Subgroups — #32991861

Order 18688.

918 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988611	C₂	1	1	yes	maximal, characteristic
2	C₂×#17988457	C₂	1	1	yes	maximal, characteristic
3	C₂×#17988355	C₂	1	4	yes	maximal
4	C₂×C₄×M₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11924012	C₂²	1	2	yes	—
6	C₂²×C₄⋊Dic₇₃	C₄	1	4	yes	—
7	C₂²×C₄⋊Dic₇₃	C₂²	1	2	yes	—
8	#17988611	—	2	2	no	—
9	#17988611	C₄	1	4	yes	complemented
10	C₂×C₄⋊#10596831	—	2	2	no	—
11	#17988457	—	2	2	no	—
12	#17988457	—	2	1	no	—
13	#17988457	—	2	1	no	—
14	#17988355	—	2	8	no	—
15	C₂²×M₁₆×C₇₃	C₂²	1	2	yes	—
16	C₂²×C₄²×C₇₃	C₂²	1	1	yes	characteristic
17	C₄×M₁₆×C₇₃	—	2	2	no	—
18	C₄×M₁₆×C₇₃	C₄	1	4	yes	complemented
19	C₂×#97×C₇₃	—	2	1	no	—
20	C₂×C₄×C₈×C₇₃	—	2	1	no	—
21	C₂³×Dic₇₃	C₂×C₄	1	4	yes	—
22	C₂×#11924012	—	2	2	no	—
23	C₂×#11924012	—	2	4	no	—
24	C₂×#11924012	—	2	4	no	—
25	C₂×#11924012	C₂×C₄	1	4	yes	—
26	C₂²×#10596831	—	2	2	no	—
27	C₂×C₄⋊Dic₇₃	—	2	4	no	—
28	C₂×C₄⋊Dic₇₃	—	2	4	no	—
29	C₂×C₄⋊Dic₇₃	—	2	4	no	—
30	C₂×C₄⋊Dic₇₃	—	2	2	no	—
31	C₂×C₄⋊Dic₇₃	—	2	2	no	—
32	C₂×C₄⋊Dic₇₃	—	2	4	no	—
33	C₂×C₄⋊Dic₇₃	—	4	8	no	—
34	C₄⋊#10596831	—	2	8	no	—
35	C₂³×C₄×C₇₃	C₂×C₄	1	2	yes	—
36	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
37	C₂×M₁₆×C₇₃	—	2	2	no	—
38	C₂×M₁₆×C₇₃	—	2	2	no	—
39	C₂×M₁₆×C₇₃	—	2	2	no	—
40	C₂×M₁₆×C₇₃	C₂×C₄	1	4	yes	—
41	C₂×M₁₆×C₇₃	D₄	1	4	yes	—
42	C₂×M₁₆×C₇₃	Q₈	1	4	yes	—
43	C₂²×C₈×C₇₃	—	2	2	no	—
44	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₇₃	D₄	1	2	yes	—
47	C₂×C₄²×C₇₃	D₄	1	2	yes	—
48	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₇₃	Q₈	1	2	yes	—
51	#97×C₇₃	—	2	4	no	—
52	C₄×C₈×C₇₃	—	2	4	no	—
53	C₂²×Dic₇₃	—	2	4	no	—
54	C₂²×Dic₇₃	—	2	8	no	—
55	C₂²×Dic₇₃	—	4	8	no	—
56	#11924012	—	2	8	no	—
57	#11924012	—	4	4	no	—
58	C₂×#10596831	—	2	4	no	—
59	C₂×#10596831	—	4	2	no	—
60	C₂×#10596831	—	4	2	no	—
61	C₄⋊Dic₇₃	—	4	8	no	—
62	C₄⋊Dic₇₃	—	4	8	no	—
63	C₄⋊Dic₇₃	—	4	4	no	—
64	C₄⋊Dic₇₃	—	4	2	no	—
65	C₄⋊Dic₇₃	—	4	2	no	—
66	C₂⁴×C₇₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₇₃	—	2	4	no	—
68	C₂²×C₄×C₇₃	—	2	4	no	—
69	C₂²×C₄×C₇₃	C₄²	1	4	yes	—
70	C₂²×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
71	C₂²×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
72	C₂²×C₄×C₇₃	C₄⋊C₄	1	4	yes	—
73	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
80	C₂²×C₄×C₇₃	C₂×Q₈	1	2	yes	—
81	C₂²×C₄×C₇₃	#41	1	2	yes	—
82	C₂²×C₄×C₇₃	#41	1	2	yes	—
83	M₁₆×C₇₃	—	2	4	no	—
84	M₁₆×C₇₃	—	2	4	no	—
85	M₁₆×C₇₃	—	2	4	no	—
86	M₁₆×C₇₃	C₄⋊C₄	1	8	yes	complemented
87	C₂×C₈×C₇₃	—	2	4	no	—
88	C₂×C₈×C₇₃	—	2	4	no	—
89	C₂×C₈×C₇₃	—	2	4	no	—
90	C₄²×C₇₃	C₂²⋊C₄	1	4	yes	—
91	C₄²×C₇₃	C₂²⋊C₄	1	2	yes	—
92	C₄²×C₇₃	C₂²⋊C₄	1	2	yes	—
93	C₄²×C₇₃	C₄⋊C₄	1	2	yes	—
94	C₄²×C₇₃	C₄⋊C₄	1	2	yes	—
95	C₄²×C₇₃	C₄⋊C₄	1	4	yes	—
96	C₂³×C₇₃	—	2	2	no	—
97	C₂³×C₇₃	—	2	2	no	—
98	C₂³×C₇₃	C₂×C₄²	1	1	yes	characteristic
99	C₂³×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂³×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂³×C₇₃	C₄×D₄	1	2	yes	—
102	C₂³×C₇₃	C₄×Q₈	1	2	yes	—
103	C₂×C₄×C₇₃	—	2	2	no	—
104	C₂×C₄×C₇₃	—	2	2	no	—
105	C₂×C₄×C₇₃	—	2	8	no	—
106	C₂×C₄×C₇₃	—	2	8	no	—
107	C₂×C₄×C₇₃	#95	1	4	yes	—
108	C₂×C₄×C₇₃	#95	1	4	yes	—
109	C₂×C₄×C₇₃	#100	1	4	yes	—
110	C₂×C₄×C₇₃	#101	1	4	yes	—
111	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
115	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₇₃	#117	1	2	yes	—
117	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
118	C₂×C₄×C₇₃	C₄×Q₈	1	2	yes	—
119	C₂×C₄×C₇₃	#121	1	2	yes	—
120	C₂×C₄×C₇₃	#121	1	2	yes	—
121	C₂×C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
122	C₂×C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
123	C₂×C₄×C₇₃	#124	1	1	yes	characteristic
124	C₂×C₄×C₇₃	#124	1	1	yes	characteristic
125	C₂×C₄×C₇₃	#125	1	2	yes	—
126	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
127	C₂×C₄×C₇₃	C₄⋊Q₈	1	2	yes	—
128	C₂×C₄×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
129	C₂×Dic₇₃	—	4	4	no	—
130	C₂×Dic₇₃	—	4	8	no	—
131	C₂×Dic₇₃	—	8	8	no	—
132	C₈×C₇₃	—	2	8	no	—
133	#10596831	—	4	4	no	—
134	C₂²×C₇₃	—	2	2	no	—
135	C₂²×C₇₃	—	2	2	no	—
136	C₂²×C₇₃	—	2	2	no	—
137	C₂²×C₇₃	—	2	2	no	—
138	C₂²×C₇₃	—	2	4	no	—
139	C₂²×C₇₃	C₂×#95	1	1	yes	characteristic
140	C₂²×C₇₃	C₄×C₄⋊C₄	1	2	yes	—
141	C₂²×C₇₃	#389	1	2	yes	—
142	C₂²×C₇₃	#390	1	1	yes	characteristic
143	C₂²×C₇₃	#391	1	1	yes	characteristic
144	C₂²×C₇₃	C₂×#100	1	2	yes	—
145	C₂²×C₇₃	C₂×#101	1	2	yes	—
146	C₄×C₇₃	—	2	4	no	—
147	C₄×C₇₃	#343	1	8	yes	complemented
148	C₄×C₇₃	#380	1	2	yes	—
149	C₄×C₇₃	#382	1	1	yes	characteristic
150	C₄×C₇₃	#384	1	1	yes	characteristic
151	C₄×C₇₃	#389	1	2	yes	—
152	C₄×C₇₃	#390	1	1	yes	characteristic
153	C₄×C₇₃	#391	1	1	yes	characteristic
154	Dic₇₃	—	8	4	no	—
155	#10857	—	73	1	no	maximal, Sylow
156	C₂×C₇₃	—	2	2	no	—
157	C₂×C₇₃	—	2	2	no	—
158	C₂×C₇₃	C₄⋊#95	1	1	yes	characteristic
159	C₂×C₇₃	C₂×#343	1	2	yes	—
160	C₂×C₇₃	#1965	1	1	yes	characteristic
161	C₂×C₇₃	#1966	1	1	yes	characteristic
162	C₂×C₇₃	#1997	1	2	yes	—
163	C₂×#423	—	73	1	no	—
164	C₂×C₄×M₁₆	D₇₃	1	1	yes	characteristic
165	C₂×#384	—	73	1	no	—
166	C₂×#343	—	73	4	no	—
167	C₇₃	#10857	1	1	yes	characteristic, complemented, Sylow
168	C₂²×M₁₆	—	73	2	no	—
169	C₂²×M₁₆	C₂×D₇₃	1	2	yes	—
170	C₂²×C₄⋊C₄	—	73	4	no	—
171	C₂²×C₄⋊C₄	—	73	2	no	—
172	C₂²×C₄²	C₂×D₇₃	1	1	yes	characteristic
173	#423	—	73	4	no	—
174	#423	—	146	2	no	—
175	C₂×C₄⋊C₈	—	146	2	no	—
176	C₄×M₁₆	—	2	2	no	—
177	C₄×M₁₆	Dic₇₃	1	4	yes	complemented
178	C₂×#97	—	2	1	no	—
179	C₂×C₄×C₈	—	2	1	no	—
180	#384	—	146	2	no	—
181	#384	—	146	1	no	—
182	#384	—	146	1	no	—
183	#343	—	146	8	no	—
184	C₂³×C₄	—	73	4	no	—
185	C₂³×C₄	C₄×D₇₃	1	2	yes	—
186	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
187	C₂×M₁₆	—	2	2	no	—
188	C₂×M₁₆	—	2	2	no	—
189	C₂×M₁₆	—	2	2	no	—
190	C₂×M₁₆	—	73	4	no	—
191	C₂×M₁₆	—	146	4	no	—
192	C₂×M₁₆	—	146	4	no	—
193	C₂×M₁₆	—	146	2	no	—
194	C₂×M₁₆	Dic₁₄₆	1	4	yes	—
195	C₂×M₁₆	D₂₉₂	1	4	yes	—
196	C₂×M₁₆	C₂×Dic₇₃	1	4	yes	—
197	C₂²×C₈	—	2	2	no	—
198	C₂²×C₈	—	146	2	no	—
199	C₂×C₄⋊C₄	—	292	8	no	—
200	C₂×C₄⋊C₄	—	146	4	no	—
201	C₂×C₄⋊C₄	—	146	4	no	—
202	C₂×C₄⋊C₄	—	146	2	no	—
203	C₂×C₄⋊C₄	—	146	4	no	—
204	C₂×C₄⋊C₄	—	146	2	no	—
205	C₂×C₄⋊C₄	—	146	4	no	—
206	C₂×C₄²	Dic₁₄₆	1	2	yes	—
207	C₂×C₄²	C₄×D₇₃	1	2	yes	—
208	C₂×C₄²	D₂₉₂	1	2	yes	—
209	C₂×C₄²	C₂×Dic₇₃	1	2	yes	—
210	C₂×C₄²	C₂²⋊D₇₃	1	2	yes	—
211	C₂×C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
212	C₂×C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
213	C₄⋊C₈	—	146	8	no	—
214	#97	—	2	4	no	—
215	C₄×C₈	—	2	4	no	—
216	C₂⁴	C₂×C₄×D₇₃	1	1	yes	characteristic
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	292	8	no	—
220	C₂²×C₄	—	146	4	no	—
221	C₂²×C₄	—	146	8	no	—
222	C₂²×C₄	C₄×Dic₇₃	1	4	yes	—
223	C₂²×C₄	#11924005	1	2	yes	—
224	C₂²×C₄	#11924005	1	2	yes	—
225	C₂²×C₄	#11924007	1	4	yes	—
226	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
227	C₂²×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
228	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
229	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
230	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
231	C₂²×C₄	C₂×D₂₉₂	1	2	yes	—
232	C₂²×C₄	C₂×Dic₁₄₆	1	1	yes	characteristic
233	C₂²×C₄	C₂×Dic₁₄₆	1	2	yes	—
234	C₂²×C₄	#11924028	1	2	yes	—
235	C₂²×C₄	#11924028	1	2	yes	—
236	M₁₆	—	2	4	no	—
237	M₁₆	—	2	4	no	—
238	M₁₆	—	2	4	no	—
239	M₁₆	—	292	4	no	—
240	M₁₆	—	146	8	no	—
241	M₁₆	C₄⋊Dic₇₃	1	8	yes	complemented
242	C₂×C₈	—	2	4	no	—
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	—	2	4	no	—
245	C₂×C₈	—	292	2	no	—
246	C₂×C₈	—	292	2	no	—
247	C₂×C₈	—	146	4	no	—
248	C₄⋊C₄	—	292	8	no	—
249	C₄⋊C₄	—	292	8	no	—
250	C₄⋊C₄	—	292	4	no	—
251	C₄⋊C₄	—	292	2	no	—
252	C₄⋊C₄	—	292	2	no	—
253	C₄²	#11924005	1	4	yes	—
254	C₄²	C₂²⋊Dic₇₃	1	2	yes	—
255	C₄²	C₂²⋊Dic₇₃	1	2	yes	—
256	C₄²	#11924007	1	2	yes	—
257	C₄²	#11924007	1	2	yes	—
258	C₄²	C₄⋊Dic₇₃	1	4	yes	—
259	C₂³	—	2	2	no	—
260	C₂³	—	2	2	no	—
261	C₂³	C₂×C₄×Dic₇₃	1	1	yes	characteristic
262	C₂³	C₂×#11924005	1	1	yes	characteristic
263	C₂³	C₂×#11924007	1	1	yes	characteristic
264	C₂³	C₄×D₂₉₂	1	2	yes	—
265	C₂³	C₄×Dic₁₄₆	1	2	yes	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	8	no	—
269	C₂×C₄	—	2	8	no	—
270	C₂×C₄	—	292	4	no	—
271	C₂×C₄	—	292	8	no	—
272	C₂×C₄	—	584	8	no	—
273	C₂×C₄	#14556498	1	4	yes	—
274	C₂×C₄	#14556498	1	4	yes	—
275	C₂×C₄	#14556507	1	4	yes	—
276	C₂×C₄	#14556509	1	4	yes	—
277	C₂×C₄	C₂×#11924005	1	1	yes	characteristic
278	C₂×C₄	C₂×C₂²⋊Dic₇₃	1	1	yes	characteristic
279	C₂×C₄	C₂×#11924007	1	1	yes	characteristic
280	C₂×C₄	C₂×C₄⋊Dic₇₃	1	2	yes	—
281	C₂×C₄	C₂×C₄⋊Dic₇₃	1	1	yes	characteristic
282	C₂×C₄	#14556550	1	2	yes	—
283	C₂×C₄	C₄×D₂₉₂	1	2	yes	—
284	C₂×C₄	C₄×Dic₁₄₆	1	2	yes	—
285	C₂×C₄	#14556565	1	2	yes	—
286	C₂×C₄	#14556565	1	2	yes	—
287	C₂×C₄	#14556570	1	2	yes	—
288	C₂×C₄	#14556570	1	2	yes	—
289	C₂×C₄	#14556579	1	1	yes	characteristic
290	C₂×C₄	#14556579	1	1	yes	characteristic
291	C₂×C₄	#14556583	1	2	yes	—
292	C₂×C₄	#14556588	1	1	yes	characteristic
293	C₂×C₄	#14556591	1	2	yes	—
294	C₂×C₄	#14556591	1	1	yes	characteristic
295	C₈	—	2	8	no	—
296	C₈	—	292	4	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	2	4	no	—
302	C₂²	C₂×#14556498	1	1	yes	characteristic
303	C₂²	C₄×C₄⋊Dic₇₃	1	2	yes	—
304	C₂²	#17988485	1	2	yes	—
305	C₂²	#17988489	1	1	yes	characteristic
306	C₂²	#17988493	1	1	yes	characteristic
307	C₂²	C₂×#14556507	1	2	yes	—
308	C₂²	C₂×#14556509	1	2	yes	—
309	C₄	—	2	4	no	—
310	C₄	—	584	4	no	—
311	C₄	#17988355	1	8	yes	complemented
312	C₄	#17988447	1	2	yes	—
313	C₄	#17988453	1	1	yes	characteristic
314	C₄	#17988457	1	1	yes	characteristic
315	C₄	#17988485	1	2	yes	—
316	C₄	#17988489	1	1	yes	characteristic
317	C₄	#17988493	1	1	yes	characteristic
318	C₂	—	2	2	no	—
319	C₂	—	2	2	no	—
320	C₂	#22608161	1	1	yes	characteristic
321	C₂	C₂×#17988355	1	2	yes	—
322	C₂	#22610771	1	1	yes	characteristic
323	C₂	#22610775	1	1	yes	characteristic
324	C₂	#22610920	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#11924005
commutator	C₂³×C₇₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₇₃
Fitting	C₂×C₄×M₁₆×C₇₃	C₂
socle	C₂³×C₇₃	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₇₃	C₂²×C₄
cosocle	C₂³×C₄×C₇₃	C₂³
hypercenter	C₂×C₄×M₁₆	D₇₃
hypocenter	C₇₃	#10857
LCS	G ▷ C₂³×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
9. #17988611	C₄	semidirect
18. C₄×M₁₆×C₇₃	C₄	semidirect
86. M₁₆×C₇₃	C₄⋊C₄	semidirect
147. C₄×C₇₃	#343	semidirect
167. C₇₃	#10857	semidirect
177. C₄×M₁₆	Dic₇₃	semidirect
241. M₁₆	C₄⋊Dic₇₃	semidirect
311. C₄	#17988355	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17988611	2
C₂×#17988457	2
C₂×#17988355	2
C₂×C₄×M₁₆×C₇₃	2
#10857	73