Subgroups — #33105790

Order 18688.

770 conjugacy classes of subgroups.

The subgroup lattice has 296 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	#22612886	C₂	1	2	yes	maximal
3	#22610802	C₂	1	2	yes	maximal
4	#22610566	C₂	1	4	yes	maximal
5	#22610088	C₂	1	1	yes	maximal, characteristic
6	#22609937	C₂	1	2	yes	maximal
7	#22609828	C₂	1	2	yes	maximal
8	C₇₃×#1770	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	1	yes	characteristic
13	C₂×#14556523	C₂²	1	2	yes	—
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	#17988474	—	2	4	no	—
19	#17988457	C₂²	1	2	yes	—
20	#17988452	—	2	2	no	—
21	C₄⋊C₄×Dic₇₃	—	2	2	no	—
22	C₄×C₄⋊Dic₇₃	C₂²	1	4	yes	—
23	#17988350	C₂²	1	4	yes	—
24	C₂×C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
25	C₂×#97×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	—	2	2	no	—
36	C₂×C₄⋊Dic₇₃	—	2	2	no	—
37	C₂×C₄⋊Dic₇₃	C₂³	1	4	yes	—
38	C₂×C₄⋊Dic₇₃	C₂³	1	2	yes	—
39	C₂×#11924007	—	2	2	no	—
40	C₂×C₄×Dic₇₃	—	2	2	no	—
41	C₂×C₄×Dic₇₃	—	2	4	no	—
42	#14556525	C₂×C₄	1	4	yes	—
43	#14556523	—	2	4	no	—
44	#14556523	—	4	2	no	—
45	#14556516	—	4	4	no	—
46	#14556516	C₂×C₄	1	8	yes	—
47	#14556498	—	2	4	no	—
48	#14556498	—	2	4	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	#97×C₇₃	—	2	4	no	—
58	C₄×C₈×C₇₃	C₂×C₄	1	4	yes	—
59	#95×C₇₃	D₄	1	2	yes	—
60	C₂×Dic₁₄₆	—	2	2	no	—
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	C₂²×C₄	1	4	yes	—
66	C₂²×Dic₇₃	—	2	2	no	—
67	C₂²×Dic₇₃	—	2	2	no	—
68	C₂²×Dic₇₃	—	2	4	no	—
69	Dic₂₉₂	—	2	8	no	—
70	C₄⋊Dic₇₃	—	2	2	no	—
71	C₄⋊Dic₇₃	—	2	4	no	—
72	C₄⋊Dic₇₃	—	2	4	no	—
73	C₄⋊Dic₇₃	—	4	4	no	—
74	C₄⋊Dic₇₃	—	8	2	no	—
75	C₄⋊Dic₇₃	C₂²×C₄	1	4	yes	—
76	#11924007	—	2	8	no	—
77	C₄×Dic₇₃	—	2	8	no	—
78	C₄×Dic₇₃	—	8	4	no	—
79	C₂²×C₄×C₇₃	C₂×D₄	1	4	yes	—
80	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
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	4	no	—
100	C₂×C₄×C₇₃	—	2	4	no	—
101	C₂×C₄×C₇₃	—	2	4	no	—
102	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
103	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
104	C₂×C₄×C₇₃	#137	1	4	yes	—
105	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
106	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
107	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
108	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
109	C₂×C₄×C₇₃	H(5,𝔽₂)	1	1	yes	characteristic
110	C₂×C₄×C₇₃	2₋¹⁺⁴	1	1	yes	characteristic
111	C₂×Dic₇₃	—	2	4	no	—
112	C₂×Dic₇₃	—	2	4	no	—
113	C₂×Dic₇₃	—	4	2	no	—
114	C₂×Dic₇₃	—	4	2	no	—
115	C₂×Dic₇₃	—	4	2	no	—
116	C₂×Dic₇₃	—	4	2	no	—
117	C₂×Dic₇₃	—	4	4	no	—
118	C₂×Dic₇₃	—	4	4	no	—
119	C₂×Dic₇₃	—	4	4	no	—
120	Dic₁₄₆	—	2	4	no	—
121	Dic₁₄₆	—	2	4	no	—
122	Dic₁₄₆	—	4	2	no	—
123	Dic₁₄₆	—	4	2	no	—
124	C₈×C₇₃	—	2	2	no	—
125	C₈×C₇₃	—	4	2	no	—
126	C₈×C₇₃	#117	1	4	yes	—
127	C₂²×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
128	C₂²×C₇₃	#547	1	1	yes	characteristic
129	C₂²×C₇₃	#550	1	1	yes	characteristic
130	C₂²×C₇₃	C₂×#137	1	2	yes	—
131	C₂²×C₇₃	#577	1	2	yes	—
132	C₄×C₇₃	—	2	2	no	—
133	C₄×C₇₃	—	4	4	no	—
134	C₄×C₇₃	#441	1	4	yes	—
135	C₄×C₇₃	C₂×#117	1	1	yes	characteristic
136	C₄×C₇₃	#520	1	1	yes	characteristic
137	C₄×C₇₃	#555	1	1	yes	characteristic
138	C₄×C₇₃	#556	1	1	yes	characteristic
139	Dic₇₃	—	4	4	no	—
140	Dic₇₃	—	4	4	no	—
141	Dic₇₃	—	8	4	no	—
142	#22814	—	73	1	no	maximal, Sylow
143	C₂×C₇₃	#2354	1	1	yes	characteristic
144	C₂×C₇₃	C₂×#441	1	1	yes	characteristic
145	C₂×C₇₃	#2952	1	1	yes	characteristic
146	C₂×C₇₃	#3356	1	2	yes	—
147	C₂×C₇₃	#3400	1	2	yes	—
148	C₂×C₄×Q₁₆	—	73	1	no	—
149	#2355	—	73	2	no	—
150	#1971	—	73	2	no	—
151	#1927	—	73	4	no	—
152	#1844	—	73	1	no	—
153	#1803	—	73	2	no	—
154	#1776	—	73	2	no	—
155	#1770	D₇₃	1	1	yes	characteristic
156	C₇₃	#22814	1	1	yes	characteristic, complemented, Sylow
157	C₂²×Q₁₆	—	73	1	no	—
158	C₂×C₄×Q₈	—	73	2	no	—
159	C₄×Q₁₆	—	73	8	no	—
160	C₂×#107	—	73	1	no	—
161	C₂×#106	—	73	2	no	—
162	C₂×#106	—	73	2	no	—
163	C₂×C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
164	C₂×#103	—	73	4	no	—
165	C₂×#103	—	73	2	no	—
166	C₂×#97	C₂×D₇₃	1	2	yes	—
167	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
168	#391	—	73	4	no	—
169	#387	—	146	4	no	—
170	#384	—	73	2	no	—
171	#382	—	146	2	no	—
172	C₄×C₄⋊C₄	—	73	4	no	—
173	C₄×C₄⋊C₄	—	146	2	no	—
174	C₄×C₄⋊C₄	C₂×D₇₃	1	1	yes	characteristic
175	#340	—	73	4	no	—
176	#336	C₂×D₇₃	1	2	yes	—
177	C₂²×Q₈	—	73	2	no	—
178	C₂×Q₁₆	—	73	4	no	—
179	C₂×Q₁₆	—	146	2	no	—
180	C₂×Q₁₆	—	146	2	no	—
181	C₂²×C₈	C₂²×D₇₃	1	2	yes	—
182	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
183	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
184	C₄×Q₈	—	73	8	no	—
185	C₄×Q₈	—	146	4	no	—
186	C₂×C₄⋊C₄	—	73	2	no	—
187	C₂×C₄⋊C₄	—	73	4	no	—
188	C₂×C₄⋊C₄	—	146	2	no	—
189	C₂×C₄⋊C₄	—	146	2	no	—
190	C₂×C₄⋊C₄	—	146	2	no	—
191	C₂×C₄⋊C₄	D₂₉₂	1	2	yes	—
192	C₂×C₄²	—	146	4	no	—
193	C₂×C₄²	—	146	2	no	—
194	C₂×C₄²	C₂²×D₇₃	1	2	yes	—
195	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
196	#107	—	73	4	no	—
197	#106	—	292	2	no	—
198	#106	—	146	4	no	—
199	C₄⋊C₈	—	2	2	no	—
200	#103	—	292	4	no	—
201	#103	—	73	8	no	—
202	#97	—	2	4	no	—
203	C₄×C₈	C₄×D₇₃	1	4	yes	—
204	#95	—	146	4	no	—
205	#95	—	146	4	no	—
206	#95	—	146	4	no	—
207	#95	D₂₉₂	1	2	yes	—
208	C₂×Q₈	—	73	4	no	—
209	C₂×Q₈	—	146	2	no	—
210	C₂×Q₈	—	146	2	no	—
211	C₂×Q₈	—	146	2	no	—
212	C₂×Q₈	—	146	2	no	—
213	C₂×Q₈	—	146	2	no	—
214	C₂²×C₄	—	146	4	no	—
215	C₂²×C₄	—	146	2	no	—
216	C₂²×C₄	—	146	2	no	—
217	C₂²×C₄	C₂×D₂₉₂	1	4	yes	—
218	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
219	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
220	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
221	Q₁₆	—	146	8	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
228	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
229	C₂×C₈	#11924027	1	2	yes	—
230	C₂×C₈	#11924029	1	2	yes	—
231	C₄⋊C₄	—	2	4	no	—
232	C₄⋊C₄	—	292	4	no	—
233	C₄⋊C₄	—	584	2	no	—
234	C₄⋊C₄	—	73	4	no	—
235	C₄⋊C₄	—	146	4	no	—
236	C₄⋊C₄	—	146	4	no	—
237	C₄⋊C₄	—	146	8	no	—
238	C₄⋊C₄	—	146	2	no	—
239	C₄²	—	2	2	no	—
240	C₄²	—	2	2	no	—
241	C₄²	—	584	4	no	—
242	C₄²	—	146	8	no	—
243	C₄²	C₂×C₄×D₇₃	1	2	yes	—
244	C₄²	#11924028	1	2	yes	—
245	C₂³	C₂²×D₂₉₂	1	1	yes	characteristic
246	Q₈	—	292	2	no	—
247	Q₈	—	292	2	no	—
248	Q₈	—	146	4	no	—
249	Q₈	—	146	4	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	292	2	no	—
255	C₂×C₄	—	292	2	no	—
256	C₂×C₄	—	292	2	no	—
257	C₂×C₄	—	292	2	no	—
258	C₂×C₄	—	292	4	no	—
259	C₂×C₄	—	292	4	no	—
260	C₂×C₄	—	292	4	no	—
261	C₂×C₄	—	146	4	no	—
262	C₂×C₄	—	146	4	no	—
263	C₂×C₄	C₄×D₂₉₂	1	2	yes	—
264	C₂×C₄	C₄×D₂₉₂	1	2	yes	—
265	C₂×C₄	#14556624	1	4	yes	—
266	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
267	C₂×C₄	C₂×#11924027	1	1	yes	characteristic
268	C₂×C₄	C₂×#11924028	1	1	yes	characteristic
269	C₂×C₄	C₂×#11924029	1	1	yes	characteristic
270	C₂×C₄	#14556637	1	1	yes	characteristic
271	C₂×C₄	#14556638	1	1	yes	characteristic
272	C₈	—	2	2	no	—
273	C₈	—	4	2	no	—
274	C₈	#14556549	1	4	yes	—
275	C₂²	C₂×C₄×D₂₉₂	1	1	yes	characteristic
276	C₂²	#17989241	1	1	yes	characteristic
277	C₂²	#17989259	1	1	yes	characteristic
278	C₂²	C₂×#14556624	1	2	yes	—
279	C₂²	#17989399	1	2	yes	—
280	C₄	—	2	2	no	—
281	C₄	—	4	4	no	—
282	C₄	—	292	4	no	—
283	C₄	—	292	4	no	—
284	C₄	—	584	4	no	—
285	C₄	#17988695	1	4	yes	—
286	C₄	C₂×#14556549	1	1	yes	characteristic
287	C₄	#17989067	1	1	yes	characteristic
288	C₄	#17989302	1	1	yes	characteristic
289	C₄	#17989310	1	1	yes	characteristic
290	C₂	#22612878	1	1	yes	characteristic
291	C₂	C₂×#17988695	1	1	yes	characteristic
292	C₂	#22618816	1	1	yes	characteristic
293	C₂	#22622547	1	2	yes	—
294	C₂	#22622986	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₂₉₂
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₃
Fitting	C₇₃×#1770	C₂
socle, prime radical	C₂³×C₇₃	C₂²×D₄
cosocle	C₂²×C₄×C₇₃	C₂⁴
hypercenter	#1770	D₇₃
hypocenter	C₇₃	#22814
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ #1770

Complements

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

N	H ≅ G/N	type
11. C₄×Dic₂₉₂	C₄	semidirect
156. C₇₃	#22814	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
#22612886	2
#22610802	2
#22610566	2
#22610088	2
#22609937	2
#22609828	2
C₇₃×#1770	2
#22814	73