Subgroups — #33105744

Order 18688.

766 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₂₉₂	C₂	1	1	yes	maximal, characteristic
2	#22612894	C₂	1	2	yes	maximal
3	#22610792	C₂	1	2	yes	maximal
4	#22610566	C₂	1	4	yes	maximal
5	#22610095	C₂	1	1	yes	maximal, characteristic
6	#22609947	C₂	1	2	yes	maximal
7	C₄×#14556525	C₂	1	2	yes	maximal
8	C₈×C₄⋊C₄×C₇₃	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₂₉₂	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₁₄₆	C₂²	1	2	yes	—
11	C₄×Dic₂₉₂	C₄	1	8	yes	complemented
12	C₂×#14556525	C₂²	1	2	yes	—
13	C₂×#14556525	C₂²	1	1	yes	characteristic
14	C₂×#14556523	C₂²	1	2	yes	—
15	C₂×#14556516	C₂²	1	4	yes	—
16	C₂×#14556516	C₂²	1	2	yes	—
17	#17988493	C₂²	1	4	yes	—
18	#17988468	—	2	4	no	—
19	#17988454	—	2	2	no	—
20	#17988453	C₂²	1	2	yes	—
21	C₄×C₄⋊Dic₇₃	C₂²	1	4	yes	—
22	#17988432	—	2	2	no	—
23	#17988350	C₂²	1	4	yes	—
24	C₂×C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₇₃	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
27	C₄×C₄⋊C₄×C₇₃	C₂²	1	1	yes	characteristic
28	#336×C₇₃	C₂²	1	2	yes	—
29	C₂²×Dic₁₄₆	C₂³	1	2	yes	—
30	C₂×Dic₂₉₂	—	2	2	no	—
31	C₂×Dic₂₉₂	—	2	2	no	—
32	C₂×Dic₂₉₂	C₂×C₄	1	4	yes	—
33	C₄×Dic₁₄₆	—	2	4	no	—
34	C₄×Dic₁₄₆	C₂×C₄	1	8	yes	—
35	C₂×C₄⋊Dic₇₃	C₂³	1	4	yes	—
36	C₂×C₄⋊Dic₇₃	C₂³	1	2	yes	—
37	C₂×#11924007	—	2	2	no	—
38	C₂×C₄×Dic₇₃	—	2	2	no	—
39	C₂×C₄×Dic₇₃	—	2	4	no	—
40	#14556525	—	2	4	no	—
41	#14556525	C₂×C₄	1	4	yes	—
42	#14556523	—	4	2	no	—
43	#14556516	—	4	4	no	—
44	#14556516	C₂×C₄	1	8	yes	—
45	#14556498	—	2	2	no	—
46	#14556498	—	2	4	no	—
47	#14556498	—	2	4	no	—
48	#14556498	—	2	2	no	—
49	#14556498	—	2	4	no	—
50	C₂²×C₈×C₇₃	C₂³	1	2	yes	—
51	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₇₃	D₄	1	2	yes	—
54	C₂×C₄²×C₇₃	C₂³	1	2	yes	—
55	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
56	C₄⋊C₈×C₇₃	—	2	2	no	—
57	C₄×C₈×C₇₃	—	2	4	no	—
58	C₄×C₈×C₇₃	C₂×C₄	1	4	yes	—
59	#95×C₇₃	D₄	1	1	yes	characteristic
60	#95×C₇₃	D₄	1	1	yes	characteristic
61	C₂×Dic₁₄₆	—	2	2	no	—
62	C₂×Dic₁₄₆	—	2	2	no	—
63	C₂×Dic₁₄₆	—	2	2	no	—
64	C₂×Dic₁₄₆	—	2	2	no	—
65	C₂×Dic₁₄₆	—	2	2	no	—
66	C₂×Dic₁₄₆	C₂²×C₄	1	4	yes	—
67	C₂²×Dic₇₃	—	2	2	no	—
68	C₂²×Dic₇₃	—	2	2	no	—
69	C₂²×Dic₇₃	—	2	4	no	—
70	Dic₂₉₂	—	2	8	no	—
71	C₄⋊Dic₇₃	—	2	2	no	—
72	C₄⋊Dic₇₃	—	2	4	no	—
73	C₄⋊Dic₇₃	—	2	4	no	—
74	C₄⋊Dic₇₃	C₂²×C₄	1	4	yes	—
75	#11924007	—	2	8	no	—
76	C₄×Dic₇₃	—	2	8	no	—
77	C₄×Dic₇₃	—	4	8	no	—
78	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
82	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
83	C₂×C₈×C₇₃	—	2	1	no	—
84	C₂×C₈×C₇₃	—	2	1	no	—
85	C₂×C₈×C₇₃	—	2	2	no	—
86	C₂×C₈×C₇₃	—	2	2	no	—
87	C₂×C₈×C₇₃	—	2	2	no	—
88	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
89	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
90	C₂×C₈×C₇₃	#41	1	2	yes	—
91	C₂×C₈×C₇₃	#41	1	2	yes	—
92	C₄⋊C₄×C₇₃	—	2	4	no	—
93	C₄²×C₇₃	—	2	2	no	—
94	C₄²×C₇₃	—	2	2	no	—
95	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
96	C₄²×C₇₃	#41	1	2	yes	—
97	C₂³×C₇₃	C₂²×D₄	1	1	yes	characteristic
98	C₂×C₄×C₇₃	—	2	2	no	—
99	C₂×C₄×C₇₃	—	2	2	no	—
100	C₂×C₄×C₇₃	—	2	2	no	—
101	C₂×C₄×C₇₃	—	2	2	no	—
102	C₂×C₄×C₇₃	—	2	2	no	—
103	C₂×C₄×C₇₃	—	2	2	no	—
104	C₂×C₄×C₇₃	—	2	2	no	—
105	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
106	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
107	C₂×C₄×C₇₃	#135	1	4	yes	—
108	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
109	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
110	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
111	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
112	C₂×C₄×C₇₃	H(5,𝔽₂)	1	1	yes	characteristic
113	C₂×C₄×C₇₃	2₋¹⁺⁴	1	1	yes	characteristic
114	C₂×Dic₇₃	—	2	4	no	—
115	C₂×Dic₇₃	—	2	4	no	—
116	C₂×Dic₇₃	—	4	2	no	—
117	C₂×Dic₇₃	—	4	2	no	—
118	C₂×Dic₇₃	—	4	4	no	—
119	C₂×Dic₇₃	—	4	4	no	—
120	C₂×Dic₇₃	—	4	4	no	—
121	C₂×Dic₇₃	—	4	4	no	—
122	Dic₁₄₆	—	2	4	no	—
123	Dic₁₄₆	—	2	4	no	—
124	Dic₁₄₆	—	4	2	no	—
125	Dic₁₄₆	—	4	2	no	—
126	C₈×C₇₃	—	2	2	no	—
127	C₈×C₇₃	—	4	2	no	—
128	C₈×C₇₃	#117	1	4	yes	—
129	C₂²×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
130	C₂²×C₇₃	#547	1	1	yes	characteristic
131	C₂²×C₇₃	#550	1	1	yes	characteristic
132	C₂²×C₇₃	C₂×#135	1	2	yes	—
133	C₂²×C₇₃	#578	1	2	yes	—
134	C₄×C₇₃	—	2	4	no	—
135	C₄×C₇₃	—	2	2	no	—
136	C₄×C₇₃	—	2	2	no	—
137	C₄×C₇₃	—	4	2	no	—
138	C₄×C₇₃	C₂×#117	1	1	yes	characteristic
139	C₄×C₇₃	#520	1	1	yes	characteristic
140	C₄×C₇₃	#555	1	1	yes	characteristic
141	C₄×C₇₃	#556	1	1	yes	characteristic
142	Dic₇₃	—	4	4	no	—
143	Dic₇₃	—	4	4	no	—
144	Dic₇₃	—	8	4	no	—
145	#22809	—	73	1	no	maximal, Sylow
146	C₂×C₇₃	#2354	1	1	yes	characteristic
147	C₂×C₇₃	C₄×#135	1	1	yes	characteristic
148	C₂×C₇₃	#2950	1	1	yes	characteristic
149	C₂×C₇₃	#3336	1	2	yes	—
150	C₂×C₇₃	#3393	1	2	yes	—
151	C₂×C₄×Q₁₆	—	73	1	no	—
152	#2356	—	73	2	no	—
153	#1969	—	73	2	no	—
154	#1927	—	73	4	no	—
155	#1846	—	73	1	no	—
156	#1805	—	73	2	no	—
157	C₄×#107	—	73	2	no	—
158	C₈×C₄⋊C₄	D₇₃	1	1	yes	characteristic
159	C₇₃	#22809	1	1	yes	characteristic, complemented, Sylow
160	C₂²×Q₁₆	—	73	1	no	—
161	C₂×C₄×Q₈	—	73	2	no	—
162	C₄×Q₁₆	—	73	8	no	—
163	C₂×#107	—	73	1	no	—
164	C₂×#107	—	73	2	no	—
165	C₂×#106	—	73	2	no	—
166	C₂×C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
167	C₂×#103	—	73	4	no	—
168	C₂×#103	—	73	2	no	—
169	C₂×C₄×C₈	C₂×D₇₃	1	2	yes	—
170	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
171	#391	—	73	4	no	—
172	#387	—	146	4	no	—
173	#383	—	146	2	no	—
174	#382	—	73	2	no	—
175	C₄×C₄⋊C₄	—	73	4	no	—
176	C₄×C₄⋊C₄	C₂×D₇₃	1	1	yes	characteristic
177	#376	—	146	2	no	—
178	#340	—	73	4	no	—
179	#336	C₂×D₇₃	1	2	yes	—
180	C₂²×Q₈	—	73	2	no	—
181	C₂×Q₁₆	—	73	4	no	—
182	C₂×Q₁₆	—	146	2	no	—
183	C₂×Q₁₆	—	146	2	no	—
184	C₂²×C₈	C₂²×D₇₃	1	2	yes	—
185	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
186	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
187	C₄×Q₈	—	73	8	no	—
188	C₄×Q₈	—	146	4	no	—
189	C₂×C₄⋊C₄	—	73	2	no	—
190	C₂×C₄⋊C₄	—	73	4	no	—
191	C₂×C₄⋊C₄	—	146	2	no	—
192	C₂×C₄⋊C₄	D₂₉₂	1	2	yes	—
193	C₂×C₄²	—	146	4	no	—
194	C₂×C₄²	—	146	2	no	—
195	C₂×C₄²	C₂²×D₇₃	1	2	yes	—
196	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
197	#107	—	73	4	no	—
198	#107	—	146	4	no	—
199	#106	—	292	2	no	—
200	C₄⋊C₈	—	2	2	no	—
201	#103	—	292	4	no	—
202	#103	—	73	8	no	—
203	C₄×C₈	—	2	4	no	—
204	C₄×C₈	C₄×D₇₃	1	4	yes	—
205	#95	—	146	2	no	—
206	#95	—	146	4	no	—
207	#95	—	146	4	no	—
208	#95	—	146	2	no	—
209	#95	—	146	4	no	—
210	#95	D₂₉₂	1	1	yes	characteristic
211	#95	D₂₉₂	1	1	yes	characteristic
212	C₂×Q₈	—	73	4	no	—
213	C₂×Q₈	—	146	2	no	—
214	C₂×Q₈	—	146	2	no	—
215	C₂×Q₈	—	146	2	no	—
216	C₂×Q₈	—	146	2	no	—
217	C₂×Q₈	—	146	2	no	—
218	C₂²×C₄	—	146	4	no	—
219	C₂²×C₄	—	146	2	no	—
220	C₂²×C₄	—	146	2	no	—
221	C₂²×C₄	C₂×D₂₉₂	1	2	yes	—
222	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
223	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
224	C₂²×C₄	C₂×D₂₉₂	1	2	yes	—
225	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
226	Q₁₆	—	146	8	no	—
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	2	1	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
233	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
234	C₂×C₈	#11924027	1	2	yes	—
235	C₂×C₈	#11924029	1	2	yes	—
236	C₄⋊C₄	—	2	4	no	—
237	C₄⋊C₄	—	73	4	no	—
238	C₄⋊C₄	—	146	4	no	—
239	C₄⋊C₄	—	146	4	no	—
240	C₄⋊C₄	—	146	8	no	—
241	C₄⋊C₄	—	146	2	no	—
242	C₄²	—	2	2	no	—
243	C₄²	—	2	2	no	—
244	C₄²	—	292	8	no	—
245	C₄²	—	146	8	no	—
246	C₄²	C₂×C₄×D₇₃	1	2	yes	—
247	C₄²	#11924028	1	2	yes	—
248	C₂³	C₂²×D₂₉₂	1	1	yes	characteristic
249	Q₈	—	292	2	no	—
250	Q₈	—	292	2	no	—
251	Q₈	—	146	4	no	—
252	Q₈	—	146	4	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	292	2	no	—
261	C₂×C₄	—	292	2	no	—
262	C₂×C₄	—	292	4	no	—
263	C₂×C₄	—	292	4	no	—
264	C₂×C₄	—	292	4	no	—
265	C₂×C₄	—	292	4	no	—
266	C₂×C₄	—	146	4	no	—
267	C₂×C₄	—	146	4	no	—
268	C₂×C₄	C₄×D₂₉₂	1	2	yes	—
269	C₂×C₄	C₄×D₂₉₂	1	2	yes	—
270	C₂×C₄	#14556613	1	4	yes	—
271	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
272	C₂×C₄	C₂×#11924027	1	1	yes	characteristic
273	C₂×C₄	C₂×#11924028	1	1	yes	characteristic
274	C₂×C₄	C₂×#11924029	1	1	yes	characteristic
275	C₂×C₄	#14556637	1	1	yes	characteristic
276	C₂×C₄	#14556638	1	1	yes	characteristic
277	C₈	—	2	2	no	—
278	C₈	—	4	2	no	—
279	C₈	#14556549	1	4	yes	—
280	C₂²	C₂×C₄×D₂₉₂	1	1	yes	characteristic
281	C₂²	#17989241	1	1	yes	characteristic
282	C₂²	#17989259	1	1	yes	characteristic
283	C₂²	C₂×#14556613	1	2	yes	—
284	C₂²	#17989404	1	2	yes	—
285	C₄	—	2	2	no	—
286	C₄	—	2	4	no	—
287	C₄	—	2	2	no	—
288	C₄	—	4	2	no	—
289	C₄	—	292	4	no	—
290	C₄	—	292	4	no	—
291	C₄	—	584	4	no	—
292	C₄	C₂×#14556549	1	1	yes	characteristic
293	C₄	#17989067	1	1	yes	characteristic
294	C₄	#17989302	1	1	yes	characteristic
295	C₄	#17989310	1	1	yes	characteristic
296	C₂	#22612878	1	1	yes	characteristic
297	C₂	C₄×#14556613	1	1	yes	characteristic
298	C₂	#22618800	1	1	yes	characteristic
299	C₂	#22622300	1	2	yes	—
300	C₂	#22622920	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₂₉₂
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₃
Fitting	C₈×C₄⋊C₄×C₇₃	C₂
socle, prime radical	C₂³×C₇₃	C₂²×D₄
cosocle	C₂²×C₄×C₇₃	C₂⁴
hypercenter	C₈×C₄⋊C₄	D₇₃
hypocenter	C₇₃	#22809
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
11. C₄×Dic₂₉₂	C₄	semidirect
159. C₇₃	#22809	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₂₉₂	2
#22612894	2
#22610792	2
#22610566	2
#22610095	2
#22609947	2
C₄×#14556525	2
C₈×C₄⋊C₄×C₇₃	2
#22809	73