Subgroups — #30856461

Order 17152.

718 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22568846	C₂	1	2	yes	maximal
2	C₂×#16863353	C₂	1	1	yes	maximal, characteristic
3	#22560384	C₂	1	1	yes	maximal, characteristic, complemented
4	#22560346	C₂	1	2	yes	maximal, complemented
5	#22560248	C₂	1	2	yes	maximal
6	#22560244	C₂	1	2	yes	maximal
7	#22560117	C₂	1	2	yes	maximal, complemented
8	#108×Dic₆₇	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1849	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#11761611	C₂²	1	1	yes	characteristic
11	C₂²×C₈×D₆₇	C₂²	1	1	yes	characteristic
12	C₂×#14524000	C₂²	1	1	yes	characteristic
13	#16863456	—	2	8	no	—
14	C₂×#14523979	C₂²	1	2	yes	—
15	C₂×#14523979	C₂²	1	2	yes	—
16	#16863420	C₂²	1	2	yes	—
17	#16863359	C₂²	1	4	yes	—
18	#16863353	C₄	1	8	yes	—
19	M₁₆×Dic₆₇	C₂²	1	4	yes	—
20	C₂×#14523953	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
22	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
23	#16863160	C₂²	1	4	yes	—
24	#16863160	C₂²	1	4	yes	—
25	C₂×#108×C₆₇	C₂²	1	2	yes	—
26	C₂×#108×C₆₇	C₂²	1	2	yes	—
27	#423×C₆₇	C₂²	1	2	yes	—
28	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
29	C₂²×C₄×D₆₇	C₂³	1	1	yes	characteristic
30	C₂×#11761612	C₂³	1	4	yes	—
31	C₂×#11761611	—	2	2	no	—
32	C₂×#11761611	—	2	2	no	—
33	C₂×#11761611	C₂×C₄	1	4	yes	—
34	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
35	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
36	C₂×C₈×D₆₇	—	2	2	no	—
37	C₂×C₈×D₆₇	—	2	2	no	—
38	C₂×C₈×D₆₇	C₂×C₄	1	4	yes	—
39	#14524000	C₂×C₄	1	4	yes	—
40	#14524000	D₄	1	4	yes	—
41	C₂×Dic₆₇⋊C₄	Q₈	1	2	yes	—
42	C₂×(C₂×D₆₇)⋊C₄	Q₈	1	2	yes	—
43	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
44	#14523979	—	2	4	no	—
45	#14523979	—	2	4	no	—
46	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
47	(C₂×D₆₇)⋊C₈	—	2	8	no	—
48	#14523953	—	2	4	no	—
49	#14523953	C₂×C₄	1	4	yes	—
50	C₄×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
51	C₈×Dic₆₇	—	2	4	no	—
52	C₈×Dic₆₇	C₂×C₄	1	4	yes	—
53	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
54	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
55	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
57	#108×C₆₇	—	2	4	no	—
58	#108×C₆₇	—	2	4	no	—
59	C₄⋊C₈×C₆₇	—	2	4	no	—
60	#97×C₆₇	C₂×C₄	1	4	yes	—
61	C₂³×D₆₇	C₂×Q₈	1	1	yes	characteristic
62	C₂²×Dic₆₇	C₂×Q₈	1	1	yes	characteristic
63	C₂²×Dic₆₇	C₂×Q₈	1	2	yes	—
64	C₂×C₄×D₆₇	—	2	1	no	—
65	C₂×C₄×D₆₇	—	2	1	no	—
66	C₂×C₄×D₆₇	—	2	1	no	—
67	C₂×C₄×D₆₇	—	2	1	no	—
68	C₂×C₄×D₆₇	C₂²×C₄	1	2	yes	—
69	C₂×C₄×D₆₇	C₂×D₄	1	2	yes	—
70	#11761612	—	2	4	no	—
71	#11761612	—	2	4	no	—
72	#11761611	—	2	8	no	—
73	C₂×C₆₇⋊C₈	—	2	1	no	—
74	C₂×C₆₇⋊C₈	—	2	1	no	—
75	C₂×C₆₇⋊C₈	—	2	1	no	—
76	C₂×C₆₇⋊C₈	—	2	1	no	—
77	C₂×C₆₇⋊C₈	—	2	4	no	—
78	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
79	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
80	C₈×D₆₇	—	2	8	no	—
81	Dic₆₇⋊C₄	C₄⋊C₄	1	8	yes	—
82	(C₂×D₆₇)⋊C₄	C₄⋊C₄	1	8	yes	—
83	C₄×Dic₆₇	C₂²×C₄	1	2	yes	—
84	C₄×Dic₆₇	C₂×D₄	1	2	yes	—
85	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
86	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
87	M₁₆×C₆₇	—	2	4	no	—
88	M₁₆×C₆₇	—	2	4	no	—
89	C₂×C₈×C₆₇	—	2	4	no	—
90	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
91	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
92	C₂×C₈×C₆₇	#41	1	2	yes	—
93	C₂×C₈×C₆₇	#41	1	2	yes	—
94	C₂×C₈×C₆₇	#41	1	2	yes	—
95	C₂×C₈×C₆₇	#41	1	2	yes	—
96	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
97	C₄²×C₆₇	C₂×D₄	1	2	yes	—
98	C₂³×C₆₇	C₂²×Q₈	1	1	yes	characteristic
99	C₂²×D₆₇	—	2	1	no	—
100	C₂²×D₆₇	—	2	1	no	—
101	C₂²×D₆₇	—	4	2	no	—
102	C₂²×D₆₇	C₂×C₄⋊C₄	1	2	yes	—
103	C₂×C₄×C₆₇	—	4	2	no	—
104	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
105	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
106	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
107	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
108	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
109	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
110	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
111	C₂×Dic₆₇	—	2	1	no	—
112	C₂×Dic₆₇	—	2	1	no	—
113	C₂×Dic₆₇	—	4	2	no	—
114	C₂×Dic₆₇	C₂×C₄⋊C₄	1	4	yes	—
115	C₂×Dic₆₇	C₂×C₄⋊C₄	1	2	yes	—
116	C₄×D₆₇	—	2	2	no	—
117	C₄×D₆₇	—	2	2	no	—
118	C₄×D₆₇	—	2	2	no	—
119	C₄×D₆₇	—	2	2	no	—
120	C₈×C₆₇	—	4	4	no	—
121	C₈×C₆₇	#117	1	4	yes	—
122	C₈×C₆₇	#117	1	4	yes	—
123	C₆₇⋊C₈	—	2	2	no	—
124	C₆₇⋊C₈	—	2	2	no	—
125	C₆₇⋊C₈	—	4	4	no	—
126	C₂²×C₆₇	—	2	2	no	—
127	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
128	C₂²×C₆₇	#533	1	1	yes	characteristic
129	C₂²×C₆₇	#533	1	1	yes	characteristic
130	C₂×D₆₇	—	2	4	no	—
131	C₂×D₆₇	—	2	2	no	—
132	C₂×D₆₇	—	4	2	no	—
133	C₂×D₆₇	—	4	2	no	—
134	C₄×C₆₇	—	2	4	no	—
135	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
136	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
137	C₄×C₆₇	#532	1	1	yes	characteristic
138	C₄×C₆₇	#532	1	1	yes	characteristic
139	Dic₆₇	—	2	4	no	—
140	Dic₆₇	—	2	2	no	—
141	#20829	—	67	1	no	maximal, Sylow
142	C₂×C₆₇	—	2	1	no	—
143	C₂×C₆₇	—	2	1	no	—
144	C₂×C₆₇	#2285	1	1	yes	characteristic
145	C₂×C₆₇	#2916	1	2	yes	—
146	D₆₇	—	4	2	no	—
147	#2921	—	67	2	no	—
148	C₂×#405	—	67	1	no	—
149	#1849	—	67	1	no	—
150	#1849	D₆₇	1	1	yes	characteristic, complemented
151	#1837	—	67	2	no	—
152	#1811	—	67	2	no	—
153	#1810	—	67	2	no	—
154	#1778	—	67	2	no	—
155	C₄×#108	—	67	2	no	—
156	C₆₇	#20829	1	1	yes	characteristic, complemented, Sylow
157	C₂²×M₁₆	—	67	1	no	—
158	C₂³×C₈	—	67	1	no	—
159	C₂×#117	—	67	1	no	—
160	#433	—	134	8	no	—
161	C₂×#108	—	67	2	no	—
162	C₂×#108	—	67	2	no	—
163	C₂×#108	C₂×D₆₇	1	2	yes	—
164	C₂×#108	C₂×D₆₇	1	2	yes	—
165	#423	—	67	2	no	—
166	#423	C₂×D₆₇	1	2	yes	—
167	#407	—	67	4	no	—
168	#405	—	67	8	no	—
169	C₄×M₁₆	—	67	4	no	—
170	C₂×#97	—	67	1	no	—
171	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
172	C₂×C₄×C₈	—	67	1	no	—
173	C₂×C₄×C₈	—	67	1	no	—
174	#341	—	67	4	no	—
175	#341	—	67	4	no	—
176	C₂³×C₄	—	67	1	no	—
177	C₂×M₁₆	—	67	4	no	—
178	C₂×M₁₆	—	67	4	no	—
179	C₂×M₁₆	—	134	2	no	—
180	C₂×M₁₆	—	134	2	no	—
181	C₂×M₁₆	C₂²×D₆₇	1	4	yes	—
182	C₂²×C₈	—	67	1	no	—
183	C₂²×C₈	—	67	1	no	—
184	C₂²×C₈	—	67	4	no	—
185	C₂²×C₈	—	134	2	no	—
186	C₂²×C₈	—	134	2	no	—
187	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
188	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
189	#117	—	67	4	no	—
190	#117	—	67	4	no	—
191	C₂×C₄⋊C₄	—	67	2	no	—
192	C₂×C₂²⋊C₄	—	67	2	no	—
193	C₂×C₄²	—	67	1	no	—
194	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
195	#108	—	2	4	no	—
196	#108	—	2	4	no	—
197	#108	—	134	4	no	—
198	#108	—	134	4	no	—
199	C₄⋊C₈	—	2	4	no	—
200	C₄⋊C₈	—	134	4	no	—
201	C₂²⋊C₈	—	134	8	no	—
202	#97	—	67	4	no	—
203	#97	—	134	4	no	—
204	#97	C₄×D₆₇	1	4	yes	—
205	C₄×C₈	—	67	4	no	—
206	C₄×C₈	—	67	4	no	—
207	C₄×C₈	—	134	4	no	—
208	C₂⁴	—	67	1	no	—
209	C₂²×C₄	—	67	2	no	—
210	C₂²×C₄	—	67	1	no	—
211	C₂²×C₄	—	67	2	no	—
212	C₂²×C₄	—	67	2	no	—
213	C₂²×C₄	—	134	1	no	—
214	C₂²×C₄	—	134	1	no	—
215	C₂²×C₄	—	134	1	no	—
216	C₂²×C₄	—	134	1	no	—
217	C₂²×C₄	Q₈×D₆₇	1	2	yes	—
218	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
219	M₁₆	—	2	4	no	—
220	M₁₆	—	2	4	no	—
221	M₁₆	—	134	4	no	—
222	M₁₆	—	134	4	no	—
223	M₁₆	—	134	8	no	—
224	C₂×C₈	—	2	4	no	—
225	C₂×C₈	—	67	2	no	—
226	C₂×C₈	—	67	2	no	—
227	C₂×C₈	—	134	4	no	—
228	C₂×C₈	—	134	1	no	—
229	C₂×C₈	—	134	1	no	—
230	C₂×C₈	—	134	1	no	—
231	C₂×C₈	—	134	1	no	—
232	C₂×C₈	—	134	8	no	—
233	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
234	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
235	C₂×C₈	#11761627	1	2	yes	—
236	C₂×C₈	#11761627	1	2	yes	—
237	C₂×C₈	#11761629	1	2	yes	—
238	C₂×C₈	#11761629	1	2	yes	—
239	C₄⋊C₄	—	67	8	no	—
240	C₂²⋊C₄	—	67	8	no	—
241	C₄²	—	67	2	no	—
242	C₄²	—	67	2	no	—
243	C₄²	C₂×C₄×D₆₇	1	2	yes	—
244	C₄²	D₄×D₆₇	1	2	yes	—
245	C₂³	—	268	2	no	—
246	C₂³	—	67	2	no	—
247	C₂³	—	134	1	no	—
248	C₂³	—	134	1	no	—
249	C₂³	C₂×Q₈×D₆₇	1	1	yes	characteristic
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	268	2	no	—
252	C₂×C₄	—	67	4	no	—
253	C₂×C₄	—	67	2	no	—
254	C₂×C₄	—	134	1	no	—
255	C₂×C₄	—	134	1	no	—
256	C₂×C₄	—	134	2	no	—
257	C₂×C₄	—	134	2	no	—
258	C₂×C₄	—	134	2	no	—
259	C₂×C₄	—	134	2	no	—
260	C₂×C₄	C₄⋊C₄×D₆₇	1	4	yes	—
261	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
262	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
263	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
264	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
265	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
266	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
267	C₈	—	4	4	no	—
268	C₈	—	268	4	no	—
269	C₈	—	134	2	no	—
270	C₈	—	134	2	no	—
271	C₈	#14524001	1	4	yes	—
272	C₈	#14524001	1	4	yes	—
273	C₂²	—	2	2	no	—
274	C₂²	—	268	2	no	—
275	C₂²	—	268	2	no	—
276	C₂²	—	134	2	no	—
277	C₂²	—	134	4	no	—
278	C₂²	C₂×C₄⋊C₄×D₆₇	1	1	yes	characteristic
279	C₂²	#16863951	1	1	yes	characteristic
280	C₂²	#16863951	1	1	yes	characteristic
281	C₄	—	2	4	no	—
282	C₄	—	134	2	no	—
283	C₄	—	134	4	no	—
284	C₄	C₂×#14524001	1	1	yes	characteristic
285	C₄	C₂×#14524001	1	1	yes	characteristic
286	C₄	#16863944	1	1	yes	characteristic
287	C₄	#16863947	1	1	yes	characteristic
288	C₂	—	2	1	no	—
289	C₂	—	2	1	no	—
290	C₂	—	268	2	no	—
291	C₂	#22562614	1	1	yes	characteristic
292	C₂	#22568819	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#1849	C₂
socle	C₂²×C₆₇	C₂²×C₄⋊C₄
prime radical	C₂³×D₆₇	C₂×Q₈
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#1849	D₆₇
hypocenter	C₆₇	#20829
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1849

Complements

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

N	H ≅ G/N	type
3. #22560384	C₂	semidirect
4. #22560346	C₂	semidirect
7. #22560117	C₂	semidirect
8. #108×Dic₆₇	C₂	semidirect
9. C₆₇×#1849	C₂	semidirect
150. #1849	D₆₇	semidirect
156. C₆₇	#20829	semidirect

Maximal subgroups

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

subgroup	index
#22568846	2
C₂×#16863353	2
#22560384	2
#22560346	2
#22560248	2
#22560244	2
#22560117	2
#108×Dic₆₇	2
C₆₇×#1849	2
#20829	67