Subgroups — #30773226

Order 17152.

484 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14523983	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863637	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863211	C₂	1	2	yes	maximal, complemented
4	#22558319	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1810	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₆₈	C₂²	1	1	yes	characteristic
7	C₂×#14524022	—	2	1	no	—
8	C₂×#14523983	—	2	4	no	—
9	C₂×#14523983	—	2	1	no	—
10	C₂×#14523983	C₂²	1	2	yes	—
11	C₂²×C₆₇⋊C₁₆	—	2	1	no	—
12	#16863637	—	2	1	no	—
13	#16863637	—	2	1	no	—
14	#16863637	C₄	1	2	yes	—
15	#16863637	C₄	1	2	yes	—
16	C₂×#14523977	C₂²	1	1	yes	characteristic
17	C₂×#14523968	—	2	1	no	—
18	#16863211	—	2	4	no	—
19	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×#108×C₆₇	C₂²	1	2	yes	—
21	#407×C₆₇	—	2	1	no	—
22	#341×C₆₇	—	2	1	no	—
23	C₂²×Dic₁₃₄	—	2	1	no	—
24	C₂×Dic₂₆₈	—	2	2	no	—
25	C₂×Dic₂₆₈	—	2	1	no	—
26	C₂×Dic₂₆₈	—	2	2	no	—
27	C₂×Dic₂₆₈	C₂×C₄	1	2	yes	—
28	#14524022	—	2	2	no	—
29	#14524022	—	2	2	no	—
30	#14524022	—	4	1	no	—
31	#14524022	—	4	1	no	—
32	C₂×C₄⋊Dic₆₇	—	2	1	no	—
33	C₂×Dic₆₇⋊C₄	—	4	1	no	—
34	C₂×C₂²⋊Dic₆₇	—	4	1	no	—
35	#14523983	—	2	4	no	—
36	#14523983	—	2	2	no	—
37	#14523983	D₄	1	4	yes	—
38	C₂×C₆₇⋊C₁₆	—	2	4	no	—
39	C₂×C₆₇⋊C₁₆	—	2	2	no	—
40	#14523977	—	2	1	no	—
41	#14523977	C₂×C₄	1	2	yes	—
42	#14523968	—	2	2	no	—
43	#14523968	—	2	2	no	—
44	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
45	C₂×M₁₆×C₆₇	—	2	2	no	—
46	C₂²×C₈×C₆₇	—	2	2	no	—
47	C₂²×C₈×C₆₇	—	2	2	no	—
48	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
51	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
52	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
53	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
54	#108×C₆₇	—	2	2	no	—
55	#108×C₆₇	—	2	2	no	—
56	C₂²⋊C₈×C₆₇	—	4	1	no	—
57	C₂²⋊C₈×C₆₇	—	4	1	no	—
58	C₂×Dic₁₃₄	—	2	2	no	—
59	C₂×Dic₁₃₄	—	4	2	no	—
60	C₂×Dic₁₃₄	—	4	2	no	—
61	C₂×Dic₁₃₄	—	4	1	no	—
62	C₂²×Dic₆₇	—	4	1	no	—
63	C₂²×Dic₆₇	—	4	1	no	—
64	Dic₂₆₈	—	2	4	no	—
65	Dic₂₆₈	—	4	1	no	—
66	Dic₂₆₈	—	4	1	no	—
67	C₄⋊Dic₆₇	—	2	2	no	—
68	C₄⋊Dic₆₇	—	4	1	no	—
69	Dic₆₇⋊C₄	—	4	2	no	—
70	Dic₆₇⋊C₄	—	4	2	no	—
71	C₂²⋊Dic₆₇	—	4	2	no	—
72	C₂²⋊Dic₆₇	—	4	2	no	—
73	C₆₇⋊C₁₆	—	2	4	no	—
74	C₂⁴×C₆₇	SD₁₆	1	1	yes	characteristic
75	C₂²×C₄×C₆₇	—	2	2	no	—
76	C₂²×C₄×C₆₇	—	2	2	no	—
77	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
78	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
79	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
80	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
81	C₂²×C₄×C₆₇	SD₁₆	1	1	yes	characteristic
82	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
83	M₁₆×C₆₇	—	4	2	no	—
84	M₁₆×C₆₇	—	4	2	no	—
85	C₂×C₈×C₆₇	—	2	2	no	—
86	C₂×C₈×C₆₇	—	2	2	no	—
87	C₂×C₈×C₆₇	—	2	2	no	—
88	C₂×C₈×C₆₇	—	2	1	no	—
89	C₂×C₈×C₆₇	—	2	1	no	—
90	C₂×C₈×C₆₇	—	2	1	no	—
91	C₂×C₈×C₆₇	—	2	1	no	—
92	C₂×C₈×C₆₇	—	4	2	no	—
93	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
97	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₆₇	#41	1	2	yes	—
99	C₂³×C₆₇	—	2	2	no	—
100	C₂³×C₆₇	—	2	2	no	—
101	C₂³×C₆₇	—	4	1	no	—
102	C₂³×C₆₇	#102	1	1	yes	characteristic
103	C₂³×C₆₇	#102	1	1	yes	characteristic
104	C₂³×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
105	C₂×C₄×C₆₇	—	2	2	no	—
106	C₂×C₄×C₆₇	—	2	1	no	—
107	C₂×C₄×C₆₇	—	2	1	no	—
108	C₂×C₄×C₆₇	—	2	1	no	—
109	C₂×C₄×C₆₇	—	2	1	no	—
110	C₂×C₄×C₆₇	—	2	2	no	—
111	C₂×C₄×C₆₇	—	2	2	no	—
112	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
113	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
114	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₆₇	#121	1	2	yes	—
116	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
117	C₂×C₄×C₆₇	#135	1	2	yes	—
118	C₂×Dic₆₇	—	4	2	no	—
119	C₂×Dic₆₇	—	4	2	no	—
120	C₂×Dic₆₇	—	8	2	no	—
121	C₂×Dic₆₇	—	8	2	no	—
122	Dic₁₃₄	—	4	2	no	—
123	Dic₁₃₄	—	4	2	no	—
124	Dic₁₃₄	—	8	2	no	—
125	C₈×C₆₇	—	2	1	no	—
126	C₈×C₆₇	—	2	1	no	—
127	C₈×C₆₇	—	8	2	no	—
128	C₈×C₆₇	C₄×D₄	1	2	yes	—
129	C₈×C₆₇	#120	1	1	yes	characteristic
130	C₈×C₆₇	#123	1	1	yes	characteristic
131	C₂²×C₆₇	—	2	1	no	—
132	C₂²×C₆₇	—	2	1	no	—
133	C₂²×C₆₇	—	2	1	no	—
134	C₂²×C₆₇	—	2	1	no	—
135	C₂²×C₆₇	—	2	1	no	—
136	C₂²×C₆₇	—	2	1	no	—
137	C₂²×C₆₇	—	2	1	no	—
138	C₂²×C₆₇	—	4	4	no	—
139	C₂²×C₆₇	#362	1	2	yes	complemented
140	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
141	C₂²×C₆₇	#465	1	2	yes	—
142	C₄×C₆₇	—	2	1	no	—
143	C₄×C₆₇	—	2	1	no	—
144	C₄×C₆₇	#386	1	1	yes	characteristic
145	C₄×C₆₇	#416	1	1	yes	characteristic
146	C₄×C₆₇	#466	1	2	yes	—
147	Dic₆₇	—	8	2	no	—
148	Dic₆₇	—	8	2	no	—
149	#13066	—	67	1	no	maximal, Sylow
150	C₂×C₆₇	—	2	2	no	—
151	C₂×C₆₇	—	2	2	no	—
152	C₂×C₆₇	—	4	1	no	—
153	C₂×C₆₇	#1893	1	1	yes	characteristic
154	C₂×C₆₇	C₂×#362	1	1	yes	characteristic
155	C₂×C₆₇	#2148	1	1	yes	characteristic
156	C₂²×M₃₂	—	67	1	no	—
157	C₂×#467	—	67	1	no	—
158	C₂×#362	—	67	2	no	—
159	#1810	D₆₇	1	1	yes	characteristic
160	#1387	—	67	2	no	—
161	C₆₇	#13066	1	1	yes	characteristic, complemented, Sylow
162	C₂²×Q₁₆	—	67	1	no	—
163	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
164	C₂×C₂²⋊Q₈	—	134	1	no	—
165	C₂×M₃₂	—	67	2	no	—
166	C₂×M₃₂	—	134	4	no	—
167	C₂×M₃₂	—	134	1	no	—
168	C₂²×C₁₆	—	134	1	no	—
169	#467	—	67	2	no	—
170	#467	—	67	2	no	—
171	#467	—	134	1	no	—
172	#467	—	134	1	no	—
173	C₂×#108	C₂×D₆₇	1	2	yes	—
174	C₂×#107	—	67	1	no	—
175	C₂×#103	—	134	1	no	—
176	#407	—	2	1	no	—
177	#362	—	134	4	no	—
178	#341	—	2	1	no	—
179	C₂²×Q₈	—	134	1	no	—
180	C₂³×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
181	C₂×Q₁₆	—	67	2	no	—
182	C₂×Q₁₆	—	134	2	no	—
183	C₂×Q₁₆	—	134	1	no	—
184	C₂×Q₁₆	—	134	2	no	—
185	C₂×M₁₆	—	2	2	no	—
186	C₂²×C₈	—	2	2	no	—
187	C₂²×C₈	—	2	2	no	—
188	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
189	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
190	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
191	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
192	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
193	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
194	C₂²⋊Q₈	—	268	1	no	—
195	C₂²⋊Q₈	—	268	1	no	—
196	C₂²⋊Q₈	—	134	2	no	—
197	C₂²⋊Q₈	—	134	2	no	—
198	C₂×C₄⋊C₄	—	268	1	no	—
199	C₂×C₄⋊C₄	—	134	1	no	—
200	C₂×C₂²⋊C₄	—	268	1	no	—
201	M₃₂	—	67	4	no	—
202	M₃₂	—	134	4	no	—
203	M₃₂	—	134	2	no	—
204	C₂×C₁₆	—	134	4	no	—
205	C₂×C₁₆	—	134	2	no	—
206	#108	—	2	2	no	—
207	#108	—	2	2	no	—
208	#107	—	67	2	no	—
209	#107	—	134	1	no	—
210	#103	—	134	2	no	—
211	#103	—	134	2	no	—
212	C₂²⋊C₈	—	4	1	no	—
213	C₂²⋊C₈	—	4	1	no	—
214	C₂⁴	#11761615	1	1	yes	characteristic
215	C₂×Q₈	—	268	1	no	—
216	C₂×Q₈	—	268	2	no	—
217	C₂×Q₈	—	268	2	no	—
218	C₂×Q₈	—	134	2	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	268	1	no	—
222	C₂²×C₄	—	268	1	no	—
223	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
224	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
225	C₂²×C₄	#11761613	1	1	yes	characteristic
226	C₂²×C₄	#11761613	1	1	yes	characteristic
227	C₂²×C₄	#11761615	1	1	yes	characteristic
228	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
229	Q₁₆	—	268	1	no	—
230	Q₁₆	—	268	1	no	—
231	Q₁₆	—	134	4	no	—
232	M₁₆	—	4	2	no	—
233	M₁₆	—	4	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	1	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	2	1	no	—
240	C₂×C₈	—	2	1	no	—
241	C₂×C₈	—	4	2	no	—
242	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
243	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
244	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
245	C₂×C₈	D₄×D₆₇	1	2	yes	—
246	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
247	C₂×C₈	#11761627	1	2	yes	—
248	C₄⋊C₄	—	268	2	no	—
249	C₄⋊C₄	—	268	2	no	—
250	C₄⋊C₄	—	268	1	no	—
251	C₄⋊C₄	—	134	2	no	—
252	C₂²⋊C₄	—	268	2	no	—
253	C₂²⋊C₄	—	268	2	no	—
254	C₁₆	—	134	4	no	—
255	C₂³	—	2	2	no	—
256	C₂³	—	2	2	no	—
257	C₂³	—	4	1	no	—
258	C₂³	#14523963	1	1	yes	characteristic
259	C₂³	#14523963	1	1	yes	characteristic
260	C₂³	C₂×#11761615	1	1	yes	characteristic
261	Q₈	—	268	2	no	—
262	Q₈	—	268	2	no	—
263	Q₈	—	536	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	268	2	no	—
272	C₂×C₄	—	268	2	no	—
273	C₂×C₄	—	536	2	no	—
274	C₂×C₄	—	536	2	no	—
275	C₂×C₄	#14523963	1	1	yes	characteristic
276	C₂×C₄	#14523963	1	1	yes	characteristic
277	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
278	C₂×C₄	#14524015	1	2	yes	—
279	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
280	C₂×C₄	#14524063	1	2	yes	—
281	C₈	—	2	1	no	—
282	C₈	—	2	1	no	—
283	C₈	—	8	2	no	—
284	C₈	(C₂²⋊D₆₇)⋊C₄	1	2	yes	—
285	C₈	#14524013	1	1	yes	characteristic
286	C₈	#14524027	1	1	yes	characteristic
287	C₂²	—	2	1	no	—
288	C₂²	—	2	1	no	—
289	C₂²	—	2	1	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	2	1	no	—
292	C₂²	—	2	1	no	—
293	C₂²	—	2	1	no	—
294	C₂²	—	4	4	no	—
295	C₂²	#16863211	1	2	yes	complemented
296	C₂²	C₂×#14523963	1	1	yes	characteristic
297	C₂²	#16863627	1	2	yes	—
298	C₄	—	2	1	no	—
299	C₄	—	2	1	no	—
300	C₄	—	536	2	no	—
301	C₄	—	536	2	no	—
302	C₄	#16863273	1	1	yes	characteristic
303	C₄	#16863390	1	1	yes	characteristic
304	C₄	#16863630	1	2	yes	—
305	C₂	—	2	2	no	—
306	C₂	—	2	2	no	—
307	C₂	—	4	1	no	—
308	C₂	#22560635	1	1	yes	characteristic
309	C₂	C₂×#16863211	1	1	yes	characteristic
310	C₂	#22561956	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14523963
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#1810	C₂
socle	C₂²×C₆₇	C₂×#102
prime radical	C₂⁴×C₆₇	SD₁₆
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#1810	D₆₇
hypocenter	C₆₇	#13066
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ #1810

Complements

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

N	H ≅ G/N	type
3. C₂×#16863211	C₂	semidirect
4. #22558319	C₂	semidirect
139. C₂²×C₆₇	#362	semidirect
161. C₆₇	#13066	semidirect
295. C₂²	#16863211	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14523983	2
C₂×#16863637	2
C₂×#16863211	2
#22558319	2
C₆₇×#1810	2
#13066	67