Subgroups — #34003529

Order 18688.

610 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22625717	C₂	1	1	yes	maximal, characteristic
2	C₄×#14556646	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₈⋊#10596833	C₂	1	2	yes	maximal
4	#22624767	C₂	1	2	yes	maximal
5	C₄×Q₈×Dic₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#10596833	C₂²	1	2	yes	—
7	C₂×#14556646	—	2	1	no	—
8	C₂×#14556646	—	2	1	no	—
9	C₂×#14556646	—	2	1	no	—
10	C₂×C₄×#10596833	C₂²	1	2	yes	—
11	#17989453	—	2	1	no	—
12	Q₈⋊#10596833	C₄	1	8	yes	complemented
13	C₂×Q₈×Dic₇₃	C₄	1	1	yes	characteristic
14	C₂×Q₈×Dic₇₃	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×Dic₇₃	—	2	1	no	—
16	C₄⋊C₄×Dic₇₃	—	2	1	no	—
17	C₄⋊C₄×Dic₇₃	C₄	1	1	yes	characteristic
18	C₄⋊C₄×Dic₇₃	C₂²	1	1	yes	characteristic
19	C₄×C₄⋊Dic₇₃	—	2	1	no	—
20	C₄×C₄⋊Dic₇₃	C₄	1	1	yes	characteristic
21	C₄²×Dic₇₃	—	2	1	no	—
22	C₄²×Dic₇₃	C₂²	1	1	yes	characteristic
23	C₂×C₄×Q₈×C₇₃	C₄	1	1	yes	characteristic
24	C₂²×#10596833	—	2	2	no	—
25	C₂²×#10596833	—	2	2	no	—
26	C₄⋊#10596833	—	2	2	no	—
27	C₄⋊#10596833	C₂×C₄	1	4	yes	—
28	#14556646	—	2	4	no	—
29	#14556646	—	2	4	no	—
30	#14556646	—	4	2	no	—
31	C₄×#10596833	—	2	2	no	—
32	C₄×#10596833	C₂×C₄	1	4	yes	—
33	Q₈×Dic₇₃	—	2	2	no	—
34	Q₈×Dic₇₃	—	2	2	no	—
35	Q₈×Dic₇₃	—	2	2	no	—
36	Q₈×Dic₇₃	C₂×C₄	1	4	yes	—
37	C₂×C₄⋊Dic₇₃	—	2	1	no	—
38	C₂×C₄⋊Dic₇₃	—	2	1	no	—
39	C₂×C₄⋊Dic₇₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄⋊Dic₇₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×Dic₇₃	—	2	1	no	—
42	C₂×C₄×Dic₇₃	—	2	1	no	—
43	C₂×C₄×Dic₇₃	—	2	1	no	—
44	C₂×C₄×Dic₇₃	—	2	1	no	—
45	C₂×C₄×Dic₇₃	—	2	1	no	—
46	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×Dic₇₃	D₄	1	1	yes	characteristic
48	C₂×C₄×Dic₇₃	D₄	1	1	yes	characteristic
49	C₂×C₄×Dic₇₃	D₄	1	1	yes	characteristic
50	C₂×C₄×Dic₇₃	D₄	1	1	yes	characteristic
51	C₂×C₄×Dic₇₃	C₂³	1	1	yes	characteristic
52	#14556498	—	2	1	no	—
53	#14556498	—	2	1	no	—
54	#14556498	—	2	1	no	—
55	C₂²×Q₈×C₇₃	C₂×C₄	1	1	yes	characteristic
56	C₄×Q₈×C₇₃	—	2	2	no	—
57	C₄×Q₈×C₇₃	—	2	2	no	—
58	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
59	C₂×C₄⋊C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄²×C₇₃	—	2	1	no	—
61	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
62	C₂×#10596833	—	2	4	no	—
63	C₂×#10596833	—	2	4	no	—
64	C₂×#10596833	—	2	4	no	—
65	C₂×#10596833	—	2	4	no	—
66	C₂×#10596833	—	4	2	no	—
67	C₂×#10596833	—	4	2	no	—
68	C₂²×Dic₇₃	—	2	1	no	—
69	C₂²×Dic₇₃	—	2	1	no	—
70	C₂²×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×Dic₇₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×Dic₇₃	C₂×D₄	1	1	yes	characteristic
74	C₄⋊Dic₇₃	—	2	4	no	—
75	C₄⋊Dic₇₃	—	2	1	no	—
76	C₄⋊Dic₇₃	—	2	1	no	—
77	C₄⋊Dic₇₃	—	2	4	no	—
78	C₄⋊Dic₇₃	C₄²	1	2	yes	—
79	C₄⋊Dic₇₃	M₁₆	1	2	yes	—
80	C₄×Dic₇₃	—	2	4	no	—
81	C₄×Dic₇₃	—	2	2	no	—
82	C₄×Dic₇₃	—	2	1	no	—
83	C₄×Dic₇₃	—	2	1	no	—
84	C₄×Dic₇₃	—	2	2	no	—
85	C₄×Dic₇₃	—	2	4	no	—
86	C₄×Dic₇₃	—	2	1	no	—
87	C₄×Dic₇₃	—	2	2	no	—
88	C₄×Dic₇₃	—	2	1	no	—
89	C₄×Dic₇₃	—	4	2	no	—
90	C₄×Dic₇₃	—	4	2	no	—
91	C₄×Dic₇₃	—	4	1	no	—
92	C₄×Dic₇₃	—	4	1	no	—
93	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
94	C₄×Dic₇₃	#41	1	2	yes	—
95	C₂×Q₈×C₇₃	—	2	1	no	—
96	C₂×Q₈×C₇₃	—	2	1	no	—
97	C₂×Q₈×C₇₃	—	2	1	no	—
98	C₂×Q₈×C₇₃	C₄²	1	2	yes	—
99	C₂×Q₈×C₇₃	M₁₆	1	2	yes	—
100	C₂×Q₈×C₇₃	M₁₆	1	2	yes	—
101	C₂²×C₄×C₇₃	—	2	1	no	—
102	C₂²×C₄×C₇₃	—	2	1	no	—
103	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
104	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
105	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
106	C₄⋊C₄×C₇₃	—	2	4	no	—
107	C₄⋊C₄×C₇₃	—	2	1	no	—
108	C₄⋊C₄×C₇₃	M₁₆	1	2	yes	—
109	C₄²×C₇₃	—	2	4	no	—
110	C₄²×C₇₃	—	2	1	no	—
111	C₄²×C₇₃	—	2	1	no	—
112	C₂³×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	Q₈×C₇₃	—	2	2	no	—
114	Q₈×C₇₃	—	2	2	no	—
115	Q₈×C₇₃	—	2	2	no	—
116	Q₈×C₇₃	#97	1	4	yes	complemented
117	C₂×C₄×C₇₃	—	2	2	no	—
118	C₂×C₄×C₇₃	—	2	1	no	—
119	C₂×C₄×C₇₃	—	2	2	no	—
120	C₂×C₄×C₇₃	—	2	2	no	—
121	C₂×C₄×C₇₃	—	2	1	no	—
122	C₂×C₄×C₇₃	—	2	1	no	—
123	C₂×C₄×C₇₃	—	2	2	no	—
124	C₂×C₄×C₇₃	—	2	1	no	—
125	C₂×C₄×C₇₃	—	2	1	no	—
126	C₂×C₄×C₇₃	—	4	1	no	—
127	C₂×C₄×C₇₃	—	4	1	no	—
128	C₂×C₄×C₇₃	#104	1	2	yes	—
129	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
130	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
131	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
132	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
133	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
134	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
135	C₂×Dic₇₃	—	2	1	no	—
136	C₂×Dic₇₃	—	2	1	no	—
137	C₂×Dic₇₃	—	2	1	no	—
138	C₂×Dic₇₃	—	2	1	no	—
139	C₂×Dic₇₃	—	2	2	no	—
140	C₂×Dic₇₃	—	2	1	no	—
141	C₂×Dic₇₃	—	2	2	no	—
142	C₂×Dic₇₃	—	2	1	no	—
143	C₂×Dic₇₃	—	2	1	no	—
144	C₂×Dic₇₃	—	4	1	no	—
145	C₂×Dic₇₃	—	4	1	no	—
146	C₂×Dic₇₃	—	4	1	no	—
147	C₂×Dic₇₃	—	4	1	no	—
148	C₂×Dic₇₃	#104	1	2	yes	—
149	C₂×Dic₇₃	C₄×D₄	1	2	yes	—
150	C₂×Dic₇₃	C₄×D₄	1	2	yes	—
151	C₂×Dic₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
152	C₂×Dic₇₃	#137	1	2	yes	—
153	C₂×Dic₇₃	#137	1	1	yes	characteristic
154	#10596833	—	2	8	no	—
155	#10596833	—	4	4	no	—
156	C₂²×C₇₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
157	C₂²×C₇₃	#407	1	1	yes	characteristic
158	C₂²×C₇₃	#407	1	1	yes	characteristic
159	C₂²×C₇₃	#417	1	1	yes	characteristic
160	C₂²×C₇₃	#419	1	1	yes	characteristic
161	C₂²×C₇₃	C₂×#104	1	1	yes	characteristic
162	C₂²×C₇₃	C₂×#104	1	1	yes	characteristic
163	C₄×C₇₃	—	2	2	no	—
164	C₄×C₇₃	—	2	4	no	—
165	C₄×C₇₃	—	4	1	no	—
166	C₄×C₇₃	—	4	2	no	—
167	C₄×C₇₃	C₂×#97	1	1	yes	characteristic
168	C₄×C₇₃	C₄×M₁₆	1	1	yes	characteristic
169	C₄×C₇₃	#432	1	1	yes	characteristic
170	C₄×C₇₃	#432	1	1	yes	characteristic
171	Dic₇₃	—	2	2	no	—
172	Dic₇₃	—	2	2	no	—
173	Dic₇₃	—	4	1	no	—
174	Dic₇₃	—	4	2	no	—
175	Dic₇₃	—	4	2	no	—
176	Dic₇₃	#440	1	2	yes	—
177	Dic₇₃	#441	1	2	yes	—
178	#8173	—	73	1	no	maximal, Sylow
179	C₂×C₇₃	#1489	1	1	yes	characteristic
180	C₂×C₇₃	#1489	1	1	yes	characteristic
181	C₂×C₇₃	#1491	1	1	yes	characteristic
182	C₂×C₇₃	#1491	1	1	yes	characteristic
183	C₂×C₇₃	#1749	1	1	yes	characteristic
184	C₂×C₇₃	C₄×#104	1	1	yes	characteristic
185	C₂×C₇₃	#1763	1	1	yes	characteristic
186	C₄²×Q₈	—	73	1	no	—
187	#1767	—	73	1	no	—
188	C₄×#97	—	73	1	no	—
189	C₂×Q₈⋊C₈	—	73	2	no	—
190	#1279	—	73	2	no	—
191	C₇₃	#8173	1	1	yes	characteristic, complemented, Sylow
192	C₂×C₄×Q₈	—	73	1	no	—
193	C₂×C₄×Q₈	—	73	1	no	—
194	C₂×C₄×Q₈	#64408	1	1	yes	characteristic
195	C₂×C₄⋊C₈	—	73	2	no	—
196	C₂×#97	—	146	1	no	—
197	C₂×#97	—	146	1	no	—
198	C₂×#97	—	146	1	no	—
199	C₂×C₄×C₈	—	73	2	no	—
200	C₄×C₄⋊C₄	—	73	1	no	—
201	C₄×C₄⋊C₄	—	73	1	no	—
202	C₄×C₄⋊C₄	—	73	1	no	—
203	C₄×C₄⋊C₄	—	146	1	no	—
204	C₄×C₄⋊C₄	—	146	1	no	—
205	C₄×C₄⋊C₄	—	146	1	no	—
206	C₄³	—	73	1	no	—
207	C₄³	—	146	1	no	—
208	#336	—	146	1	no	—
209	Q₈⋊C₈	—	73	8	no	—
210	C₂²×Q₈	C₂×#64408	1	1	yes	characteristic
211	C₂²×C₈	—	146	2	no	—
212	C₂²×C₈	—	146	2	no	—
213	C₄×Q₈	—	2	2	no	—
214	C₄×Q₈	—	2	2	no	—
215	C₄×Q₈	—	73	4	no	—
216	C₄×Q₈	—	146	2	no	—
217	C₄×Q₈	—	146	2	no	—
218	C₄×Q₈	—	146	2	no	—
219	C₂×C₄⋊C₄	—	2	1	no	—
220	C₂×C₄⋊C₄	—	73	1	no	—
221	C₂×C₄⋊C₄	—	73	1	no	—
222	C₂×C₄⋊C₄	—	146	1	no	—
223	C₂×C₄⋊C₄	—	146	1	no	—
224	C₂×C₄⋊C₄	C₂×#64408	1	1	yes	characteristic
225	C₂×C₄²	—	2	1	no	—
226	C₂×C₄²	—	73	1	no	—
227	C₂×C₄²	—	73	1	no	—
228	C₂×C₄²	—	73	1	no	—
229	C₂×C₄²	—	73	1	no	—
230	C₂×C₄²	—	73	1	no	—
231	C₂×C₄²	—	73	1	no	—
232	C₂×C₄²	—	146	1	no	—
233	C₂×C₄²	—	146	1	no	—
234	C₂×C₄²	—	146	1	no	—
235	C₂×C₄²	—	146	1	no	—
236	C₂×C₄²	—	146	1	no	—
237	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
238	C₄⋊C₈	—	73	4	no	—
239	C₄⋊C₈	—	146	2	no	—
240	#97	—	292	2	no	—
241	#97	—	146	4	no	—
242	#97	—	146	4	no	—
243	C₄×C₈	—	73	4	no	—
244	C₄×C₈	—	146	2	no	—
245	#95	—	146	1	no	—
246	#95	—	146	1	no	—
247	#95	—	146	1	no	—
248	C₂×Q₈	—	2	1	no	—
249	C₂×Q₈	—	2	1	no	—
250	C₂×Q₈	—	2	1	no	—
251	C₂×Q₈	C₄×#64408	1	2	yes	—
252	C₂×Q₈	#11924037	1	2	yes	—
253	C₂×Q₈	#11924038	1	2	yes	—
254	C₂²×C₄	—	2	1	no	—
255	C₂²×C₄	—	2	1	no	—
256	C₂²×C₄	—	73	1	no	—
257	C₂²×C₄	—	73	1	no	—
258	C₂²×C₄	—	73	1	no	—
259	C₂²×C₄	—	73	1	no	—
260	C₂²×C₄	—	146	1	no	—
261	C₂²×C₄	—	146	1	no	—
262	C₂²×C₄	C₂²⋊#64408	1	1	yes	characteristic
263	C₂²×C₄	C₂²⋊#64408	1	1	yes	characteristic
264	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
265	C₂×C₈	—	292	2	no	—
266	C₂×C₈	—	292	2	no	—
267	C₂×C₈	—	146	4	no	—
268	C₂×C₈	—	146	4	no	—
269	C₂×C₈	—	146	4	no	—
270	C₂×C₈	—	146	4	no	—
271	C₄⋊C₄	—	2	4	no	—
272	C₄⋊C₄	—	2	1	no	—
273	C₄⋊C₄	—	73	2	no	—
274	C₄⋊C₄	—	73	2	no	—
275	C₄⋊C₄	—	146	4	no	—
276	C₄⋊C₄	—	146	4	no	—
277	C₄⋊C₄	—	146	1	no	—
278	C₄⋊C₄	—	146	1	no	—
279	C₄⋊C₄	#11924037	1	2	yes	—
280	C₄²	—	2	4	no	—
281	C₄²	—	2	1	no	—
282	C₄²	—	2	1	no	—
283	C₄²	—	292	1	no	—
284	C₄²	—	292	1	no	—
285	C₄²	—	292	2	no	—
286	C₄²	—	292	2	no	—
287	C₄²	—	73	2	no	—
288	C₄²	—	73	2	no	—
289	C₄²	—	146	1	no	—
290	C₄²	—	146	2	no	—
291	C₄²	—	146	2	no	—
292	C₄²	—	146	1	no	—
293	C₄²	—	146	4	no	—
294	C₄²	—	146	4	no	—
295	C₄²	—	146	1	no	—
296	C₄²	—	146	1	no	—
297	C₄²	—	146	2	no	—
298	C₂³	C₂×C₂²⋊#64408	1	1	yes	characteristic
299	Q₈	—	2	2	no	—
300	Q₈	—	2	2	no	—
301	Q₈	—	2	2	no	—
302	Q₈	#14556646	1	4	yes	complemented
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	4	1	no	—
313	C₂×C₄	—	4	1	no	—
314	C₂×C₄	—	292	1	no	—
315	C₂×C₄	—	292	1	no	—
316	C₂×C₄	—	292	1	no	—
317	C₂×C₄	—	292	1	no	—
318	C₂×C₄	—	73	2	no	—
319	C₂×C₄	—	73	1	no	—
320	C₂×C₄	—	73	1	no	—
321	C₂×C₄	—	73	2	no	—
322	C₂×C₄	—	73	2	no	—
323	C₂×C₄	—	73	2	no	—
324	C₂×C₄	—	146	2	no	—
325	C₂×C₄	—	146	2	no	—
326	C₂×C₄	—	146	1	no	—
327	C₂×C₄	—	146	1	no	—
328	C₂×C₄	—	146	1	no	—
329	C₂×C₄	—	146	1	no	—
330	C₂×C₄	—	146	1	no	—
331	C₂×C₄	—	146	1	no	—
332	C₂×C₄	—	146	1	no	—
333	C₂×C₄	#14556656	1	2	yes	—
334	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
335	C₂×C₄	#14556670	1	1	yes	characteristic
336	C₂×C₄	C₂×#11924037	1	1	yes	characteristic
337	C₂×C₄	C₂×#11924037	1	1	yes	characteristic
338	C₂×C₄	D₇₃⋊M₁₆	1	1	yes	characteristic
339	C₂×C₄	C₂×#11924038	1	1	yes	characteristic
340	C₈	—	292	4	no	—
341	C₈	—	146	8	no	—
342	C₂²	C₄×C₂²⋊#64408	1	1	yes	characteristic
343	C₂²	#17989557	1	1	yes	characteristic
344	C₂²	#17989558	1	1	yes	characteristic
345	C₂²	#17989578	1	1	yes	characteristic
346	C₂²	#17989580	1	1	yes	characteristic
347	C₂²	D₇₃⋊#104	1	1	yes	characteristic
348	C₂²	C₂×#14556656	1	1	yes	characteristic
349	C₄	—	2	4	no	—
350	C₄	—	2	2	no	—
351	C₄	—	4	1	no	—
352	C₄	—	4	2	no	—
353	C₄	—	292	2	no	—
354	C₄	—	292	2	no	—
355	C₄	—	292	1	no	—
356	C₄	—	73	2	no	—
357	C₄	—	73	2	no	—
358	C₄	—	146	2	no	—
359	C₄	—	146	2	no	—
360	C₄	C₂×#14556646	1	1	yes	characteristic
361	C₄	C₄×#11924037	1	1	yes	characteristic
362	C₄	#17989607	1	1	yes	characteristic
363	C₄	#17989608	1	1	yes	characteristic
364	C₂	#22625266	1	1	yes	characteristic
365	C₂	#22625269	1	1	yes	characteristic
366	C₂	#22625273	1	1	yes	characteristic
367	C₂	#22625274	1	1	yes	characteristic
368	C₂	#22625657	1	1	yes	characteristic
369	C₂	C₄×#14556656	1	1	yes	characteristic
370	C₂	#22625703	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊#64408
commutator	C₂×C₄×C₇₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×#64408
Fitting	C₂×C₄×Q₈×C₇₃	C₄
socle, prime radical	C₂³×C₇₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄×Dic₇₃	C₂³
hypercenter	C₂×C₄×Q₈	#64408
hypocenter	C₇₃	#8173
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
12. Q₈⋊#10596833	C₄	semidirect
116. Q₈×C₇₃	#97	semidirect
191. C₇₃	#8173	semidirect
302. Q₈	#14556646	semidirect

Maximal subgroups

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

subgroup	index
#22625717	2
C₄×#14556646	2
C₂×Q₈⋊#10596833	2
#22624767	2
C₄×Q₈×Dic₇₃	2
#8173	73