172	C₂²×C₈	—	73	1	no	—
173	C₂²×C₈	—	73	2	no	—
174	C₂²×C₈	—	73	1	no	—
175	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
176	#117	—	292	2	no	—
177	#117	—	292	2	no	—
178	#117	—	146	4	no	—
179	#117	—	146	4	no	—
180	C₂×C₄⋊C₄	—	146	4	no	—
181	C₂×C₂²⋊C₄	—	292	2	no	—
182	C₂×C₄²	—	292	2	no	—
183	M₃₂	—	73	4	no	—
184	M₃₂	—	73	8	no	—
185	M₃₂	C₂×#64408	1	4	yes	complemented
186	C₂×C₁₆	—	73	2	no	—
187	C₂×C₁₆	—	73	4	no	—
188	C₂×C₁₆	—	73	4	no	—
189	C₂×C₁₆	C₂×#64408	1	2	yes	complemented
190	#107	—	73	8	no	—
191	#107	—	146	4	no	—
192	#106	—	146	4	no	—
193	#106	—	146	4	no	—
194	C₂⁴	—	73	1	no	—
195	C₂²×C₄	—	292	4	no	—
196	C₂²×C₄	—	73	1	no	—
197	C₂²×C₄	—	73	2	no	—
198	C₂²×C₄	—	73	1	no	—
199	C₂²×C₄	—	73	1	no	—
200	C₂²×C₄	—	73	2	no	—
201	C₂²×C₄	—	73	2	no	—
202	C₂²×C₄	—	73	1	no	—
203	C₂²×C₄	—	73	2	no	—
204	C₂²×C₄	—	73	1	no	—
205	C₂²×C₄	C₄⋊#64408	1	1	yes	characteristic
206	C₂×C₈	—	73	2	no	—
207	C₂×C₈	—	73	1	no	—
208	C₂×C₈	—	73	2	no	—
209	C₂×C₈	—	73	1	no	—
210	C₂×C₈	—	73	1	no	—
211	C₂×C₈	—	73	2	no	—
212	C₂×C₈	—	73	1	no	—
213	C₂×C₈	—	73	2	no	—
214	C₂×C₈	—	73	2	no	—
215	C₂×C₈	—	73	2	no	—
216	C₂×C₈	—	73	1	no	—
217	C₂×C₈	—	73	2	no	—
218	C₂×C₈	—	73	2	no	—
219	C₂×C₈	—	73	1	no	—
220	C₂×C₈	C₄⋊#64408	1	2	yes	—
221	C₂×C₈	C₄⋊#64408	1	1	yes	characteristic
222	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
223	C₂×C₈	C₂²×#64408	1	2	yes	—
224	C₄⋊C₄	—	292	4	no	—
225	C₄⋊C₄	—	146	8	no	—
226	C₂²⋊C₄	—	292	4	no	—
227	C₂²⋊C₄	—	292	4	no	—
228	C₄²	—	292	4	no	—
229	C₄²	—	292	4	no	—
230	C₁₆	—	73	4	no	—
231	C₁₆	C₂²×#64408	1	4	yes	—
232	C₂³	—	292	2	no	—
233	C₂³	—	73	2	no	—
234	C₂³	—	73	1	no	—
235	C₂³	—	73	1	no	—
236	C₂³	—	73	2	no	—
237	C₂³	#14556660	1	1	yes	characteristic
238	C₂×C₄	—	292	8	no	—
239	C₂×C₄	—	584	8	no	—
240	C₂×C₄	—	73	2	no	—
241	C₂×C₄	—	73	1	no	—
242	C₂×C₄	—	73	2	no	—
243	C₂×C₄	—	73	1	no	—
244	C₂×C₄	—	73	1	no	—
245	C₂×C₄	—	73	1	no	—
246	C₂×C₄	—	73	2	no	—
247	C₂×C₄	—	73	2	no	—
248	C₂×C₄	—	73	2	no	—
249	C₂×C₄	—	73	2	no	—
250	C₂×C₄	—	73	2	no	—
251	C₂×C₄	—	73	1	no	—
252	C₂×C₄	—	73	2	no	—
253	C₂×C₄	—	73	1	no	—
254	C₂×C₄	#14556660	1	1	yes	characteristic
255	C₂×C₄	#14556660	1	2	yes	—
256	C₂×C₄	C₂×C₄⋊#64408	1	2	yes	—
257	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
258	C₈	—	73	1	no	—
259	C₈	—	73	2	no	—
260	C₈	—	73	1	no	—
261	C₈	C₂×C₄⋊#64408	1	1	yes	characteristic
262	C₈	C₂×C₄⋊#64408	1	2	yes	—
263	C₈	C₂³×#64408	1	1	yes	characteristic
264	C₂²	—	4	1	no	—
265	C₂²	—	292	2	no	—
266	C₂²	—	292	1	no	—
267	C₂²	—	73	1	no	—
268	C₂²	—	73	1	no	—
269	C₂²	—	73	2	no	—
270	C₂²	—	146	4	no	—
271	C₂²	—	146	2	no	—
272	C₂²	C₂×#14556660	1	1	yes	characteristic
273	C₂²	C₂×#14556660	1	2	yes	—
274	C₄	—	584	8	no	—
275	C₄	—	73	1	no	—
276	C₄	—	73	2	no	—
277	C₄	—	73	1	no	—
278	C₄	C₂×#14556660	1	2	yes	—
279	C₄	C₂×#14556660	1	1	yes	characteristic
280	C₄	C₂²×C₄⋊#64408	1	1	yes	characteristic
281	C₂	—	2	2	no	—
282	C₂	—	2	1	no	—
283	C₂	—	73	2	no	—
284	C₂	—	146	2	no	—
285	C₂	—	146	1	no	—
286	C₂	C₂²×#14556660	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#14556660
commutator	C₈×C₇₃	C₂³×C₄
Frattini	C₈	C₂³×#64408
Fitting	C₂×M₃₂×C₇₃	C₄
socle	C₂×C₇₃	C₂²×#107
prime radical	C₂³×D₇₃	D₈
cosocle	C₈×D₇₃	C₂⁴
hypercenter	C₂×M₃₂	#64408
hypocenter	C₇₃	#33409
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

Complements

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

N	H ≅ G/N	type
2. #22626283	C₂	semidirect
3. #22626282	C₂	semidirect
12. #17989514	C₂²	semidirect
13. #17989513	C₂²	semidirect
15. C₂×#14556531	C₄	semidirect
16. C₂×#14556530	C₄	semidirect
17. M₃₂×D₇₃	C₄	semidirect
20. C₂×M₃₂×C₇₃	C₄	semidirect
42. #14556531	C₂×C₄	semidirect
43. #14556530	C₂×C₄	semidirect
44. C₂×#11924003	C₂×C₄	semidirect
45. C₁₆×D₇₃	C₂×C₄	semidirect
48. M₃₂×C₇₃	C₂×C₄	semidirect
49. C₂×C₁₆×C₇₃	C₂×C₄	semidirect
148. C₇₃	#33409	semidirect
155. C₂×M₃₂	#64408	semidirect
185. M₃₂	C₂×#64408	semidirect
189. C₂×C₁₆	C₂×#64408	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17989597	2
#22626283	2
#22626282	2
C₂×M₃₂×D₇₃	2
#33409	73