Subgroups — #34015659

Order 18688.

890 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22626735	C₂	1	2	yes	maximal
2	#22625135	C₂	1	2	yes	maximal, complemented
3	#22625134	C₂	1	1	yes	maximal, characteristic, complemented
4	#22625129	C₂	1	4	yes	maximal, complemented
5	M₁₆×#10596833	C₂	1	2	yes	maximal, complemented
6	#22625125	C₂	1	2	yes	maximal, complemented
7	C₂×#17989428	C₂	1	1	yes	maximal, characteristic, complemented
8	#22618353	C₂	1	1	yes	maximal, characteristic
9	#17989636	C₄	1	4	yes	—
10	#17989615	C₄	1	4	yes	—
11	#17989614	C₄	1	8	yes	—
12	#17989605	C₂²	1	4	yes	—
13	#17989603	C₂²	1	2	yes	—
14	C₄×#11924036	C₂²	1	4	yes	—
15	C₂×C₄×#10596833	C₂²	1	2	yes	—
16	#17989428	C₂²	1	1	yes	characteristic
17	#17989428	C₂²	1	2	yes	—
18	#17989428	C₂²	1	1	yes	characteristic
19	#17989428	C₂²	1	4	yes	—
20	#17989428	C₂²	1	2	yes	—
21	C₈×#10596833	C₂²	1	4	yes	—
22	C₈×#10596833	C₂²	1	2	yes	—
23	C₂×#14556599	C₄	1	1	yes	characteristic
24	C₂×#14556597	C₄	1	1	yes	characteristic
25	#17989055	C₂²	1	1	yes	characteristic
26	#17988544	C₄	1	2	yes	—
27	#17988543	C₄	1	1	yes	characteristic
28	#17988541	C₄	1	2	yes	—
29	#17988539	C₂²	1	2	yes	—
30	M₁₆×Dic₇₃	C₂²	1	1	yes	characteristic
31	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
32	C₄×C₈×D₇₃	C₂²	1	2	yes	—
33	#405×C₇₃	C₄	1	1	yes	characteristic
34	C₂×#11924036	C₂×C₄	1	8	yes	—
35	C₂²×#10596833	C₂×C₄	1	4	yes	—
36	#14556658	C₂×C₄	1	8	yes	—
37	C₄⋊#10596833	C₂×C₄	1	4	yes	—
38	#14556648	C₂×C₄	1	8	yes	—
39	C₂²⋊#10596833	C₂×C₄	1	4	yes	—
40	C₄×#10596833	C₂×C₄	1	4	yes	—
41	C₄×#10596833	C₂×C₄	1	8	yes	—
42	C₄×#10596833	C₂³	1	2	yes	—
43	C₄×#10596833	C₂³	1	4	yes	—
44	C₄×#10596833	C₂³	1	2	yes	—
45	C₂×#11924028	C₂×C₄	1	1	yes	characteristic
46	#14556599	C₈	1	8	yes	complemented
47	#14556597	C₈	1	8	yes	complemented
48	C₂×#11924012	C₂×C₄	1	1	yes	characteristic
49	C₂×#11924011	C₂×C₄	1	2	yes	—
50	C₂×#11924011	C₂×C₄	1	2	yes	—
51	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
52	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
53	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
54	#14556556	C₂×C₄	1	2	yes	—
55	#14556553	C₂×C₄	1	2	yes	—
56	#14556551	C₂×C₄	1	4	yes	—
57	#14556548	C₂×C₄	1	2	yes	—
58	C₂×C₄×Dic₇₃	C₂³	1	1	yes	characteristic
59	C₄²×D₇₃	C₂³	1	2	yes	—
60	#14556502	C₂×C₄	1	2	yes	—
61	#14556501	C₂×C₄	1	2	yes	—
62	C₄×#10596831	C₂×C₄	1	2	yes	—
63	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
64	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
65	C₈×Dic₇₃	C₂³	1	2	yes	—
66	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
67	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
68	C₂×M₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
69	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
70	#117×C₇₃	C₂×C₄	1	1	yes	characteristic
71	#97×C₇₃	C₂×C₄	1	2	yes	—
72	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
73	#11924036	—	4	8	no	—
74	C₂×#10596833	—	4	4	no	—
75	C₂×#10596833	C₂²×C₄	1	4	yes	—
76	C₂×#10596833	C₂²×C₄	1	4	yes	—
77	C₂×#10596833	C₂²×C₄	1	8	yes	—
78	#11924028	C₂×C₈	1	4	yes	—
79	#11924028	C₂×C₈	1	4	yes	—
80	C₂×Dic₁₄₆	C₄²	1	1	yes	characteristic
81	C₂×D₂₉₂	C₄²	1	1	yes	characteristic
82	C₂×C₂²⋊D₇₃	C₄²	1	2	yes	—
83	C₂²×Dic₇₃	C₂²×C₄	1	2	yes	—
84	C₂×C₄×D₇₃	C₂²×C₄	1	4	yes	—
85	C₂×C₄×D₇₃	C₂²×C₄	1	2	yes	—
86	#11924012	C₂×C₈	1	4	yes	complemented
87	#11924011	C₂×C₈	1	8	yes	complemented
88	#11924011	C₂×C₈	1	8	yes	complemented
89	C₂×#10596831	C₂×C₈	1	4	yes	complemented
90	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
91	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
92	C₂×#10596831	C₂²×C₄	1	2	yes	—
93	C₈×D₇₃	C₂×C₈	1	8	yes	complemented
94	C₈×D₇₃	C₂×C₈	1	8	yes	complemented
95	#11924007	C₄²	1	4	yes	—
96	#11924005	C₄²	1	4	yes	—
97	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
98	C₄×Dic₇₃	C₂²×C₄	1	4	yes	—
99	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
100	C₄×Dic₇₃	C₂⁴	1	1	yes	characteristic
101	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
102	M₁₆×C₇₃	C₂×C₈	1	4	yes	complemented
103	C₂×C₈×C₇₃	C₂×C₈	1	4	yes	complemented
104	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
106	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
107	C₄⋊C₄×C₇₃	C₄²	1	2	yes	—
108	C₂²⋊C₄×C₇₃	C₄²	1	2	yes	—
109	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
110	C₂³×C₇₃	C₂×C₄²	1	1	yes	characteristic
111	C₂²×D₇₃	C₂×C₄²	1	2	yes	—
112	C₂×C₄×C₇₃	C₂×C₄²	1	4	yes	—
113	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
114	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
115	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
116	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
117	C₂²⋊D₇₃	C₄×C₈	1	8	yes	—
118	C₂×Dic₇₃	—	2	4	no	—
119	C₂×Dic₇₃	C₂×C₄²	1	2	yes	—
120	C₂×Dic₇₃	C₂×C₄²	1	2	yes	—
121	C₂×Dic₇₃	C₂³×C₄	1	2	yes	—
122	D₂₉₂	C₄×C₈	1	4	yes	—
123	C₄×D₇₃	—	2	8	no	—
124	C₄×D₇₃	C₂²×C₈	1	4	yes	—
125	C₄×D₇₃	C₂²×C₈	1	4	yes	—
126	Dic₁₄₆	C₄×C₈	1	4	yes	—
127	#10596833	—	2	4	no	—
128	#10596833	—	2	4	no	—
129	#10596833	—	2	8	no	—
130	C₈×C₇₃	C₂²×C₈	1	2	yes	—
131	C₈×C₇₃	C₂²×C₈	1	2	yes	—
132	C₈×C₇₃	C₂²×C₈	1	4	yes	—
133	#10596831	C₂²×C₈	1	2	yes	—
134	#10596831	C₂²×C₈	1	4	yes	—
135	#10596831	C₂²×C₈	1	2	yes	—
136	C₂²×C₇₃	—	2	2	no	—
137	C₂²×C₇₃	C₂×C₄×C₈	1	2	yes	—
138	C₂²×C₇₃	C₂²×C₄²	1	1	yes	characteristic
139	C₂×D₇₃	—	2	4	no	—
140	C₂×D₇₃	C₂×C₄×C₈	1	4	yes	—
141	C₄×C₇₃	—	2	4	no	—
142	C₄×C₇₃	C₂×C₄×C₈	1	2	yes	—
143	C₄×C₇₃	C₂³×C₈	1	1	yes	characteristic
144	C₄×C₇₃	C₂³×C₈	1	1	yes	characteristic
145	Dic₇₃	—	2	2	no	—
146	Dic₇₃	C₂×C₄×C₈	1	4	yes	—
147	Dic₇₃	#568	1	4	yes	—
148	#17316	—	73	1	no	maximal, Sylow
149	C₂×C₇₃	—	2	2	no	—
150	C₂×C₇₃	C₂²×C₄×C₈	1	1	yes	characteristic
151	C₂×C₇₃	#2875	1	2	yes	—
152	D₇₃	—	2	4	no	—
153	C₈×#41	—	73	2	no	—
154	C₄×#131	—	73	1	no	—
155	#1450	—	73	1	no	—
156	#1450	—	73	2	no	—
157	C₈×M₁₆	—	73	2	no	—
158	C₈×M₁₆	—	73	4	no	—
159	C₂×#322	—	73	1	no	—
160	C₂×#322	—	73	2	no	—
161	C₇₃	#17316	1	1	yes	characteristic, complemented, Sylow
162	C₂×#131	—	73	1	no	—
163	C₂×#131	—	73	1	no	—
164	C₄×#41	—	73	1	no	—
165	C₈×Q₈	—	73	4	no	—
166	C₈×D₄	—	73	8	no	—
167	C₈×D₄	—	73	4	no	—
168	#431	—	73	2	no	—
169	#431	—	73	4	no	—
170	#405	—	73	1	no	—
171	#405	—	73	2	no	—
172	#405	—	73	2	no	—
173	#405	#64408	1	1	yes	characteristic
174	C₄×M₁₆	—	73	1	no	—
175	C₄×M₁₆	—	73	2	no	—
176	C₂×C₄×C₈	—	73	2	no	—
177	C₂×C₄×C₈	—	73	4	no	—
178	C₂×C₄×C₈	—	73	1	no	—
179	C₂×C₄×C₈	—	73	2	no	—
180	#322	—	73	1	no	—
181	#322	—	73	2	no	—
182	#322	—	73	4	no	—
183	#322	—	73	1	no	—
184	#322	—	73	2	no	—
185	C₈²	—	73	4	no	—
186	C₈²	—	73	2	no	—
187	C₂×#41	—	73	1	no	—
188	#131	—	73	8	no	—
189	#131	—	73	8	no	—
190	C₂×M₁₆	—	73	1	no	—
191	C₂×M₁₆	—	73	2	no	—
192	C₂×M₁₆	—	73	2	no	—
193	C₂×M₁₆	C₂×#64408	1	1	yes	characteristic
194	C₂²×C₈	—	73	4	no	—
195	C₂²×C₈	—	73	8	no	—
196	C₂²×C₈	—	73	1	no	—
197	C₂²×C₈	—	73	2	no	—
198	C₂²×C₈	—	73	2	no	—
199	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
200	C₄×Q₈	—	73	2	no	—
201	C₄×D₄	—	73	2	no	—
202	C₄×D₄	—	73	4	no	—
203	#117	—	73	2	no	—
204	#117	C₂×#64408	1	1	yes	characteristic
205	C₂×C₄²	—	73	1	no	—
206	C₂×C₄²	—	73	2	no	—
207	C₄⋊C₈	—	73	4	no	—
208	C₄⋊C₈	—	73	8	no	—
209	C₂²⋊C₈	—	73	4	no	—
210	C₂²⋊C₈	—	73	8	no	—
211	#97	—	73	2	no	—
212	#97	—	73	2	no	—
213	#97	C₂×#64408	1	2	yes	—
214	C₄×C₈	—	73	2	no	—
215	C₄×C₈	—	73	4	no	—
216	C₄×C₈	—	73	2	no	—
217	C₄×C₈	—	73	4	no	—
218	C₄×C₈	—	73	8	no	—
219	C₄×C₈	—	73	1	no	—
220	C₄×C₈	—	73	2	no	—
221	C₄×C₈	—	73	1	no	—
222	C₄×C₈	—	73	1	no	—
223	C₄×C₈	—	73	1	no	—
224	C₄×C₈	—	73	2	no	—
225	C₄×C₈	C₂×#64408	1	2	yes	—
226	#41	—	73	4	no	—
227	#41	—	73	4	no	—
228	C₂×Q₈	—	73	1	no	—
229	C₂×D₄	—	73	2	no	—
230	C₂×D₄	—	73	1	no	—
231	C₂²×C₄	—	73	2	no	—
232	C₂²×C₄	—	73	2	no	—
233	C₂²×C₄	—	73	4	no	—
234	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
235	M₁₆	—	73	4	no	—
236	M₁₆	—	73	8	no	—
237	M₁₆	—	73	8	no	—
238	M₁₆	#11924036	1	4	yes	complemented
239	C₂×C₈	—	292	4	no	—
240	C₂×C₈	—	292	8	no	—
241	C₂×C₈	—	73	4	no	—
242	C₂×C₈	—	73	1	no	—
243	C₂×C₈	—	73	1	no	—
244	C₂×C₈	—	73	2	no	—
245	C₂×C₈	—	73	8	no	—
246	C₂×C₈	—	73	8	no	—
247	C₂×C₈	—	73	4	no	—
248	C₂×C₈	—	73	4	no	—
249	C₂×C₈	—	73	8	no	—
250	C₂×C₈	#11924036	1	4	yes	complemented
251	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
252	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
253	C₂×C₈	C₂²×#64408	1	2	yes	—
254	C₄⋊C₄	—	73	4	no	—
255	C₄⋊C₄	C₄×#64408	1	2	yes	—
256	C₂²⋊C₄	—	73	4	no	—
257	C₂²⋊C₄	C₄×#64408	1	2	yes	—
258	C₄²	—	73	1	no	—
259	C₄²	—	73	1	no	—
260	C₄²	—	73	4	no	—
261	C₄²	—	73	2	no	—
262	C₄²	C₂²×#64408	1	2	yes	—
263	C₂³	—	73	2	no	—
264	C₂³	C₂×C₄×#64408	1	1	yes	characteristic
265	Q₈	—	73	4	no	—
266	D₄	—	73	8	no	—
267	D₄	—	73	4	no	—
268	C₂×C₄	—	73	2	no	—
269	C₂×C₄	—	73	2	no	—
270	C₂×C₄	—	73	4	no	—
271	C₂×C₄	—	73	4	no	—
272	C₂×C₄	—	73	2	no	—
273	C₂×C₄	—	146	8	no	—
274	C₂×C₄	—	146	4	no	—
275	C₂×C₄	C₂×C₄×#64408	1	4	yes	—
276	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
277	C₂×C₄	C₂×#11924036	1	2	yes	—
278	C₂×C₄	C₂×#11924036	1	2	yes	—
279	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
280	C₈	—	73	2	no	—
281	C₈	—	73	2	no	—
282	C₈	—	73	4	no	—
283	C₈	—	146	4	no	—
284	C₈	—	146	4	no	—
285	C₈	—	146	8	no	—
286	C₈	C₂×#11924036	1	2	yes	—
287	C₈	C₂×#11924036	1	4	yes	—
288	C₈	C₂×#11924036	1	2	yes	—
289	C₂²	—	2	2	no	—
290	C₂²	—	73	4	no	—
291	C₂²	—	146	4	no	—
292	C₂²	C₄×#11924036	1	2	yes	—
293	C₂²	C₂²×C₄×#64408	1	1	yes	characteristic
294	C₄	—	2	4	no	—
295	C₄	—	73	4	no	—
296	C₄	—	73	4	no	—
297	C₄	—	146	2	no	—
298	C₄	C₄×#11924036	1	2	yes	—
299	C₄	C₂²×#11924036	1	1	yes	characteristic
300	C₄	C₂²×#11924036	1	1	yes	characteristic
301	C₂	—	2	2	no	—
302	C₂	—	146	4	no	—
303	C₂	C₂×C₄×#11924036	1	1	yes	characteristic
304	C₂	#22626501	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#64408
commutator	C₂×C₇₃	C₂²×C₄×C₈
Fitting	#405×C₇₃	C₄
socle	C₂²×C₇₃	C₂²×C₄²
prime radical	C₂×#11924028	C₂×C₄
cosocle	C₄×Dic₇₃	C₂⁴
hypercenter	#405	#64408
hypocenter	C₇₃	#17316
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ #405

Complements

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

N	H ≅ G/N	type
2. #22625135	C₂	semidirect
3. #22625134	C₂	semidirect
4. #22625129	C₂	semidirect
5. M₁₆×#10596833	C₂	semidirect
6. #22625125	C₂	semidirect
7. C₂×#17989428	C₂	semidirect
46. #14556599	C₈	semidirect
47. #14556597	C₈	semidirect
86. #11924012	C₂×C₈	semidirect
87. #11924011	C₂×C₈	semidirect
88. #11924011	C₂×C₈	semidirect
89. C₂×#10596831	C₂×C₈	semidirect
93. C₈×D₇₃	C₂×C₈	semidirect
94. C₈×D₇₃	C₂×C₈	semidirect
102. M₁₆×C₇₃	C₂×C₈	semidirect
103. C₂×C₈×C₇₃	C₂×C₈	semidirect
161. C₇₃	#17316	semidirect
238. M₁₆	#11924036	semidirect
250. C₂×C₈	#11924036	semidirect

Maximal subgroups

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

subgroup	index
#22626735	2
#22625135	2
#22625134	2
#22625129	2
M₁₆×#10596833	2
#22625125	2
C₂×#17989428	2
#22618353	2
#17316	73