Subgroups — #33243714

Order 18688.

508 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22622920	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×D₅₈₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#22612168	C₂	1	2	yes	maximal, complemented
4	#22612159	C₂	1	1	yes	maximal, characteristic, complemented
5	#22612084	C₂	1	2	yes	maximal, complemented
6	#22612069	C₂	1	1	yes	maximal, characteristic, complemented
7	#22611987	C₂	1	2	yes	maximal
8	#22611977	C₂	1	1	yes	maximal, characteristic, complemented
9	#22611797	C₂	1	2	yes	maximal, complemented
10	C₄×D₁₁₆₈	C₂	1	1	yes	maximal, characteristic
11	C₇₃×#2183	C₂	1	1	yes	maximal, characteristic, complemented
12	#17989310	—	2	1	no	—
13	Q₈×D₂₉₂	—	2	1	no	—
14	C₂×#14556536	C₂²	1	2	yes	—
15	C₂×D₁₁₆₈	C₂²	1	1	yes	characteristic
16	#17988992	C₂²	1	1	yes	characteristic
17	#17988992	C₂²	1	2	yes	—
18	#17988971	C₂²	1	2	yes	—
19	#17988967	C₂²	1	1	yes	characteristic
20	#17988868	—	2	1	no	—
21	#17988868	—	2	2	no	—
22	#17988810	—	2	2	no	—
23	#17988793	—	2	1	no	—
24	C₄×Dic₂₉₂	C₂²	1	2	yes	—
25	C₄×D₅₈₄	C₂²	1	1	yes	characteristic
26	C₄×D₅₈₄	C₂²	1	2	yes	—
27	C₄×D₅₈₄	C₂²	1	1	yes	characteristic
28	#17988413	C₂²	1	1	yes	characteristic
29	#17988413	C₂²	1	2	yes	—
30	#17988392	C₂²	1	2	yes	—
31	#17988392	C₂²	1	2	yes	—
32	#17988392	C₂²	1	2	yes	—
33	#17988389	C₂²	1	2	yes	—
34	#17988389	C₂²	1	1	yes	characteristic
35	#17988389	C₂²	1	1	yes	characteristic
36	C₈×Q₈×C₇₃	C₂²	1	1	yes	characteristic
37	C₄⋊C₁₆×C₇₃	C₂²	1	1	yes	characteristic
38	C₄⋊C₁₆×C₇₃	C₂²	1	2	yes	—
39	#346×C₇₃	C₂²	1	2	yes	—
40	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
41	C₂×Q₈×D₇₃	—	4	1	no	—
42	C₂×Dic₂₉₂	—	2	1	no	—
43	C₂×Dic₂₉₂	C₂³	1	2	yes	—
44	C₂×#11924017	—	2	2	no	—
45	C₂×D₅₈₄	C₂³	1	1	yes	characteristic
46	C₂×D₅₈₄	C₂³	1	1	yes	characteristic
47	#14556591	—	2	1	no	—
48	#14556591	—	2	2	no	—
49	#14556591	—	2	1	no	—
50	#14556586	—	4	2	no	—
51	#14556579	—	2	2	no	—
52	#14556571	—	4	1	no	—
53	#14556571	—	4	2	no	—
54	#14556567	—	4	1	no	—
55	C₄×Dic₁₄₆	—	2	2	no	—
56	C₄×D₂₉₂	—	2	1	no	—
57	C₄×D₂₉₂	—	2	2	no	—
58	C₄×D₂₉₂	—	2	1	no	—
59	#14556549	—	4	1	no	—
60	#14556536	—	2	4	no	—
61	D₁₁₆₈	—	2	2	no	—
62	#14556525	—	2	1	no	—
63	#14556525	—	2	2	no	—
64	#14556525	C₂³	1	2	yes	—
65	#14556525	C₂³	1	1	yes	characteristic
66	#14556525	C₂³	1	1	yes	characteristic
67	#14556516	—	2	2	no	—
68	#14556513	—	2	1	no	—
69	#14556513	—	2	2	no	—
70	#14556513	—	2	1	no	—
71	C₄×Q₈×C₇₃	D₄	1	1	yes	characteristic
72	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
73	C₂×C₁₆×C₇₃	C₂³	1	2	yes	—
74	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
75	C₄⋊C₈×C₇₃	D₄	1	1	yes	characteristic
76	C₄⋊C₈×C₇₃	D₄	1	2	yes	—
77	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₇₃	C₂³	1	2	yes	—
79	C₂×Dic₁₄₆	—	2	2	no	—
80	C₂×Dic₁₄₆	—	4	2	no	—
81	C₂×Dic₁₄₆	—	4	1	no	—
82	Q₈×D₇₃	—	8	4	no	—
83	C₂×D₂₉₂	—	2	1	no	—
84	C₂×D₂₉₂	—	2	1	no	—
85	C₂×C₄×D₇₃	—	4	1	no	—
86	C₂×C₄×D₇₃	—	4	2	no	—
87	C₂×C₄×D₇₃	—	4	1	no	—
88	Dic₂₉₂	—	2	2	no	—
89	Dic₂₉₂	—	2	2	no	—
90	Dic₂₉₂	—	8	1	no	—
91	#11924017	—	8	2	no	—
92	D₅₈₄	—	2	1	no	—
93	D₅₈₄	—	2	1	no	—
94	D₅₈₄	—	2	1	no	—
95	D₅₈₄	#41	1	2	yes	—
96	C₄⋊Dic₇₃	—	2	1	no	—
97	C₄⋊Dic₇₃	—	2	1	no	—
98	C₄⋊Dic₇₃	—	2	2	no	—
99	C₄⋊Dic₇₃	—	4	1	no	—
100	C₄⋊Dic₇₃	—	4	2	no	—
101	C₄⋊Dic₇₃	—	4	1	no	—
102	C₄⋊Dic₇₃	—	4	1	no	—
103	#11924007	—	4	2	no	—
104	#11924005	—	4	2	no	—
105	#11924005	—	4	2	no	—
106	#11924005	—	4	1	no	—
107	#11924005	—	4	1	no	—
108	#11924005	—	4	2	no	—
109	C₄×Dic₇₃	—	4	2	no	—
110	C₂×Q₈×C₇₃	D₈	1	1	yes	characteristic
111	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
113	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
114	C₄⋊C₄×C₇₃	D₈	1	1	yes	characteristic
115	C₄⋊C₄×C₇₃	D₈	1	2	yes	—
116	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₇₃	C₂×D₄	1	2	yes	—
118	C₁₆×C₇₃	—	2	1	no	—
119	C₁₆×C₇₃	—	2	2	no	—
120	C₁₆×C₇₃	—	2	1	no	—
121	C₂²×D₇₃	—	4	1	no	—
122	C₂²×D₇₃	—	4	1	no	—
123	Q₈×C₇₃	—	2	2	no	—
124	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
125	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
127	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
129	C₂×Dic₇₃	—	4	1	no	—
130	C₂×Dic₇₃	—	4	2	no	—
131	C₂×Dic₇₃	—	4	1	no	—
132	C₂×Dic₇₃	—	4	2	no	—
133	D₂₉₂	—	2	2	no	—
134	D₂₉₂	—	4	1	no	—
135	D₂₉₂	—	4	1	no	—
136	D₂₉₂	—	4	1	no	—
137	C₄×D₇₃	—	8	4	no	—
138	C₄×D₇₃	—	8	2	no	—
139	C₄×D₇₃	—	8	2	no	—
140	Dic₁₄₆	—	4	2	no	—
141	Dic₁₄₆	—	4	2	no	—
142	Dic₁₄₆	—	8	4	no	—
143	Dic₁₄₆	—	8	2	no	—
144	C₈×C₇₃	—	2	2	no	—
145	C₈×C₇₃	—	2	1	no	—
146	C₈×C₇₃	C₂×#41	1	1	yes	characteristic
147	C₈×C₇₃	2₋¹⁺⁴	1	1	yes	characteristic
148	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
149	C₂×D₇₃	—	4	2	no	—
150	C₂×D₇₃	—	8	1	no	—
151	C₂×D₇₃	—	8	1	no	—
152	C₂×D₇₃	—	8	1	no	—
153	C₂×D₇₃	—	8	1	no	—
154	C₂×D₇₃	—	8	1	no	—
155	C₄×C₇₃	—	2	2	no	—
156	C₄×C₇₃	—	2	2	no	—
157	C₄×C₇₃	—	2	1	no	—
158	C₄×C₇₃	#509	1	2	yes	—
159	C₄×C₇₃	#548	1	1	yes	characteristic
160	C₄×C₇₃	#577	1	1	yes	characteristic
161	Dic₇₃	—	8	2	no	—
162	Dic₇₃	—	8	1	no	—
163	Dic₇₃	—	8	2	no	—
164	Dic₇₃	—	8	1	no	—
165	#33934	—	73	1	no	maximal, Sylow
166	C₂×C₇₃	#3358	1	1	yes	characteristic
167	C₂×C₇₃	C₂×#509	1	1	yes	characteristic
168	C₂×C₇₃	#3417	1	1	yes	characteristic
169	D₇₃	—	8	2	no	—
170	D₇₃	—	16	1	no	—
171	#3393	—	73	1	no	—
172	Q₈×D₈	—	73	1	no	—
173	#2243	—	73	2	no	—
174	#2241	—	73	1	no	—
175	#2227	—	73	2	no	—
176	#2224	—	73	1	no	—
177	#2210	—	73	2	no	—
178	#2208	—	73	1	no	—
179	#2183	D₇₃	1	1	yes	characteristic, complemented
180	#2175	—	73	2	no	—
181	C₄×D₁₆	—	73	1	no	—
182	C₇₃	#33934	1	1	yes	characteristic, complemented, Sylow
183	#556	—	146	1	no	—
184	D₄×Q₈	—	146	1	no	—
185	C₂×SD₃₂	—	73	2	no	—
186	C₂×D₁₆	—	73	1	no	—
187	#500	—	73	1	no	—
188	#500	—	73	2	no	—
189	#495	—	73	2	no	—
190	#494	—	73	1	no	—
191	#474	—	146	1	no	—
192	#474	—	146	2	no	—
193	#464	—	146	2	no	—
194	#461	—	146	1	no	—
195	C₈×Q₈	C₂×D₇₃	1	1	yes	characteristic
196	C₄×Q₁₆	—	73	2	no	—
197	C₄×D₈	—	73	1	no	—
198	C₄×D₈	—	73	2	no	—
199	C₄×D₈	—	73	1	no	—
200	#366	—	73	1	no	—
201	#366	—	73	2	no	—
202	C₄⋊C₁₆	C₂×D₇₃	1	1	yes	characteristic
203	C₄⋊C₁₆	C₂×D₇₃	1	2	yes	—
204	#358	—	73	2	no	—
205	#358	—	73	2	no	—
206	#358	—	73	2	no	—
207	#357	—	73	1	no	—
208	#357	—	73	1	no	—
209	#357	—	73	2	no	—
210	#346	C₂×D₇₃	1	2	yes	—
211	C₄×C₁₆	C₂×D₇₃	1	1	yes	characteristic
212	C₂²×Q₈	—	292	1	no	—
213	C₂×Q₁₆	—	73	2	no	—
214	C₂×Q₁₆	—	146	1	no	—
215	C₂×SD₁₆	—	146	2	no	—
216	C₂×D₈	—	73	1	no	—
217	C₂×D₈	—	73	1	no	—
218	C₄⋊Q₈	—	146	1	no	—
219	C₄⋊Q₈	—	146	1	no	—
220	C₄⋊Q₈	—	146	2	no	—
221	#126	—	292	2	no	—
222	#124	—	146	2	no	—
223	C₂²⋊Q₈	—	292	1	no	—
224	C₂²⋊Q₈	—	292	1	no	—
225	C₂²⋊Q₈	—	292	2	no	—
226	C₄×Q₈	—	146	2	no	—
227	C₄×Q₈	D₂₉₂	1	1	yes	characteristic
228	C₄×D₄	—	146	1	no	—
229	C₄×D₄	—	146	2	no	—
230	C₄×D₄	—	146	1	no	—
231	#117	—	292	1	no	—
232	SD₃₂	—	146	4	no	—
233	D₁₆	—	146	2	no	—
234	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
235	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
236	C₂×C₁₆	C₂²×D₇₃	1	2	yes	—
237	#107	—	73	1	no	—
238	#107	—	73	2	no	—
239	#107	—	73	1	no	—
240	#107	—	146	1	no	—
241	#107	—	146	2	no	—
242	C₄⋊C₈	D₂₉₂	1	1	yes	characteristic
243	C₄⋊C₈	D₂₉₂	1	2	yes	—
244	#103	—	146	2	no	—
245	#102	—	146	1	no	—
246	#102	—	146	1	no	—
247	#102	—	146	2	no	—
248	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
249	C₄×C₈	C₂²×D₇₃	1	2	yes	—
250	C₂×Q₈	—	292	2	no	—
251	C₂×Q₈	—	292	1	no	—
252	C₂×Q₈	—	584	4	no	—
253	C₂×Q₈	—	146	2	no	—
254	C₂×Q₈	D₅₈₄	1	1	yes	characteristic
255	C₂×D₄	—	146	1	no	—
256	C₂×D₄	—	146	1	no	—
257	C₂²×C₄	—	292	1	no	—
258	C₂²×C₄	—	292	2	no	—
259	C₂²×C₄	—	292	1	no	—
260	Q₁₆	—	584	1	no	—
261	Q₁₆	—	146	2	no	—
262	Q₁₆	—	146	2	no	—
263	SD₁₆	—	584	2	no	—
264	D₈	—	73	2	no	—
265	D₈	—	146	1	no	—
266	D₈	—	146	1	no	—
267	D₈	—	146	1	no	—
268	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
269	C₂×C₈	C₂×D₂₉₂	1	2	yes	—
270	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
271	C₄⋊C₄	—	292	2	no	—
272	C₄⋊C₄	—	292	1	no	—
273	C₄⋊C₄	—	292	2	no	—
274	C₄⋊C₄	—	292	1	no	—
275	C₄⋊C₄	—	292	1	no	—
276	C₄⋊C₄	—	146	1	no	—
277	C₄⋊C₄	—	146	1	no	—
278	C₄⋊C₄	—	146	2	no	—
279	C₄⋊C₄	D₅₈₄	1	1	yes	characteristic
280	C₄⋊C₄	D₅₈₄	1	2	yes	—
281	C₂²⋊C₄	—	292	2	no	—
282	C₂²⋊C₄	—	292	2	no	—
283	C₂²⋊C₄	—	292	1	no	—
284	C₂²⋊C₄	—	292	1	no	—
285	C₂²⋊C₄	—	292	2	no	—
286	C₄²	—	292	2	no	—
287	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
288	C₄²	C₂×D₂₉₂	1	2	yes	—
289	C₁₆	—	2	1	no	—
290	C₁₆	—	2	2	no	—
291	C₁₆	—	2	1	no	—
292	C₂³	—	292	1	no	—
293	C₂³	—	292	1	no	—
294	Q₈	—	2	2	no	—
295	Q₈	—	292	2	no	—
296	Q₈	—	292	2	no	—
297	Q₈	—	584	4	no	—
298	Q₈	—	584	2	no	—
299	D₄	—	292	1	no	—
300	D₄	—	292	1	no	—
301	D₄	—	292	1	no	—
302	D₄	—	146	2	no	—
303	C₂×C₄	—	292	1	no	—
304	C₂×C₄	—	292	2	no	—
305	C₂×C₄	—	292	2	no	—
306	C₂×C₄	—	292	1	no	—
307	C₂×C₄	—	584	4	no	—
308	C₂×C₄	—	584	2	no	—
309	C₂×C₄	—	584	2	no	—
310	C₂×C₄	C₂×D₅₈₄	1	2	yes	—
311	C₂×C₄	C₂×D₅₈₄	1	1	yes	characteristic
312	C₂×C₄	C₂×D₅₈₄	1	2	yes	—
313	C₂×C₄	C₂×D₅₈₄	1	1	yes	characteristic
314	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
315	C₈	—	2	2	no	—
316	C₈	—	2	1	no	—
317	C₈	#41×D₇₃	1	1	yes	characteristic
318	C₈	#14556639	1	1	yes	characteristic
319	C₂²	—	292	2	no	—
320	C₂²	—	584	1	no	—
321	C₂²	—	584	1	no	—
322	C₂²	—	584	1	no	—
323	C₂²	—	584	1	no	—
324	C₂²	—	584	1	no	—
325	C₂²	C₂²×D₅₈₄	1	1	yes	characteristic
326	C₄	—	2	2	no	—
327	C₄	—	2	2	no	—
328	C₄	—	2	1	no	—
329	C₄	—	584	2	no	—
330	C₄	—	584	1	no	—
331	C₄	—	584	1	no	—
332	C₄	—	584	2	no	—
333	C₄	#17989025	1	2	yes	—
334	C₄	#17989249	1	1	yes	characteristic
335	C₄	#17989399	1	1	yes	characteristic
336	C₂	—	584	2	no	—
337	C₂	—	1168	1	no	—
338	C₂	#22622567	1	1	yes	characteristic
339	C₂	C₂×#17989025	1	1	yes	characteristic
340	C₂	#22623084	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₅₈₄
commutator, cosocle	C₂×C₈×C₇₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₇₃×#2183	C₂
socle	C₂²×C₇₃	C₂²×D₈
prime radical	C₂×D₁₁₆₈	C₂²
hypercenter	#2183	D₇₃
hypocenter	C₇₃	#33934
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ #2183

Complements

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

N	H ≅ G/N	type
1. #22622920	C₂	semidirect
2. Q₈×D₅₈₄	C₂	semidirect
3. #22612168	C₂	semidirect
4. #22612159	C₂	semidirect
5. #22612084	C₂	semidirect
6. #22612069	C₂	semidirect
8. #22611977	C₂	semidirect
9. #22611797	C₂	semidirect
11. C₇₃×#2183	C₂	semidirect
179. #2183	D₇₃	semidirect
182. C₇₃	#33934	semidirect

Maximal subgroups

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

subgroup	index
#22622920	2
Q₈×D₅₈₄	2
#22612168	2
#22612159	2
#22612084	2
#22612069	2
#22611987	2
#22611977	2
#22611797	2
C₄×D₁₁₆₈	2
C₇₃×#2183	2
#33934	73