Subgroups — #34021335

Order 18688.

1126 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17989597	C₂	1	4	yes	maximal
2	#22626380	C₂	1	2	yes	maximal
3	#22625868	C₂	1	4	yes	maximal, complemented
4	#22625867	C₂	1	2	yes	maximal, complemented
5	#22625866	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×C₈×D₇₃	C₂	1	1	yes	maximal, characteristic
7	C₂×#14556670	C₂²	1	4	yes	—
8	#17989597	—	2	16	no	—
9	C₂×#14556660	C₂²	1	8	yes	—
10	C₂×#14556659	C₂²	1	8	yes	—
11	#17989533	C₂²	1	4	yes	—
12	#17989531	C₂²	1	4	yes	—
13	#17989529	—	2	4	no	—
14	C₄×C₄⋊#64408	—	2	4	no	—
15	C₂²×C₈×D₇₃	C₂²	1	2	yes	—
16	C₂×C₄²×D₇₃	C₂²	1	1	yes	characteristic
17	C₂×C₄×#10596831	C₄	1	1	yes	characteristic, complemented
18	C₂×C₈×Dic₇₃	C₄	1	2	yes	complemented
19	C₄×C₈×D₇₃	C₄	1	1	yes	characteristic, complemented
20	C₄×C₈×D₇₃	C₄	1	1	yes	characteristic, complemented
21	C₄×C₈×D₇₃	C₄	1	2	yes	complemented
22	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
23	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
24	C₄×C₈×D₇₃	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₇₃	C₄	1	1	yes	characteristic, complemented
26	#14556670	—	2	16	no	—
27	C₂×C₄⋊#64408	—	2	8	no	—
28	C₂×C₄⋊#64408	C₂³	1	8	yes	—
29	C₂×C₂²⋊#64408	—	2	4	no	—
30	C₂×C₄×#64408	—	2	4	no	—
31	#14556660	—	2	16	no	—
32	#14556659	—	2	16	no	—
33	(C₂×#64408)⋊C₄	—	2	8	no	—
34	C₂²×C₄×D₇₃	D₄	1	2	yes	—
35	C₂²×C₄×D₇₃	C₂³	1	1	yes	characteristic
36	C₂²×#10596831	C₂×C₄	1	2	yes	—
37	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
38	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
39	C₂×C₈×D₇₃	D₄	1	8	yes	—
40	C₂×C₈×D₇₃	Q₈	1	8	yes	—
41	C₂×C₈×D₇₃	C₂³	1	2	yes	—
42	C₂×C₈×D₇₃	C₂³	1	2	yes	—
43	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×Dic₇₃	Q₈	1	2	yes	—
45	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
46	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
47	C₄²×D₇₃	D₄	1	2	yes	—
48	C₄²×D₇₃	Q₈	1	2	yes	—
49	C₄²×D₇₃	C₂³	1	1	yes	characteristic
50	C₄²×D₇₃	C₂³	1	1	yes	characteristic
51	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
52	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
53	C₄×#10596831	C₂×C₄	1	2	yes	—
54	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
55	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
56	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
57	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
58	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
59	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
60	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
61	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
62	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
63	C₂²×#64408	—	2	8	no	—
64	C₄⋊#64408	—	2	16	no	—
65	C₄⋊#64408	—	4	16	no	—
66	C₂²⋊#64408	—	4	8	no	—
67	C₄×#64408	—	4	8	no	—
68	C₂³×D₇₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×Dic₇₃	C₄⋊C₄	1	2	yes	—
70	C₂²×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
71	C₂×C₄×D₇₃	—	2	8	no	—
72	C₂×C₄×D₇₃	C₄⋊C₄	1	2	yes	—
73	C₂×C₄×D₇₃	C₄⋊C₄	1	2	yes	—
74	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₄×D₇₃	C₂×D₄	1	2	yes	—
76	C₂×C₄×D₇₃	C₂×D₄	1	4	yes	—
77	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
78	C₂×C₄×D₇₃	C₂×D₄	1	2	yes	—
79	C₂×C₄×D₇₃	C₂×Q₈	1	4	yes	—
80	C₂×C₄×D₇₃	C₂×Q₈	1	1	yes	characteristic
81	C₂×C₄×D₇₃	C₂⁴	1	1	yes	characteristic
82	C₂×#10596831	C₄⋊C₄	1	8	yes	complemented
83	C₂×#10596831	C₂²×C₄	1	2	yes	—
84	C₂×#10596831	C₂²×C₄	1	2	yes	—
85	C₈×D₇₃	C₄⋊C₄	1	8	yes	complemented
86	C₈×D₇₃	C₄⋊C₄	1	8	yes	complemented
87	C₈×D₇₃	C₂×D₄	1	4	yes	—
88	C₈×D₇₃	C₂×D₄	1	4	yes	—
89	C₈×D₇₃	C₂×Q₈	1	4	yes	—
90	C₈×D₇₃	C₂×Q₈	1	4	yes	—
91	C₄×Dic₇₃	C₄⋊C₄	1	2	yes	—
92	C₄×Dic₇₃	C₄⋊C₄	1	2	yes	—
93	C₄×Dic₇₃	C₄⋊C₄	1	2	yes	—
94	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
95	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
96	C₄×Dic₇₃	C₂×Q₈	1	2	yes	—
97	C₄×Dic₇₃	C₂×Q₈	1	2	yes	—
98	C₂²×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
99	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₈×C₇₃	C₄⋊C₄	1	8	yes	complemented
101	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
102	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
103	C₄²×C₇₃	C₄⋊C₄	1	2	yes	—
104	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
105	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
106	C₂³×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂²×D₇₃	—	2	4	no	—
108	C₂²×D₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂²×D₇₃	#127	1	2	yes	—
110	C₂²×D₇₃	C₄⋊Q₈	1	2	yes	—
111	C₂²×D₇₃	C₂²×D₄	1	1	yes	characteristic
112	C₂×#64408	—	4	8	no	—
113	C₂×#64408	—	4	16	no	—
114	C₂×C₄×C₇₃	—	2	4	no	—
115	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
116	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	4	yes	—
117	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
118	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
120	C₂×Dic₇₃	—	2	4	no	—
121	C₂×Dic₇₃	C₂×C₄⋊C₄	1	2	yes	—
122	C₂×Dic₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂×Dic₇₃	C₂×C₄⋊C₄	1	2	yes	—
124	C₂×Dic₇₃	C₄⋊Q₈	1	4	yes	—
125	C₂×Dic₇₃	C₂²×Q₈	1	1	yes	characteristic
126	C₄×D₇₃	—	2	4	no	—
127	C₄×D₇₃	—	2	4	no	—
128	C₄×D₇₃	—	2	4	no	—
129	C₄×D₇₃	C₂×C₄⋊C₄	1	4	yes	—
130	C₄×D₇₃	#127	1	2	yes	—
131	C₄×D₇₃	C₄⋊Q₈	1	4	yes	—
132	C₄×D₇₃	C₄⋊Q₈	1	2	yes	—
133	C₄×D₇₃	#135	1	8	yes	—
134	C₄×D₇₃	C₂²×D₄	1	2	yes	—
135	C₄×D₇₃	C₂²×Q₈	1	2	yes	—
136	C₈×C₇₃	C₂×C₄⋊C₄	1	4	yes	—
137	C₈×C₇₃	C₂×C₄⋊C₄	1	4	yes	—
138	#10596831	C₂×C₄⋊C₄	1	4	yes	—
139	#10596831	C₂×C₄⋊C₄	1	4	yes	—
140	C₂²×C₇₃	—	2	2	no	—
141	C₂²×C₇₃	#384	1	2	yes	—
142	C₂²×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
143	C₂×D₇₃	—	2	2	no	—
144	C₂×D₇₃	—	2	4	no	—
145	C₂×D₇₃	—	2	4	no	—
146	C₂×D₇₃	#384	1	2	yes	—
147	C₂×D₇₃	C₂×#127	1	1	yes	characteristic
148	C₂×D₇₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
149	C₂×D₇₃	C₂×#135	1	4	yes	—
150	#64408	—	8	8	no	—
151	C₄×C₇₃	—	2	2	no	—
152	C₄×C₇₃	#384	1	2	yes	—
153	C₄×C₇₃	#427	1	4	yes	complemented
154	C₄×C₇₃	C₂²×C₄⋊C₄	1	2	yes	—
155	Dic₇₃	—	2	2	no	—
156	Dic₇₃	#384	1	2	yes	—
157	Dic₇₃	#427	1	4	yes	complemented
158	Dic₇₃	C₂×C₄⋊Q₈	1	2	yes	—
159	#20797	—	73	1	no	maximal, Sylow
160	C₂×C₇₃	—	2	2	no	—
161	C₂×C₇₃	C₂×#384	1	1	yes	characteristic
162	C₂×C₇₃	C₂×#427	1	2	yes	—
163	D₇₃	—	2	2	no	—
164	D₇₃	#3147	1	2	yes	—
165	D₇₃	#3160	1	2	yes	—
166	C₂×#427	—	73	4	no	—
167	C₂²×C₄×C₈	—	73	1	no	—
168	#2285	—	73	2	no	—
169	#1846	—	73	4	no	—
170	#1845	—	73	2	no	—
171	#1844	—	73	2	no	—
172	C₇₃	#20797	1	1	yes	characteristic, complemented, Sylow
173	C₂³×C₈	—	73	2	no	—
174	C₂×#117	—	73	4	no	—
175	C₂²×C₄²	—	73	1	no	—
176	#427	—	146	16	no	—
177	C₂×#107	—	73	8	no	—
178	C₂×#106	—	73	8	no	—
179	C₂×C₄×C₈	—	73	2	no	—
180	C₂×C₄×C₈	—	73	1	no	—
181	C₂×C₄×C₈	—	73	1	no	—
182	C₂×C₄×C₈	—	73	1	no	—
183	C₂×C₄×C₈	—	73	2	no	—
184	C₂×C₄×C₈	—	73	2	no	—
185	C₂×C₄×C₈	—	73	1	no	—
186	C₂×C₄×C₈	—	73	1	no	—
187	C₂×C₄×C₈	#64408	1	1	yes	characteristic, complemented
188	#384	—	73	4	no	—
189	#382	—	73	4	no	—
190	#380	—	146	4	no	—
191	C₄×C₄⋊C₄	—	146	4	no	—
192	C₂³×C₄	—	73	1	no	—
193	C₂³×C₄	—	73	2	no	—
194	C₂²×C₈	—	73	2	no	—
195	C₂²×C₈	—	73	8	no	—
196	C₂²×C₈	—	73	8	no	—
197	C₂²×C₈	—	73	2	no	—
198	C₂²×C₈	—	73	2	no	—
199	C₂²×C₈	—	73	2	no	—
200	C₂²×C₈	—	73	2	no	—
201	C₂²×C₈	C₂×#64408	1	2	yes	—
202	#117	—	146	16	no	—
203	C₂×C₄⋊C₄	—	73	8	no	—
204	C₂×C₄⋊C₄	—	146	8	no	—
205	C₂×C₂²⋊C₄	—	146	4	no	—
206	C₂×C₄²	—	73	2	no	—
207	C₂×C₄²	—	73	1	no	—
208	C₂×C₄²	—	73	2	no	—
209	C₂×C₄²	—	73	2	no	—
210	C₂×C₄²	—	73	1	no	—
211	C₂×C₄²	—	73	1	no	—
212	C₂×C₄²	—	73	1	no	—
213	C₂×C₄²	—	73	1	no	—
214	C₂×C₄²	—	146	4	no	—
215	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
216	#107	—	146	16	no	—
217	#106	—	146	16	no	—
218	C₄×C₈	—	73	2	no	—
219	C₄×C₈	—	73	2	no	—
220	C₄×C₈	—	73	2	no	—
221	C₄×C₈	—	73	2	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	1	yes	characteristic
226	C₄×C₈	C₂×#64408	1	1	yes	characteristic
227	C₄×C₈	C₂×#64408	1	2	yes	—
228	#95	—	146	8	no	—
229	C₂⁴	—	73	1	no	—
230	C₂²×C₄	—	73	1	no	—
231	C₂²×C₄	—	73	1	no	—
232	C₂²×C₄	—	73	1	no	—
233	C₂²×C₄	—	73	2	no	—
234	C₂²×C₄	—	73	4	no	—
235	C₂²×C₄	—	73	4	no	—
236	C₂²×C₄	—	73	1	no	—
237	C₂²×C₄	—	73	1	no	—
238	C₂²×C₄	—	73	2	no	—
239	C₂²×C₄	—	73	2	no	—
240	C₂²×C₄	—	73	2	no	—
241	C₂²×C₄	—	73	2	no	—
242	C₂²×C₄	—	146	8	no	—
243	C₂²×C₄	—	146	8	no	—
244	C₂²×C₄	C₄⋊#64408	1	2	yes	—
245	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
246	C₂×C₈	—	73	2	no	—
247	C₂×C₈	—	73	2	no	—
248	C₂×C₈	—	73	8	no	—
249	C₂×C₈	—	73	8	no	—
250	C₂×C₈	—	73	8	no	—
251	C₂×C₈	—	73	4	no	—
252	C₂×C₈	—	73	4	no	—
253	C₂×C₈	—	73	4	no	—
254	C₂×C₈	—	73	4	no	—
255	C₂×C₈	C₄⋊#64408	1	8	yes	complemented
256	C₂×C₈	C₂²×#64408	1	2	yes	—
257	C₂×C₈	C₂²×#64408	1	2	yes	—
258	C₄⋊C₄	—	292	16	no	—
259	C₄⋊C₄	—	146	16	no	—
260	C₂²⋊C₄	—	292	8	no	—
261	C₄²	—	292	8	no	—
262	C₄²	—	73	1	no	—
263	C₄²	—	73	1	no	—
264	C₄²	—	73	2	no	—
265	C₄²	—	73	2	no	—
266	C₄²	—	73	2	no	—
267	C₄²	—	73	2	no	—
268	C₄²	—	73	2	no	—
269	C₄²	C₄⋊#64408	1	2	yes	—
270	C₄²	C₂²×#64408	1	1	yes	characteristic
271	C₄²	C₂²×#64408	1	1	yes	characteristic
272	C₂³	—	73	1	no	—
273	C₂³	—	73	1	no	—
274	C₂³	—	73	2	no	—
275	C₂³	—	73	2	no	—
276	C₂³	—	146	4	no	—
277	C₂³	C₂×C₄⋊#64408	1	1	yes	characteristic
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	292	8	no	—
280	C₂×C₄	—	292	16	no	—
281	C₂×C₄	—	73	2	no	—
282	C₂×C₄	—	73	1	no	—
283	C₂×C₄	—	73	1	no	—
284	C₂×C₄	—	73	4	no	—
285	C₂×C₄	—	73	2	no	—
286	C₂×C₄	—	73	4	no	—
287	C₂×C₄	—	73	2	no	—
288	C₂×C₄	—	73	2	no	—
289	C₂×C₄	—	73	4	no	—
290	C₂×C₄	—	73	2	no	—
291	C₂×C₄	—	73	8	no	—
292	C₂×C₄	—	73	2	no	—
293	C₂×C₄	—	146	4	no	—
294	C₂×C₄	—	146	4	no	—
295	C₂×C₄	—	146	4	no	—
296	C₂×C₄	—	146	4	no	—
297	C₂×C₄	C₂×C₄⋊#64408	1	2	yes	—
298	C₂×C₄	C₂×C₄⋊#64408	1	4	yes	—
299	C₂×C₄	C₂×C₄⋊#64408	1	2	yes	—
300	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
301	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
302	C₈	—	73	4	no	—
303	C₈	—	73	4	no	—
304	C₈	C₂×C₄⋊#64408	1	4	yes	—
305	C₈	C₂×C₄⋊#64408	1	4	yes	—
306	C₂²	—	2	2	no	—
307	C₂²	—	73	4	no	—
308	C₂²	—	73	1	no	—
309	C₂²	—	73	1	no	—
310	C₂²	—	73	2	no	—
311	C₂²	—	146	4	no	—
312	C₂²	—	146	4	no	—
313	C₂²	—	146	2	no	—
314	C₂²	#17989533	1	2	yes	—
315	C₂²	C₂²×C₄⋊#64408	1	1	yes	characteristic
316	C₄	—	2	2	no	—
317	C₄	—	584	8	no	—
318	C₄	—	73	4	no	—
319	C₄	—	73	2	no	—
320	C₄	—	73	2	no	—
321	C₄	—	146	2	no	—
322	C₄	#17989533	1	2	yes	—
323	C₄	#17989597	1	4	yes	complemented
324	C₄	C₂²×C₄⋊#64408	1	2	yes	—
325	C₂	—	2	2	no	—
326	C₂	—	73	2	no	—
327	C₂	—	73	2	no	—
328	C₂	—	146	2	no	—
329	C₂	C₂×#17989533	1	1	yes	characteristic
330	C₂	C₂×#17989597	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₄⋊#64408
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#64408
Fitting	C₂×C₄×C₈×C₇₃	C₄
socle	C₂²×C₇₃	C₂²×C₄⋊C₄
prime radical	C₂³×D₇₃	C₂×D₄
cosocle	C₂×C₄×D₇₃	C₂⁴
hypercenter	C₂×C₄×C₈	#64408
hypocenter	C₇₃	#20797
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. #22625868	C₂	semidirect
4. #22625867	C₂	semidirect
5. #22625866	C₂	semidirect
17. C₂×C₄×#10596831	C₄	semidirect
18. C₂×C₈×Dic₇₃	C₄	semidirect
19. C₄×C₈×D₇₃	C₄	semidirect
20. C₄×C₈×D₇₃	C₄	semidirect
21. C₄×C₈×D₇₃	C₄	semidirect
25. C₂×C₄×C₈×C₇₃	C₄	semidirect
82. C₂×#10596831	C₄⋊C₄	semidirect
85. C₈×D₇₃	C₄⋊C₄	semidirect
86. C₈×D₇₃	C₄⋊C₄	semidirect
100. C₂×C₈×C₇₃	C₄⋊C₄	semidirect
153. C₄×C₇₃	#427	semidirect
157. Dic₇₃	#427	semidirect
172. C₇₃	#20797	semidirect
187. C₂×C₄×C₈	#64408	semidirect
255. C₂×C₈	C₄⋊#64408	semidirect
323. C₄	#17989597	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17989597	2
#22626380	2
#22625868	2
#22625867	2
#22625866	2
C₂×C₄×C₈×D₇₃	2
#20797	73