Subgroups — #34034868

Order 18688.

530 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22627158	C₂	1	4	yes	maximal, complemented
2	#22627157	C₂	1	2	yes	maximal, complemented
3	#22627155	C₂	1	2	yes	maximal, complemented
4	#22627121	C₂	1	2	yes	maximal
5	#22627120	C₂	1	1	yes	maximal, characteristic
6	C₄×#14556688	C₂	1	1	yes	maximal, characteristic
7	C₄×#14556688	C₂	1	2	yes	maximal
8	Q₈×#11924036	C₂	1	1	yes	maximal, characteristic
9	C₂×#14556688	C₂²	1	2	yes	—
10	C₂×#14556688	C₂²	1	1	yes	characteristic
11	C₂×#14556688	C₂²	1	1	yes	characteristic
12	#17989690	C₂²	1	4	yes	—
13	#17989690	C₂²	1	4	yes	—
14	C₄⋊#11924040	C₂²	1	2	yes	—
15	C₄⋊#11924040	C₂²	1	2	yes	—
16	#17989681	C₂²	1	4	yes	—
17	#17989681	C₂²	1	2	yes	—
18	C₄×#11924040	C₂²	1	2	yes	—
19	C₄×#11924040	C₂²	1	4	yes	—
20	#17989636	C₄	1	2	yes	—
21	#17989636	C₄	1	1	yes	characteristic
22	#17989636	C₂²	1	1	yes	characteristic
23	#17989636	C₂²	1	2	yes	—
24	Q₈×#10596833	C₄	1	1	yes	characteristic
25	Q₈×#10596833	C₂²	1	1	yes	characteristic
26	#17989588	C₄	1	2	yes	—
27	#17989588	C₄	1	1	yes	characteristic
28	C₄×#11924036	C₂²	1	1	yes	characteristic
29	C₄×#11924036	C₂²	1	2	yes	—
30	C₄×Q₈×D₇₃	C₄	1	1	yes	characteristic
31	#14556688	—	2	4	no	—
32	#14556688	—	2	2	no	—
33	#14556688	Q₈	1	4	yes	complemented
34	C₂×#11924040	C₂³	1	4	yes	—
35	C₂×#11924040	C₂³	1	2	yes	—
36	C₂×#11924040	C₂³	1	2	yes	—
37	C₂×#11924036	C₂×C₄	1	1	yes	characteristic
38	C₂×#11924036	C₂×C₄	1	2	yes	—
39	C₂×#11924036	C₂³	1	1	yes	characteristic
40	#14556658	C₂×C₄	1	2	yes	—
41	#14556658	C₂×C₄	1	2	yes	—
42	#14556658	C₂×C₄	1	2	yes	—
43	C₄⋊#10596833	C₂×C₄	1	1	yes	characteristic
44	C₄⋊#10596833	C₂×C₄	1	2	yes	—
45	C₄⋊#10596833	C₂×C₄	1	2	yes	—
46	C₄⋊#10596833	C₂×C₄	1	1	yes	characteristic
47	C₄×#10596833	C₂×C₄	1	1	yes	characteristic
48	C₄×#10596833	C₂×C₄	1	2	yes	—
49	C₄×#10596833	C₂×C₄	1	1	yes	characteristic
50	C₄×#10596833	C₂×C₄	1	2	yes	—
51	C₄×#10596833	C₂³	1	1	yes	characteristic
52	C₄×#10596833	C₂³	1	2	yes	—
53	C₄×#10596833	C₂³	1	2	yes	—
54	C₄×#10596833	C₂³	1	1	yes	characteristic
55	C₂×Q₈×D₇₃	C₈	1	1	yes	characteristic
56	Q₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
57	C₄×Dic₁₄₆	C₈	1	2	yes	—
58	C₄×Dic₁₄₆	C₈	1	1	yes	characteristic
59	#14556556	C₂×C₄	1	2	yes	—
60	#14556556	C₂×C₄	1	1	yes	characteristic
61	C₄⋊C₄×D₇₃	C₈	1	2	yes	—
62	C₄⋊C₄×D₇₃	C₈	1	1	yes	characteristic
63	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
64	C₄²×D₇₃	C₂×C₄	1	2	yes	—
65	C₄×Q₈×C₇₃	C₈	1	1	yes	characteristic
66	#11924040	—	2	4	no	—
67	#11924040	—	2	2	no	—
68	#11924040	C₂×Q₈	1	4	yes	complemented
69	#11924036	—	2	2	no	—
70	#11924036	—	2	4	no	—
71	#11924036	C₂×Q₈	1	2	yes	—
72	#11924036	#41	1	2	yes	—
73	C₂×#10596833	C₂²×C₄	1	2	yes	—
74	C₂×#10596833	C₂²×C₄	1	1	yes	characteristic
75	C₂×#10596833	C₂²×C₄	1	1	yes	characteristic
76	C₂×#10596833	C₂²×C₄	1	2	yes	—
77	C₂×#10596833	C₂²×C₄	1	1	yes	characteristic
78	C₂×#10596833	C₂⁴	1	1	yes	characteristic
79	C₂×Dic₁₄₆	C₂×C₈	1	2	yes	—
80	C₂×Dic₁₄₆	C₂×C₈	1	1	yes	characteristic
81	Q₈×D₇₃	—	2	4	no	—
82	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
83	C₂×C₄×D₇₃	C₂×C₈	1	2	yes	—
84	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
85	C₂×C₄×D₇₃	C₂×C₈	1	2	yes	—
86	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
87	C₄⋊Dic₇₃	C₂×C₈	1	2	yes	—
88	C₄⋊Dic₇₃	C₂×C₈	1	1	yes	characteristic
89	#11924007	C₂×C₈	1	2	yes	—
90	#11924007	C₂×C₈	1	2	yes	—
91	#11924007	C₂×C₈	1	2	yes	—
92	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
93	C₄×Dic₇₃	C₂×C₈	1	2	yes	—
94	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
95	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
96	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
97	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
98	C₂×Q₈×C₇₃	C₂×C₈	1	1	yes	characteristic
99	C₄⋊C₄×C₇₃	C₂×C₈	1	2	yes	—
100	C₄⋊C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
101	C₄²×C₇₃	C₂×C₈	1	2	yes	—
102	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
103	C₂²×D₇₃	C₂²×C₈	1	1	yes	characteristic
104	Q₈×C₇₃	—	2	2	no	—
105	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
106	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
107	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
108	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
109	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
110	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
111	C₂×Dic₇₃	C₂²×C₈	1	2	yes	—
112	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
113	C₂×Dic₇₃	C₂²×C₈	1	2	yes	—
114	C₂×Dic₇₃	C₂³×C₄	1	1	yes	characteristic
115	C₄×D₇₃	—	2	2	no	—
116	C₄×D₇₃	—	2	4	no	—
117	C₄×D₇₃	—	4	1	no	—
118	C₄×D₇₃	—	4	2	no	—
119	C₄×D₇₃	C₄×Q₈	1	2	yes	—
120	C₄×D₇₃	#131	1	2	yes	—
121	Dic₁₄₆	—	2	4	no	—
122	Dic₁₄₆	M₃₂	1	4	yes	complemented
123	#10596833	—	2	2	no	—
124	#10596833	—	2	1	no	—
125	#10596833	—	2	1	no	—
126	#10596833	—	2	2	no	—
127	#10596833	C₄×Q₈	1	2	yes	—
128	#10596833	C₂²×Q₈	1	1	yes	characteristic
129	#10596833	C₂×#41	1	1	yes	characteristic
130	C₂²×C₇₃	C₂³×C₈	1	1	yes	characteristic
131	C₂×D₇₃	—	2	1	no	—
132	C₂×D₇₃	—	2	1	no	—
133	C₂×D₇₃	C₈×Q₈	1	2	yes	—
134	C₄×C₇₃	—	2	2	no	—
135	C₄×C₇₃	—	2	2	no	—
136	C₄×C₇₃	—	2	1	no	—
137	C₄×C₇₃	C₈×Q₈	1	2	yes	—
138	C₄×C₇₃	C₂×M₃₂	1	2	yes	—
139	Dic₇₃	—	2	1	no	—
140	Dic₇₃	—	2	1	no	—
141	Dic₇₃	—	2	2	no	—
142	Dic₇₃	C₂×M₃₂	1	4	yes	—
143	Dic₇₃	C₂×C₄×Q₈	1	1	yes	characteristic
144	Dic₇₃	C₂×#131	1	1	yes	characteristic
145	Q₈×M₃₂	—	73	1	no	maximal, Sylow
146	C₂×C₇₃	C₂×C₈×Q₈	1	1	yes	characteristic
147	C₂×C₇₃	C₂²×M₃₂	1	1	yes	characteristic
148	C₂×C₇₃	#3407	1	1	yes	characteristic
149	D₇₃	—	2	2	no	—
150	C₂×C₈×Q₈	—	73	1	no	—
151	#2183	—	73	2	no	—
152	#2183	—	73	4	no	—
153	Q₈×C₁₆	—	73	2	no	—
154	#2150	—	73	1	no	—
155	#2150	—	73	2	no	—
156	C₄×M₃₂	—	73	1	no	—
157	C₄×M₃₂	—	73	2	no	—
158	C₇₃	Q₈×M₃₂	1	1	yes	characteristic, complemented, Sylow
159	C₂×C₄×Q₈	—	73	1	no	—
160	C₂×M₃₂	—	73	1	no	—
161	C₂×M₃₂	—	73	2	no	—
162	C₂×M₃₂	—	73	1	no	—
163	C₈×Q₈	—	73	1	no	—
164	C₈×Q₈	—	73	1	no	—
165	C₈×Q₈	—	73	2	no	—
166	C₈×Q₈	—	73	1	no	—
167	C₈×Q₈	—	73	1	no	—
168	C₈×Q₈	—	73	2	no	—
169	C₂×C₄⋊C₈	—	73	1	no	—
170	C₂×C₄⋊C₈	—	73	2	no	—
171	C₂×C₄×C₈	—	73	1	no	—
172	C₂×C₄×C₈	—	73	2	no	—
173	C₄⋊C₁₆	—	73	2	no	—
174	C₄⋊C₁₆	—	73	2	no	—
175	C₄⋊C₁₆	—	73	4	no	—
176	C₄⋊C₁₆	—	73	4	no	—
177	#346	—	73	4	no	—
178	#346	—	73	2	no	—
179	C₄×C₁₆	—	73	2	no	—
180	C₄×C₁₆	—	73	4	no	—
181	C₂²×Q₈	—	73	1	no	—
182	C₂²×C₈	—	73	1	no	—
183	C₂²×C₈	—	73	1	no	—
184	C₂²×C₈	—	73	2	no	—
185	C₄×Q₈	—	73	1	no	—
186	C₄×Q₈	—	73	1	no	—
187	C₄×Q₈	—	73	2	no	—
188	C₄×Q₈	—	73	1	no	—
189	C₄×Q₈	—	73	2	no	—
190	C₄×Q₈	#10596842	1	1	yes	characteristic
191	C₂×C₄⋊C₄	—	73	2	no	—
192	C₂×C₄⋊C₄	—	73	1	no	—
193	C₂×C₄²	—	73	2	no	—
194	C₂×C₄²	—	73	1	no	—
195	M₃₂	—	73	4	no	—
196	M₃₂	—	146	4	no	—
197	M₃₂	—	146	2	no	—
198	C₂×C₁₆	—	73	2	no	—
199	C₂×C₁₆	—	73	4	no	—
200	C₂×C₁₆	—	73	2	no	—
201	C₄⋊C₈	—	73	1	no	—
202	C₄⋊C₈	—	73	2	no	—
203	C₄⋊C₈	—	73	2	no	—
204	C₄⋊C₈	—	73	1	no	—
205	C₄⋊C₈	—	73	2	no	—
206	C₄⋊C₈	—	73	2	no	—
207	C₄⋊C₈	—	73	2	no	—
208	C₄×C₈	—	73	1	no	—
209	C₄×C₈	—	73	2	no	—
210	C₄×C₈	—	73	1	no	—
211	C₄×C₈	—	73	2	no	—
212	C₄×C₈	—	73	2	no	—
213	C₄×C₈	—	73	1	no	—
214	C₄×C₈	—	73	2	no	—
215	C₄×C₈	—	73	1	no	—
216	C₂×Q₈	—	73	1	no	—
217	C₂×Q₈	—	73	2	no	—
218	C₂×Q₈	—	146	4	no	—
219	C₂×Q₈	C₂×#10596842	1	1	yes	characteristic
220	C₂²×C₄	—	73	2	no	—
221	C₂²×C₄	—	73	1	no	—
222	C₂²×C₄	—	73	1	no	—
223	C₂²×C₄	—	73	2	no	—
224	C₂²×C₄	—	73	1	no	—
225	C₂×C₈	—	73	1	no	—
226	C₂×C₈	—	73	1	no	—
227	C₂×C₈	—	73	2	no	—
228	C₂×C₈	—	73	2	no	—
229	C₂×C₈	—	73	2	no	—
230	C₂×C₈	—	73	1	no	—
231	C₂×C₈	—	73	1	no	—
232	C₂×C₈	—	73	2	no	—
233	C₂×C₈	—	146	2	no	—
234	C₂×C₈	—	146	4	no	—
235	C₄⋊C₄	—	73	2	no	—
236	C₄⋊C₄	—	73	2	no	—
237	C₄⋊C₄	—	73	2	no	—
238	C₄⋊C₄	—	73	2	no	—
239	C₄⋊C₄	—	73	1	no	—
240	C₄⋊C₄	C₂×#10596842	1	2	yes	—
241	C₄⋊C₄	C₂×#10596842	1	1	yes	characteristic
242	C₄²	—	73	2	no	—
243	C₄²	—	73	1	no	—
244	C₄²	—	73	2	no	—
245	C₄²	—	73	2	no	—
246	C₄²	—	73	1	no	—
247	C₄²	—	73	1	no	—
248	C₄²	C₂×#10596842	1	2	yes	—
249	C₄²	C₂×#10596842	1	1	yes	characteristic
250	C₁₆	—	73	4	no	—
251	C₁₆	—	146	4	no	—
252	C₁₆	—	146	2	no	—
253	C₂³	—	73	1	no	—
254	Q₈	—	2	2	no	—
255	Q₈	—	73	4	no	—
256	Q₈	—	146	4	no	—
257	C₂×C₄	—	292	2	no	—
258	C₂×C₄	—	292	1	no	—
259	C₂×C₄	—	73	1	no	—
260	C₂×C₄	—	73	2	no	—
261	C₂×C₄	—	73	2	no	—
262	C₂×C₄	—	73	2	no	—
263	C₂×C₄	—	73	2	no	—
264	C₂×C₄	—	73	1	no	—
265	C₂×C₄	—	73	1	no	—
266	C₂×C₄	—	146	4	no	—
267	C₂×C₄	—	146	2	no	—
268	C₂×C₄	C₂²×#10596842	1	2	yes	—
269	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
270	C₂×C₄	C₂²×#10596842	1	2	yes	—
271	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
272	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
273	C₈	—	73	1	no	—
274	C₈	—	73	1	no	—
275	C₈	—	73	2	no	—
276	C₈	—	146	2	no	—
277	C₈	—	146	1	no	—
278	C₈	—	146	1	no	—
279	C₈	—	146	2	no	—
280	C₂²	—	73	2	no	—
281	C₂²	—	146	1	no	—
282	C₂²	—	146	1	no	—
283	C₂²	C₂³×#10596842	1	1	yes	characteristic
284	C₄	—	2	2	no	—
285	C₄	—	2	2	no	—
286	C₄	—	2	1	no	—
287	C₄	—	73	1	no	—
288	C₄	—	73	1	no	—
289	C₄	—	73	4	no	—
290	C₄	—	146	1	no	—
291	C₄	—	146	2	no	—
292	C₄	—	146	1	no	—
293	C₄	Q₈×#10596842	1	2	yes	—
294	C₄	#17989706	1	2	yes	—
295	C₂	—	146	2	no	—
296	C₂	C₂×Q₈×#10596842	1	1	yes	characteristic
297	C₂	C₂×#17989706	1	1	yes	characteristic
298	C₂	#22627185	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#10596842
commutator, socle	C₂²×C₇₃	C₂³×C₈
Fitting	C₄×Q₈×C₇₃	C₈
prime radical	C₂²×D₇₃	C₂²×C₈
cosocle	C₂×#10596833	C₂⁴
hypercenter	C₄×Q₈	#10596842
hypocenter	C₇₃	Q₈×M₃₂
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. #22627158	C₂	semidirect
2. #22627157	C₂	semidirect
3. #22627155	C₂	semidirect
33. #14556688	Q₈	semidirect
68. #11924040	C₂×Q₈	semidirect
122. Dic₁₄₆	M₃₂	semidirect
158. C₇₃	Q₈×M₃₂	semidirect

Maximal subgroups

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

subgroup	index
#22627158	2
#22627157	2
#22627155	2
#22627121	2
#22627120	2
C₄×#14556688	2
Q₈×#11924036	2
Q₈×M₃₂	73