Subgroups — #34019437

Order 18688.

630 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	Q₈×#11924038	C₂	1	2	yes	maximal
2	#22625577	C₂	1	2	yes	maximal, complemented
3	#22625554	C₂	1	4	yes	maximal, complemented
4	#22625524	C₂	1	2	yes	maximal, complemented
5	#22625486	C₂	1	1	yes	maximal, characteristic
6	#22625260	C₂	1	2	yes	maximal
7	#22625141	C₂	1	1	yes	maximal, characteristic
8	C₄×Q₁₆×D₇₃	C₂	1	1	yes	maximal, characteristic
9	#17989637	—	2	4	no	—
10	#17989637	C₂²	1	4	yes	—
11	Q₈×#10596833	C₂²	1	4	yes	—
12	#17989606	C₂²	1	1	yes	characteristic
13	#17989591	—	2	2	no	—
14	#17989591	C₂²	1	2	yes	—
15	C₄×#11924038	—	2	2	no	—
16	C₄×#11924038	C₂²	1	1	yes	characteristic
17	#17989448	C₂²	1	2	yes	—
18	#17989448	C₂²	1	2	yes	—
19	#17989437	C₂²	1	4	yes	—
20	Q₈⋊#10596833	C₂²	1	4	yes	—
21	#17989429	C₂²	1	2	yes	—
22	#17989428	C₂²	1	2	yes	—
23	C₂×Q₁₆×D₇₃	C₄	1	1	yes	characteristic
24	C₄×Q₈×D₇₃	C₂²	1	2	yes	—
25	C₄×Dic₂₉₂	C₄	1	1	yes	characteristic
26	Q₁₆×Dic₇₃	C₂²	1	1	yes	characteristic
27	#17988681	C₂²	1	1	yes	characteristic
28	C₄×#11924018	C₄	1	2	yes	—
29	#17988679	C₂²	1	2	yes	—
30	#107×D₇₃	C₄	1	1	yes	characteristic
31	#103×D₇₃	C₄	1	2	yes	—
32	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
33	C₄×Q₁₆×C₇₃	C₄	1	1	yes	characteristic
34	C₂×#11924038	—	2	2	no	—
35	C₂×#11924038	D₄	1	2	yes	—
36	C₄⋊#10596833	—	2	4	no	—
37	C₄⋊#10596833	—	2	4	no	—
38	C₄⋊#10596833	—	2	2	no	—
39	C₄⋊#10596833	—	2	1	no	—
40	C₄⋊#10596833	C₂³	1	4	yes	—
41	#14556648	—	2	1	no	—
42	#14556646	—	2	4	no	—
43	#14556646	D₄	1	2	yes	—
44	C₄×#10596833	—	2	4	no	—
45	C₄×#10596833	C₂³	1	2	yes	—
46	C₄×#10596833	C₂³	1	2	yes	—
47	C₂×Q₈×D₇₃	C₂×C₄	1	2	yes	—
48	C₂×Dic₂₉₂	C₂×C₄	1	1	yes	characteristic
49	C₂×#11924018	C₂×C₄	1	2	yes	—
50	Q₁₆×D₇₃	—	2	4	no	—
51	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
53	Q₈×Dic₇₃	C₂³	1	2	yes	—
54	C₄×Dic₁₄₆	—	2	2	no	—
55	C₄×Dic₁₄₆	C₂×C₄	1	2	yes	—
56	#14556556	—	2	2	no	—
57	#14556556	C₂³	1	2	yes	—
58	C₄⋊C₄×D₇₃	—	2	2	no	—
59	C₄⋊C₄×D₇₃	C₂×C₄	1	2	yes	—
60	C₄²×D₇₃	—	2	2	no	—
61	C₄²×D₇₃	C₂³	1	1	yes	characteristic
62	#14556525	C₂×C₄	1	1	yes	characteristic
63	#14556524	C₂×C₄	1	2	yes	—
64	#14556516	C₂×C₄	1	2	yes	—
65	#14556515	C₂×C₄	1	2	yes	—
66	#14556514	C₂×C₄	1	2	yes	—
67	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
68	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
69	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
70	C₂×Q₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
71	C₄×Q₈×C₇₃	C₂×C₄	1	2	yes	—
72	#107×C₇₃	C₂×C₄	1	1	yes	characteristic
73	#103×C₇₃	C₂×C₄	1	2	yes	—
74	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
75	#11924038	—	2	4	no	—
76	#11924038	—	4	4	no	—
77	C₂×#10596833	—	2	4	no	—
78	C₂×#10596833	—	2	2	no	—
79	C₂×#10596833	C₂×D₄	1	4	yes	—
80	C₂×Dic₁₄₆	—	2	2	no	—
81	C₂×Dic₁₄₆	C₂²×C₄	1	2	yes	—
82	Q₈×D₇₃	—	2	4	no	—
83	Q₈×D₇₃	—	4	2	no	—
84	C₂×C₄×D₇₃	—	2	2	no	—
85	C₂×C₄×D₇₃	—	2	2	no	—
86	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
88	C₂×C₄×D₇₃	#41	1	1	yes	characteristic
89	Dic₂₉₂	—	2	2	no	—
90	#11924018	—	2	4	no	—
91	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
92	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
93	C₈×D₇₃	—	2	2	no	—
94	C₈×D₇₃	—	4	1	no	—
95	C₄⋊Dic₇₃	—	2	2	no	—
96	C₄⋊Dic₇₃	C₂²×C₄	1	2	yes	—
97	#11924007	—	2	2	no	—
98	#11924007	—	2	2	no	—
99	#11924007	—	2	2	no	—
100	C₄×Dic₇₃	—	2	2	no	—
101	C₄×Dic₇₃	—	2	2	no	—
102	C₄×Dic₇₃	—	2	2	no	—
103	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
104	C₄×Dic₇₃	#41	1	1	yes	characteristic
105	C₄×Dic₇₃	C₂⁴	1	1	yes	characteristic
106	C₂×Q₈×C₇₃	C₂²×C₄	1	2	yes	—
107	Q₁₆×C₇₃	M₁₆	1	4	yes	complemented
108	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
110	C₄⋊C₄×C₇₃	—	2	2	no	—
111	C₄⋊C₄×C₇₃	C₂²×C₄	1	2	yes	—
112	C₄²×C₇₃	—	2	2	no	—
113	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
114	C₂²×D₇₃	C₄×D₄	1	1	yes	characteristic
115	Q₈×C₇₃	—	2	2	no	—
116	Q₈×C₇₃	C₂×M₁₆	1	4	yes	—
117	C₂×C₄×C₇₃	—	2	2	no	—
118	C₂×C₄×C₇₃	—	2	2	no	—
119	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
122	C₂×Dic₇₃	—	2	2	no	—
123	C₂×Dic₇₃	—	2	2	no	—
124	C₂×Dic₇₃	C₄×D₄	1	1	yes	characteristic
125	C₂×Dic₇₃	C₂²×D₄	1	1	yes	characteristic
126	C₂×Dic₇₃	C₂×#41	1	1	yes	characteristic
127	C₄×D₇₃	—	2	1	no	—
128	C₄×D₇₃	—	2	1	no	—
129	C₄×D₇₃	—	4	1	no	—
130	C₄×D₇₃	—	4	4	no	—
131	C₄×D₇₃	—	8	2	no	—
132	C₄×D₇₃	#137	1	4	yes	—
133	Dic₁₄₆	—	2	2	no	—
134	Dic₁₄₆	—	2	2	no	—
135	Dic₁₄₆	—	4	4	no	—
136	#10596833	—	2	4	no	—
137	#10596833	—	4	2	no	—
138	#10596833	—	4	4	no	—
139	C₈×C₇₃	—	2	1	no	—
140	C₈×C₇₃	C₂×M₁₆	1	2	yes	—
141	#10596831	—	2	1	no	—
142	#10596831	—	2	1	no	—
143	C₂²×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
144	C₂×D₇₃	—	2	1	no	—
145	C₂×D₇₃	—	2	1	no	—
146	C₂×D₇₃	#441	1	2	yes	—
147	C₄×C₇₃	—	2	4	no	—
148	C₄×C₇₃	—	2	1	no	—
149	C₄×C₇₃	—	4	2	no	—
150	C₄×C₇₃	#441	1	2	yes	—
151	C₄×C₇₃	C₂²×M₁₆	1	1	yes	characteristic
152	C₄×C₇₃	#568	1	1	yes	characteristic
153	Dic₇₃	—	2	1	no	—
154	Dic₇₃	—	2	1	no	—
155	Dic₇₃	—	4	2	no	—
156	Dic₇₃	—	4	2	no	—
157	Dic₇₃	C₂×#137	1	2	yes	—
158	#18042	—	73	1	no	maximal, Sylow
159	C₂×C₇₃	D₄×M₁₆	1	1	yes	characteristic
160	C₂×C₇₃	C₂×#441	1	1	yes	characteristic
161	C₂×C₇₃	#2957	1	1	yes	characteristic
162	D₇₃	—	2	2	no	—
163	Q₈×M₁₆	—	73	2	no	—
164	C₂×C₄×Q₁₆	—	73	1	no	—
165	#1593	—	73	2	no	—
166	#1587	—	73	4	no	—
167	#1579	—	73	2	no	—
168	#1566	—	73	1	no	—
169	#1487	—	73	2	no	—
170	#1451	—	73	1	no	—
171	C₇₃	#18042	1	1	yes	characteristic, complemented, Sylow
172	C₂²×Q₁₆	—	73	1	no	—
173	C₂×C₄×Q₈	—	73	2	no	—
174	#446	—	73	4	no	—
175	#446	—	146	4	no	—
176	C₈×Q₈	—	73	4	no	—
177	C₄×Q₁₆	—	73	1	no	—
178	C₄×Q₁₆	—	73	2	no	—
179	C₄×Q₁₆	—	73	1	no	—
180	C₄×Q₁₆	—	73	2	no	—
181	C₄×Q₁₆	—	73	1	no	—
182	C₄×Q₁₆	#64408	1	1	yes	characteristic
183	#431	—	73	1	no	—
184	C₂×#107	—	73	1	no	—
185	#423	—	73	2	no	—
186	#423	—	146	2	no	—
187	C₂×#103	—	73	2	no	—
188	C₄×M₁₆	—	73	1	no	—
189	C₄×M₁₆	—	146	2	no	—
190	C₂×C₄×C₈	—	73	1	no	—
191	#334	—	73	2	no	—
192	#334	—	73	2	no	—
193	Q₈⋊C₈	—	73	4	no	—
194	Q₈⋊C₈	—	73	4	no	—
195	#322	—	73	2	no	—
196	#322	—	73	2	no	—
197	C₂²×Q₈	—	73	2	no	—
198	C₂×Q₁₆	—	73	2	no	—
199	C₂×Q₁₆	—	73	1	no	—
200	C₂×Q₁₆	—	146	4	no	—
201	C₂×Q₁₆	C₂×#64408	1	1	yes	characteristic
202	C₂×M₁₆	—	73	2	no	—
203	C₂×M₁₆	—	146	2	no	—
204	C₂²×C₈	—	73	1	no	—
205	C₂²×C₈	—	73	1	no	—
206	C₄×Q₈	—	73	2	no	—
207	C₄×Q₈	—	73	2	no	—
208	C₄×Q₈	—	73	2	no	—
209	C₄×Q₈	—	146	2	no	—
210	C₄×Q₈	—	146	2	no	—
211	C₄×Q₈	C₂×#64408	1	2	yes	—
212	C₂×C₄⋊C₄	—	73	2	no	—
213	C₂×C₄⋊C₄	—	146	2	no	—
214	C₂×C₄²	—	73	1	no	—
215	C₂×C₄²	—	146	2	no	—
216	#107	—	73	1	no	—
217	#107	—	73	2	no	—
218	#107	C₂×#64408	1	1	yes	characteristic
219	C₄⋊C₈	—	73	4	no	—
220	C₄⋊C₈	—	146	2	no	—
221	C₄⋊C₈	—	146	1	no	—
222	C₄⋊C₈	—	146	4	no	—
223	C₄⋊C₈	—	146	4	no	—
224	#103	—	73	2	no	—
225	#103	—	73	2	no	—
226	#103	—	73	2	no	—
227	#103	C₂×#64408	1	2	yes	—
228	C₂²⋊C₈	—	146	1	no	—
229	#97	—	73	2	no	—
230	#97	—	146	4	no	—
231	C₄×C₈	—	73	2	no	—
232	C₄×C₈	—	73	1	no	—
233	C₄×C₈	—	73	1	no	—
234	C₄×C₈	—	73	2	no	—
235	C₄×C₈	—	73	1	no	—
236	C₄×C₈	—	146	4	no	—
237	C₄×C₈	C₂×#64408	1	1	yes	characteristic
238	C₂×Q₈	—	292	2	no	—
239	C₂×Q₈	—	73	2	no	—
240	C₂×Q₈	—	146	2	no	—
241	C₂×Q₈	—	146	4	no	—
242	C₂×Q₈	C₂²×#64408	1	2	yes	—
243	C₂²×C₄	—	73	1	no	—
244	C₂²×C₄	—	73	1	no	—
245	C₂²×C₄	—	73	1	no	—
246	C₂²×C₄	—	146	2	no	—
247	C₂²×C₄	—	146	2	no	—
248	Q₁₆	—	146	4	no	—
249	Q₁₆	—	146	2	no	—
250	Q₁₆	#11924038	1	4	yes	complemented
251	M₁₆	—	292	4	no	—
252	M₁₆	—	146	4	no	—
253	C₂×C₈	—	292	1	no	—
254	C₂×C₈	—	73	4	no	—
255	C₂×C₈	—	73	1	no	—
256	C₂×C₈	—	73	1	no	—
257	C₂×C₈	—	146	2	no	—
258	C₂×C₈	—	146	4	no	—
259	C₂×C₈	—	146	2	no	—
260	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
261	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	73	2	no	—
264	C₄⋊C₄	—	146	2	no	—
265	C₄⋊C₄	—	146	2	no	—
266	C₄⋊C₄	—	146	2	no	—
267	C₄⋊C₄	—	146	2	no	—
268	C₄⋊C₄	C₂²×#64408	1	2	yes	—
269	C₄²	—	2	2	no	—
270	C₄²	—	73	1	no	—
271	C₄²	—	73	1	no	—
272	C₄²	—	73	1	no	—
273	C₄²	—	146	2	no	—
274	C₄²	—	146	2	no	—
275	C₄²	—	146	2	no	—
276	C₄²	C₂²×#64408	1	1	yes	characteristic
277	C₂³	—	73	1	no	—
278	Q₈	—	2	2	no	—
279	Q₈	—	292	4	no	—
280	Q₈	—	146	2	no	—
281	Q₈	—	146	2	no	—
282	Q₈	C₂×#11924038	1	4	yes	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	292	4	no	—
286	C₂×C₄	—	292	1	no	—
287	C₂×C₄	—	584	2	no	—
288	C₂×C₄	—	73	1	no	—
289	C₂×C₄	—	73	4	no	—
290	C₂×C₄	—	73	1	no	—
291	C₂×C₄	—	73	1	no	—
292	C₂×C₄	—	146	2	no	—
293	C₂×C₄	—	146	2	no	—
294	C₂×C₄	—	146	1	no	—
295	C₂×C₄	—	146	1	no	—
296	C₂×C₄	D₄×#64408	1	1	yes	characteristic
297	C₂×C₄	D₄×#64408	1	1	yes	characteristic
298	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
299	C₈	—	2	1	no	—
300	C₈	—	292	4	no	—
301	C₈	—	292	2	no	—
302	C₈	—	146	4	no	—
303	C₈	—	146	1	no	—
304	C₈	—	146	1	no	—
305	C₈	C₂×#11924038	1	2	yes	—
306	C₂²	—	73	2	no	—
307	C₂²	—	146	1	no	—
308	C₂²	—	146	1	no	—
309	C₂²	C₂×D₄×#64408	1	1	yes	characteristic
310	C₄	—	2	4	no	—
311	C₄	—	2	1	no	—
312	C₄	—	4	2	no	—
313	C₄	—	292	2	no	—
314	C₄	—	292	2	no	—
315	C₄	—	73	2	no	—
316	C₄	—	146	1	no	—
317	C₄	—	146	1	no	—
318	C₄	#17989627	1	2	yes	—
319	C₄	C₂²×#11924038	1	1	yes	characteristic
320	C₄	#17989666	1	1	yes	characteristic
321	C₂	—	146	2	no	—
322	C₂	D₄×#11924038	1	1	yes	characteristic
323	C₂	C₂×#17989627	1	1	yes	characteristic
324	C₂	#22626710	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×#64408
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#64408
Fitting	C₄×Q₁₆×C₇₃	C₄
socle	C₂²×C₇₃	C₂×C₄×D₄
prime radical	C₂²×D₇₃	C₄×D₄
cosocle	C₄×Dic₇₃	C₂⁴
hypercenter	C₄×Q₁₆	#64408
hypocenter	C₇₃	#18042
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
2. #22625577	C₂	semidirect
3. #22625554	C₂	semidirect
4. #22625524	C₂	semidirect
107. Q₁₆×C₇₃	M₁₆	semidirect
171. C₇₃	#18042	semidirect
250. Q₁₆	#11924038	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#11924038	2
#22625577	2
#22625554	2
#22625524	2
#22625486	2
#22625260	2
#22625141	2
C₄×Q₁₆×D₇₃	2
#18042	73