Subgroups — #34010416

Order 18688.

382 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22626261	C₂	1	1	yes	maximal, characteristic
2	C₄×#14556664	C₂	1	1	yes	maximal, characteristic
3	#22624898	C₂	1	2	yes	maximal, complemented
4	Q₈⋊#11924031	C₂	1	2	yes	maximal, complemented
5	C₈×Q₈×D₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556664	—	2	1	no	—
7	C₂×#14556664	—	2	1	no	—
8	#17989508	—	2	1	no	—
9	C₄⋊#11924031	C₂²	1	2	yes	—
10	#17989473	—	2	1	no	—
11	C₄×#11924031	C₂²	1	2	yes	—
12	C₄×Q₈×D₇₃	C₄	1	1	yes	characteristic
13	Q₈×#10596831	C₂²	1	1	yes	characteristic
14	C₈×Dic₁₄₆	—	2	1	no	—
15	C₈×Dic₁₄₆	C₄	1	1	yes	characteristic
16	#17988711	—	2	1	no	—
17	#17988711	C₂²	1	1	yes	characteristic
18	C₄⋊C₈×D₇₃	—	2	1	no	—
19	C₄⋊C₈×D₇₃	C₄	1	1	yes	characteristic
20	C₄×C₈×D₇₃	—	2	1	no	—
21	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
22	C₈×Q₈×C₇₃	C₄	1	1	yes	characteristic
23	#14556664	—	2	4	no	—
24	#14556664	—	4	2	no	—
25	C₂×#11924031	—	2	2	no	—
26	C₂×#11924031	—	2	2	no	—
27	C₂×Q₈×D₇₃	C₈	1	1	yes	characteristic
28	C₂×C₈×D₇₃	—	2	1	no	—
29	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
30	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
31	Q₈×Dic₇₃	C₈	1	1	yes	characteristic
32	C₄×Dic₁₄₆	—	2	1	no	—
33	C₄×Dic₁₄₆	C₂×C₄	1	1	yes	characteristic
34	#14556556	—	2	1	no	—
35	#14556556	C₈	1	1	yes	characteristic
36	C₄⋊C₄×D₇₃	—	2	1	no	—
37	C₄⋊C₄×D₇₃	C₈	1	1	yes	characteristic
38	C₄²×D₇₃	—	2	1	no	—
39	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
40	C₄⋊#10596831	—	2	1	no	—
41	C₄⋊#10596831	C₂×C₄	1	1	yes	characteristic
42	#14556520	—	2	1	no	—
43	#14556520	—	2	1	no	—
44	#14556520	—	2	1	no	—
45	C₄×#10596831	—	2	1	no	—
46	C₄×#10596831	C₂³	1	1	yes	characteristic
47	C₈×Dic₇₃	—	2	1	no	—
48	C₈×Dic₇₃	—	2	1	no	—
49	C₈×Dic₇₃	D₄	1	1	yes	characteristic
50	C₈×Dic₇₃	D₄	1	1	yes	characteristic
51	C₄×Q₈×C₇₃	C₂×C₄	1	1	yes	characteristic
52	C₄⋊C₈×C₇₃	—	2	1	no	—
53	C₄⋊C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₇₃	—	2	1	no	—
55	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
56	#11924031	—	2	4	no	—
57	#11924031	—	4	2	no	—
58	C₂×Dic₁₄₆	—	2	1	no	—
59	C₂×Dic₁₄₆	C₂×C₈	1	1	yes	characteristic
60	Q₈×D₇₃	—	2	2	no	—
61	Q₈×D₇₃	C₁₆	1	4	yes	complemented
62	C₂×C₄×D₇₃	—	2	1	no	—
63	C₂×C₄×D₇₃	—	2	1	no	—
64	C₂×C₄×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
66	C₂×C₄×D₇₃	M₁₆	1	1	yes	characteristic
67	C₂×#10596831	—	2	1	no	—
68	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
69	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
70	C₈×D₇₃	—	2	2	no	—
71	C₈×D₇₃	—	4	2	no	—
72	C₈×D₇₃	SD₁₆	1	2	yes	—
73	C₈×D₇₃	Q₁₆	1	2	yes	—
74	C₄⋊Dic₇₃	—	2	1	no	—
75	C₄⋊Dic₇₃	C₂×C₈	1	1	yes	characteristic
76	#11924007	—	2	1	no	—
77	#11924007	—	2	1	no	—
78	#11924007	—	2	1	no	—
79	C₄×Dic₇₃	—	2	1	no	—
80	C₄×Dic₇₃	—	2	1	no	—
81	C₄×Dic₇₃	—	2	1	no	—
82	C₄×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
83	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
84	C₄×Dic₇₃	M₁₆	1	1	yes	characteristic
85	C₂×Q₈×C₇₃	C₂×C₈	1	1	yes	characteristic
86	C₂×C₈×C₇₃	—	2	1	no	—
87	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
89	C₄⋊C₄×C₇₃	—	2	1	no	—
90	C₄⋊C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
91	C₄²×C₇₃	—	2	1	no	—
92	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
93	C₂²×D₇₃	C₂²⋊C₈	1	1	yes	characteristic
94	Q₈×C₇₃	—	2	1	no	—
95	Q₈×C₇₃	C₂×C₁₆	1	2	yes	complemented
96	C₂×C₄×C₇₃	—	2	1	no	—
97	C₂×C₄×C₇₃	—	2	1	no	—
98	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
100	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
101	C₂×Dic₇₃	—	2	1	no	—
102	C₂×Dic₇₃	—	2	1	no	—
103	C₂×Dic₇₃	C₂²⋊C₈	1	1	yes	characteristic
104	C₂×Dic₇₃	C₂²⋊C₈	1	1	yes	characteristic
105	C₂×Dic₇₃	C₂²⋊C₈	1	1	yes	characteristic
106	C₄×D₇₃	—	2	4	no	—
107	C₄×D₇₃	—	2	2	no	—
108	C₄×D₇₃	—	4	2	no	—
109	C₄×D₇₃	#103	1	2	yes	—
110	C₄×D₇₃	#104	1	2	yes	—
111	C₄×D₇₃	C₂×C₁₆	1	2	yes	—
112	C₄×D₇₃	M₃₂	1	2	yes	—
113	Dic₁₄₆	—	2	1	no	—
114	Dic₁₄₆	—	2	4	no	—
115	Dic₁₄₆	C₂×C₁₆	1	2	yes	complemented
116	C₈×C₇₃	—	2	1	no	—
117	C₈×C₇₃	—	4	1	no	—
118	C₈×C₇₃	#103	1	2	yes	—
119	#10596831	—	2	1	no	—
120	#10596831	—	4	1	no	—
121	#10596831	C₂×SD₁₆	1	1	yes	characteristic
122	#10596831	C₂×Q₁₆	1	1	yes	characteristic
123	C₂²×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
124	C₂×D₇₃	Q₈⋊C₈	1	2	yes	—
125	C₂×D₇₃	C₂²⋊C₁₆	1	2	yes	—
126	C₂×D₇₃	#350	1	2	yes	—
127	C₄×C₇₃	—	2	2	no	—
128	C₄×C₇₃	—	2	1	no	—
129	C₄×C₇₃	—	4	1	no	—
130	C₄×C₇₃	C₂×#103	1	1	yes	characteristic
131	C₄×C₇₃	C₂×#104	1	1	yes	characteristic
132	C₄×C₇₃	C₂²×C₁₆	1	1	yes	characteristic
133	C₄×C₇₃	C₂×M₃₂	1	1	yes	characteristic
134	Dic₇₃	—	2	2	no	—
135	Dic₇₃	—	2	1	no	—
136	Dic₇₃	—	4	1	no	—
137	Dic₇₃	Q₈⋊C₈	1	2	yes	—
138	Dic₇₃	C₂²⋊C₁₆	1	2	yes	—
139	C₂×Q₈⋊C₁₆	—	73	1	no	maximal, Sylow
140	C₂×C₇₃	C₂×Q₈⋊C₈	1	1	yes	characteristic
141	C₂×C₇₃	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
142	C₂×C₇₃	C₂×#350	1	1	yes	characteristic
143	D₇₃	Q₈⋊C₁₆	1	4	yes	complemented
144	C₂×C₈×Q₈	—	73	1	no	—
145	C₂×C₄⋊C₁₆	—	73	1	no	—
146	C₂×C₄×C₁₆	—	73	1	no	—
147	Q₈⋊C₁₆	—	73	2	no	—
148	Q₈⋊C₁₆	—	73	2	no	—
149	C₇₃	C₂×Q₈⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
150	C₂×C₄×Q₈	—	73	1	no	—
151	C₂²×C₁₆	—	146	1	no	—
152	C₂²×C₁₆	—	146	1	no	—
153	C₈×Q₈	—	73	1	no	—
154	C₈×Q₈	—	73	1	no	—
155	C₈×Q₈	—	73	1	no	—
156	C₈×Q₈	—	146	1	no	—
157	C₈×Q₈	—	146	1	no	—
158	C₈×Q₈	#64408	1	1	yes	characteristic
159	C₂×C₄⋊C₈	—	73	1	no	—
160	C₂×C₄⋊C₈	—	146	1	no	—
161	C₂×C₄×C₈	—	73	1	no	—
162	C₂×C₄×C₈	—	146	1	no	—
163	C₄⋊C₁₆	—	73	2	no	—
164	C₄⋊C₁₆	—	146	1	no	—
165	C₄×C₁₆	—	73	2	no	—
166	C₄×C₁₆	—	146	1	no	—
167	C₂²×Q₈	—	73	1	no	—
168	C₂²×C₈	—	73	1	no	—
169	C₂²×C₈	—	73	1	no	—
170	C₂²×C₈	—	146	1	no	—
171	C₄×Q₈	—	73	1	no	—
172	C₄×Q₈	—	73	1	no	—
173	C₄×Q₈	—	73	1	no	—
174	C₄×Q₈	—	146	1	no	—
175	C₄×Q₈	—	146	1	no	—
176	C₄×Q₈	C₂×#64408	1	1	yes	characteristic
177	C₂×C₄⋊C₄	—	73	1	no	—
178	C₂×C₄⋊C₄	—	146	1	no	—
179	C₂×C₄²	—	73	1	no	—
180	C₂×C₄²	—	146	1	no	—
181	C₂×C₁₆	—	292	2	no	—
182	C₂×C₁₆	—	146	2	no	—
183	C₂×C₁₆	—	146	2	no	—
184	C₂×C₁₆	—	146	4	no	—
185	C₄⋊C₈	—	2	1	no	—
186	C₄⋊C₈	—	73	1	no	—
187	C₄⋊C₈	—	146	1	no	—
188	C₄⋊C₈	—	146	1	no	—
189	C₄⋊C₈	—	146	1	no	—
190	C₄⋊C₈	—	146	1	no	—
191	C₄⋊C₈	C₂×#64408	1	1	yes	characteristic
192	C₄×C₈	—	2	1	no	—
193	C₄×C₈	—	73	1	no	—
194	C₄×C₈	—	73	1	no	—
195	C₄×C₈	—	73	1	no	—
196	C₄×C₈	—	146	1	no	—
197	C₄×C₈	—	146	1	no	—
198	C₄×C₈	—	146	1	no	—
199	C₄×C₈	C₂×#64408	1	1	yes	characteristic
200	C₂×Q₈	—	73	4	no	—
201	C₂×Q₈	—	73	1	no	—
202	C₂×Q₈	—	146	1	no	—
203	C₂×Q₈	—	146	2	no	—
204	C₂×Q₈	#11924036	1	1	yes	characteristic
205	C₂²×C₄	—	73	1	no	—
206	C₂²×C₄	—	73	1	no	—
207	C₂²×C₄	—	73	1	no	—
208	C₂²×C₄	—	146	1	no	—
209	C₂²×C₄	—	146	1	no	—
210	C₂×C₈	—	2	1	no	—
211	C₂×C₈	—	292	2	no	—
212	C₂×C₈	—	73	1	no	—
213	C₂×C₈	—	73	2	no	—
214	C₂×C₈	—	73	2	no	—
215	C₂×C₈	—	73	1	no	—
216	C₂×C₈	—	146	2	no	—
217	C₂×C₈	—	146	1	no	—
218	C₂×C₈	C₂²⋊#64408	1	1	yes	characteristic
219	C₂×C₈	C₂²⋊#64408	1	1	yes	characteristic
220	C₄⋊C₄	—	2	1	no	—
221	C₄⋊C₄	—	73	1	no	—
222	C₄⋊C₄	—	146	1	no	—
223	C₄⋊C₄	—	146	1	no	—
224	C₄⋊C₄	—	146	1	no	—
225	C₄⋊C₄	—	146	1	no	—
226	C₄⋊C₄	#11924036	1	1	yes	characteristic
227	C₄²	—	2	1	no	—
228	C₄²	—	73	1	no	—
229	C₄²	—	73	1	no	—
230	C₄²	—	73	1	no	—
231	C₄²	—	146	1	no	—
232	C₄²	—	146	1	no	—
233	C₄²	—	146	1	no	—
234	C₄²	C₂²×#64408	1	1	yes	characteristic
235	C₁₆	—	292	2	no	—
236	C₁₆	—	146	4	no	—
237	C₂³	—	73	1	no	—
238	Q₈	—	2	1	no	—
239	Q₈	—	73	2	no	—
240	Q₈	—	146	4	no	—
241	Q₈	—	146	1	no	—
242	Q₈	#14556664	1	2	yes	complemented
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	292	2	no	—
246	C₂×C₄	—	73	1	no	—
247	C₂×C₄	—	73	1	no	—
248	C₂×C₄	—	73	2	no	—
249	C₂×C₄	—	73	2	no	—
250	C₂×C₄	—	73	2	no	—
251	C₂×C₄	—	73	2	no	—
252	C₂×C₄	—	73	1	no	—
253	C₂×C₄	—	146	2	no	—
254	C₂×C₄	—	146	4	no	—
255	C₂×C₄	—	146	1	no	—
256	C₂×C₄	—	146	1	no	—
257	C₂×C₄	C₂×C₂²⋊#64408	1	1	yes	characteristic
258	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
259	C₂×C₄	D₇₃⋊M₁₆	1	1	yes	characteristic
260	C₈	—	2	1	no	—
261	C₈	—	4	1	no	—
262	C₈	—	292	1	no	—
263	C₈	—	73	1	no	—
264	C₈	—	73	1	no	—
265	C₈	—	146	1	no	—
266	C₈	#14556654	1	2	yes	—
267	C₂²	—	73	2	no	—
268	C₂²	—	73	2	no	—
269	C₂²	—	73	2	no	—
270	C₂²	#17989555	1	1	yes	characteristic
271	C₄	—	2	2	no	—
272	C₄	—	2	1	no	—
273	C₄	—	4	1	no	—
274	C₄	—	292	1	no	—
275	C₄	—	73	2	no	—
276	C₄	—	73	2	no	—
277	C₄	—	146	2	no	—
278	C₄	—	146	1	no	—
279	C₄	C₂×#14556654	1	1	yes	characteristic
280	C₄	D₇₃⋊#104	1	1	yes	characteristic
281	C₄	C₂×#14556664	1	1	yes	characteristic
282	C₄	#17989643	1	1	yes	characteristic
283	C₂	—	73	4	no	—
284	C₂	#22625220	1	1	yes	characteristic
285	C₂	#22626156	1	1	yes	characteristic
286	C₂	#22626171	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²⋊#64408
commutator	C₄×C₇₃	C₂²×C₁₆
Frattini	C₄²	C₂²×#64408
Fitting	C₈×Q₈×C₇₃	C₄
socle	C₂²×C₇₃	C₂×C₂²⋊C₈
prime radical	C₂²×D₇₃	C₂²⋊C₈
cosocle	C₄×#10596831	C₂³
hypercenter	C₈×Q₈	#64408
hypocenter	C₇₃	C₂×Q₈⋊C₁₆
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈×Q₈

Complements

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

N	H ≅ G/N	type
3. #22624898	C₂	semidirect
4. Q₈⋊#11924031	C₂	semidirect
61. Q₈×D₇₃	C₁₆	semidirect
95. Q₈×C₇₃	C₂×C₁₆	semidirect
115. Dic₁₄₆	C₂×C₁₆	semidirect
143. D₇₃	Q₈⋊C₁₆	semidirect
149. C₇₃	C₂×Q₈⋊C₁₆	semidirect
242. Q₈	#14556664	semidirect

Maximal subgroups

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

subgroup	index
#22626261	2
C₄×#14556664	2
#22624898	2
Q₈⋊#11924031	2
C₈×Q₈×D₇₃	2
C₂×Q₈⋊C₁₆	73