Subgroups — #33055459

Order 18688.

678 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22619011	C₂	1	2	yes	maximal
2	C₂×C₄×Dic₂₉₂	C₂	1	1	yes	maximal, characteristic
3	#22608751	C₂	1	4	yes	maximal, complemented
4	C₈×Dic₂₉₂	C₂	1	4	yes	maximal, complemented
5	#22608679	C₂	1	1	yes	maximal, characteristic
6	#22608327	C₂	1	2	yes	maximal
7	C₇₃×#1450	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₂₉₂	C₄	1	1	yes	characteristic
9	C₂×C₄×Dic₁₄₆	C₂²	1	2	yes	—
10	C₈×Dic₁₄₆	C₂²	1	8	yes	—
11	#17988711	—	2	4	no	—
12	C₄×Dic₂₉₂	C₄	1	2	yes	—
13	C₄×Dic₂₉₂	C₄	1	2	yes	—
14	C₄×Dic₂₉₂	C₂²	1	2	yes	—
15	C₄×Dic₂₉₂	C₂²	1	2	yes	—
16	#17988640	C₂²	1	2	yes	—
17	#17988637	—	2	4	no	—
18	C₂×#14556525	C₄	1	1	yes	characteristic
19	C₂×#14556516	C₄	1	2	yes	—
20	#17988340	C₂²	1	2	yes	—
21	#17988338	C₂²	1	2	yes	—
22	#17988319	C₂²	1	8	yes	—
23	#431×C₇₃	C₂²	1	1	yes	characteristic
24	#431×C₇₃	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
26	#322×C₇₃	C₂²	1	2	yes	—
27	C₈²×C₇₃	C₂²	1	2	yes	—
28	C₂²×Dic₁₄₆	C₂×C₄	1	2	yes	—
29	C₂×Dic₂₉₂	C₈	1	8	yes	complemented
30	C₂×Dic₂₉₂	C₂×C₄	1	2	yes	—
31	C₂×Dic₂₉₂	C₂×C₄	1	2	yes	—
32	C₄×Dic₁₄₆	—	2	2	no	—
33	C₄×Dic₁₄₆	—	2	2	no	—
34	C₄×Dic₁₄₆	C₂×C₄	1	4	yes	—
35	C₄×Dic₁₄₆	C₂³	1	4	yes	—
36	C₂×C₄⋊Dic₇₃	C₂×C₄	1	2	yes	—
37	C₂×#11924007	—	2	2	no	—
38	C₂×C₄×Dic₇₃	—	2	2	no	—
39	#14556525	C₂×C₄	1	2	yes	—
40	#14556525	C₂×C₄	1	2	yes	—
41	C₄⋊#10596831	C₂³	1	4	yes	—
42	#14556520	—	2	8	no	—
43	#14556516	C₂×C₄	1	4	yes	—
44	#14556516	C₂×C₄	1	4	yes	—
45	C₂²⋊#10596831	—	2	2	no	—
46	C₄×#10596831	—	2	2	no	—
47	C₈×Dic₇₃	—	2	8	no	—
48	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
51	C₄⋊C₈×C₇₃	—	2	1	no	—
52	C₄⋊C₈×C₇₃	D₄	1	2	yes	—
53	C₂²⋊C₈×C₇₃	—	2	1	no	—
54	C₂²⋊C₈×C₇₃	D₄	1	2	yes	—
55	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
56	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
57	C₄×C₈×C₇₃	C₂³	1	2	yes	—
58	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
59	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
60	C₄×C₈×C₇₃	C₂³	1	2	yes	—
61	C₂×Dic₁₄₆	—	2	2	no	—
62	C₂×Dic₁₄₆	—	2	4	no	—
63	C₂×Dic₁₄₆	C₂×C₈	1	8	yes	—
64	C₂×Dic₁₄₆	C₂²×C₄	1	4	yes	—
65	C₂²×Dic₇₃	—	2	2	no	—
66	C₂²×Dic₇₃	—	2	2	no	—
67	Dic₂₉₂	C₂×C₈	1	8	yes	—
68	Dic₂₉₂	C₂×C₈	1	8	yes	—
69	C₂×#10596831	—	2	4	no	—
70	C₄⋊Dic₇₃	—	2	2	no	—
71	C₄⋊Dic₇₃	C₂²×C₄	1	4	yes	—
72	#11924007	—	2	4	no	—
73	#11924007	—	2	4	no	—
74	C₄×Dic₇₃	—	2	4	no	—
75	C₄×Dic₇₃	—	2	4	no	—
76	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
79	C₂×C₈×C₇₃	—	2	2	no	—
80	C₂×C₈×C₇₃	—	2	2	no	—
81	C₂×C₈×C₇₃	C₂×C₈	1	4	yes	—
82	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₇₃	C₂×D₄	1	4	yes	—
87	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₇₃	#41	1	1	yes	characteristic
90	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
91	C₂³×C₇₃	C₄×D₄	1	1	yes	characteristic
92	C₂×C₄×C₇₃	—	2	2	no	—
93	C₂×C₄×C₇₃	—	4	1	no	—
94	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
95	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
96	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
97	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
98	C₂×C₄×C₇₃	#131	1	2	yes	—
99	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
100	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
101	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
102	C₂×Dic₇₃	—	2	4	no	—
103	C₂×Dic₇₃	—	2	8	no	—
104	C₂×Dic₇₃	—	2	4	no	—
105	C₂×Dic₇₃	—	8	2	no	—
106	Dic₁₄₆	—	2	4	no	—
107	Dic₁₄₆	—	2	8	no	—
108	Dic₁₄₆	C₂²×C₈	1	8	yes	—
109	C₈×C₇₃	—	2	1	no	—
110	C₈×C₇₃	—	2	1	no	—
111	C₈×C₇₃	—	2	4	no	—
112	C₈×C₇₃	—	4	2	no	—
113	C₈×C₇₃	C₂²×C₈	1	2	yes	—
114	C₈×C₇₃	C₂²×C₈	1	2	yes	—
115	#10596831	—	4	4	no	—
116	C₂²×C₇₃	—	2	1	no	—
117	C₂²×C₇₃	—	2	1	no	—
118	C₂²×C₇₃	C₈×D₄	1	2	yes	—
119	C₂²×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
120	C₄×C₇₃	—	2	1	no	—
121	C₄×C₇₃	—	2	1	no	—
122	C₄×C₇₃	—	2	1	no	—
123	C₄×C₇₃	C₈×D₄	1	2	yes	—
124	C₄×C₇₃	C₂³×C₈	1	1	yes	characteristic
125	C₄×C₇₃	C₂×#131	1	1	yes	characteristic
126	C₄×C₇₃	#578	1	2	yes	—
127	Dic₇₃	—	2	8	no	—
128	Dic₇₃	—	4	4	no	—
129	#17856	—	73	1	no	maximal, Sylow
130	C₂×C₇₃	—	2	2	no	—
131	C₂×C₇₃	C₂×C₈×D₄	1	1	yes	characteristic
132	C₂×C₇₃	#2950	1	1	yes	characteristic
133	C₂×C₇₃	#2957	1	1	yes	characteristic
134	#2982	—	73	2	no	—
135	C₂×C₄×Q₁₆	—	73	1	no	—
136	#1579	—	73	4	no	—
137	C₈×Q₁₆	—	73	4	no	—
138	#1564	—	73	1	no	—
139	#1479	—	73	2	no	—
140	#1450	D₇₃	1	1	yes	characteristic
141	C₇₃	#17856	1	1	yes	characteristic, complemented, Sylow
142	C₂²×Q₁₆	—	73	1	no	—
143	C₂×C₄×Q₈	—	73	2	no	—
144	C₈×Q₈	—	73	8	no	—
145	C₈×Q₈	—	146	4	no	—
146	C₄×Q₁₆	—	73	2	no	—
147	C₄×Q₁₆	—	73	2	no	—
148	C₄×Q₁₆	—	73	2	no	—
149	C₄×Q₁₆	—	73	2	no	—
150	#431	—	73	2	no	—
151	#431	—	146	4	no	—
152	#431	C₂×D₇₃	1	1	yes	characteristic
153	#431	C₂×D₇₃	1	1	yes	characteristic
154	C₂×#107	—	73	1	no	—
155	C₂×#103	—	73	2	no	—
156	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
157	#335	—	73	2	no	—
158	#334	—	73	2	no	—
159	Q₈⋊C₈	—	73	8	no	—
160	#322	C₂×D₇₃	1	2	yes	—
161	C₈²	C₂×D₇₃	1	2	yes	—
162	C₂²×Q₈	—	73	2	no	—
163	C₂×Q₁₆	—	73	8	no	—
164	C₂×Q₁₆	—	73	2	no	—
165	C₂×Q₁₆	—	73	2	no	—
166	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
167	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
168	C₄×Q₈	—	73	4	no	—
169	C₄×Q₈	—	73	4	no	—
170	C₄×Q₈	—	146	2	no	—
171	C₄×Q₈	—	146	2	no	—
172	C₂×C₄⋊C₄	—	73	2	no	—
173	C₂×C₄⋊C₄	—	146	2	no	—
174	C₂×C₄²	—	146	2	no	—
175	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
176	#107	—	73	2	no	—
177	#107	—	73	2	no	—
178	C₄⋊C₈	—	2	1	no	—
179	C₄⋊C₈	—	73	4	no	—
180	C₄⋊C₈	—	146	8	no	—
181	C₄⋊C₈	D₂₉₂	1	2	yes	—
182	#103	—	73	4	no	—
183	#103	—	73	4	no	—
184	C₂²⋊C₈	—	2	1	no	—
185	C₂²⋊C₈	—	146	2	no	—
186	C₂²⋊C₈	D₂₉₂	1	2	yes	—
187	C₄×C₈	—	146	8	no	—
188	C₄×C₈	—	146	2	no	—
189	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
190	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
191	C₄×C₈	C₂²×D₇₃	1	2	yes	—
192	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
193	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
194	C₄×C₈	C₂²×D₇₃	1	2	yes	—
195	C₂×Q₈	—	73	8	no	—
196	C₂×Q₈	—	73	4	no	—
197	C₂×Q₈	—	146	2	no	—
198	C₂×Q₈	—	146	4	no	—
199	C₂²×C₄	—	146	2	no	—
200	C₂²×C₄	—	146	2	no	—
201	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
202	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
203	C₂²×C₄	#11924028	1	1	yes	characteristic
204	Q₁₆	—	73	8	no	—
205	Q₁₆	—	73	8	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	146	4	no	—
209	C₂×C₈	C₈×D₇₃	1	4	yes	—
210	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
211	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
212	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
213	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
214	C₂×C₈	C₂×D₂₉₂	1	4	yes	—
215	C₄⋊C₄	—	73	4	no	—
216	C₄⋊C₄	—	146	4	no	—
217	C₄⋊C₄	—	146	2	no	—
218	C₄⋊C₄	—	146	4	no	—
219	C₄²	—	146	4	no	—
220	C₄²	—	146	4	no	—
221	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
222	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
223	C₄²	#11924028	1	1	yes	characteristic
224	C₄²	C₂³×D₇₃	1	1	yes	characteristic
225	C₂³	C₄×D₂₉₂	1	1	yes	characteristic
226	Q₈	—	73	8	no	—
227	Q₈	—	146	8	no	—
228	Q₈	—	146	4	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	4	1	no	—
231	C₂×C₄	—	584	2	no	—
232	C₂×C₄	—	146	4	no	—
233	C₂×C₄	—	146	8	no	—
234	C₂×C₄	—	146	4	no	—
235	C₂×C₄	C₄×D₂₉₂	1	1	yes	characteristic
236	C₂×C₄	C₄×D₂₉₂	1	1	yes	characteristic
237	C₂×C₄	C₄×D₂₉₂	1	1	yes	characteristic
238	C₂×C₄	C₂×C₈×D₇₃	1	2	yes	—
239	C₂×C₄	#14556597	1	2	yes	—
240	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
241	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
242	C₂×C₄	C₂×#11924028	1	1	yes	characteristic
243	C₈	—	2	4	no	—
244	C₈	—	2	1	no	—
245	C₈	—	2	1	no	—
246	C₈	—	4	2	no	—
247	C₈	—	292	4	no	—
248	C₈	C₂×C₈×D₇₃	1	2	yes	—
249	C₈	C₂×C₈×D₇₃	1	2	yes	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	C₈×D₂₉₂	1	2	yes	—
253	C₂²	C₂×C₄×D₂₉₂	1	1	yes	characteristic
254	C₄	—	2	1	no	—
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	292	4	no	—
258	C₄	—	146	8	no	—
259	C₄	C₈×D₂₉₂	1	2	yes	—
260	C₄	C₂²×C₈×D₇₃	1	1	yes	characteristic
261	C₄	C₂×#14556597	1	1	yes	characteristic
262	C₄	#17989404	1	2	yes	—
263	C₂	—	2	2	no	—
264	C₂	C₂×C₈×D₂₉₂	1	1	yes	characteristic
265	C₂	#22618800	1	1	yes	characteristic
266	C₂	#22618849	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₉₂
commutator	C₄×C₇₃	C₂³×C₈
Frattini	C₄²	C₂³×D₇₃
Fitting	C₇₃×#1450	C₂
socle	C₂²×C₇₃	C₂×C₄×D₄
prime radical	C₂³×C₇₃	C₄×D₄
cosocle	C₄²×C₇₃	C₂⁴
hypercenter	#1450	D₇₃
hypocenter	C₇₃	#17856
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #1450

Complements

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

N	H ≅ G/N	type
3. #22608751	C₂	semidirect
4. C₈×Dic₂₉₂	C₂	semidirect
29. C₂×Dic₂₉₂	C₈	semidirect
141. C₇₃	#17856	semidirect

Maximal subgroups

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

subgroup	index
#22619011	2
C₂×C₄×Dic₂₉₂	2
#22608751	2
C₈×Dic₂₉₂	2
#22608679	2
#22608327	2
C₇₃×#1450	2
#17856	73