Subgroups — #30856207

Order 17152.

608 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	#22568635	C₂	1	1	yes	maximal, characteristic
2	#22560359	C₂	1	2	yes	maximal, complemented
3	#22560334	C₂	1	2	yes	maximal, complemented
4	#22560314	C₂	1	2	yes	maximal
5	#22560255	C₂	1	1	yes	maximal, characteristic
6	#22560096	C₂	1	2	yes	maximal, complemented
7	#22560012	C₂	1	2	yes	maximal
8	#22559965	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1813	C₂	1	1	yes	maximal, characteristic
10	C₂×#14524050	C₂²	1	1	yes	characteristic
11	#41×Dic₆₇	C₂²	1	1	yes	characteristic
12	#16863434	C₂²	1	2	yes	—
13	#16863370	C₂²	1	2	yes	—
14	#16863354	C₄	1	2	yes	complemented
15	#16863354	C₂²	1	4	yes	—
16	#16863354	C₂²	1	1	yes	characteristic
17	#16863351	C₄	1	2	yes	complemented
18	#16863351	C₂²	1	1	yes	characteristic
19	M₁₆×Dic₆₇	C₄	1	2	yes	complemented
20	M₁₆×Dic₆₇	C₂²	1	1	yes	characteristic
21	C₂×#14523953	C₄	1	2	yes	complemented
22	C₂×#14523953	C₂²	1	1	yes	characteristic
23	#16863168	C₂²	1	4	yes	—
24	#16863157	C₂²	1	4	yes	—
25	#16863153	C₂²	1	4	yes	—
26	#16863153	C₂²	1	2	yes	—
27	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
28	C₂×#108×C₆₇	C₂²	1	2	yes	—
29	#413×C₆₇	C₂²	1	2	yes	—
30	#338×C₆₇	C₂²	1	2	yes	—
31	#14524050	—	2	4	no	—
32	C₂×#11761612	C₂×C₄	1	2	yes	—
33	C₂×#11761612	C₂³	1	1	yes	characteristic
34	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
35	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
36	Q₈×Dic₆₇	—	2	1	no	—
37	D₄×Dic₆₇	—	2	1	no	—
38	D₄×Dic₆₇	—	2	2	no	—
39	#14524002	C₂×C₄	1	2	yes	—
40	#14524002	C₂³	1	4	yes	—
41	#14524002	C₂³	1	1	yes	characteristic
42	C₂×C₄×Dic₆₇	C₂×C₄	1	2	yes	—
43	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
44	#14523977	—	2	2	no	—
45	#14523975	—	2	2	no	—
46	#14523958	—	2	4	no	—
47	#14523953	—	2	4	no	—
48	#14523953	C₂×C₄	1	1	yes	characteristic
49	#14523953	C₂×C₄	1	2	yes	—
50	#14523953	C₂×C₄	1	2	yes	—
51	#14523953	C₂×C₄	1	2	yes	—
52	#14523953	C₂×C₄	1	1	yes	characteristic
53	#14523953	C₂×C₄	1	1	yes	characteristic
54	#14523953	C₂×C₄	1	1	yes	characteristic
55	C₈×Dic₆₇	—	2	4	no	—
56	C₈×Dic₆₇	C₂×C₄	1	2	yes	—
57	C₈×Dic₆₇	C₂×C₄	1	2	yes	—
58	C₈×Dic₆₇	C₂×C₄	1	2	yes	—
59	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
60	#131×C₆₇	—	2	2	no	—
61	#131×C₆₇	—	2	2	no	—
62	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
63	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
64	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
65	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
66	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
67	#108×C₆₇	—	2	2	no	—
68	#108×C₆₇	—	2	2	no	—
69	#101×C₆₇	—	2	2	no	—
70	#100×C₆₇	—	2	2	no	—
71	C₂²×Dic₆₇	—	2	2	no	—
72	C₂²×Dic₆₇	—	2	1	no	—
73	#11761612	—	2	2	no	—
74	#11761612	—	2	4	no	—
75	C₂×C₆₇⋊C₈	—	2	4	no	—
76	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
77	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₆₇⋊C₈	#41	1	4	yes	—
80	C₄⋊Dic₆₇	—	2	2	no	—
81	C₄⋊Dic₆₇	—	2	1	no	—
82	C₄⋊Dic₆₇	—	2	4	no	—
83	C₂²⋊Dic₆₇	—	2	2	no	—
84	C₂²⋊Dic₆₇	—	2	1	no	—
85	C₂²⋊Dic₆₇	—	2	4	no	—
86	C₄×Dic₆₇	—	2	2	no	—
87	C₄×Dic₆₇	—	2	1	no	—
88	C₄×Dic₆₇	—	2	4	no	—
89	C₄×Dic₆₇	C₂²×C₄	1	2	yes	—
90	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
91	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
92	#41×C₆₇	—	2	2	no	—
93	#41×C₆₇	—	2	2	no	—
94	C₂×Q₈×C₆₇	#41	1	1	yes	characteristic
95	C₂×D₄×C₆₇	#41	1	1	yes	characteristic
96	C₂×D₄×C₆₇	#41	1	2	yes	—
97	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
98	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
99	M₁₆×C₆₇	—	2	2	no	—
100	M₁₆×C₆₇	—	2	4	no	—
101	M₁₆×C₆₇	—	2	4	no	—
102	M₁₆×C₆₇	—	2	4	no	—
103	C₂×C₈×C₆₇	—	2	2	no	—
104	C₂×C₈×C₆₇	—	2	2	no	—
105	C₂×C₈×C₆₇	—	2	4	no	—
106	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
107	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₈×C₆₇	#41	1	2	yes	—
110	C₂×C₈×C₆₇	#41	1	2	yes	—
111	C₂³×C₆₇	#117	1	2	yes	—
112	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
113	Q₈×C₆₇	—	2	2	no	—
114	D₄×C₆₇	—	2	4	no	—
115	D₄×C₆₇	—	2	2	no	—
116	C₂×C₄×C₆₇	—	2	2	no	—
117	C₂×C₄×C₆₇	—	2	2	no	—
118	C₂×C₄×C₆₇	#117	1	2	yes	—
119	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
120	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
121	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
122	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
123	C₂×Dic₆₇	—	2	1	no	—
124	C₂×Dic₆₇	—	2	2	no	—
125	C₂×Dic₆₇	—	2	1	no	—
126	C₂×Dic₆₇	—	4	4	no	—
127	C₂×Dic₆₇	—	8	1	no	—
128	C₂×Dic₆₇	—	8	2	no	—
129	C₈×C₆₇	—	2	2	no	—
130	C₈×C₆₇	—	4	4	no	—
131	C₈×C₆₇	#117	1	1	yes	characteristic
132	C₈×C₆₇	#117	1	1	yes	characteristic
133	C₈×C₆₇	#117	1	2	yes	—
134	C₆₇⋊C₈	—	2	2	no	—
135	C₆₇⋊C₈	#117	1	2	yes	—
136	C₆₇⋊C₈	#117	1	2	yes	—
137	C₂²×C₆₇	—	2	2	no	—
138	C₂²×C₆₇	—	4	1	no	—
139	C₂²×C₆₇	—	4	2	no	—
140	C₂²×C₆₇	C₂×#117	1	1	yes	characteristic
141	C₂²×C₆₇	#529	1	2	yes	—
142	C₄×C₆₇	—	2	2	no	—
143	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
144	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
145	C₄×C₆₇	#529	1	2	yes	—
146	Dic₆₇	—	4	2	no	—
147	Dic₆₇	—	4	1	no	—
148	Dic₆₇	—	4	1	no	—
149	Dic₆₇	—	8	4	no	—
150	#20788	—	67	1	no	maximal, Sylow
151	C₂×C₆₇	—	2	1	no	—
152	C₂×C₆₇	—	2	2	no	—
153	C₂×C₆₇	—	4	2	no	—
154	C₂×C₆₇	#2283	1	1	yes	characteristic
155	#2875	—	67	1	no	—
156	#1841	—	67	2	no	—
157	#1833	—	67	2	no	—
158	#1827	—	67	2	no	—
159	#1813	D₆₇	1	1	yes	characteristic
160	#1812	—	67	1	no	—
161	#1773	—	67	2	no	—
162	#1757	—	67	2	no	—
163	#1747	—	67	2	no	—
164	C₆₇	#20788	1	1	yes	characteristic, complemented, Sylow
165	C₂×#131	—	67	1	no	—
166	C₂×#131	C₂×D₆₇	1	1	yes	characteristic
167	C₄×#41	—	67	1	no	—
168	C₂×#108	C₂×D₆₇	1	2	yes	—
169	#427	—	67	2	no	—
170	#413	C₂×D₆₇	1	2	yes	—
171	#410	—	67	2	no	—
172	#405	—	67	2	no	—
173	#405	—	67	1	no	—
174	#405	—	67	2	no	—
175	#405	—	67	1	no	—
176	#405	—	67	4	no	—
177	C₄×M₁₆	—	67	2	no	—
178	C₄×M₁₆	—	67	1	no	—
179	C₂×#97	—	67	2	no	—
180	C₂×#97	—	67	1	no	—
181	#344	—	67	4	no	—
182	#339	—	67	4	no	—
183	#338	C₂×D₆₇	1	2	yes	—
184	H(3,ℤ/4)	—	67	4	no	—
185	H(3,ℤ/4)	—	67	2	no	—
186	C₂×#41	C₂²×D₆₇	1	1	yes	characteristic
187	#131	—	2	2	no	—
188	#131	—	2	2	no	—
189	#131	—	134	4	no	—
190	C₂×M₁₆	—	67	2	no	—
191	C₂×M₁₆	—	67	1	no	—
192	C₂×M₁₆	C₄×D₆₇	1	2	yes	—
193	C₂×M₁₆	C₂²×D₆₇	1	4	yes	—
194	C₂×M₁₆	C₂²×D₆₇	1	1	yes	characteristic
195	C₂²×C₈	—	67	1	no	—
196	C₂²×C₈	—	67	2	no	—
197	C₂²×C₈	C₄×D₆₇	1	2	yes	—
198	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
199	C₄×Q₈	—	134	1	no	—
200	C₄×D₄	—	134	1	no	—
201	C₄×D₄	—	134	2	no	—
202	#117	—	67	2	no	—
203	#117	—	67	1	no	—
204	#117	—	67	4	no	—
205	C₂×C₄²	—	67	2	no	—
206	C₂×C₄²	—	67	1	no	—
207	#108	—	2	2	no	—
208	#108	—	2	2	no	—
209	#107	—	134	2	no	—
210	#106	—	134	2	no	—
211	#101	—	2	2	no	—
212	#100	—	2	2	no	—
213	#99	—	134	4	no	—
214	#97	—	67	1	no	—
215	#97	—	67	1	no	—
216	#97	—	67	2	no	—
217	#97	—	67	1	no	—
218	#97	—	67	2	no	—
219	#97	—	67	1	no	—
220	#97	—	67	2	no	—
221	#97	—	134	4	no	—
222	C₄×C₈	—	67	2	no	—
223	C₄×C₈	—	67	2	no	—
224	C₄×C₈	—	67	2	no	—
225	C₄×C₈	—	134	4	no	—
226	#41	—	2	2	no	—
227	#41	—	2	2	no	—
228	C₂×Q₈	#11761627	1	1	yes	characteristic
229	C₂×D₄	#11761627	1	1	yes	characteristic
230	C₂×D₄	#11761629	1	2	yes	—
231	C₂²×C₄	—	134	2	no	—
232	C₂²×C₄	—	134	1	no	—
233	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
234	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
235	M₁₆	—	2	2	no	—
236	M₁₆	—	2	4	no	—
237	M₁₆	—	2	4	no	—
238	M₁₆	—	2	4	no	—
239	M₁₆	—	134	4	no	—
240	M₁₆	—	134	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	—	67	1	no	—
245	C₂×C₈	—	67	1	no	—
246	C₂×C₈	—	67	4	no	—
247	C₂×C₈	—	67	2	no	—
248	C₂×C₈	—	134	4	no	—
249	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
250	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
251	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
252	C₂×C₈	#11761627	1	2	yes	—
253	C₂×C₈	#11761627	1	2	yes	—
254	C₄⋊C₄	—	134	4	no	—
255	C₄⋊C₄	—	134	2	no	—
256	C₄⋊C₄	—	134	1	no	—
257	C₂²⋊C₄	—	134	4	no	—
258	C₂²⋊C₄	—	134	2	no	—
259	C₂²⋊C₄	—	134	1	no	—
260	C₄²	—	67	2	no	—
261	C₄²	—	67	1	no	—
262	C₄²	—	67	1	no	—
263	C₄²	—	134	4	no	—
264	C₄²	—	134	2	no	—
265	C₄²	—	134	1	no	—
266	C₂³	#14524001	1	2	yes	—
267	C₂³	C₂×#11761629	1	1	yes	characteristic
268	Q₈	—	2	2	no	—
269	D₄	—	2	4	no	—
270	D₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	268	4	no	—
274	C₂×C₄	—	536	1	no	—
275	C₂×C₄	—	536	2	no	—
276	C₂×C₄	—	134	1	no	—
277	C₂×C₄	—	134	2	no	—
278	C₂×C₄	—	134	1	no	—
279	C₂×C₄	#14524001	1	2	yes	—
280	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
281	C₂×C₄	#41×D₆₇	1	2	yes	—
282	C₂×C₄	C₂×#11761627	1	2	yes	—
283	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
284	C₈	—	2	2	no	—
285	C₈	—	4	4	no	—
286	C₈	—	67	2	no	—
287	C₈	—	67	2	no	—
288	C₈	—	134	2	no	—
289	C₈	#14523999	1	1	yes	characteristic
290	C₈	#14523999	1	1	yes	characteristic
291	C₈	#14523999	1	2	yes	—
292	C₂²	—	2	2	no	—
293	C₂²	—	4	1	no	—
294	C₂²	—	4	2	no	—
295	C₂²	C₂×#14524001	1	1	yes	characteristic
296	C₂²	#16863932	1	2	yes	—
297	C₄	—	2	2	no	—
298	C₄	—	268	1	no	—
299	C₄	—	268	1	no	—
300	C₄	—	268	2	no	—
301	C₄	—	536	4	no	—
302	C₄	#117×D₆₇	1	1	yes	characteristic
303	C₄	C₂×#14523999	1	1	yes	characteristic
304	C₄	#16863927	1	2	yes	—
305	C₂	—	2	1	no	—
306	C₂	—	2	2	no	—
307	C₂	—	4	2	no	—
308	C₂	#22562604	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#117×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#1813	C₂
socle	C₂×C₆₇	#2283
prime radical	C₂×#41×C₆₇	C₂³
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#1813	D₆₇
hypocenter	C₆₇	#20788
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #1813

Complements

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

N	H ≅ G/N	type
2. #22560359	C₂	semidirect
3. #22560334	C₂	semidirect
6. #22560096	C₂	semidirect
8. #22559965	C₂	semidirect
14. #16863354	C₄	semidirect
17. #16863351	C₄	semidirect
19. M₁₆×Dic₆₇	C₄	semidirect
21. C₂×#14523953	C₄	semidirect
164. C₆₇	#20788	semidirect

Maximal subgroups

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

subgroup	index
#22568635	2
#22560359	2
#22560334	2
#22560314	2
#22560255	2
#22560096	2
#22560012	2
#22559965	2
C₆₇×#1813	2
#20788	67