Subgroups — #34012125

Order 18688.

586 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17989643	C₂	1	2	yes	maximal
2	#22624973	C₂	1	2	yes	maximal, complemented
3	#22624972	C₂	1	2	yes	maximal, complemented
4	C₂×#17988541	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556666	C₄	1	4	yes	complemented
6	#17989643	—	2	4	no	—
7	#17989643	—	2	4	no	—
8	C₂×#14556665	C₂²	1	4	yes	—
9	C₂×#14556664	C₄	1	4	yes	complemented
10	C₂²×#11924011	C₄	1	1	yes	characteristic
11	C₂²×C₈×D₇₃	C₂²	1	1	yes	characteristic
12	C₂×#14556548	C₄	1	1	yes	characteristic
13	#17988541	—	2	4	no	—
14	C₂×#14556502	C₂²	1	1	yes	characteristic
15	C₂×#14556501	C₂²	1	1	yes	characteristic
16	C₂×C₈×Dic₇₃	C₄	1	1	yes	characteristic
17	C₂×C₄×C₈×C₇₃	C₄	1	1	yes	characteristic
18	#14556666	—	2	8	no	—
19	#14556665	—	2	4	no	—
20	#14556665	—	2	4	no	—
21	#14556664	—	2	8	no	—
22	C₂×#11924031	C₂×C₄	1	8	yes	—
23	C₂²×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
24	C₂×#11924011	—	2	2	no	—
25	C₂×#11924011	—	2	2	no	—
26	C₂×#11924011	C₈	1	4	yes	—
27	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
28	C₂²×#10596831	C₂³	1	1	yes	characteristic
29	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
30	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
31	C₂×C₈×D₇₃	D₄	1	4	yes	—
32	C₂×C₈×D₇₃	D₄	1	2	yes	—
33	C₂×C₈×D₇₃	Q₈	1	2	yes	—
34	#14556548	—	2	2	no	—
35	#14556548	C₈	1	4	yes	—
36	C₂×#11924007	—	2	1	no	—
37	C₂×#11924005	—	2	1	no	—
38	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
39	#14556502	—	2	2	no	—
40	#14556501	—	2	2	no	—
41	C₈×Dic₇₃	—	2	2	no	—
42	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₇₃	—	2	2	no	—
46	#11924031	—	2	8	no	—
47	C₂³×D₇₃	M₁₆	1	1	yes	characteristic
48	C₂²×Dic₇₃	—	2	1	no	—
49	C₂²×Dic₇₃	M₁₆	1	1	yes	characteristic
50	C₂×C₄×D₇₃	C₄²	1	2	yes	—
51	C₂×C₄×D₇₃	C₂²⋊C₄	1	2	yes	—
52	C₂×C₄×D₇₃	C₄⋊C₄	1	2	yes	—
53	C₂×C₄×D₇₃	C₂×C₈	1	2	yes	—
54	C₂×C₄×D₇₃	C₂×C₈	1	2	yes	—
55	C₂×C₄×D₇₃	M₁₆	1	2	yes	—
56	#11924011	—	2	8	no	—
57	C₂×#10596831	—	2	1	no	—
58	C₂×#10596831	—	2	1	no	—
59	C₂×#10596831	C₂×C₈	1	2	yes	—
60	C₂×#10596831	C₂²×C₄	1	2	yes	—
61	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
62	C₂×#10596831	C₂×D₄	1	2	yes	—
63	C₂×#10596831	C₂×Q₈	1	1	yes	characteristic
64	C₈×D₇₃	C₂²⋊C₄	1	4	yes	—
65	C₈×D₇₃	C₂²⋊C₄	1	4	yes	—
66	C₈×D₇₃	C₄⋊C₄	1	4	yes	—
67	C₈×D₇₃	C₄⋊C₄	1	4	yes	—
68	#11924007	—	2	4	no	—
69	#11924005	—	2	4	no	—
70	C₄×Dic₇₃	—	2	1	no	—
71	C₄×Dic₇₃	C₂×C₈	1	2	yes	—
72	C₂²×C₄×C₇₃	—	2	1	no	—
73	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₈×C₇₃	—	2	1	no	—
75	C₂×C₈×C₇₃	—	2	1	no	—
76	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
77	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
78	C₂×C₈×C₇₃	C₂²⋊C₄	1	2	yes	—
79	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
80	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
81	C₄²×C₇₃	—	2	1	no	—
82	C₄²×C₇₃	C₂×C₈	1	2	yes	—
83	C₂³×C₇₃	C₂×M₁₆	1	1	yes	characteristic
84	C₂²×D₇₃	—	4	2	no	—
85	C₂²×D₇₃	#97	1	2	yes	—
86	C₂²×D₇₃	C₂²⋊C₈	1	2	yes	—
87	C₂²×D₇₃	C₄⋊C₈	1	2	yes	—
88	C₂×C₄×C₇₃	—	2	2	no	—
89	C₂×C₄×C₇₃	—	4	2	no	—
90	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
91	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
94	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
95	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
96	C₂×Dic₇₃	—	2	2	no	—
97	C₂×Dic₇₃	—	4	2	no	—
98	C₂×Dic₇₃	#97	1	2	yes	—
99	C₂×Dic₇₃	C₂²⋊C₈	1	2	yes	—
100	C₂×Dic₇₃	C₄⋊C₈	1	2	yes	—
101	C₄×D₇₃	#95	1	4	yes	—
102	C₄×D₇₃	C₄×C₈	1	4	yes	—
103	C₄×D₇₃	C₂²⋊C₈	1	4	yes	—
104	C₄×D₇₃	C₄⋊C₈	1	4	yes	—
105	C₈×C₇₃	—	2	2	no	—
106	C₈×C₇₃	#95	1	2	yes	—
107	C₈×C₇₃	#95	1	2	yes	—
108	#10596831	—	2	2	no	—
109	#10596831	C₂×C₂²⋊C₄	1	2	yes	—
110	#10596831	C₂×C₄⋊C₄	1	2	yes	—
111	C₂²×C₇₃	—	2	2	no	—
112	C₂²×C₇₃	C₂×#97	1	1	yes	characteristic
113	C₂²×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
114	C₂²×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
115	C₂×D₇₃	—	2	4	no	—
116	C₂×D₇₃	—	4	2	no	—
117	C₂×D₇₃	—	4	2	no	—
118	C₂×D₇₃	#336	1	4	yes	—
119	C₄×C₇₃	—	4	2	no	—
120	C₄×C₇₃	C₂×#95	1	1	yes	characteristic
121	C₄×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
122	C₄×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
123	C₄×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
124	Dic₇₃	—	4	2	no	—
125	Dic₇₃	#336	1	4	yes	—
126	#12547	—	73	1	no	maximal, Sylow
127	C₂×C₇₃	—	2	1	no	—
128	C₂×C₇₃	—	2	1	no	—
129	C₂×C₇₃	C₂×#336	1	1	yes	characteristic
130	C₂×C₇₃	#2110	1	2	yes	—
131	D₇₃	—	2	4	no	—
132	C₂²×M₃₂	—	73	2	no	—
133	C₂×#405	—	73	1	no	—
134	#1375	—	73	2	no	—
135	#1375	—	73	2	no	—
136	C₇₃	#12547	1	1	yes	characteristic, complemented, Sylow
137	C₂²×M₁₆	—	73	1	no	—
138	C₂³×C₈	—	73	1	no	—
139	C₂×#117	—	73	1	no	—
140	C₂×M₃₂	—	73	4	no	—
141	C₂×M₃₂	—	73	4	no	—
142	C₂×M₃₂	—	146	4	no	—
143	C₂×M₃₂	—	146	4	no	—
144	C₂²×C₁₆	—	73	4	no	—
145	#405	—	146	4	no	—
146	C₂×#97	—	73	1	no	—
147	C₂×#97	—	73	1	no	—
148	C₂×C₄×C₈	—	73	1	no	—
149	C₂×C₄×C₈	#64408	1	1	yes	characteristic
150	C₂³×C₄	—	73	1	no	—
151	C₂×M₁₆	—	73	4	no	—
152	C₂×M₁₆	—	146	2	no	—
153	C₂×M₁₆	—	146	2	no	—
154	C₂²×C₈	—	73	1	no	—
155	C₂²×C₈	—	73	2	no	—
156	C₂²×C₈	—	73	2	no	—
157	C₂²×C₈	—	73	4	no	—
158	C₂²×C₈	—	73	2	no	—
159	C₂²×C₈	—	73	2	no	—
160	C₂²×C₈	—	73	1	no	—
161	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
162	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
163	#117	—	73	4	no	—
164	#117	—	146	2	no	—
165	C₂×C₄⋊C₄	—	146	1	no	—
166	C₂×C₂²⋊C₄	—	146	1	no	—
167	C₂×C₄²	—	73	1	no	—
168	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
169	M₃₂	—	146	4	no	—
170	M₃₂	—	146	4	no	—
171	M₃₂	—	146	8	no	—
172	C₂×C₁₆	—	73	8	no	—
173	C₂×C₁₆	—	146	8	no	—
174	#97	—	146	2	no	—
175	#97	—	146	2	no	—
176	C₄×C₈	—	2	2	no	—
177	C₄×C₈	—	146	2	no	—
178	C₂⁴	—	73	1	no	—
179	C₂²×C₄	—	2	1	no	—
180	C₂²×C₄	—	73	1	no	—
181	C₂²×C₄	—	73	2	no	—
182	C₂²×C₄	—	73	2	no	—
183	C₂²×C₄	—	73	2	no	—
184	C₂²×C₄	—	73	2	no	—
185	C₂²×C₄	—	73	2	no	—
186	C₂²×C₄	—	73	2	no	—
187	C₂²×C₄	—	146	1	no	—
188	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
189	M₁₆	—	146	8	no	—
190	C₂×C₈	—	2	1	no	—
191	C₂×C₈	—	2	1	no	—
192	C₂×C₈	—	73	2	no	—
193	C₂×C₈	—	73	2	no	—
194	C₂×C₈	—	73	1	no	—
195	C₂×C₈	—	73	1	no	—
196	C₂×C₈	—	73	4	no	—
197	C₂×C₈	—	73	4	no	—
198	C₂×C₈	—	73	4	no	—
199	C₂×C₈	—	73	4	no	—
200	C₂×C₈	—	73	2	no	—
201	C₂×C₈	—	146	1	no	—
202	C₂×C₈	—	146	1	no	—
203	C₂×C₈	C₄×#64408	1	1	yes	characteristic
204	C₂×C₈	C₄×#64408	1	1	yes	characteristic
205	C₂×C₈	C₂²⋊#64408	1	2	yes	—
206	C₂×C₈	C₄⋊#64408	1	2	yes	—
207	C₂×C₈	#11924036	1	2	yes	—
208	C₄⋊C₄	—	146	4	no	—
209	C₂²⋊C₄	—	146	4	no	—
210	C₄²	—	2	1	no	—
211	C₄²	—	73	2	no	—
212	C₄²	—	146	1	no	—
213	C₄²	#11924036	1	2	yes	—
214	C₁₆	—	146	8	no	—
215	C₂³	—	292	2	no	—
216	C₂³	—	73	2	no	—
217	C₂³	—	73	2	no	—
218	C₂³	—	73	2	no	—
219	C₂³	D₇₃⋊M₁₆	1	1	yes	characteristic
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	—	292	2	no	—
223	C₂×C₄	—	73	2	no	—
224	C₂×C₄	—	73	2	no	—
225	C₂×C₄	—	73	2	no	—
226	C₂×C₄	—	73	4	no	—
227	C₂×C₄	—	73	4	no	—
228	C₂×C₄	—	73	4	no	—
229	C₂×C₄	—	73	4	no	—
230	C₂×C₄	—	146	2	no	—
231	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
232	C₂×C₄	C₂×C₂²⋊#64408	1	1	yes	characteristic
233	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
234	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
235	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
236	C₂×C₄	D₇₃⋊M₁₆	1	1	yes	characteristic
237	C₈	—	2	2	no	—
238	C₈	—	73	2	no	—
239	C₈	—	73	2	no	—
240	C₈	—	146	2	no	—
241	C₈	(C₂×#64408)⋊C₄	1	2	yes	—
242	C₈	(C₂×#64408)⋊C₄	1	2	yes	—
243	C₂²	—	2	2	no	—
244	C₂²	—	292	2	no	—
245	C₂²	—	292	2	no	—
246	C₂²	—	73	4	no	—
247	C₂²	—	146	4	no	—
248	C₂²	#17989546	1	1	yes	characteristic
249	C₂²	#17989555	1	1	yes	characteristic
250	C₂²	#17989588	1	1	yes	characteristic
251	C₄	—	4	2	no	—
252	C₄	—	292	2	no	—
253	C₄	—	73	4	no	—
254	C₄	C₂×(C₂×#64408)⋊C₄	1	1	yes	characteristic
255	C₄	C₄×#11924036	1	1	yes	characteristic
256	C₄	#17989555	1	1	yes	characteristic
257	C₄	#17989588	1	1	yes	characteristic
258	C₂	—	2	1	no	—
259	C₂	—	2	1	no	—
260	C₂	—	146	4	no	—
261	C₂	#22625592	1	1	yes	characteristic
262	C₂	#22626153	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₄×#64408
commutator	C₄×C₇₃	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×#64408
Fitting	C₂×C₄×C₈×C₇₃	C₄
socle	C₂²×C₇₃	C₂×#97
prime radical	C₂³×D₇₃	M₁₆
cosocle	C₂²×#10596831	C₂³
hypercenter	C₂×C₄×C₈	#64408
hypocenter	C₇₃	#12547
LCS	G ▷ C₄×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #22624973	C₂	semidirect
3. #22624972	C₂	semidirect
5. C₂×#14556666	C₄	semidirect
9. C₂×#14556664	C₄	semidirect
136. C₇₃	#12547	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17989643	2
#22624973	2
#22624972	2
C₂×#17988541	2
#12547	73