Subgroups — #34003523

Order 18688.

598 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊#10596833	C₂	1	1	yes	maximal, characteristic
2	#22625645	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×C₄×#10596833	—	2	1	no	—
8	C₂×C₄×#10596833	C₂²	1	2	yes	—
9	#17989453	—	2	1	no	—
10	#17989453	—	2	1	no	—
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	C₄×#10596833	—	2	2	no	—
29	C₄×#10596833	—	4	2	no	—
30	C₄×#10596833	C₂×C₄	1	4	yes	—
31	Q₈×Dic₇₃	—	2	2	no	—
32	Q₈×Dic₇₃	—	2	2	no	—
33	Q₈×Dic₇₃	C₈	1	4	yes	complemented
34	Q₈×Dic₇₃	C₂×C₄	1	4	yes	—
35	C₂×C₄⋊Dic₇₃	—	2	1	no	—
36	C₂×C₄⋊Dic₇₃	—	2	1	no	—
37	C₂×C₄⋊Dic₇₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄⋊Dic₇₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×Dic₇₃	—	2	1	no	—
40	C₂×C₄×Dic₇₃	—	2	1	no	—
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	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×Dic₇₃	D₄	1	1	yes	characteristic
46	C₂×C₄×Dic₇₃	D₄	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	C₂³	1	1	yes	characteristic
50	#14556498	—	2	1	no	—
51	#14556498	—	2	1	no	—
52	#14556498	—	2	1	no	—
53	C₂²×Q₈×C₇₃	C₂×C₄	1	1	yes	characteristic
54	C₄×Q₈×C₇₃	—	2	2	no	—
55	C₄×Q₈×C₇₃	C₈	1	4	yes	complemented
56	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
57	C₂×C₄⋊C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄²×C₇₃	—	2	1	no	—
59	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
60	C₂×#10596833	—	2	4	no	—
61	C₂×#10596833	—	2	4	no	—
62	C₂×#10596833	—	4	2	no	—
63	C₂×#10596833	—	4	2	no	—
64	C₂×#10596833	—	4	2	no	—
65	C₂×#10596833	—	4	2	no	—
66	C₂²×Dic₇₃	—	2	1	no	—
67	C₂²×Dic₇₃	—	2	1	no	—
68	C₂²×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×Dic₇₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×Dic₇₃	C₂×D₄	1	1	yes	characteristic
72	C₄⋊Dic₇₃	—	2	1	no	—
73	C₄⋊Dic₇₃	—	2	1	no	—
74	C₄⋊Dic₇₃	—	2	4	no	—
75	C₄⋊Dic₇₃	—	2	4	no	—
76	C₄⋊Dic₇₃	C₄²	1	2	yes	—
77	C₄⋊Dic₇₃	C₂×C₈	1	2	yes	—
78	C₄×Dic₇₃	—	2	4	no	—
79	C₄×Dic₇₃	—	2	4	no	—
80	C₄×Dic₇₃	—	2	1	no	—
81	C₄×Dic₇₃	—	2	1	no	—
82	C₄×Dic₇₃	—	2	2	no	—
83	C₄×Dic₇₃	—	4	1	no	—
84	C₄×Dic₇₃	—	4	1	no	—
85	C₄×Dic₇₃	—	4	2	no	—
86	C₄×Dic₇₃	—	4	2	no	—
87	C₄×Dic₇₃	—	4	1	no	—
88	C₄×Dic₇₃	—	4	1	no	—
89	C₄×Dic₇₃	C₂×C₈	1	2	yes	—
90	C₄×Dic₇₃	M₁₆	1	2	yes	—
91	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
92	C₄×Dic₇₃	#41	1	2	yes	—
93	C₂×Q₈×C₇₃	—	2	1	no	—
94	C₂×Q₈×C₇₃	—	2	1	no	—
95	C₂×Q₈×C₇₃	—	2	1	no	—
96	C₂×Q₈×C₇₃	C₄²	1	2	yes	—
97	C₂×Q₈×C₇₃	C₂×C₈	1	2	yes	—
98	C₂×Q₈×C₇₃	C₂×C₈	1	2	yes	—
99	C₂²×C₄×C₇₃	—	2	1	no	—
100	C₂²×C₄×C₇₃	—	2	1	no	—
101	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
102	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
103	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
104	C₄⋊C₄×C₇₃	—	2	4	no	—
105	C₄⋊C₄×C₇₃	—	2	1	no	—
106	C₄⋊C₄×C₇₃	C₂×C₈	1	2	yes	—
107	C₄²×C₇₃	—	2	4	no	—
108	C₄²×C₇₃	C₂×C₈	1	2	yes	—
109	C₄²×C₇₃	M₁₆	1	2	yes	—
110	C₂³×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	Q₈×C₇₃	—	2	2	no	—
112	Q₈×C₇₃	—	2	2	no	—
113	Q₈×C₇₃	—	2	2	no	—
114	Q₈×C₇₃	C₄×C₈	1	4	yes	complemented
115	C₂×C₄×C₇₃	—	2	2	no	—
116	C₂×C₄×C₇₃	—	2	2	no	—
117	C₂×C₄×C₇₃	—	2	2	no	—
118	C₂×C₄×C₇₃	—	2	1	no	—
119	C₂×C₄×C₇₃	—	2	1	no	—
120	C₂×C₄×C₇₃	—	2	1	no	—
121	C₂×C₄×C₇₃	—	2	1	no	—
122	C₂×C₄×C₇₃	—	2	2	no	—
123	C₂×C₄×C₇₃	—	4	1	no	—
124	C₂×C₄×C₇₃	—	4	1	no	—
125	C₂×C₄×C₇₃	C₂²⋊C₈	1	2	yes	—
126	C₂×C₄×C₇₃	C₂²⋊C₈	1	2	yes	—
127	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
128	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
129	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
130	C₂×C₄×C₇₃	C₂²×C₈	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×Dic₇₃	—	2	1	no	—
134	C₂×Dic₇₃	—	2	1	no	—
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	2	no	—
141	C₂×Dic₇₃	—	4	1	no	—
142	C₂×Dic₇₃	—	4	1	no	—
143	C₂×Dic₇₃	—	4	1	no	—
144	C₂×Dic₇₃	—	4	1	no	—
145	C₂×Dic₇₃	C₂²⋊C₈	1	2	yes	—
146	C₂×Dic₇₃	C₂²⋊C₈	1	2	yes	—
147	C₂×Dic₇₃	C₄×D₄	1	2	yes	—
148	C₂×Dic₇₃	C₄×D₄	1	2	yes	—
149	C₂×Dic₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
150	C₂×Dic₇₃	#137	1	2	yes	—
151	C₂×Dic₇₃	#137	1	1	yes	characteristic
152	#10596833	—	4	4	no	—
153	#10596833	—	4	4	no	—
154	C₂²×C₇₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
155	C₂²×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
156	C₂²×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
157	C₂²×C₇₃	#417	1	1	yes	characteristic
158	C₂²×C₇₃	#419	1	1	yes	characteristic
159	C₂²×C₇₃	#421	1	1	yes	characteristic
160	C₂²×C₇₃	#421	1	1	yes	characteristic
161	C₄×C₇₃	—	2	2	no	—
162	C₄×C₇₃	—	2	2	no	—
163	C₄×C₇₃	—	2	4	no	—
164	C₄×C₇₃	—	4	2	no	—
165	C₄×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
166	C₄×C₇₃	C₄×M₁₆	1	1	yes	characteristic
167	C₄×C₇₃	#431	1	1	yes	characteristic
168	C₄×C₇₃	#431	1	1	yes	characteristic
169	Dic₇₃	—	2	2	no	—
170	Dic₇₃	—	2	2	no	—
171	Dic₇₃	—	2	2	no	—
172	Dic₇₃	—	4	2	no	—
173	Dic₇₃	—	4	2	no	—
174	Dic₇₃	#440	1	2	yes	—
175	Dic₇₃	#441	1	2	yes	—
176	#8171	—	73	1	no	maximal, Sylow
177	C₂×C₇₃	#1477	1	1	yes	characteristic
178	C₂×C₇₃	#1477	1	1	yes	characteristic
179	C₂×C₇₃	#1479	1	1	yes	characteristic
180	C₂×C₇₃	#1479	1	1	yes	characteristic
181	C₂×C₇₃	C₄×C₂²⋊C₈	1	1	yes	characteristic
182	C₂×C₇₃	#1759	1	1	yes	characteristic
183	C₂×C₇₃	#1763	1	1	yes	characteristic
184	C₄²×Q₈	—	73	1	no	—
185	C₄×C₄⋊C₈	—	73	1	no	—
186	#1744	—	73	1	no	—
187	C₂×Q₈⋊C₈	—	73	2	no	—
188	#1279	—	73	2	no	—
189	C₇₃	#8171	1	1	yes	characteristic, complemented, Sylow
190	C₂×C₄×Q₈	—	73	1	no	—
191	C₂×C₄×Q₈	—	73	1	no	—
192	C₂×C₄×Q₈	#64408	1	1	yes	characteristic
193	C₂×C₄⋊C₈	—	73	2	no	—
194	C₂×C₄×C₈	—	73	2	no	—
195	C₂×C₄×C₈	—	146	1	no	—
196	C₄×C₄⋊C₄	—	73	1	no	—
197	C₄×C₄⋊C₄	—	73	1	no	—
198	C₄×C₄⋊C₄	—	73	1	no	—
199	C₄×C₄⋊C₄	—	146	1	no	—
200	C₄×C₄⋊C₄	—	146	1	no	—
201	C₄×C₄⋊C₄	—	146	1	no	—
202	C₄³	—	73	1	no	—
203	C₄³	—	146	1	no	—
204	#336	—	146	1	no	—
205	#336	—	146	1	no	—
206	#336	—	146	1	no	—
207	Q₈⋊C₈	—	73	8	no	—
208	C₂²×Q₈	C₂×#64408	1	1	yes	characteristic
209	C₂²×C₈	—	146	2	no	—
210	C₂²×C₈	—	146	2	no	—
211	C₄×Q₈	—	2	2	no	—
212	C₄×Q₈	—	73	4	no	—
213	C₄×Q₈	—	73	4	no	—
214	C₄×Q₈	—	146	2	no	—
215	C₄×Q₈	—	146	2	no	—
216	C₄×Q₈	#10596833	1	4	yes	complemented
217	C₂×C₄⋊C₄	—	2	1	no	—
218	C₂×C₄⋊C₄	—	73	1	no	—
219	C₂×C₄⋊C₄	—	73	1	no	—
220	C₂×C₄⋊C₄	—	146	1	no	—
221	C₂×C₄⋊C₄	—	146	1	no	—
222	C₂×C₄⋊C₄	C₂×#64408	1	1	yes	characteristic
223	C₂×C₄²	—	2	1	no	—
224	C₂×C₄²	—	73	1	no	—
225	C₂×C₄²	—	73	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	—	146	1	no	—
231	C₂×C₄²	—	146	1	no	—
232	C₂×C₄²	—	146	1	no	—
233	C₂×C₄²	—	146	1	no	—
234	C₂×C₄²	—	146	1	no	—
235	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
236	C₄⋊C₈	—	73	4	no	—
237	C₄⋊C₈	—	146	2	no	—
238	C₄×C₈	—	292	2	no	—
239	C₄×C₈	—	73	4	no	—
240	C₄×C₈	—	146	2	no	—
241	#95	—	146	1	no	—
242	#95	—	146	1	no	—
243	#95	—	146	1	no	—
244	C₂×Q₈	—	2	1	no	—
245	C₂×Q₈	—	2	1	no	—
246	C₂×Q₈	—	2	1	no	—
247	C₂×Q₈	C₄×#64408	1	2	yes	—
248	C₂×Q₈	C₂×#10596833	1	2	yes	—
249	C₂×Q₈	#11924036	1	2	yes	—
250	C₂²×C₄	—	2	1	no	—
251	C₂²×C₄	—	2	1	no	—
252	C₂²×C₄	—	73	1	no	—
253	C₂²×C₄	—	73	1	no	—
254	C₂²×C₄	—	73	1	no	—
255	C₂²×C₄	—	73	1	no	—
256	C₂²×C₄	—	146	1	no	—
257	C₂²×C₄	—	146	1	no	—
258	C₂²×C₄	C₂²⋊#64408	1	1	yes	characteristic
259	C₂²×C₄	C₂²⋊#64408	1	1	yes	characteristic
260	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
261	C₂×C₈	—	292	2	no	—
262	C₂×C₈	—	292	2	no	—
263	C₂×C₈	—	292	2	no	—
264	C₂×C₈	—	292	2	no	—
265	C₂×C₈	—	146	4	no	—
266	C₂×C₈	—	146	4	no	—
267	C₄⋊C₄	—	2	1	no	—
268	C₄⋊C₄	—	2	4	no	—
269	C₄⋊C₄	—	73	2	no	—
270	C₄⋊C₄	—	73	2	no	—
271	C₄⋊C₄	—	146	4	no	—
272	C₄⋊C₄	—	146	4	no	—
273	C₄⋊C₄	—	146	1	no	—
274	C₄⋊C₄	—	146	1	no	—
275	C₄⋊C₄	C₂×#10596833	1	2	yes	—
276	C₄²	—	2	4	no	—
277	C₄²	—	292	2	no	—
278	C₄²	—	292	1	no	—
279	C₄²	—	292	1	no	—
280	C₄²	—	292	1	no	—
281	C₄²	—	292	1	no	—
282	C₄²	—	292	2	no	—
283	C₄²	—	73	2	no	—
284	C₄²	—	73	2	no	—
285	C₄²	—	73	2	no	—
286	C₄²	—	73	2	no	—
287	C₄²	—	146	1	no	—
288	C₄²	—	146	1	no	—
289	C₄²	—	146	2	no	—
290	C₄²	—	146	4	no	—
291	C₄²	—	146	4	no	—
292	C₄²	C₂×#10596833	1	2	yes	—
293	C₄²	#11924037	1	2	yes	—
294	C₂³	C₂×C₂²⋊#64408	1	1	yes	characteristic
295	Q₈	—	2	2	no	—
296	Q₈	—	2	2	no	—
297	Q₈	—	2	2	no	—
298	Q₈	C₄×#10596833	1	4	yes	complemented
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	4	1	no	—
309	C₂×C₄	—	292	1	no	—
310	C₂×C₄	—	292	1	no	—
311	C₂×C₄	—	292	1	no	—
312	C₂×C₄	—	292	1	no	—
313	C₂×C₄	—	73	2	no	—
314	C₂×C₄	—	73	1	no	—
315	C₂×C₄	—	73	1	no	—
316	C₂×C₄	—	73	2	no	—
317	C₂×C₄	—	73	2	no	—
318	C₂×C₄	—	73	2	no	—
319	C₂×C₄	—	73	2	no	—
320	C₂×C₄	—	146	2	no	—
321	C₂×C₄	—	146	2	no	—
322	C₂×C₄	—	146	1	no	—
323	C₂×C₄	—	146	1	no	—
324	C₂×C₄	—	146	1	no	—
325	C₂×C₄	—	146	1	no	—
326	C₂×C₄	—	146	1	no	—
327	C₂×C₄	—	146	1	no	—
328	C₂×C₄	C₂²⋊#10596833	1	2	yes	—
329	C₂×C₄	C₂²⋊#10596833	1	2	yes	—
330	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
331	C₂×C₄	#14556670	1	1	yes	characteristic
332	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
333	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
334	C₂×C₄	C₂×#11924037	1	1	yes	characteristic
335	C₂×C₄	D₇₃⋊M₁₆	1	1	yes	characteristic
336	C₈	—	292	4	no	—
337	C₈	—	292	4	no	—
338	C₂²	C₄×C₂²⋊#64408	1	1	yes	characteristic
339	C₂²	C₂×C₂²⋊#10596833	1	1	yes	characteristic
340	C₂²	#17989555	1	1	yes	characteristic
341	C₂²	#17989578	1	1	yes	characteristic
342	C₂²	#17989580	1	1	yes	characteristic
343	C₂²	#17989584	1	1	yes	characteristic
344	C₂²	#17989586	1	1	yes	characteristic
345	C₄	—	2	4	no	—
346	C₄	—	2	2	no	—
347	C₄	—	2	2	no	—
348	C₄	—	4	2	no	—
349	C₄	—	292	2	no	—
350	C₄	—	292	2	no	—
351	C₄	—	73	2	no	—
352	C₄	—	73	2	no	—
353	C₄	—	146	2	no	—
354	C₄	—	146	2	no	—
355	C₄	—	146	2	no	—
356	C₄	C₂×C₄×#10596833	1	1	yes	characteristic
357	C₄	C₄×#11924037	1	1	yes	characteristic
358	C₄	#17989603	1	1	yes	characteristic
359	C₄	#17989604	1	1	yes	characteristic
360	C₂	#22625226	1	1	yes	characteristic
361	C₂	#22625229	1	1	yes	characteristic
362	C₂	#22625233	1	1	yes	characteristic
363	C₂	#22625234	1	1	yes	characteristic
364	C₂	C₄×C₂²⋊#10596833	1	1	yes	characteristic
365	C₂	#22625695	1	1	yes	characteristic
366	C₂	#22625703	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊#64408
commutator	C₄×C₇₃	C₂×C₄×C₈
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	#8171
LCS	G ▷ 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
33. Q₈×Dic₇₃	C₈	semidirect
55. C₄×Q₈×C₇₃	C₈	semidirect
114. Q₈×C₇₃	C₄×C₈	semidirect
189. C₇₃	#8171	semidirect
216. C₄×Q₈	#10596833	semidirect
298. Q₈	C₄×#10596833	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊#10596833	2
#22625645	2
C₂×Q₈⋊#10596833	2
#22624767	2
C₄×Q₈×Dic₇₃	2
#8171	73