Subgroups — #34030352

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	#22626348	C₂	1	2	yes	maximal, complemented
2	#22626316	C₂	1	2	yes	maximal, complemented
3	#22626314	C₂	1	2	yes	maximal, complemented
4	#22626312	C₂	1	2	yes	maximal, complemented
5	#22626298	C₂	1	2	yes	maximal, complemented
6	#22626292	C₂	1	1	yes	maximal, characteristic, complemented
7	#22626141	C₂	1	2	yes	maximal, complemented
8	C₄×#14556665	C₂	1	1	yes	maximal, characteristic, complemented
9	#22618900	C₂	1	1	yes	maximal, characteristic
10	C₂×#14556666	C₂²	1	2	yes	—
11	C₂×#14556665	C₂²	1	2	yes	—
12	C₂×#14556664	C₂²	1	2	yes	—
13	#17989508	C₂²	1	2	yes	—
14	#17989508	C₂²	1	2	yes	—
15	C₄⋊#11924031	C₂²	1	2	yes	—
16	#17989482	C₂²	1	2	yes	—
17	#17989482	C₂²	1	2	yes	—
18	C₂²⋊#11924031	C₂²	1	1	yes	characteristic
19	C₂²⋊#11924031	C₂²	1	1	yes	characteristic
20	#17989476	C₂²	1	2	yes	—
21	#17989475	C₂²	1	2	yes	—
22	#17989475	C₂²	1	2	yes	—
23	#17989473	C₂²	1	2	yes	—
24	C₄×#11924031	C₂²	1	1	yes	characteristic
25	C₄×#11924031	C₂²	1	1	yes	characteristic
26	#17989121	C₄	1	1	yes	characteristic
27	#17988714	C₄	1	1	yes	characteristic
28	#17988711	C₂²	1	1	yes	characteristic
29	#17988664	C₄	1	1	yes	characteristic
30	#17988655	C₄	1	2	yes	—
31	#17988652	C₂²	1	1	yes	characteristic
32	#17988650	C₂²	1	2	yes	—
33	#17988649	C₄	1	2	yes	—
34	#17988544	C₂²	1	2	yes	—
35	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
36	#432×C₇₃	C₄	1	1	yes	characteristic
37	#14556666	—	2	4	no	—
38	#14556665	—	2	4	no	—
39	#14556664	—	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×#11924011	C₂×C₄	1	2	yes	—
45	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
46	C₂²×#10596831	C₂³	1	1	yes	characteristic
47	C₂×C₈×D₇₃	C₂³	1	2	yes	—
48	#14556585	C₈	1	2	yes	—
49	#14556583	C₈	1	1	yes	characteristic
50	#14556579	C₈	1	1	yes	characteristic
51	#14556574	C₈	1	2	yes	—
52	#14556570	C₈	1	1	yes	characteristic
53	#14556565	C₈	1	1	yes	characteristic
54	C₄×Dic₁₄₆	C₂×C₄	1	1	yes	characteristic
55	C₄×D₂₉₂	C₂×C₄	1	1	yes	characteristic
56	C₄×C₂²⋊D₇₃	C₂×C₄	1	2	yes	—
57	#14556548	C₂×C₄	1	2	yes	—
58	#14556520	C₂×C₄	1	2	yes	—
59	#14556520	C₂×C₄	1	2	yes	—
60	#14556503	C₂×C₄	1	2	yes	—
61	#14556503	C₂×C₄	1	2	yes	—
62	#14556501	C₂×C₄	1	2	yes	—
63	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
64	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
65	C₄×#10596831	C₂³	1	1	yes	characteristic
66	C₄×#10596831	C₂³	1	1	yes	characteristic
67	C₈×Dic₇₃	C₂³	1	2	yes	—
68	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
69	C₄⋊C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
70	C₄⋊C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
71	C₂²⋊C₈×C₇₃	C₂×C₄	1	2	yes	—
72	#97×C₇₃	C₂×C₄	1	2	yes	—
73	#11924031	—	2	2	no	—
74	#11924031	—	2	2	no	—
75	#11924031	—	2	2	no	—
76	#11924031	#41	1	4	yes	—
77	C₂×Dic₁₄₆	C₂×C₈	1	1	yes	characteristic
78	C₂×D₂₉₂	C₂×C₈	1	1	yes	characteristic
79	C₂×C₂²⋊D₇₃	C₂×C₈	1	2	yes	—
80	C₂×C₄×D₇₃	C₂²×C₄	1	2	yes	—
81	#11924011	—	2	4	no	—
82	C₂×#10596831	—	2	2	no	—
83	C₂×#10596831	—	2	2	no	—
84	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
85	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
86	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
87	C₂×#10596831	C₂⁴	1	1	yes	characteristic
88	C₈×D₇₃	—	2	2	no	—
89	C₈×D₇₃	—	2	2	no	—
90	C₄⋊Dic₇₃	C₂×C₈	1	1	yes	characteristic
91	C₄⋊Dic₇₃	C₂×C₈	1	1	yes	characteristic
92	#11924007	C₂×C₈	1	2	yes	—
93	#11924007	C₂×C₈	1	2	yes	—
94	#11924007	C₂×C₈	1	2	yes	—
95	#11924007	C₂×C₈	1	2	yes	—
96	C₂²⋊Dic₇₃	C₂×C₈	1	2	yes	—
97	#11924005	C₂×C₈	1	2	yes	—
98	#11924005	C₂×C₈	1	2	yes	—
99	#11924005	C₂×C₈	1	2	yes	—
100	#11924005	C₂×C₈	1	2	yes	—
101	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
102	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
103	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
104	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
106	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
107	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
108	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
109	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
110	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
111	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
112	C₂²×D₇₃	C₂²×C₈	1	2	yes	—
113	C₂×C₄×C₇₃	—	2	1	no	—
114	C₂×C₄×C₇₃	—	2	1	no	—
115	C₂×C₄×C₇₃	—	4	1	no	—
116	C₂×C₄×C₇₃	—	4	1	no	—
117	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
118	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
119	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
120	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
121	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
122	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
123	C₂²⋊D₇₃	—	4	2	no	—
124	C₂×Dic₇₃	C₂²×C₈	1	2	yes	—
125	C₂×Dic₇₃	C₂²×C₈	1	2	yes	—
126	C₂×Dic₇₃	C₂²×C₈	1	2	yes	—
127	D₂₉₂	—	4	1	no	—
128	C₄×D₇₃	—	2	2	no	—
129	C₄×D₇₃	—	2	2	no	—
130	Dic₁₄₆	—	4	1	no	—
131	C₈×C₇₃	—	2	2	no	—
132	C₈×C₇₃	—	2	2	no	—
133	#10596831	—	2	1	no	—
134	#10596831	—	2	1	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	1	no	—
140	C₂²×C₇₃	C₂³×C₈	1	1	yes	characteristic
141	C₂×D₇₃	—	2	2	no	—
142	C₂×D₇₃	—	2	2	no	—
143	C₂×D₇₃	—	2	2	no	—
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	1	no	—
149	C₄×C₇₃	C₄×#41	1	1	yes	characteristic
150	C₄×C₇₃	C₂×#131	1	1	yes	characteristic
151	Dic₇₃	—	2	2	no	—
152	Dic₇₃	—	2	2	no	—
153	Dic₇₃	—	2	2	no	—
154	#33481	—	73	1	no	maximal, Sylow
155	C₂×C₇₃	—	4	1	no	—
156	C₂×C₇₃	C₈×#41	1	1	yes	characteristic
157	C₂×C₇₃	#3407	1	1	yes	characteristic
158	C₂×C₇₃	#3407	1	1	yes	characteristic
159	D₇₃	—	4	2	no	—
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	—	73	1	no	—
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	#64408	1	1	yes	characteristic
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	1	no	—
200	C₂²×C₈	—	73	2	no	—
201	#126	—	73	2	no	—
202	#125	—	73	1	no	—
203	#124	—	73	1	no	—
204	#123	—	73	2	no	—
205	C₂²⋊Q₈	—	73	1	no	—
206	#121	—	73	1	no	—
207	C₄×Q₈	—	73	1	no	—
208	C₄×D₄	—	73	2	no	—
209	C₄×D₄	—	73	1	no	—
210	#117	—	73	2	no	—
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	2	no	—
220	C₄⋊C₈	—	73	2	no	—
221	C₄⋊C₈	C₂×#64408	1	1	yes	characteristic
222	C₄⋊C₈	C₂×#64408	1	1	yes	characteristic
223	C₂²⋊C₈	—	73	2	no	—
224	C₂²⋊C₈	—	73	2	no	—
225	C₂²⋊C₈	C₂×#64408	1	2	yes	—
226	#97	—	73	2	no	—
227	#97	C₂×#64408	1	2	yes	—
228	C₄×C₈	—	73	2	no	—
229	C₄×C₈	—	73	1	no	—
230	C₄×C₈	—	73	1	no	—
231	C₄×C₈	—	73	1	no	—
232	C₄×C₈	—	73	1	no	—
233	C₂×Q₈	—	73	1	no	—
234	C₂×D₄	—	73	2	no	—
235	C₂×D₄	—	73	1	no	—
236	C₂²×C₄	—	73	2	no	—
237	C₂²×C₄	#11924036	1	1	yes	characteristic
238	C₂²×C₄	#11924036	1	1	yes	characteristic
239	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
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	—	146	2	no	—
246	C₂×C₈	—	146	2	no	—
247	C₂×C₈	—	146	2	no	—
248	C₂×C₈	—	146	2	no	—
249	C₂×C₈	C₂²×#64408	1	2	yes	—
250	C₂×C₈	C₂²×#64408	1	2	yes	—
251	C₄⋊C₄	—	73	2	no	—
252	C₄⋊C₄	—	73	2	no	—
253	C₄⋊C₄	—	73	2	no	—
254	C₄⋊C₄	—	73	2	no	—
255	C₄⋊C₄	—	73	1	no	—
256	C₄⋊C₄	—	73	1	no	—
257	C₂²⋊C₄	—	73	2	no	—
258	C₂²⋊C₄	—	73	2	no	—
259	C₂²⋊C₄	—	73	2	no	—
260	C₂²⋊C₄	—	73	2	no	—
261	C₂²⋊C₄	—	73	2	no	—
262	C₄²	—	73	2	no	—
263	C₄²	#11924036	1	1	yes	characteristic
264	C₄²	#11924036	1	1	yes	characteristic
265	C₄²	C₂²×#64408	1	1	yes	characteristic
266	C₄²	C₂²×#64408	1	1	yes	characteristic
267	C₁₆	—	73	4	no	—
268	C₁₆	—	146	2	no	—
269	C₁₆	—	146	2	no	—
270	C₁₆	—	146	2	no	—
271	C₂³	—	73	2	no	—
272	C₂³	C₂×#11924036	1	1	yes	characteristic
273	Q₈	—	292	1	no	—
274	D₄	—	292	1	no	—
275	D₄	—	292	2	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	4	1	no	—
279	C₂×C₄	—	4	1	no	—
280	C₂×C₄	—	73	2	no	—
281	C₂×C₄	—	73	2	no	—
282	C₂×C₄	—	73	2	no	—
283	C₂×C₄	—	146	2	no	—
284	C₂×C₄	—	146	2	no	—
285	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
286	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
287	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
288	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
289	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
290	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
291	C₈	—	2	2	no	—
292	C₈	—	2	2	no	—
293	C₈	—	73	2	no	—
294	C₈	—	73	2	no	—
295	C₈	—	146	1	no	—
296	C₈	—	146	1	no	—
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	2	1	no	—
300	C₂²	—	146	2	no	—
301	C₂²	—	146	2	no	—
302	C₂²	—	146	2	no	—
303	C₂²	C₂²×#11924036	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	1	no	—
309	C₄	—	146	2	no	—
310	C₄	—	146	2	no	—
311	C₄	—	146	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	—	292	2	no	—
316	C₂	#22626735	1	1	yes	characteristic
317	C₂	#22626900	1	1	yes	characteristic
318	C₂	#22626900	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#64408
commutator, socle	C₂²×C₇₃	C₂³×C₈
Fitting	#432×C₇₃	C₄
prime radical	#14556565	C₈
cosocle	C₂×#10596831	C₂⁴
hypercenter	#432	#64408
hypocenter	C₇₃	#33481
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ #432

Complements

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

N	H ≅ G/N	type
1. #22626348	C₂	semidirect
2. #22626316	C₂	semidirect
3. #22626314	C₂	semidirect
4. #22626312	C₂	semidirect
5. #22626298	C₂	semidirect
6. #22626292	C₂	semidirect
7. #22626141	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
#22626348	2
#22626316	2
#22626314	2
#22626312	2
#22626298	2
#22626292	2
#22626141	2
C₄×#14556665	2
#22618900	2
#33481	73