Subgroups — #32991854

Order 18688.

802 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#11924012	C₂	1	1	yes	maximal, characteristic
2	#22612338	C₂	1	1	yes	maximal, characteristic
3	C₂×#17988358	C₂	1	4	yes	maximal
4	C₂×#423×C₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11924012	C₂²	1	2	yes	—
6	C₂×#14556550	C₂²	1	2	yes	—
7	C₄×#11924012	—	2	1	no	—
8	C₄×#11924012	—	2	1	no	—
9	C₄×#11924012	C₄	1	2	yes	complemented
10	C₄×#11924012	C₄	1	2	yes	complemented
11	C₂×#14556502	—	2	1	no	—
12	C₂×C₄×#10596831	—	2	1	no	—
13	#17988457	—	2	1	no	—
14	#17988453	—	2	1	no	—
15	#17988447	C₄	1	4	yes	—
16	C₄×C₄⋊Dic₇₃	—	2	2	no	—
17	#17988358	—	2	8	no	—
18	C₂²×M₁₆×C₇₃	C₂²	1	2	yes	—
19	C₂²×C₄²×C₇₃	C₂²	1	1	yes	characteristic
20	#423×C₇₃	—	2	1	no	—
21	#423×C₇₃	—	2	1	no	—
22	#423×C₇₃	—	2	2	no	—
23	C₂×C₄⋊C₈×C₇₃	C₄	1	4	yes	complemented
24	C₂×#11924012	—	2	2	no	—
25	C₂×#11924012	—	2	2	no	—
26	C₂×#11924012	—	2	2	no	—
27	C₂×#11924012	C₂×C₄	1	4	yes	—
28	C₂×#11924012	D₄	1	4	yes	—
29	C₂×#11924012	Q₈	1	4	yes	—
30	C₂²×#10596831	—	2	2	no	—
31	#14556550	—	2	4	no	—
32	#14556550	—	2	4	no	—
33	C₂×C₄⋊Dic₇₃	—	2	4	no	—
34	C₂×C₄⋊Dic₇₃	—	2	2	no	—
35	C₂×C₂²⋊Dic₇₃	C₂×C₄	1	4	yes	—
36	C₂×C₄×Dic₇₃	—	2	2	no	—
37	#14556502	—	2	2	no	—
38	#14556502	—	2	2	no	—
39	C₄×#10596831	—	2	2	no	—
40	C₄×#10596831	—	2	2	no	—
41	#14556498	—	2	4	no	—
42	C₂³×C₄×C₇₃	C₂×C₄	1	2	yes	—
43	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
44	C₂×M₁₆×C₇₃	—	2	4	no	—
45	C₂×M₁₆×C₇₃	—	2	2	no	—
46	C₂×M₁₆×C₇₃	—	2	2	no	—
47	C₂×M₁₆×C₇₃	—	2	4	no	—
48	C₂²×C₈×C₇₃	C₂×C₄	1	4	yes	—
49	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
53	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
54	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
55	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
56	C₂×C₄²×C₇₃	D₄	1	2	yes	—
57	C₂×C₄²×C₇₃	Q₈	1	2	yes	—
58	C₄⋊C₈×C₇₃	—	2	4	no	—
59	C₄⋊C₈×C₇₃	—	2	4	no	—
60	C₂²×Dic₇₃	—	2	4	no	—
61	#11924012	—	2	4	no	—
62	#11924012	—	2	4	no	—
63	#11924012	—	2	4	no	—
64	#11924012	C₄⋊C₄	1	8	yes	complemented
65	C₂×#10596831	—	2	4	no	—
66	C₂×#10596831	—	2	4	no	—
67	C₂×#10596831	—	2	4	no	—
68	C₄⋊Dic₇₃	—	4	8	no	—
69	C₄⋊Dic₇₃	—	4	4	no	—
70	C₂²⋊Dic₇₃	—	2	8	no	—
71	C₄×Dic₇₃	—	4	2	no	—
72	C₄×Dic₇₃	—	4	2	no	—
73	C₂⁴×C₇₃	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₇₃	—	4	4	no	—
75	C₂²×C₄×C₇₃	C₄²	1	2	yes	—
76	C₂²×C₄×C₇₃	C₄²	1	2	yes	—
77	C₂²×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
78	C₂²×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
79	C₂²×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
80	C₂²×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
81	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
84	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₇₃	C₂×Q₈	1	2	yes	—
88	C₂²×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
89	C₂²×C₄×C₇₃	#41	1	2	yes	—
90	C₂²×C₄×C₇₃	#41	1	2	yes	—
91	M₁₆×C₇₃	—	4	4	no	—
92	M₁₆×C₇₃	—	4	2	no	—
93	M₁₆×C₇₃	—	4	2	no	—
94	C₂×C₈×C₇₃	—	2	4	no	—
95	C₂×C₈×C₇₃	—	2	8	no	—
96	C₄²×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
97	C₄²×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
98	C₄²×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
99	C₄²×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
100	C₄²×C₇₃	C₂²⋊C₄	1	2	yes	—
101	C₄²×C₇₃	C₂²⋊C₄	1	2	yes	—
102	C₄²×C₇₃	C₄⋊C₄	1	2	yes	—
103	C₄²×C₇₃	C₄⋊C₄	1	2	yes	—
104	C₄²×C₇₃	C₄⋊C₄	1	2	yes	—
105	C₄²×C₇₃	C₄⋊C₄	1	2	yes	—
106	C₂³×C₇₃	—	4	2	no	—
107	C₂³×C₇₃	C₂×C₄²	1	1	yes	characteristic
108	C₂³×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂³×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂³×C₇₃	C₄×D₄	1	2	yes	—
111	C₂³×C₇₃	C₄×Q₈	1	2	yes	—
112	C₂×C₄×C₇₃	—	2	4	no	—
113	C₂×C₄×C₇₃	—	2	4	no	—
114	C₂×C₄×C₇₃	—	2	4	no	—
115	C₂×C₄×C₇₃	—	4	2	no	—
116	C₂×C₄×C₇₃	—	4	2	no	—
117	C₂×C₄×C₇₃	—	4	2	no	—
118	C₂×C₄×C₇₃	#95	1	2	yes	—
119	C₂×C₄×C₇₃	#95	1	2	yes	—
120	C₂×C₄×C₇₃	#95	1	2	yes	—
121	C₂×C₄×C₇₃	#95	1	2	yes	—
122	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
125	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₇₃	#117	1	2	yes	—
128	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
129	C₂×C₄×C₇₃	C₄×Q₈	1	2	yes	—
130	C₂×C₄×C₇₃	#121	1	2	yes	—
131	C₂×C₄×C₇₃	#121	1	2	yes	—
132	C₂×C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
133	C₂×C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
134	C₂×C₄×C₇₃	#124	1	1	yes	characteristic
135	C₂×C₄×C₇₃	#124	1	1	yes	characteristic
136	C₂×C₄×C₇₃	#125	1	2	yes	—
137	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
138	C₂×C₄×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
139	C₂×C₄×C₇₃	C₄⋊Q₈	1	2	yes	—
140	C₂×Dic₇₃	—	4	8	no	—
141	C₂×Dic₇₃	—	4	4	no	—
142	C₈×C₇₃	—	4	4	no	—
143	#10596831	—	2	8	no	—
144	C₂²×C₇₃	—	2	2	no	—
145	C₂²×C₇₃	—	2	2	no	—
146	C₂²×C₇₃	—	2	2	no	—
147	C₂²×C₇₃	—	4	2	no	—
148	C₂²×C₇₃	—	4	2	no	—
149	C₂²×C₇₃	C₂×#95	1	1	yes	characteristic
150	C₂²×C₇₃	C₄×C₄⋊C₄	1	2	yes	—
151	C₂²×C₇₃	#389	1	2	yes	—
152	C₂²×C₇₃	#390	1	1	yes	characteristic
153	C₂²×C₇₃	#391	1	1	yes	characteristic
154	C₄×C₇₃	—	2	2	no	—
155	C₄×C₇₃	—	2	2	no	—
156	C₄×C₇₃	—	2	2	no	—
157	C₄×C₇₃	#344	1	4	yes	complemented
158	C₄×C₇₃	#380	1	2	yes	—
159	C₄×C₇₃	#382	1	1	yes	characteristic
160	C₄×C₇₃	#384	1	1	yes	characteristic
161	C₄×C₇₃	#389	1	2	yes	—
162	C₄×C₇₃	#390	1	1	yes	characteristic
163	C₄×C₇₃	#391	1	1	yes	characteristic
164	Dic₇₃	—	8	4	no	—
165	#10855	—	73	1	no	maximal, Sylow
166	C₂×C₇₃	—	2	1	no	—
167	C₂×C₇₃	—	2	2	no	—
168	C₂×C₇₃	—	2	2	no	—
169	C₂×C₇₃	—	2	1	no	—
170	C₂×C₇₃	C₄⋊#95	1	1	yes	characteristic
171	C₂×C₇₃	C₂×#344	1	2	yes	—
172	C₂×#423	D₇₃	1	1	yes	characteristic
173	C₂×C₄×M₁₆	—	73	1	no	—
174	#2285	—	73	1	no	—
175	C₂×#344	—	73	4	no	—
176	C₇₃	#10855	1	1	yes	characteristic, complemented, Sylow
177	C₂²×M₁₆	—	73	2	no	—
178	C₂²×M₁₆	C₂×D₇₃	1	2	yes	—
179	C₂×#117	—	73	2	no	—
180	C₂²×C₄²	C₂×D₇₃	1	1	yes	characteristic
181	#423	—	2	1	no	—
182	#423	—	2	1	no	—
183	#423	—	2	2	no	—
184	C₂×C₄⋊C₈	Dic₇₃	1	4	yes	complemented
185	C₄×M₁₆	—	73	2	no	—
186	C₄×M₁₆	—	73	2	no	—
187	C₄×M₁₆	—	146	1	no	—
188	C₄×M₁₆	—	146	1	no	—
189	C₂×#97	—	146	1	no	—
190	C₂×C₄×C₈	—	146	1	no	—
191	#384	—	146	1	no	—
192	#382	—	146	1	no	—
193	#380	—	73	4	no	—
194	C₄×C₄⋊C₄	—	146	2	no	—
195	#344	—	146	8	no	—
196	C₂³×C₄	C₄×D₇₃	1	2	yes	—
197	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
198	C₂×M₁₆	—	2	4	no	—
199	C₂×M₁₆	—	2	2	no	—
200	C₂×M₁₆	—	2	2	no	—
201	C₂×M₁₆	—	2	4	no	—
202	C₂×M₁₆	—	73	4	no	—
203	C₂×M₁₆	—	73	4	no	—
204	C₂×M₁₆	—	73	4	no	—
205	C₂×M₁₆	—	146	2	no	—
206	C₂×M₁₆	—	146	2	no	—
207	C₂×M₁₆	—	146	2	no	—
208	C₂²×C₈	—	146	2	no	—
209	C₂²×C₈	C₂×Dic₇₃	1	4	yes	—
210	#117	—	146	4	no	—
211	#117	—	146	4	no	—
212	C₂×C₄⋊C₄	—	146	4	no	—
213	C₂×C₄⋊C₄	—	146	2	no	—
214	C₂×C₂²⋊C₄	—	73	4	no	—
215	C₂×C₄²	—	146	2	no	—
216	C₂×C₄²	Dic₁₄₆	1	2	yes	—
217	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
218	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
219	C₂×C₄²	D₂₉₂	1	1	yes	characteristic
220	C₂×C₄²	D₂₉₂	1	1	yes	characteristic
221	C₂×C₄²	C₂×Dic₇₃	1	2	yes	—
222	C₂×C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
223	C₂×C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
224	C₂×C₄²	C₂²⋊D₇₃	1	2	yes	—
225	C₄⋊C₈	—	2	4	no	—
226	C₄⋊C₈	—	2	4	no	—
227	#97	—	146	2	no	—
228	#97	—	146	2	no	—
229	C₄×C₈	—	146	2	no	—
230	C₄×C₈	—	146	2	no	—
231	#95	—	146	4	no	—
232	C₂⁴	C₂×C₄×D₇₃	1	1	yes	characteristic
233	C₂²×C₄	—	4	4	no	—
234	C₂²×C₄	—	146	4	no	—
235	C₂²×C₄	C₄×Dic₇₃	1	2	yes	—
236	C₂²×C₄	C₄×Dic₇₃	1	2	yes	—
237	C₂²×C₄	#11924005	1	2	yes	—
238	C₂²×C₄	#11924005	1	2	yes	—
239	C₂²×C₄	#11924007	1	2	yes	—
240	C₂²×C₄	#11924007	1	2	yes	—
241	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
242	C₂²×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
243	C₂²×C₄	D₄×D₇₃	1	2	yes	—
244	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
245	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
246	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
247	C₂²×C₄	Q₈×D₇₃	1	2	yes	—
248	C₂²×C₄	C₂×Dic₁₄₆	1	1	yes	characteristic
249	C₂²×C₄	#11924027	1	2	yes	—
250	C₂²×C₄	#11924029	1	2	yes	—
251	M₁₆	—	4	4	no	—
252	M₁₆	—	4	2	no	—
253	M₁₆	—	4	2	no	—
254	M₁₆	—	73	8	no	—
255	M₁₆	—	146	4	no	—
256	M₁₆	—	146	4	no	—
257	M₁₆	—	146	4	no	—
258	C₂×C₈	—	2	4	no	—
259	C₂×C₈	—	2	8	no	—
260	C₂×C₈	—	146	4	no	—
261	C₂×C₈	—	146	4	no	—
262	C₂×C₈	—	146	4	no	—
263	C₄⋊C₄	—	292	8	no	—
264	C₄⋊C₄	—	292	4	no	—
265	C₂²⋊C₄	—	146	8	no	—
266	C₄²	—	292	2	no	—
267	C₄²	—	292	2	no	—
268	C₄²	#11924005	1	1	yes	characteristic
269	C₄²	#11924005	1	1	yes	characteristic
270	C₄²	#11924005	1	1	yes	characteristic
271	C₄²	#11924005	1	1	yes	characteristic
272	C₄²	C₂²⋊Dic₇₃	1	2	yes	—
273	C₄²	C₂²⋊Dic₇₃	1	2	yes	—
274	C₄²	#11924007	1	2	yes	—
275	C₄²	#11924007	1	2	yes	—
276	C₄²	C₄⋊Dic₇₃	1	2	yes	—
277	C₄²	C₄⋊Dic₇₃	1	2	yes	—
278	C₂³	—	4	2	no	—
279	C₂³	C₂×C₄×Dic₇₃	1	1	yes	characteristic
280	C₂³	C₂×#11924005	1	1	yes	characteristic
281	C₂³	C₂×#11924007	1	1	yes	characteristic
282	C₂³	#14556553	1	2	yes	—
283	C₂³	#14556556	1	2	yes	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	2	4	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	292	8	no	—
291	C₂×C₄	—	292	4	no	—
292	C₂×C₄	#14556498	1	2	yes	—
293	C₂×C₄	#14556498	1	2	yes	—
294	C₂×C₄	#14556498	1	2	yes	—
295	C₂×C₄	#14556498	1	2	yes	—
296	C₂×C₄	C₂×#11924005	1	1	yes	characteristic
297	C₂×C₄	C₂×C₂²⋊Dic₇₃	1	1	yes	characteristic
298	C₂×C₄	C₄⋊C₄×D₇₃	1	2	yes	—
299	C₂×C₄	C₂×#11924007	1	1	yes	characteristic
300	C₂×C₄	C₂×C₄⋊Dic₇₃	1	1	yes	characteristic
301	C₂×C₄	#14556549	1	2	yes	—
302	C₂×C₄	D₄×Dic₇₃	1	2	yes	—
303	C₂×C₄	Q₈×Dic₇₃	1	2	yes	—
304	C₂×C₄	#14556562	1	2	yes	—
305	C₂×C₄	#14556566	1	2	yes	—
306	C₂×C₄	#14556567	1	2	yes	—
307	C₂×C₄	#14556571	1	2	yes	—
308	C₂×C₄	#14556578	1	1	yes	characteristic
309	C₂×C₄	#14556580	1	1	yes	characteristic
310	C₂×C₄	#14556582	1	2	yes	—
311	C₂×C₄	#14556587	1	1	yes	characteristic
312	C₂×C₄	#14556589	1	1	yes	characteristic
313	C₂×C₄	#14556590	1	2	yes	—
314	C₈	—	4	4	no	—
315	C₈	—	146	8	no	—
316	C₂²	—	2	2	no	—
317	C₂²	—	2	2	no	—
318	C₂²	—	2	2	no	—
319	C₂²	—	4	2	no	—
320	C₂²	—	4	2	no	—
321	C₂²	C₂×#14556498	1	1	yes	characteristic
322	C₂²	C₄⋊C₄×Dic₇₃	1	2	yes	—
323	C₂²	#17988483	1	2	yes	—
324	C₂²	#17988488	1	1	yes	characteristic
325	C₂²	#17988492	1	1	yes	characteristic
326	C₄	—	2	2	no	—
327	C₄	—	2	2	no	—
328	C₄	—	2	2	no	—
329	C₄	—	584	4	no	—
330	C₄	#17988358	1	4	yes	complemented
331	C₄	#17988444	1	2	yes	—
332	C₄	#17988452	1	1	yes	characteristic
333	C₄	#17988456	1	1	yes	characteristic
334	C₄	#17988486	1	2	yes	—
335	C₄	#17988490	1	1	yes	characteristic
336	C₄	#17988494	1	1	yes	characteristic
337	C₂	—	2	1	no	—
338	C₂	—	2	2	no	—
339	C₂	—	2	2	no	—
340	C₂	—	2	1	no	—
341	C₂	#22608160	1	1	yes	characteristic
342	C₂	C₂×#17988358	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₇₃
commutator	C₂³×C₇₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₇₃
Fitting	C₂×#423×C₇₃	C₂
socle	C₂²×C₇₃	C₂×#95
prime radical	C₂⁴×C₇₃	C₂²×C₄
cosocle	C₂³×C₄×C₇₃	C₂³
hypercenter	C₂×#423	D₇₃
hypocenter	C₇₃	#10855
LCS	G ▷ C₂³×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
9. C₄×#11924012	C₄	semidirect
10. C₄×#11924012	C₄	semidirect
23. C₂×C₄⋊C₈×C₇₃	C₄	semidirect
64. #11924012	C₄⋊C₄	semidirect
157. C₄×C₇₃	#344	semidirect
176. C₇₃	#10855	semidirect
184. C₂×C₄⋊C₈	Dic₇₃	semidirect
330. C₄	#17988358	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#11924012	2
#22612338	2
C₂×#17988358	2
C₂×#423×C₇₃	2
#10855	73