Subgroups — #34030350

Order 18688.

508 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22626346	C₂	1	2	yes	maximal, complemented
2	#22626315	C₂	1	2	yes	maximal, complemented
3	#22626311	C₂	1	2	yes	maximal, complemented
4	#22626311	C₂	1	2	yes	maximal, complemented
5	#22626296	C₂	1	2	yes	maximal, complemented
6	#22626292	C₂	1	1	yes	maximal, characteristic, complemented
7	#22626140	C₂	1	2	yes	maximal, complemented
8	C₄×#14556665	C₂	1	1	yes	maximal, characteristic, complemented
9	#22618907	C₂	1	1	yes	maximal, characteristic
10	C₂×#14556665	C₂²	1	2	yes	—
11	C₂×#14556665	C₂²	1	2	yes	—
12	C₂²×#11924031	C₂²	1	2	yes	—
13	C₄⋊#11924031	C₂²	1	2	yes	—
14	C₄⋊#11924031	C₂²	1	2	yes	—
15	C₄⋊#11924031	C₂²	1	2	yes	—
16	C₂²⋊#11924031	C₂²	1	1	yes	characteristic
17	C₂²⋊#11924031	C₂²	1	2	yes	—
18	C₂²⋊#11924031	C₂²	1	2	yes	—
19	C₂²⋊#11924031	C₂²	1	1	yes	characteristic
20	#17989475	C₂²	1	2	yes	—
21	#17989475	C₂²	1	2	yes	—
22	#17989475	C₂²	1	2	yes	—
23	C₄×#11924031	C₂²	1	2	yes	—
24	C₄×#11924031	C₂²	1	1	yes	characteristic
25	C₄×#11924031	C₂²	1	1	yes	characteristic
26	#17988718	C₄	1	1	yes	characteristic
27	Q₈×#10596831	C₂²	1	1	yes	characteristic
28	#17988665	C₄	1	1	yes	characteristic
29	#17988659	C₄	1	2	yes	—
30	D₄×#10596831	C₂²	1	1	yes	characteristic
31	D₄×#10596831	C₂²	1	2	yes	—
32	#17988648	C₄	1	2	yes	—
33	#17988644	C₄	1	1	yes	characteristic
34	#17988543	C₂²	1	2	yes	—
35	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
36	#529×C₇₃	C₄	1	1	yes	characteristic
37	#14556665	—	2	4	no	—
38	#14556665	—	2	4	no	—
39	C₂×#11924031	—	2	4	no	—
40	C₂×#11924031	C₂³	1	2	yes	—
41	C₂×#11924031	C₂³	1	2	yes	—
42	C₂×#11924031	C₂³	1	2	yes	—
43	C₂×#11924031	C₂³	1	2	yes	—
44	C₂×#11924012	C₂×C₄	1	2	yes	—
45	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
46	C₂²×#10596831	C₂³	1	2	yes	—
47	C₂²×#10596831	C₂³	1	1	yes	characteristic
48	C₄⋊#10596831	C₂×C₄	1	1	yes	characteristic
49	C₄⋊#10596831	C₂×C₄	1	1	yes	characteristic
50	C₄⋊#10596831	C₂×C₄	1	2	yes	—
51	C₄⋊#10596831	C₂×C₄	1	2	yes	—
52	C₂²⋊#10596831	C₂×C₄	1	2	yes	—
53	C₂²⋊#10596831	C₂×C₄	1	2	yes	—
54	C₂²⋊#10596831	C₂×C₄	1	2	yes	—
55	#14556502	C₂×C₄	1	2	yes	—
56	#14556502	C₂×C₄	1	2	yes	—
57	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
58	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
59	C₄×#10596831	C₂³	1	2	yes	—
60	C₄×#10596831	C₂³	1	1	yes	characteristic
61	C₄×#10596831	C₂³	1	1	yes	characteristic
62	#126×C₇₃	C₈	1	2	yes	—
63	#125×C₇₃	C₈	1	1	yes	characteristic
64	#124×C₇₃	C₈	1	1	yes	characteristic
65	#123×C₇₃	C₈	1	2	yes	—
66	C₂²⋊Q₈×C₇₃	C₈	1	1	yes	characteristic
67	#121×C₇₃	C₈	1	1	yes	characteristic
68	C₄×Q₈×C₇₃	C₂×C₄	1	1	yes	characteristic
69	C₄×D₄×C₇₃	C₂×C₄	1	2	yes	—
70	C₄×D₄×C₇₃	C₂×C₄	1	1	yes	characteristic
71	#117×C₇₃	C₂×C₄	1	2	yes	—
72	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
73	#11924031	—	2	2	no	—
74	#11924031	—	2	2	no	—
75	#11924031	—	2	2	no	—
76	#11924031	#41	1	4	yes	—
77	#11924012	—	2	4	no	—
78	C₂×#10596831	—	2	2	no	—
79	C₂×#10596831	—	2	2	no	—
80	C₂×#10596831	—	2	2	no	—
81	C₂×#10596831	—	2	2	no	—
82	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
83	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
84	C₂×#10596831	C₂²×C₄	1	2	yes	—
85	C₂×#10596831	C₂²×C₄	1	2	yes	—
86	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
87	C₂×#10596831	C₂⁴	1	1	yes	characteristic
88	C₂×Q₈×C₇₃	C₂×C₈	1	1	yes	characteristic
89	C₂×D₄×C₇₃	C₂×C₈	1	2	yes	—
90	C₂×D₄×C₇₃	C₂×C₈	1	1	yes	characteristic
91	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
92	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
93	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
95	C₄⋊C₄×C₇₃	C₂×C₈	1	2	yes	—
96	C₄⋊C₄×C₇₃	C₂×C₈	1	2	yes	—
97	C₄⋊C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
98	C₄⋊C₄×C₇₃	C₂×C₈	1	2	yes	—
99	C₄⋊C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
100	C₄⋊C₄×C₇₃	C₂×C₈	1	2	yes	—
101	C₂²⋊C₄×C₇₃	C₂×C₈	1	2	yes	—
102	C₂²⋊C₄×C₇₃	C₂×C₈	1	2	yes	—
103	C₂²⋊C₄×C₇₃	C₂×C₈	1	2	yes	—
104	C₂²⋊C₄×C₇₃	C₂×C₈	1	2	yes	—
105	C₂²⋊C₄×C₇₃	C₂×C₈	1	2	yes	—
106	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
107	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
108	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
109	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
110	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
111	C₂³×C₇₃	C₂²×C₈	1	2	yes	—
112	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
113	Q₈×C₇₃	—	4	1	no	—
114	D₄×C₇₃	—	4	1	no	—
115	D₄×C₇₃	—	4	2	no	—
116	C₂×C₄×C₇₃	—	2	1	no	—
117	C₂×C₄×C₇₃	—	2	2	no	—
118	C₂×C₄×C₇₃	—	2	1	no	—
119	C₂×C₄×C₇₃	—	2	2	no	—
120	C₂×C₄×C₇₃	—	4	1	no	—
121	C₂×C₄×C₇₃	—	4	1	no	—
122	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
123	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
124	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
125	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
126	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
127	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
128	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
129	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
130	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
131	#10596831	—	2	2	no	—
132	#10596831	—	2	1	no	—
133	#10596831	—	2	1	no	—
134	#10596831	—	2	2	no	—
135	#10596831	#131	1	2	yes	—
136	#10596831	C₂×#41	1	2	yes	—
137	C₂²×C₇₃	—	2	1	no	—
138	C₂²×C₇₃	—	2	1	no	—
139	C₂²×C₇₃	—	2	2	no	—
140	C₂²×C₇₃	—	2	2	no	—
141	C₂²×C₇₃	—	2	1	no	—
142	C₂²×C₇₃	—	2	2	no	—
143	C₂²×C₇₃	C₂³×C₈	1	1	yes	characteristic
144	C₄×C₇₃	—	2	1	no	—
145	C₄×C₇₃	—	2	1	no	—
146	C₄×C₇₃	—	2	1	no	—
147	C₄×C₇₃	—	2	1	no	—
148	C₄×C₇₃	—	2	2	no	—
149	C₄×C₇₃	—	2	2	no	—
150	C₄×C₇₃	—	2	1	no	—
151	C₄×C₇₃	—	2	2	no	—
152	C₄×C₇₃	C₄×#41	1	1	yes	characteristic
153	C₄×C₇₃	C₂×#131	1	1	yes	characteristic
154	#33481	—	73	1	no	maximal, Sylow
155	C₂×C₇₃	—	4	1	no	—
156	C₂×C₇₃	—	4	2	no	—
157	C₂×C₇₃	C₈×#41	1	1	yes	characteristic
158	C₂×C₇₃	#3407	1	1	yes	characteristic
159	C₂×C₇₃	#3407	1	1	yes	characteristic
160	#2966	—	73	1	no	—
161	#2183	—	73	2	no	—
162	#2169	—	73	2	no	—
163	#2168	—	73	2	no	—
164	#2168	—	73	2	no	—
165	#2164	—	73	2	no	—
166	#2163	—	73	1	no	—
167	#2108	—	73	2	no	—
168	C₄×M₃₂	—	73	1	no	—
169	C₇₃	#33481	1	1	yes	characteristic, complemented, Sylow
170	#529	#64408	1	1	yes	characteristic
171	C₂×M₃₂	—	73	2	no	—
172	C₂×M₃₂	—	73	2	no	—
173	C₂²×C₁₆	—	73	2	no	—
174	#446	—	73	1	no	—
175	C₈×Q₈	—	73	1	no	—
176	#436	—	73	1	no	—
177	#435	—	73	2	no	—
178	C₈×D₄	—	73	1	no	—
179	C₈×D₄	—	73	2	no	—
180	#433	—	73	2	no	—
181	#432	—	73	1	no	—
182	#405	—	73	2	no	—
183	C₂×C₄×C₈	—	73	1	no	—
184	C₄⋊C₁₆	—	73	2	no	—
185	C₄⋊C₁₆	—	73	2	no	—
186	C₄⋊C₁₆	—	73	2	no	—
187	C₂²⋊C₁₆	—	73	1	no	—
188	C₂²⋊C₁₆	—	73	1	no	—
189	C₂²⋊C₁₆	—	73	2	no	—
190	C₂²⋊C₁₆	—	73	2	no	—
191	#346	—	73	2	no	—
192	#346	—	73	2	no	—
193	#346	—	73	2	no	—
194	C₄×C₁₆	—	73	1	no	—
195	C₄×C₁₆	—	73	1	no	—
196	C₄×C₁₆	—	73	2	no	—
197	C₂×M₁₆	—	73	2	no	—
198	C₂²×C₈	—	73	1	no	—
199	C₂²×C₈	—	73	2	no	—
200	C₂²×C₈	—	73	1	no	—
201	#126	#10596833	1	2	yes	—
202	#125	#10596833	1	1	yes	characteristic
203	#124	#10596833	1	1	yes	characteristic
204	#123	#10596833	1	2	yes	—
205	C₂²⋊Q₈	#10596833	1	1	yes	characteristic
206	#121	#10596833	1	1	yes	characteristic
207	C₄×Q₈	C₂×#64408	1	1	yes	characteristic
208	C₄×D₄	C₂×#64408	1	2	yes	—
209	C₄×D₄	C₂×#64408	1	1	yes	characteristic
210	#117	C₂×#64408	1	2	yes	—
211	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
212	M₃₂	—	146	4	no	—
213	M₃₂	—	146	4	no	—
214	C₂×C₁₆	—	73	2	no	—
215	C₂×C₁₆	—	73	2	no	—
216	C₂×C₁₆	—	73	2	no	—
217	C₂×C₁₆	—	73	2	no	—
218	C₂×C₁₆	—	146	4	no	—
219	C₄⋊C₈	—	73	1	no	—
220	C₄⋊C₈	—	73	1	no	—
221	C₄⋊C₈	—	73	2	no	—
222	C₄⋊C₈	—	73	2	no	—
223	C₂²⋊C₈	—	73	2	no	—
224	C₂²⋊C₈	—	73	2	no	—
225	C₂²⋊C₈	—	73	2	no	—
226	#97	—	73	2	no	—
227	#97	—	73	2	no	—
228	C₄×C₈	—	73	1	no	—
229	C₄×C₈	—	73	1	no	—
230	C₄×C₈	—	73	1	no	—
231	C₄×C₈	—	73	2	no	—
232	C₄×C₈	—	73	1	no	—
233	C₂×Q₈	C₂×#10596833	1	1	yes	characteristic
234	C₂×D₄	C₂×#10596833	1	2	yes	—
235	C₂×D₄	C₂×#10596833	1	1	yes	characteristic
236	C₂²×C₄	C₂×#10596833	1	1	yes	characteristic
237	C₂²×C₄	C₂×#10596833	1	1	yes	characteristic
238	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
239	C₂²×C₄	C₂²×#64408	1	2	yes	—
240	M₁₆	—	146	4	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	2	no	—
246	C₂×C₈	—	73	2	no	—
247	C₂×C₈	—	146	2	no	—
248	C₂×C₈	—	146	2	no	—
249	C₂×C₈	—	146	2	no	—
250	C₂×C₈	—	146	2	no	—
251	C₄⋊C₄	C₂×#10596833	1	2	yes	—
252	C₄⋊C₄	C₂×#10596833	1	2	yes	—
253	C₄⋊C₄	C₂×#10596833	1	1	yes	characteristic
254	C₄⋊C₄	C₂×#10596833	1	2	yes	—
255	C₄⋊C₄	C₂×#10596833	1	1	yes	characteristic
256	C₄⋊C₄	C₂×#10596833	1	2	yes	—
257	C₂²⋊C₄	C₂×#10596833	1	2	yes	—
258	C₂²⋊C₄	C₂×#10596833	1	2	yes	—
259	C₂²⋊C₄	C₂×#10596833	1	2	yes	—
260	C₂²⋊C₄	C₂×#10596833	1	2	yes	—
261	C₂²⋊C₄	C₂×#10596833	1	2	yes	—
262	C₄²	C₂×#10596833	1	1	yes	characteristic
263	C₄²	C₂×#10596833	1	1	yes	characteristic
264	C₄²	C₂²×#64408	1	1	yes	characteristic
265	C₄²	C₂²×#64408	1	1	yes	characteristic
266	C₄²	C₂²×#64408	1	2	yes	—
267	C₁₆	—	73	4	no	—
268	C₁₆	—	146	2	no	—
269	C₁₆	—	146	2	no	—
270	C₁₆	—	146	2	no	—
271	C₂³	C₂²×#10596833	1	2	yes	—
272	C₂³	C₂²×#10596833	1	1	yes	characteristic
273	Q₈	—	4	1	no	—
274	D₄	—	4	1	no	—
275	D₄	—	4	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	4	1	no	—
281	C₂×C₄	—	4	1	no	—
282	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
283	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
284	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
285	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
286	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
287	C₂×C₄	C₂²×#10596833	1	2	yes	—
288	C₂×C₄	C₂²×#10596833	1	2	yes	—
289	C₂×C₄	C₂²×#10596833	1	2	yes	—
290	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
291	C₈	—	73	2	no	—
292	C₈	—	73	2	no	—
293	C₈	—	146	1	no	—
294	C₈	—	146	2	no	—
295	C₈	—	146	1	no	—
296	C₈	—	146	2	no	—
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	2	2	no	—
303	C₂²	C₂³×#10596833	1	1	yes	characteristic
304	C₄	—	2	1	no	—
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	2	2	no	—
309	C₄	—	2	2	no	—
310	C₄	—	2	1	no	—
311	C₄	—	2	2	no	—
312	C₄	#41×#64408	1	1	yes	characteristic
313	C₄	D₇₃⋊#131	1	1	yes	characteristic
314	C₂	—	4	1	no	—
315	C₂	—	4	2	no	—
316	C₂	#41×#10596833	1	1	yes	characteristic
317	C₂	#22626898	1	1	yes	characteristic
318	C₂	#22626898	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#64408
commutator, socle	C₂²×C₇₃	C₂³×C₈
Fitting	#529×C₇₃	C₄
prime radical	#121×C₇₃	C₈
cosocle	C₂×#10596831	C₂⁴
hypercenter	#529	#64408
hypocenter	C₇₃	#33481
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ #529

Complements

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

N	H ≅ G/N	type
1. #22626346	C₂	semidirect
2. #22626315	C₂	semidirect
3. #22626311	C₂	semidirect
4. #22626311	C₂	semidirect
5. #22626296	C₂	semidirect
6. #22626292	C₂	semidirect
7. #22626140	C₂	semidirect
8. C₄×#14556665	C₂	semidirect
169. C₇₃	#33481	semidirect

Maximal subgroups

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

subgroup	index
#22626346	2
#22626315	2
#22626311	2
#22626296	2
#22626292	2
#22626140	2
C₄×#14556665	2
#22618907	2
#33481	73