Subgroups — #34034857

Order 18688.

490 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22627158	C₂	1	2	yes	maximal, complemented
2	#22627158	C₂	1	2	yes	maximal, complemented
3	#22627157	C₂	1	2	yes	maximal, complemented
4	#22627156	C₂	1	2	yes	maximal, complemented
5	#22627122	C₂	1	2	yes	maximal
6	#22627119	C₂	1	1	yes	maximal, characteristic
7	#22627092	C₂	1	2	yes	maximal
8	C₂×#17989681	C₂	1	1	yes	maximal, characteristic
9	#22626719	C₂	1	1	yes	maximal, characteristic
10	C₂×#14556687	C₂²	1	2	yes	—
11	C₂²×#11924040	C₂²	1	1	yes	characteristic
12	C₂²×#11924040	C₂²	1	1	yes	characteristic
13	#17989690	C₂²	1	2	yes	—
14	#17989690	C₂²	1	2	yes	—
15	#17989690	C₂²	1	2	yes	—
16	C₄⋊#11924040	C₂²	1	1	yes	characteristic
17	C₄⋊#11924040	C₂²	1	1	yes	characteristic
18	C₄⋊#11924040	C₂²	1	2	yes	—
19	C₄⋊#11924040	C₂²	1	2	yes	—
20	#17989681	C₂²	1	2	yes	—
21	#17989681	C₂²	1	2	yes	—
22	#17989681	C₂²	1	2	yes	—
23	#17989681	C₂²	1	2	yes	—
24	C₄×#11924040	C₂²	1	2	yes	—
25	C₄×#11924040	C₂²	1	2	yes	—
26	#17989639	C₄	1	2	yes	—
27	#17989638	C₄	1	1	yes	characteristic
28	#17989637	C₄	1	1	yes	characteristic
29	#17989636	C₂²	1	2	yes	—
30	#17989636	C₂²	1	1	yes	characteristic
31	Q₈×#10596833	C₂²	1	1	yes	characteristic
32	#17989593	C₄	1	2	yes	—
33	#17989590	C₄	1	1	yes	characteristic
34	#17989551	C₂²	1	2	yes	—
35	C₂×C₄×#10596833	C₂²	1	1	yes	characteristic
36	#17989139	C₄	1	1	yes	characteristic
37	#14556687	—	2	4	no	—
38	C₂×#11924040	—	2	2	no	—
39	C₂×#11924040	Q₈	1	4	yes	complemented
40	C₂×#11924040	C₂³	1	2	yes	—
41	C₂×#11924040	C₂³	1	2	yes	—
42	C₂×#11924040	C₂³	1	2	yes	—
43	C₂×#11924040	C₂³	1	2	yes	—
44	C₂×#11924037	C₂×C₄	1	2	yes	—
45	C₂²×#10596833	C₂×C₄	1	1	yes	characteristic
46	C₂²×#10596833	C₂³	1	1	yes	characteristic
47	#14556658	C₂×C₄	1	2	yes	—
48	#14556658	C₂×C₄	1	2	yes	—
49	#14556658	C₂×C₄	1	2	yes	—
50	C₄⋊#10596833	C₂×C₄	1	1	yes	characteristic
51	C₄⋊#10596833	C₂×C₄	1	2	yes	—
52	C₄⋊#10596833	C₂×C₄	1	2	yes	—
53	C₄⋊#10596833	C₂×C₄	1	1	yes	characteristic
54	#14556646	C₂×C₄	1	2	yes	—
55	#14556646	C₂×C₄	1	2	yes	—
56	C₄×#10596833	C₂×C₄	1	1	yes	characteristic
57	C₄×#10596833	C₂×C₄	1	1	yes	characteristic
58	C₄×#10596833	C₂³	1	2	yes	—
59	C₄×#10596833	C₂³	1	1	yes	characteristic
60	C₄×#10596833	C₂³	1	2	yes	—
61	C₄×#10596833	C₂³	1	1	yes	characteristic
62	#14556590	C₈	1	1	yes	characteristic
63	#14556589	C₈	1	1	yes	characteristic
64	#14556581	C₈	1	2	yes	—
65	#14556570	C₈	1	1	yes	characteristic
66	#14556568	C₈	1	2	yes	—
67	Q₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
68	#14556556	C₂×C₄	1	1	yes	characteristic
69	#14556556	C₂×C₄	1	2	yes	—
70	#14556547	C₂×C₄	1	2	yes	—
71	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
72	C₂²⋊Q₈×C₇₃	C₈	1	1	yes	characteristic
73	#11924040	—	2	2	no	—
74	#11924040	—	2	2	no	—
75	#11924040	—	2	2	no	—
76	#11924040	C₂×Q₈	1	4	yes	—
77	#11924037	—	2	4	no	—
78	C₂×#10596833	—	2	2	no	—
79	C₂×#10596833	C₂²×C₄	1	1	yes	characteristic
80	C₂×#10596833	C₂²×C₄	1	1	yes	characteristic
81	C₂×#10596833	C₂²×C₄	1	2	yes	—
82	C₂×#10596833	C₂²×C₄	1	2	yes	—
83	C₂×#10596833	C₂²×C₄	1	1	yes	characteristic
84	C₂×#10596833	C₂×Q₈	1	2	yes	—
85	C₂×#10596833	#41	1	2	yes	—
86	C₂×#10596833	C₂⁴	1	1	yes	characteristic
87	C₂×Dic₁₄₆	C₂×C₈	1	2	yes	—
88	C₂×Dic₁₄₆	C₂×C₈	1	1	yes	characteristic
89	C₂²×Dic₇₃	C₂×C₈	1	1	yes	characteristic
90	C₂²×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
91	C₄⋊Dic₇₃	C₂×C₈	1	2	yes	—
92	C₄⋊Dic₇₃	C₂×C₈	1	1	yes	characteristic
93	#11924007	C₂×C₈	1	2	yes	—
94	#11924007	C₂×C₈	1	2	yes	—
95	#11924007	C₂×C₈	1	2	yes	—
96	#11924007	C₂×C₈	1	2	yes	—
97	#11924007	C₂×C₈	1	2	yes	—
98	C₂²⋊Dic₇₃	C₂×C₈	1	2	yes	—
99	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
100	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
101	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
102	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
103	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
104	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
105	C₂×Q₈×C₇₃	C₂×C₈	1	1	yes	characteristic
106	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
107	C₄⋊C₄×C₇₃	C₂×C₈	1	2	yes	—
108	C₄⋊C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
109	C₂²⋊C₄×C₇₃	C₂×C₈	1	2	yes	—
110	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
111	Q₈×C₇₃	—	4	1	no	—
112	C₂×C₄×C₇₃	—	4	1	no	—
113	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
114	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
115	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
116	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
117	C₂×Dic₇₃	—	4	1	no	—
118	C₂×Dic₇₃	C₄×Q₈	1	2	yes	—
119	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
120	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
121	C₂×Dic₇₃	C₂²×C₈	1	2	yes	—
122	C₂×Dic₇₃	C₂²×C₈	1	2	yes	—
123	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
124	C₂×Dic₇₃	#131	1	2	yes	—
125	C₂×Dic₇₃	C₂³×C₄	1	1	yes	characteristic
126	Dic₁₄₆	—	4	2	no	—
127	Dic₁₄₆	—	4	1	no	—
128	#10596833	—	2	1	no	—
129	#10596833	—	2	1	no	—
130	#10596833	—	2	2	no	—
131	#10596833	—	2	2	no	—
132	#10596833	C₄×Q₈	1	2	yes	—
133	#10596833	C₂²×Q₈	1	1	yes	characteristic
134	#10596833	C₂×#41	1	1	yes	characteristic
135	C₂²×C₇₃	—	2	1	no	—
136	C₂²×C₇₃	—	2	1	no	—
137	C₂²×C₇₃	C₈×Q₈	1	2	yes	—
138	C₂²×C₇₃	C₂³×C₈	1	1	yes	characteristic
139	C₄×C₇₃	—	2	1	no	—
140	C₄×C₇₃	—	2	2	no	—
141	C₄×C₇₃	—	2	1	no	—
142	C₄×C₇₃	—	2	2	no	—
143	Dic₇₃	—	2	1	no	—
144	Dic₇₃	—	2	2	no	—
145	Dic₇₃	—	2	2	no	—
146	Dic₇₃	—	2	1	no	—
147	Dic₇₃	C₈×Q₈	1	2	yes	—
148	Dic₇₃	C₂×C₄×Q₈	1	1	yes	characteristic
149	Dic₇₃	C₂×#131	1	1	yes	characteristic
150	#33475	—	73	1	no	maximal, Sylow
151	C₂×C₇₃	—	2	2	no	—
152	C₂×C₇₃	C₂×C₈×Q₈	1	1	yes	characteristic
153	C₂×C₇₃	#3407	1	1	yes	characteristic
154	C₂×C₇₃	#3407	1	1	yes	characteristic
155	#2960	—	73	1	no	—
156	#2183	—	73	2	no	—
157	#2183	—	73	2	no	—
158	#2183	—	73	2	no	—
159	#2183	—	73	2	no	—
160	#2151	—	73	2	no	—
161	#2150	—	73	1	no	—
162	#2108	—	73	2	no	—
163	C₂×#346	—	73	1	no	—
164	C₇₃	#33475	1	1	yes	characteristic, complemented, Sylow
165	#533	—	73	1	no	—
166	C₂×M₃₂	—	73	2	no	—
167	C₂²×C₁₆	—	73	1	no	—
168	C₂²×C₁₆	—	73	1	no	—
169	#446	—	73	1	no	—
170	#446	—	73	1	no	—
171	#446	—	73	2	no	—
172	C₈×Q₈	—	73	1	no	—
173	C₈×Q₈	—	73	1	no	—
174	C₈×Q₈	—	73	2	no	—
175	#424	—	73	2	no	—
176	#423	—	73	1	no	—
177	#405	—	73	2	no	—
178	C₂×C₄×C₈	—	73	1	no	—
179	C₄⋊C₁₆	—	73	1	no	—
180	C₄⋊C₁₆	—	73	1	no	—
181	C₄⋊C₁₆	—	73	2	no	—
182	C₄⋊C₁₆	—	73	2	no	—
183	C₄⋊C₁₆	—	73	2	no	—
184	C₄⋊C₁₆	—	73	2	no	—
185	C₄⋊C₁₆	—	73	2	no	—
186	#346	—	73	2	no	—
187	#346	—	73	2	no	—
188	#346	—	73	2	no	—
189	#346	—	73	2	no	—
190	C₄×C₁₆	—	73	2	no	—
191	C₄×C₁₆	—	73	2	no	—
192	C₂×M₁₆	—	73	2	no	—
193	C₂²×C₈	—	73	1	no	—
194	C₂²×C₈	—	73	1	no	—
195	C₄⋊Q₈	—	73	1	no	—
196	C₄⋊Q₈	—	73	1	no	—
197	#125	—	73	2	no	—
198	C₂²⋊Q₈	—	73	1	no	—
199	C₂²⋊Q₈	—	73	2	no	—
200	C₂²⋊Q₈	#10596842	1	1	yes	characteristic
201	C₄×Q₈	—	73	1	no	—
202	C₄×Q₈	—	73	1	no	—
203	C₄×Q₈	—	73	2	no	—
204	#117	—	73	2	no	—
205	C₂×C₄²	—	73	1	no	—
206	M₃₂	—	146	4	no	—
207	C₂×C₁₆	—	73	4	no	—
208	C₂×C₁₆	—	73	2	no	—
209	C₂×C₁₆	—	73	2	no	—
210	C₂×C₁₆	—	73	2	no	—
211	C₂×C₁₆	—	73	2	no	—
212	C₂×C₁₆	—	146	2	no	—
213	C₄⋊C₈	—	73	1	no	—
214	C₄⋊C₈	—	73	1	no	—
215	C₄⋊C₈	—	73	2	no	—
216	C₄⋊C₈	—	73	2	no	—
217	C₄⋊C₈	—	73	2	no	—
218	C₄⋊C₈	—	73	2	no	—
219	C₄⋊C₈	—	73	2	no	—
220	#97	—	73	2	no	—
221	#97	—	73	2	no	—
222	C₄×C₈	—	73	2	no	—
223	C₄×C₈	—	73	2	no	—
224	C₄×C₈	—	73	1	no	—
225	C₄×C₈	—	73	1	no	—
226	C₄×C₈	—	73	1	no	—
227	C₄×C₈	—	73	1	no	—
228	C₂×Q₈	—	73	2	no	—
229	C₂×Q₈	—	73	1	no	—
230	C₂×Q₈	C₂×#10596842	1	1	yes	characteristic
231	C₂²×C₄	—	73	1	no	—
232	C₂²×C₄	—	73	1	no	—
233	C₂²×C₄	C₂×#10596842	1	1	yes	characteristic
234	M₁₆	—	146	4	no	—
235	C₂×C₈	—	73	1	no	—
236	C₂×C₈	—	73	2	no	—
237	C₂×C₈	—	73	2	no	—
238	C₂×C₈	—	73	1	no	—
239	C₂×C₈	—	73	1	no	—
240	C₂×C₈	—	73	1	no	—
241	C₂×C₈	—	73	2	no	—
242	C₂×C₈	—	73	2	no	—
243	C₂×C₈	—	146	2	no	—
244	C₄⋊C₄	—	73	2	no	—
245	C₄⋊C₄	—	73	2	no	—
246	C₄⋊C₄	—	73	2	no	—
247	C₄⋊C₄	—	73	2	no	—
248	C₄⋊C₄	—	73	2	no	—
249	C₄⋊C₄	—	73	2	no	—
250	C₄⋊C₄	—	73	1	no	—
251	C₄⋊C₄	C₂×#10596842	1	2	yes	—
252	C₄⋊C₄	C₂×#10596842	1	1	yes	characteristic
253	C₂²⋊C₄	—	73	2	no	—
254	C₂²⋊C₄	C₂×#10596842	1	2	yes	—
255	C₄²	—	73	1	no	—
256	C₄²	—	73	1	no	—
257	C₄²	—	73	1	no	—
258	C₄²	—	73	1	no	—
259	C₄²	—	73	2	no	—
260	C₄²	—	73	2	no	—
261	C₁₆	—	73	4	no	—
262	C₁₆	—	146	2	no	—
263	C₁₆	—	146	2	no	—
264	C₁₆	—	146	2	no	—
265	C₂³	C₂²×#10596842	1	1	yes	characteristic
266	Q₈	—	4	1	no	—
267	Q₈	—	292	1	no	—
268	Q₈	—	292	2	no	—
269	C₂×C₄	—	4	1	no	—
270	C₂×C₄	—	292	1	no	—
271	C₂×C₄	—	73	2	no	—
272	C₂×C₄	—	73	2	no	—
273	C₂×C₄	—	73	1	no	—
274	C₂×C₄	—	73	1	no	—
275	C₂×C₄	—	73	1	no	—
276	C₂×C₄	—	73	1	no	—
277	C₂×C₄	—	73	2	no	—
278	C₂×C₄	—	73	2	no	—
279	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
280	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
281	C₂×C₄	C₂²×#10596842	1	2	yes	—
282	C₂×C₄	C₂²×#10596842	1	2	yes	—
283	C₈	—	73	1	no	—
284	C₈	—	73	1	no	—
285	C₈	—	73	2	no	—
286	C₈	—	146	1	no	—
287	C₈	—	146	1	no	—
288	C₈	—	146	2	no	—
289	C₈	—	146	2	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	2	1	no	—
292	C₂²	Q₈×#10596842	1	2	yes	—
293	C₂²	C₂³×#10596842	1	1	yes	characteristic
294	C₄	—	2	1	no	—
295	C₄	—	2	1	no	—
296	C₄	—	2	2	no	—
297	C₄	—	2	2	no	—
298	C₄	—	73	1	no	—
299	C₄	—	73	1	no	—
300	C₄	—	73	2	no	—
301	C₄	—	146	2	no	—
302	C₄	—	146	1	no	—
303	C₄	—	146	1	no	—
304	C₄	—	146	2	no	—
305	C₂	—	2	2	no	—
306	C₂	C₂×Q₈×#10596842	1	1	yes	characteristic
307	C₂	#22627184	1	1	yes	characteristic
308	C₂	#22627185	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#10596842
commutator, socle	C₂²×C₇₃	C₂³×C₈
Fitting	C₂²⋊Q₈×C₇₃	C₈
prime radical	C₂³×C₇₃	C₂²×C₈
cosocle	C₂×#10596833	C₂⁴
hypercenter	C₂²⋊Q₈	#10596842
hypocenter	C₇₃	#33475
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
1. #22627158	C₂	semidirect
2. #22627158	C₂	semidirect
3. #22627157	C₂	semidirect
4. #22627156	C₂	semidirect
39. C₂×#11924040	Q₈	semidirect
164. C₇₃	#33475	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
#22627156	2
#22627122	2
#22627119	2
#22627092	2
C₂×#17989681	2
#22626719	2
#33475	73