Subgroups — #30874359

Order 17152.

808 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569125	C₂	1	2	yes	maximal
2	C₂×#16863519	C₂	1	1	yes	maximal, characteristic
3	#22568817	C₂	1	2	yes	maximal, complemented
4	#22560997	C₂	1	2	yes	maximal, complemented
5	#22560448	C₂	1	4	yes	maximal, complemented
6	#22560264	C₂	1	2	yes	maximal, complemented
7	#22560093	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1849	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14524065	C₂²	1	1	yes	characteristic
10	#16864162	C₂²	1	4	yes	—
11	C₂×#14524049	C₂²	1	2	yes	—
12	#16863519	C₄	1	8	yes	complemented
13	#16863517	—	2	4	no	—
14	#16863512	—	2	4	no	—
15	#16863444	C₂²	1	4	yes	—
16	#16863443	—	2	4	no	—
17	C₂×#14523979	C₂²	1	1	yes	characteristic
18	C₂×#14523979	C₂²	1	1	yes	characteristic
19	#108×D₆₇	—	2	4	no	—
20	#16863434	C₂²	1	1	yes	characteristic
21	#16863412	C₂²	1	4	yes	—
22	C₂×#14523971	C₂²	1	2	yes	—
23	#16863351	C₂²	1	2	yes	—
24	#16863167	C₂²	1	4	yes	—
25	#16863156	C₂²	1	2	yes	—
26	C₂×#108×C₆₇	C₂²	1	2	yes	—
27	C₂×#108×C₆₇	C₂²	1	2	yes	—
28	#423×C₆₇	C₂²	1	2	yes	—
29	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
30	C₂×#11761628	C₂³	1	2	yes	—
31	#14524065	C₂×C₄	1	4	yes	—
32	#14524065	D₄	1	4	yes	—
33	C₂×Dic₂₆₈	Q₈	1	1	yes	characteristic
34	C₂×#11761617	Q₈	1	2	yes	—
35	C₂×D₅₃₆	Q₈	1	1	yes	characteristic
36	#14524051	—	2	8	no	—
37	#14524049	—	2	2	no	—
38	#14524049	—	2	2	no	—
39	#14524049	—	4	4	no	—
40	#14524049	C₂×C₄	1	4	yes	—
41	#14524049	C₂×C₄	1	4	yes	—
42	C₂×#11761612	C₂³	1	2	yes	—
43	C₂×#11761612	C₂³	1	2	yes	—
44	C₂×#11761611	—	2	2	no	—
45	M₁₆×D₆₇	—	2	8	no	—
46	C₂×C₈×D₆₇	—	2	2	no	—
47	#14524002	C₂³	1	2	yes	—
48	#14523979	C₂×C₄	1	4	yes	—
49	#14523979	D₄	1	4	yes	—
50	#14523978	—	2	8	no	—
51	#14523977	—	2	1	no	—
52	#14523975	—	2	1	no	—
53	#14523971	C₂×C₄	1	8	yes	—
54	#14523969	—	2	8	no	—
55	#14523953	—	2	2	no	—
56	C₈×Dic₆₇	—	2	2	no	—
57	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
58	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
59	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
60	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
61	#108×C₆₇	—	2	2	no	—
62	#108×C₆₇	—	2	2	no	—
63	#108×C₆₇	—	2	2	no	—
64	#108×C₆₇	—	2	2	no	—
65	C₄⋊C₈×C₆₇	—	4	2	no	—
66	#97×C₆₇	C₂×C₄	1	4	yes	—
67	#11761628	—	2	2	no	—
68	#11761628	—	2	2	no	—
69	#11761628	C₂²×C₄	1	4	yes	—
70	#11761628	C₂×D₄	1	4	yes	—
71	C₂×Dic₁₃₄	C₂×Q₈	1	2	yes	—
72	C₂×D₂₆₈	C₂×Q₈	1	2	yes	—
73	C₂×C₂²⋊D₆₇	—	2	2	no	—
74	C₂×C₄×D₆₇	—	2	2	no	—
75	Dic₂₆₈	C₄⋊C₄	1	4	yes	—
76	#11761617	C₄⋊C₄	1	8	yes	—
77	D₅₃₆	C₄⋊C₄	1	4	yes	—
78	#11761612	—	2	2	no	—
79	#11761612	—	2	2	no	—
80	#11761612	C₂²×C₄	1	4	yes	—
81	#11761612	C₂×D₄	1	4	yes	—
82	#11761611	—	2	4	no	—
83	#11761611	—	2	4	no	—
84	#11761611	—	4	8	no	—
85	C₂×C₆₇⋊C₈	—	2	2	no	—
86	C₂×C₆₇⋊C₈	—	2	2	no	—
87	C₈×D₆₇	—	2	4	no	—
88	C₈×D₆₇	—	2	4	no	—
89	C₈×D₆₇	—	4	8	no	—
90	C₄⋊Dic₆₇	—	2	2	no	—
91	C₂²⋊Dic₆₇	—	2	2	no	—
92	C₄×Dic₆₇	—	2	2	no	—
93	C₂²×C₄×C₆₇	—	2	1	no	—
94	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
95	M₁₆×C₆₇	—	2	4	no	—
96	M₁₆×C₆₇	—	2	4	no	—
97	C₂×C₈×C₆₇	—	2	4	no	—
98	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
99	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
100	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
101	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
102	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
103	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
104	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
105	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
106	C₄²×C₆₇	—	2	1	no	—
107	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
108	C₂³×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
109	C₂²×D₆₇	—	2	2	no	—
110	C₂×C₄×C₆₇	—	2	2	no	—
111	C₂×C₄×C₆₇	—	8	1	no	—
112	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
113	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
116	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
117	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
118	C₂²⋊D₆₇	—	2	4	no	—
119	C₂²⋊D₆₇	—	2	4	no	—
120	C₂×Dic₆₇	—	2	2	no	—
121	C₂×Dic₆₇	—	4	2	no	—
122	D₂₆₈	—	2	2	no	—
123	D₂₆₈	C₂×C₄⋊C₄	1	4	yes	—
124	C₄×D₆₇	—	2	4	no	—
125	C₄×D₆₇	—	2	4	no	—
126	Dic₁₃₄	—	2	2	no	—
127	Dic₁₃₄	C₂×C₄⋊C₄	1	4	yes	—
128	C₈×C₆₇	—	4	4	no	—
129	C₈×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
130	C₈×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
131	C₈×C₆₇	C₄×D₄	1	2	yes	—
132	C₈×C₆₇	C₄×D₄	1	2	yes	—
133	C₆₇⋊C₈	—	2	4	no	—
134	C₆₇⋊C₈	—	2	4	no	—
135	C₂²×C₆₇	—	4	1	no	—
136	C₂²×C₆₇	#520	1	1	yes	characteristic
137	C₂²×C₆₇	#547	1	1	yes	characteristic
138	C₂²×C₆₇	D₄×Q₈	1	1	yes	characteristic
139	C₂×D₆₇	—	2	4	no	—
140	C₂×D₆₇	—	8	2	no	—
141	C₄×C₆₇	—	4	2	no	—
142	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
143	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
144	C₄×C₆₇	D₄²	1	1	yes	characteristic
145	C₄×C₆₇	#554	1	1	yes	characteristic
146	Dic₆₇	—	2	4	no	—
147	Dic₆₇	—	8	2	no	—
148	#22695	—	67	1	no	maximal, Sylow
149	C₂×C₆₇	—	2	1	no	—
150	C₂×C₆₇	—	2	1	no	—
151	C₂×C₆₇	—	2	1	no	—
152	C₂×C₆₇	D₄×C₄⋊C₄	1	1	yes	characteristic
153	D₆₇	—	4	4	no	—
154	#2957	—	67	2	no	—
155	C₂×#444	—	67	1	no	—
156	#2916	—	67	2	no	—
157	#1953	—	67	2	no	—
158	#1861	—	67	4	no	—
159	#1849	D₆₇	1	1	yes	characteristic, complemented
160	#1814	—	67	2	no	—
161	#1773	—	67	1	no	—
162	C₆₇	#22695	1	1	yes	characteristic, complemented, Sylow
163	C₂×#135	—	67	1	no	—
164	#568	—	67	4	no	—
165	C₂×#131	—	67	2	no	—
166	#444	—	67	8	no	—
167	#444	—	134	4	no	—
168	#443	—	134	4	no	—
169	#430	—	67	4	no	—
170	#430	—	134	4	no	—
171	C₂×#108	—	67	1	no	—
172	C₂×#108	—	67	1	no	—
173	C₂×#108	—	134	4	no	—
174	C₂×#108	C₂×D₆₇	1	2	yes	—
175	C₂×#108	C₂×D₆₇	1	2	yes	—
176	#427	—	67	1	no	—
177	#423	C₂×D₆₇	1	2	yes	—
178	#421	—	67	4	no	—
179	C₂×#104	—	67	2	no	—
180	#405	—	67	2	no	—
181	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
182	#344	—	67	4	no	—
183	#339	—	67	2	no	—
184	C₂×#41	—	67	2	no	—
185	#135	—	67	4	no	—
186	#135	—	67	4	no	—
187	C₂×Q₁₆	—	67	1	no	—
188	C₂×SD₁₆	—	67	2	no	—
189	C₂×D₈	—	67	1	no	—
190	#131	—	268	4	no	—
191	#131	—	67	4	no	—
192	#131	—	67	4	no	—
193	#131	—	134	8	no	—
194	#131	—	134	2	no	—
195	#131	—	134	2	no	—
196	C₂×M₁₆	—	67	2	no	—
197	C₂×M₁₆	—	67	2	no	—
198	C₂×M₁₆	—	134	8	no	—
199	C₂×M₁₆	—	134	2	no	—
200	C₂×M₁₆	C₂²×D₆₇	1	4	yes	—
201	C₂²×C₈	—	134	2	no	—
202	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
203	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
204	#117	—	67	2	no	—
205	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
206	#108	—	2	2	no	—
207	#108	—	2	2	no	—
208	#108	—	2	2	no	—
209	#108	—	2	2	no	—
210	#108	—	67	4	no	—
211	#108	—	67	4	no	—
212	#108	—	134	8	no	—
213	#107	—	134	1	no	—
214	#106	—	134	1	no	—
215	C₄⋊C₈	—	4	2	no	—
216	#104	—	67	8	no	—
217	#104	—	134	8	no	—
218	#97	—	134	2	no	—
219	#97	C₄×D₆₇	1	4	yes	—
220	C₄×C₈	—	134	2	no	—
221	#41	—	67	4	no	—
222	#41	—	67	4	no	—
223	#41	—	134	2	no	—
224	#41	—	134	2	no	—
225	C₂×Q₈	—	67	2	no	—
226	C₂×D₄	—	67	2	no	—
227	C₂×D₄	—	134	2	no	—
228	C₂²×C₄	—	2	1	no	—
229	C₂²×C₄	—	134	2	no	—
230	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
231	Q₁₆	—	67	4	no	—
232	SD₁₆	—	67	8	no	—
233	D₈	—	67	4	no	—
234	M₁₆	—	2	4	no	—
235	M₁₆	—	2	4	no	—
236	M₁₆	—	268	8	no	—
237	M₁₆	—	67	4	no	—
238	M₁₆	—	67	4	no	—
239	M₁₆	—	134	2	no	—
240	M₁₆	—	134	2	no	—
241	M₁₆	—	134	4	no	—
242	M₁₆	—	134	4	no	—
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	—	268	8	no	—
245	C₂×C₈	—	134	2	no	—
246	C₂×C₈	—	134	2	no	—
247	C₂×C₈	—	134	4	no	—
248	C₂×C₈	—	134	4	no	—
249	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
250	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
251	C₂×C₈	D₄×D₆₇	1	2	yes	—
252	C₂×C₈	D₄×D₆₇	1	2	yes	—
253	C₂×C₈	Q₈×D₆₇	1	1	yes	characteristic
254	C₂×C₈	Q₈×D₆₇	1	1	yes	characteristic
255	C₂×C₈	#11761629	1	1	yes	characteristic
256	C₂×C₈	#11761629	1	1	yes	characteristic
257	C₄⋊C₄	—	134	2	no	—
258	C₂²⋊C₄	—	134	2	no	—
259	C₄²	—	2	1	no	—
260	C₄²	—	134	2	no	—
261	C₄²	C₂×C₄×D₆₇	1	2	yes	—
262	C₂³	—	134	2	no	—
263	C₂³	#14524091	1	1	yes	characteristic
264	Q₈	—	67	4	no	—
265	Q₈	—	134	2	no	—
266	D₄	—	67	4	no	—
267	D₄	—	134	4	no	—
268	D₄	—	134	4	no	—
269	D₄	—	134	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	8	1	no	—
272	C₂×C₄	—	268	2	no	—
273	C₂×C₄	—	134	2	no	—
274	C₂×C₄	—	134	4	no	—
275	C₂×C₄	—	134	4	no	—
276	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
277	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
278	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
279	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
280	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
281	C₂×C₄	#14524088	1	1	yes	characteristic
282	C₈	—	4	4	no	—
283	C₈	—	134	4	no	—
284	C₈	—	134	4	no	—
285	C₈	C₄⋊C₄×D₆₇	1	2	yes	—
286	C₈	C₄⋊C₄×D₆₇	1	2	yes	—
287	C₈	#14524005	1	2	yes	—
288	C₈	#14524005	1	2	yes	—
289	C₂²	—	4	1	no	—
290	C₂²	—	536	2	no	—
291	C₂²	—	134	4	no	—
292	C₂²	#16863877	1	1	yes	characteristic
293	C₂²	#16864049	1	1	yes	characteristic
294	C₂²	#16864061	1	1	yes	characteristic
295	C₄	—	4	2	no	—
296	C₄	—	536	2	no	—
297	C₄	—	134	4	no	—
298	C₄	C₂×C₄⋊C₄×D₆₇	1	1	yes	characteristic
299	C₄	C₂×#14524005	1	1	yes	characteristic
300	C₄	#16864031	1	1	yes	characteristic
301	C₄	#16864107	1	1	yes	characteristic
302	C₂	—	2	1	no	—
303	C₂	—	2	1	no	—
304	C₂	—	2	1	no	—
305	C₂	—	268	4	no	—
306	C₂	#22563109	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16864031
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#1849	C₂
socle	C₂×C₆₇	D₄×C₄⋊C₄
prime radical	C₂×#14524065	C₂²
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#1849	D₆₇
hypocenter	C₆₇	#22695
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1849

Complements

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

N	H ≅ G/N	type
3. #22568817	C₂	semidirect
4. #22560997	C₂	semidirect
5. #22560448	C₂	semidirect
6. #22560264	C₂	semidirect
7. #22560093	C₂	semidirect
8. C₆₇×#1849	C₂	semidirect
12. #16863519	C₄	semidirect
159. #1849	D₆₇	semidirect
162. C₆₇	#22695	semidirect

Maximal subgroups

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

subgroup	index
#22569125	2
C₂×#16863519	2
#22568817	2
#22560997	2
#22560448	2
#22560264	2
#22560093	2
C₆₇×#1849	2
#22695	67