Subgroups — #33055835

Order 18688.

646 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22619049	C₂	1	2	yes	maximal, complemented
2	C₄×#14556613	C₂	1	1	yes	maximal, characteristic
3	#22608832	C₂	1	4	yes	maximal, complemented
4	#22608827	C₂	1	2	yes	maximal, complemented
5	#22608815	C₂	1	2	yes	maximal, complemented
6	#22608316	C₂	1	2	yes	maximal, complemented
7	#22608214	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#335×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14556613	C₄	1	1	yes	characteristic
10	C₄×#11924028	C₂²	1	2	yes	—
11	#17988714	C₂²	1	4	yes	—
12	C₄×Dic₂₉₂	C₄	1	1	yes	characteristic
13	C₄×Dic₂₉₂	C₂²	1	1	yes	characteristic
14	C₄×#11924017	C₄	1	2	yes	—
15	C₄×#11924017	C₂²	1	2	yes	—
16	C₄×D₅₈₄	C₄	1	1	yes	characteristic
17	C₄×D₅₈₄	C₂²	1	1	yes	characteristic
18	#17988664	C₂²	1	4	yes	—
19	#17988658	—	2	4	no	—
20	#17988643	—	2	2	no	—
21	#17988640	C₂²	1	1	yes	characteristic
22	#17988640	C₂²	1	1	yes	characteristic
23	#17988625	C₄	1	1	yes	characteristic
24	#17988609	—	2	2	no	—
25	#17988584	C₄	1	2	yes	—
26	#17988317	C₂²	1	4	yes	—
27	#17988314	C₂²	1	4	yes	—
28	#17988308	C₂²	1	2	yes	—
29	C₈×#10596831	C₂²	1	2	yes	—
30	C₂×C₄⋊C₈×C₇₃	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
32	#335×C₇₃	C₂²	1	2	yes	—
33	#335×C₇₃	C₂²	1	2	yes	—
34	C₂×#11924028	C₂×C₄	1	2	yes	—
35	#14556613	—	2	4	no	—
36	C₂×Dic₂₉₂	C₂×C₄	1	1	yes	characteristic
37	C₂×#11924017	C₂×C₄	1	2	yes	—
38	C₂×D₅₈₄	C₂×C₄	1	1	yes	characteristic
39	C₂×#11924011	—	2	4	no	—
40	C₄×Dic₁₄₆	C₂×C₄	1	2	yes	—
41	C₄×Dic₁₄₆	C₂³	1	2	yes	—
42	C₄×D₂₉₂	C₂×C₄	1	2	yes	—
43	C₄×D₂₉₂	C₂³	1	2	yes	—
44	C₄×C₂²⋊D₇₃	—	2	2	no	—
45	C₄×C₂²⋊D₇₃	—	2	2	no	—
46	#14556550	C₂×C₄	1	2	yes	—
47	#14556548	—	2	2	no	—
48	C₄²×D₇₃	—	2	2	no	—
49	#14556525	C₂×C₄	1	1	yes	characteristic
50	#14556525	C₂×C₄	1	1	yes	characteristic
51	#14556523	C₂×C₄	1	2	yes	—
52	C₄⋊#10596831	—	2	1	no	—
53	C₄⋊#10596831	D₄	1	1	yes	characteristic
54	C₄⋊#10596831	D₄	1	1	yes	characteristic
55	#14556520	—	2	4	no	—
56	#14556516	C₂×C₄	1	2	yes	—
57	#14556516	C₂×C₄	1	2	yes	—
58	#14556513	C₂×C₄	1	2	yes	—
59	#14556513	C₂×C₄	1	2	yes	—
60	C₂²⋊#10596831	—	2	1	no	—
61	C₂²⋊#10596831	D₄	1	2	yes	—
62	#14556503	—	2	4	no	—
63	#14556501	—	2	4	no	—
64	C₄×#10596831	C₂³	1	2	yes	—
65	C₄×#10596831	C₂³	1	2	yes	—
66	C₂²×C₈×C₇₃	—	2	2	no	—
67	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
68	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
70	C₄⋊C₈×C₇₃	—	2	2	no	—
71	C₄⋊C₈×C₇₃	C₂³	1	4	yes	—
72	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
73	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
74	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
76	#11924028	—	2	4	no	—
77	#11924028	—	2	4	no	—
78	C₂×Dic₁₄₆	C₂²×C₄	1	2	yes	—
79	C₂×D₂₉₂	C₂²×C₄	1	2	yes	—
80	C₂×C₂²⋊D₇₃	—	2	2	no	—
81	C₂×C₄×D₇₃	—	2	2	no	—
82	C₂×C₄×D₇₃	—	2	2	no	—
83	C₂×C₄×D₇₃	—	2	2	no	—
84	Dic₂₉₂	—	2	2	no	—
85	#11924017	—	2	4	no	—
86	D₅₈₄	—	2	2	no	—
87	#11924011	—	8	4	no	—
88	C₂×#10596831	—	2	2	no	—
89	C₂×#10596831	C₂×D₄	1	2	yes	—
90	C₂×#10596831	C₂×D₄	1	2	yes	—
91	C₄⋊Dic₇₃	C₂²×C₄	1	2	yes	—
92	C₄⋊Dic₇₃	C₂²×C₄	1	2	yes	—
93	#11924007	—	2	2	no	—
94	#11924007	—	2	2	no	—
95	C₂²⋊Dic₇₃	—	2	2	no	—
96	#11924005	—	2	2	no	—
97	#11924005	—	2	2	no	—
98	C₄×Dic₇₃	—	2	2	no	—
99	C₄×Dic₇₃	—	2	2	no	—
100	C₄×Dic₇₃	—	2	2	no	—
101	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
102	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
104	C₂×C₈×C₇₃	—	2	2	no	—
105	C₂×C₈×C₇₃	—	2	4	no	—
106	C₂×C₈×C₇₃	—	8	2	no	—
107	C₂×C₈×C₇₃	M₁₆	1	4	yes	complemented
108	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
112	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
113	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₇₃	#41	1	1	yes	characteristic
115	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
116	C₂³×C₇₃	C₄×D₄	1	1	yes	characteristic
117	C₂²×D₇₃	—	2	2	no	—
118	C₂×C₄×C₇₃	—	2	1	no	—
119	C₂×C₄×C₇₃	—	2	1	no	—
120	C₂×C₄×C₇₃	—	4	1	no	—
121	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
125	C₂×C₄×C₇₃	#131	1	2	yes	—
126	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
127	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
129	C₂²⋊D₇₃	—	4	4	no	—
130	C₂×Dic₇₃	—	2	2	no	—
131	C₂×Dic₇₃	—	2	2	no	—
132	C₂×Dic₇₃	—	2	2	no	—
133	D₂₉₂	—	2	2	no	—
134	D₂₉₂	—	2	2	no	—
135	C₄×D₇₃	—	4	2	no	—
136	C₄×D₇₃	—	4	2	no	—
137	C₄×D₇₃	—	4	4	no	—
138	C₄×D₇₃	—	8	2	no	—
139	Dic₁₄₆	—	2	2	no	—
140	Dic₁₄₆	—	2	2	no	—
141	C₈×C₇₃	—	2	1	no	—
142	C₈×C₇₃	—	2	1	no	—
143	C₈×C₇₃	—	4	4	no	—
144	C₈×C₇₃	C₂×M₁₆	1	2	yes	—
145	C₈×C₇₃	C₂×M₁₆	1	2	yes	—
146	#10596831	—	2	2	no	—
147	#10596831	—	2	2	no	—
148	#10596831	—	4	2	no	—
149	C₂²×C₇₃	—	2	1	no	—
150	C₂²×C₇₃	—	2	1	no	—
151	C₂²×C₇₃	#436	1	2	yes	—
152	C₂²×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
153	C₂×D₇₃	—	4	2	no	—
154	C₂×D₇₃	—	4	2	no	—
155	C₂×D₇₃	—	4	2	no	—
156	C₄×C₇₃	—	2	1	no	—
157	C₄×C₇₃	—	2	1	no	—
158	C₄×C₇₃	—	2	1	no	—
159	C₄×C₇₃	#436	1	2	yes	—
160	C₄×C₇₃	C₂²×M₁₆	1	1	yes	characteristic
161	C₄×C₇₃	C₂×#131	1	1	yes	characteristic
162	C₄×C₇₃	#576	1	1	yes	characteristic
163	C₄×C₇₃	#578	1	1	yes	characteristic
164	Dic₇₃	—	4	2	no	—
165	Dic₇₃	—	4	2	no	—
166	Dic₇₃	—	4	2	no	—
167	#17898	—	73	1	no	maximal, Sylow
168	C₂×C₇₃	—	2	2	no	—
169	C₂×C₇₃	C₂×#436	1	1	yes	characteristic
170	C₂×C₇₃	#2951	1	1	yes	characteristic
171	C₂×C₇₃	#2957	1	1	yes	characteristic
172	D₇₃	—	8	2	no	—
173	#2989	—	73	2	no	—
174	C₄×#135	—	73	1	no	—
175	#1592	—	73	4	no	—
176	#1591	—	73	2	no	—
177	#1589	—	73	2	no	—
178	C₂×#335	D₇₃	1	1	yes	characteristic, complemented
179	#1477	—	73	2	no	—
180	#1450	—	73	1	no	—
181	C₇₃	#17898	1	1	yes	characteristic, complemented, Sylow
182	C₂×#135	—	73	1	no	—
183	C₄×#41	—	73	2	no	—
184	#446	—	73	4	no	—
185	C₄×Q₁₆	—	73	1	no	—
186	C₄×Q₁₆	—	73	1	no	—
187	C₄×SD₁₆	—	73	2	no	—
188	C₄×SD₁₆	—	73	2	no	—
189	C₄×D₈	—	73	1	no	—
190	C₄×D₈	—	73	1	no	—
191	#436	—	73	4	no	—
192	#435	—	146	4	no	—
193	#432	—	146	2	no	—
194	#431	—	73	1	no	—
195	#431	—	73	1	no	—
196	#427	—	73	1	no	—
197	#423	—	146	2	no	—
198	C₂×C₄⋊C₈	C₂×D₇₃	1	2	yes	—
199	#416	—	73	2	no	—
200	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
201	#335	C₂×D₇₃	1	2	yes	—
202	#335	C₂×D₇₃	1	2	yes	—
203	Q₈⋊C₈	—	73	4	no	—
204	D₄⋊C₈	—	73	4	no	—
205	#322	—	73	2	no	—
206	C₈²	—	73	2	no	—
207	C₂×#41	—	73	2	no	—
208	#135	—	146	4	no	—
209	C₂×Q₁₆	—	73	1	no	—
210	C₂×SD₁₆	—	73	2	no	—
211	C₂×D₈	—	73	1	no	—
212	C₂×M₁₆	—	146	4	no	—
213	C₂²×C₈	—	2	2	no	—
214	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
215	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
216	C₄×Q₈	—	73	2	no	—
217	C₄×Q₈	—	73	2	no	—
218	C₄×D₄	—	73	2	no	—
219	C₄×D₄	—	73	2	no	—
220	C₄×D₄	—	146	2	no	—
221	C₄×D₄	—	146	2	no	—
222	#117	—	73	2	no	—
223	#117	—	146	2	no	—
224	C₂×C₄²	—	146	2	no	—
225	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
226	#107	—	73	1	no	—
227	#107	—	73	1	no	—
228	#106	—	73	2	no	—
229	C₄⋊C₈	—	2	2	no	—
230	C₄⋊C₈	—	73	1	no	—
231	C₄⋊C₈	—	73	1	no	—
232	C₄⋊C₈	—	146	4	no	—
233	C₄⋊C₈	—	146	1	no	—
234	C₄⋊C₈	C₂²×D₇₃	1	4	yes	—
235	#103	—	73	2	no	—
236	#103	—	73	2	no	—
237	#102	—	73	2	no	—
238	#102	—	73	2	no	—
239	C₂²⋊C₈	—	73	2	no	—
240	C₂²⋊C₈	—	146	4	no	—
241	C₂²⋊C₈	—	146	1	no	—
242	#97	—	146	4	no	—
243	C₄×C₈	—	73	2	no	—
244	C₄×C₈	—	73	2	no	—
245	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
246	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
247	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
248	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
249	#41	—	146	4	no	—
250	#41	—	146	4	no	—
251	C₂×Q₈	—	73	2	no	—
252	C₂×D₄	—	73	2	no	—
253	C₂×D₄	—	146	2	no	—
254	C₂²×C₄	—	146	2	no	—
255	C₂²×C₄	—	146	2	no	—
256	C₂²×C₄	—	146	2	no	—
257	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
258	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
259	C₂²×C₄	#11924029	1	1	yes	characteristic
260	Q₁₆	—	146	2	no	—
261	SD₁₆	—	146	4	no	—
262	D₈	—	146	2	no	—
263	M₁₆	—	584	4	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	2	4	no	—
266	C₂×C₈	—	8	2	no	—
267	C₂×C₈	—	73	2	no	—
268	C₂×C₈	—	73	2	no	—
269	C₂×C₈	—	146	2	no	—
270	C₂×C₈	#11924011	1	4	yes	complemented
271	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
272	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
273	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
274	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
275	C₄⋊C₄	—	73	2	no	—
276	C₄⋊C₄	—	73	2	no	—
277	C₄⋊C₄	—	146	2	no	—
278	C₄⋊C₄	—	146	2	no	—
279	C₂²⋊C₄	—	146	2	no	—
280	C₂²⋊C₄	—	146	2	no	—
281	C₂²⋊C₄	—	146	2	no	—
282	C₄²	—	146	2	no	—
283	C₄²	—	146	2	no	—
284	C₄²	—	146	2	no	—
285	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
286	C₄²	D₄×D₇₃	1	1	yes	characteristic
287	C₄²	#11924029	1	1	yes	characteristic
288	C₄²	C₂³×D₇₃	1	1	yes	characteristic
289	C₂³	—	146	2	no	—
290	C₂³	#14556553	1	1	yes	characteristic
291	Q₈	—	146	2	no	—
292	Q₈	—	146	2	no	—
293	D₄	—	292	4	no	—
294	D₄	—	146	2	no	—
295	D₄	—	146	2	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	292	2	no	—
300	C₂×C₄	—	292	2	no	—
301	C₂×C₄	—	292	4	no	—
302	C₂×C₄	—	584	2	no	—
303	C₂×C₄	—	146	2	no	—
304	C₂×C₄	—	146	2	no	—
305	C₂×C₄	—	146	2	no	—
306	C₂×C₄	#14556553	1	1	yes	characteristic
307	C₂×C₄	#14556553	1	1	yes	characteristic
308	C₂×C₄	#14556553	1	1	yes	characteristic
309	C₂×C₄	C₂×#11924011	1	2	yes	—
310	C₂×C₄	#14556599	1	2	yes	—
311	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
312	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
313	C₂×C₄	C₂×#11924029	1	1	yes	characteristic
314	C₈	—	2	1	no	—
315	C₈	—	2	1	no	—
316	C₈	—	4	4	no	—
317	C₈	—	292	2	no	—
318	C₈	—	146	2	no	—
319	C₈	—	146	2	no	—
320	C₈	C₂×#11924011	1	2	yes	—
321	C₈	C₂×#11924011	1	2	yes	—
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	292	2	no	—
325	C₂²	—	292	2	no	—
326	C₂²	—	292	2	no	—
327	C₂²	#17988664	1	2	yes	—
328	C₂²	C₂×#14556553	1	1	yes	characteristic
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	292	2	no	—
333	C₄	—	292	2	no	—
334	C₄	—	292	2	no	—
335	C₄	#17988664	1	2	yes	—
336	C₄	C₂²×#11924011	1	1	yes	characteristic
337	C₄	C₂×#14556599	1	1	yes	characteristic
338	C₄	#17989388	1	1	yes	characteristic
339	C₄	#17989400	1	1	yes	characteristic
340	C₂	—	2	2	no	—
341	C₂	—	584	2	no	—
342	C₂	C₂×#17988664	1	1	yes	characteristic
343	C₂	#22618805	1	1	yes	characteristic
344	C₂	#22618847	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₇₃
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₄²	C₂³×D₇₃
Fitting	C₂×#335×C₇₃	C₂
socle	C₂²×C₇₃	C₂×C₄×D₄
prime radical	C₂×#14556613	C₄
cosocle	C₄²×C₇₃	C₂⁴
hypercenter	C₂×#335	D₇₃
hypocenter	C₇₃	#17898
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
1. #22619049	C₂	semidirect
3. #22608832	C₂	semidirect
4. #22608827	C₂	semidirect
5. #22608815	C₂	semidirect
6. #22608316	C₂	semidirect
7. #22608214	C₂	semidirect
8. C₂×#335×C₇₃	C₂	semidirect
107. C₂×C₈×C₇₃	M₁₆	semidirect
178. C₂×#335	D₇₃	semidirect
181. C₇₃	#17898	semidirect
270. C₂×C₈	#11924011	semidirect

Maximal subgroups

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

subgroup	index
#22619049	2
C₄×#14556613	2
#22608832	2
#22608827	2
#22608815	2
#22608316	2
#22608214	2
C₂×#335×C₇₃	2
#17898	73