Subgroups — #30748793

Order 17152.

418 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22558972	C₂	1	1	yes	maximal, characteristic, complemented
2	#22558972	C₂	1	1	yes	maximal, characteristic, complemented
3	#22558550	C₂	1	2	yes	maximal
4	#22558079	C₂	1	2	yes	maximal, complemented
5	C₂×#436×C₆₇	C₂	1	1	yes	maximal, characteristic
6	#16863453	C₂²	1	2	yes	—
7	#16863453	C₂²	1	2	yes	—
8	#16863149	—	2	1	no	—
9	#16863149	—	2	1	no	—
10	#16863147	—	2	1	no	—
11	#16863147	—	2	1	no	—
12	#16863120	—	2	4	no	—
13	C₂²×M₁₆×C₆₇	C₄	1	2	yes	—
14	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
15	#436×C₆₇	—	2	1	no	—
16	#436×C₆₇	—	2	1	no	—
17	#436×C₆₇	—	2	1	no	—
18	#436×C₆₇	—	2	1	no	—
19	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊C₈×C₆₇	C₄	1	2	yes	—
21	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
23	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
24	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
25	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
26	#14523954	—	2	2	no	—
27	#14523954	—	2	2	no	—
28	C₂²×D₄×C₆₇	Q₈	1	1	yes	characteristic
29	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
30	C₂×M₁₆×C₆₇	—	2	2	no	—
31	C₂×M₁₆×C₆₇	—	2	2	no	—
32	C₂×M₁₆×C₆₇	—	4	4	no	—
33	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
35	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
36	C₄×D₄×C₆₇	—	2	1	no	—
37	C₄×D₄×C₆₇	—	2	1	no	—
38	C₄×D₄×C₆₇	—	2	1	no	—
39	C₄×D₄×C₆₇	—	2	1	no	—
40	C₂×C₄⋊C₄×C₆₇	Q₈	1	1	yes	characteristic
41	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
42	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
43	C₄⋊C₈×C₆₇	—	2	1	no	—
44	C₄⋊C₈×C₆₇	—	2	1	no	—
45	C₂²⋊C₈×C₆₇	—	2	2	no	—
46	C₂²⋊C₈×C₆₇	—	2	2	no	—
47	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
48	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
49	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
50	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
51	C₂×C₆₇⋊C₈	—	4	2	no	—
52	C₂×C₆₇⋊C₈	—	4	2	no	—
53	C₂⁴×C₆₇	C₄⋊C₄	1	2	yes	—
54	C₂×D₄×C₆₇	—	2	1	no	—
55	C₂×D₄×C₆₇	—	2	1	no	—
56	C₂×D₄×C₆₇	—	4	2	no	—
57	C₂²×C₄×C₆₇	—	2	2	no	—
58	C₂²×C₄×C₆₇	—	2	2	no	—
59	C₂²×C₄×C₆₇	—	4	2	no	—
60	C₂²×C₄×C₆₇	—	4	2	no	—
61	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
62	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
65	M₁₆×C₆₇	—	4	4	no	—
66	M₁₆×C₆₇	—	4	4	no	—
67	C₂×C₈×C₆₇	—	2	2	no	—
68	C₂×C₈×C₆₇	—	2	2	no	—
69	C₂×C₈×C₆₇	—	4	1	no	—
70	C₂×C₈×C₆₇	—	4	2	no	—
71	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
75	C₄⋊C₄×C₆₇	—	2	1	no	—
76	C₄⋊C₄×C₆₇	—	2	1	no	—
77	C₂²⋊C₄×C₆₇	—	2	2	no	—
78	C₂²⋊C₄×C₆₇	—	2	2	no	—
79	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₆₇	#41	1	1	yes	characteristic
82	C₄²×C₆₇	#41	1	1	yes	characteristic
83	C₂³×C₆₇	—	2	2	no	—
84	C₂³×C₆₇	—	2	4	no	—
85	C₂³×C₆₇	—	4	2	no	—
86	C₂³×C₆₇	—	4	2	no	—
87	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
88	D₄×C₆₇	—	4	2	no	—
89	D₄×C₆₇	—	4	2	no	—
90	C₂×C₄×C₆₇	—	2	2	no	—
91	C₂×C₄×C₆₇	—	2	1	no	—
92	C₂×C₄×C₆₇	—	2	1	no	—
93	C₂×C₄×C₆₇	—	2	2	no	—
94	C₂×C₄×C₆₇	—	4	1	no	—
95	C₂×C₄×C₆₇	—	4	4	no	—
96	C₂×C₄×C₆₇	—	4	2	no	—
97	C₂×C₄×C₆₇	—	4	2	no	—
98	C₂×C₄×C₆₇	—	4	2	no	—
99	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
101	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
102	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
103	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
104	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
105	C₈×C₆₇	—	2	1	no	—
106	C₈×C₆₇	—	2	1	no	—
107	C₈×C₆₇	—	2	1	no	—
108	C₈×C₆₇	—	4	2	no	—
109	C₈×C₆₇	#100	1	2	yes	—
110	C₆₇⋊C₈	—	8	2	no	—
111	C₆₇⋊C₈	—	8	2	no	—
112	C₂²×C₆₇	—	2	1	no	—
113	C₂²×C₆₇	—	2	1	no	—
114	C₂²×C₆₇	—	4	2	no	—
115	C₂²×C₆₇	—	4	2	no	—
116	C₂²×C₆₇	—	4	2	no	—
117	C₂²×C₆₇	—	8	4	no	—
118	C₂²×C₆₇	#380	1	1	yes	characteristic
119	C₂²×C₆₇	#430	1	2	yes	—
120	C₄×C₆₇	—	2	1	no	—
121	C₄×C₆₇	—	2	1	no	—
122	C₄×C₆₇	—	2	1	no	—
123	C₄×C₆₇	—	2	1	no	—
124	C₄×C₆₇	—	2	1	no	—
125	C₄×C₆₇	—	4	2	no	—
126	C₄×C₆₇	C₂×#100	1	1	yes	characteristic
127	C₄×C₆₇	#413	1	1	yes	characteristic
128	#8624	—	67	1	no	maximal, Sylow
129	C₂×C₆₇	—	2	2	no	—
130	C₂×C₆₇	—	8	2	no	—
131	C₂×C₆₇	#1800	1	1	yes	characteristic
132	C₂×C₆₇	#1811	1	1	yes	characteristic
133	C₂×C₆₇	#1812	1	1	yes	characteristic
134	C₂×#436	D₆₇	1	1	yes	characteristic
135	#1568	—	67	1	no	—
136	#1568	—	67	1	no	—
137	#1466	—	67	2	no	—
138	#1290	—	67	2	no	—
139	C₆₇	#8624	1	1	yes	characteristic, complemented, Sylow
140	C₂²×M₁₆	Dic₆₇	1	2	yes	—
141	C₂×C₄×D₄	C₂×D₆₇	1	1	yes	characteristic
142	#436	—	2	1	no	—
143	#436	—	2	1	no	—
144	#436	—	2	1	no	—
145	#436	—	2	1	no	—
146	#432	—	67	2	no	—
147	#432	—	67	2	no	—
148	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
149	C₂×C₂²⋊C₈	Dic₆₇	1	2	yes	—
150	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
151	#335	—	134	1	no	—
152	#335	—	134	1	no	—
153	#334	—	134	1	no	—
154	#334	—	134	1	no	—
155	#323	—	134	4	no	—
156	C₂²×D₄	Dic₁₃₄	1	1	yes	characteristic
157	C₂³×C₄	C₂×Dic₆₇	1	2	yes	—
158	C₂×M₁₆	—	2	2	no	—
159	C₂×M₁₆	—	2	2	no	—
160	C₂×M₁₆	—	4	4	no	—
161	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
162	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
163	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
164	C₄×D₄	—	2	1	no	—
165	C₄×D₄	—	2	1	no	—
166	C₄×D₄	—	2	1	no	—
167	C₄×D₄	—	2	1	no	—
168	C₂×C₄⋊C₄	Dic₁₃₄	1	1	yes	characteristic
169	C₂×C₂²⋊C₄	D₂₆₈	1	2	yes	—
170	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
171	C₄⋊C₈	—	2	1	no	—
172	C₄⋊C₈	—	2	1	no	—
173	C₄⋊C₈	—	134	2	no	—
174	C₄⋊C₈	—	134	2	no	—
175	C₂²⋊C₈	—	2	2	no	—
176	C₂²⋊C₈	—	2	2	no	—
177	C₂²⋊C₈	—	134	2	no	—
178	C₂²⋊C₈	—	134	2	no	—
179	#97	—	134	2	no	—
180	#97	—	134	2	no	—
181	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
182	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
183	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
184	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
185	C₂⁴	C₄⋊Dic₆₇	1	2	yes	—
186	C₂×D₄	—	2	1	no	—
187	C₂×D₄	—	2	1	no	—
188	C₂×D₄	—	4	2	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	—	4	2	no	—
192	C₂²×C₄	—	4	2	no	—
193	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
194	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
195	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
196	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
197	M₁₆	—	4	4	no	—
198	M₁₆	—	4	4	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	4	1	no	—
202	C₂×C₈	—	4	2	no	—
203	C₂×C₈	—	268	2	no	—
204	C₂×C₈	—	268	2	no	—
205	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
206	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
207	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
208	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
209	C₄⋊C₄	—	2	1	no	—
210	C₄⋊C₄	—	2	1	no	—
211	C₂²⋊C₄	—	2	2	no	—
212	C₂²⋊C₄	—	2	2	no	—
213	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
214	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
215	C₄²	#11761628	1	1	yes	characteristic
216	C₄²	#11761628	1	1	yes	characteristic
217	C₂³	—	2	2	no	—
218	C₂³	—	2	4	no	—
219	C₂³	—	4	2	no	—
220	C₂³	—	4	2	no	—
221	C₂³	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
222	D₄	—	4	2	no	—
223	D₄	—	4	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	4	1	no	—
229	C₂×C₄	—	4	4	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
234	C₂×C₄	#14524002	1	1	yes	characteristic
235	C₂×C₄	#14524017	1	1	yes	characteristic
236	C₂×C₄	#14524017	1	1	yes	characteristic
237	C₂×C₄	#14524022	1	1	yes	characteristic
238	C₂×C₄	#14524022	1	1	yes	characteristic
239	C₈	—	2	1	no	—
240	C₈	—	2	1	no	—
241	C₈	—	2	1	no	—
242	C₈	—	4	2	no	—
243	C₈	—	536	2	no	—
244	C₈	—	536	2	no	—
245	C₈	#14523960	1	2	yes	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	—	4	2	no	—
249	C₂²	—	4	2	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	8	4	no	—
252	C₂²	#16863256	1	1	yes	characteristic
253	C₂²	#16863445	1	2	yes	—
254	C₄	—	2	1	no	—
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	2	1	no	—
258	C₄	—	2	1	no	—
259	C₄	—	4	2	no	—
260	C₄	C₂×#14523960	1	1	yes	characteristic
261	C₄	#16863380	1	1	yes	characteristic
262	C₂	—	2	2	no	—
263	C₂	—	8	2	no	—
264	C₂	#22560201	1	1	yes	characteristic
265	C₂	#22560251	1	1	yes	characteristic
266	C₂	#22560255	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16863256
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₂×#436×C₆₇	C₂
socle	C₂²×C₆₇	#380
prime radical	C₂²×D₄×C₆₇	Q₈
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₂×#436	D₆₇
hypocenter	C₆₇	#8624
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#436

Complements

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

N	H ≅ G/N	type
1. #22558972	C₂	semidirect
2. #22558972	C₂	semidirect
4. #22558079	C₂	semidirect
139. C₆₇	#8624	semidirect

Maximal subgroups

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

subgroup	index
#22558972	2
#22558550	2
#22558079	2
C₂×#436×C₆₇	2
#8624	67