Subgroups — #30770950

Order 17152.

382 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863812	C₂	1	1	yes	maximal, characteristic
2	#22561829	C₂	1	1	yes	maximal, characteristic
3	#22561814	C₂	1	1	yes	maximal, characteristic, complemented
4	#22561813	C₂	1	1	yes	maximal, characteristic, complemented
5	#22560010	C₂	1	1	yes	maximal, characteristic
6	#22558297	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1379	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14524049	C₂²	1	1	yes	characteristic
9	#16863812	—	2	2	no	—
10	#16863812	—	2	2	no	—
11	C₂×#14523983	C₂²	1	1	yes	characteristic
12	C₂×#14523983	C₂²	1	1	yes	characteristic
13	C₂×#14523982	C₄	1	2	yes	complemented
14	C₂×C₁₆×D₆₇	C₄	1	2	yes	complemented
15	#16863378	C₄	1	1	yes	characteristic
16	#16863369	C₄	1	1	yes	characteristic
17	#16863351	C₂²	1	1	yes	characteristic
18	#16863177	C₄	1	4	yes	complemented
19	#16863174	—	2	2	no	—
20	#16863173	—	2	2	no	—
21	#16863168	C₄	1	1	yes	characteristic
22	#16863166	C₄	1	1	yes	characteristic
23	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
24	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
25	#405×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×#11761628	C₂×C₄	1	1	yes	characteristic
27	#14524049	—	2	1	no	—
28	#14524049	—	2	1	no	—
29	#14524049	—	2	1	no	—
30	#14524049	—	2	1	no	—
31	C₂×#11761612	C₂×C₄	1	1	yes	characteristic
32	C₂×#11761612	C₂×C₄	1	1	yes	characteristic
33	C₂×#11761611	C₂×C₄	1	2	yes	—
34	C₂×C₈×D₆₇	C₂×C₄	1	2	yes	—
35	#14524002	C₂×C₄	1	1	yes	characteristic
36	#14523983	—	2	1	no	—
37	#14523983	C₂×C₄	1	2	yes	—
38	#14523983	D₄	1	2	yes	—
39	#14523983	D₄	1	2	yes	—
40	#14523982	—	2	4	no	—
41	C₂×C₆₇⋊C₁₆	—	2	1	no	—
42	C₂×C₆₇⋊C₁₆	C₂×C₄	1	2	yes	—
43	C₁₆×D₆₇	—	2	4	no	—
44	#14523961	—	2	1	no	—
45	#14523959	—	2	1	no	—
46	#14523957	—	2	2	no	—
47	#14523953	—	2	1	no	—
48	C₈×Dic₆₇	—	2	1	no	—
49	C₂×M₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
51	#117×C₆₇	C₂×C₄	1	1	yes	characteristic
52	M₃₂×C₆₇	—	2	1	no	—
53	M₃₂×C₆₇	C₂×C₄	1	2	yes	—
54	C₂×C₁₆×C₆₇	—	2	1	no	—
55	C₂×C₁₆×C₆₇	—	2	1	no	—
56	C₂×C₁₆×C₆₇	—	2	1	no	—
57	C₂×C₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
58	C₂×C₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
59	#97×C₆₇	—	2	1	no	—
60	C₄×C₈×C₆₇	—	2	1	no	—
61	#11761628	—	2	1	no	—
62	#11761628	—	2	1	no	—
63	#11761628	—	2	1	no	—
64	#11761628	—	2	1	no	—
65	C₂×Dic₁₃₄	M₁₆	1	1	yes	characteristic
66	C₂×D₂₆₈	M₁₆	1	1	yes	characteristic
67	C₂×C₂²⋊D₆₇	M₁₆	1	2	yes	—
68	C₂×C₄×D₆₇	C₄²	1	2	yes	—
69	#11761612	—	2	1	no	—
70	#11761612	—	2	1	no	—
71	#11761612	—	2	1	no	—
72	#11761612	—	2	1	no	—
73	#11761611	—	2	2	no	—
74	#11761611	—	2	2	no	—
75	C₂×C₆₇⋊C₈	—	2	1	no	—
76	C₂×C₆₇⋊C₈	C₄²	1	2	yes	—
77	C₈×D₆₇	—	2	2	no	—
78	C₈×D₆₇	—	2	2	no	—
79	C₄⋊Dic₆₇	—	2	1	no	—
80	C₂²⋊Dic₆₇	—	2	1	no	—
81	C₄×Dic₆₇	—	2	1	no	—
82	C₆₇⋊C₁₆	—	2	2	no	—
83	C₆₇⋊C₁₆	C₂²⋊C₄	1	4	yes	—
84	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
85	M₁₆×C₆₇	—	2	1	no	—
86	M₁₆×C₆₇	—	2	1	no	—
87	C₂×C₈×C₆₇	—	2	1	no	—
88	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
93	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
94	C₄⋊C₄×C₆₇	—	2	1	no	—
95	C₂²⋊C₄×C₆₇	—	2	1	no	—
96	C₄²×C₆₇	—	2	1	no	—
97	C₁₆×C₆₇	—	2	1	no	—
98	C₁₆×C₆₇	—	2	1	no	—
99	C₁₆×C₆₇	—	2	2	no	—
100	C₂³×C₆₇	C₂×M₁₆	1	1	yes	characteristic
101	C₂²×D₆₇	#97	1	2	yes	—
102	C₂×C₄×C₆₇	—	4	1	no	—
103	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
104	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
107	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
108	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
109	C₂²⋊D₆₇	—	2	2	no	—
110	C₂²⋊D₆₇	—	2	2	no	—
111	C₂×Dic₆₇	—	4	1	no	—
112	C₂×Dic₆₇	#97	1	2	yes	—
113	D₂₆₈	—	2	1	no	—
114	D₂₆₈	—	2	1	no	—
115	C₄×D₆₇	—	2	2	no	—
116	C₄×D₆₇	—	2	2	no	—
117	Dic₁₃₄	—	2	1	no	—
118	Dic₁₃₄	—	2	1	no	—
119	C₈×C₆₇	—	4	1	no	—
120	C₈×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₈×C₆₇	#117	1	1	yes	characteristic
122	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
123	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
124	C₆₇⋊C₈	—	2	2	no	—
125	C₆₇⋊C₈	—	4	1	no	—
126	C₂²×C₆₇	—	4	1	no	—
127	C₂²×C₆₇	C₂×#97	1	1	yes	characteristic
128	C₂²×C₆₇	#435	1	1	yes	characteristic
129	C₂²×C₆₇	#435	1	1	yes	characteristic
130	C₂×D₆₇	—	2	2	no	—
131	C₂×D₆₇	—	4	2	no	—
132	C₄×C₆₇	—	8	1	no	—
133	C₄×C₆₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
134	C₄×C₆₇	#405	1	1	yes	characteristic
135	C₄×C₆₇	#435	1	1	yes	characteristic
136	C₄×C₆₇	#435	1	1	yes	characteristic
137	Dic₆₇	—	2	2	no	—
138	Dic₆₇	—	8	1	no	—
139	#12617	—	67	1	no	maximal, Sylow
140	C₂×C₆₇	—	2	1	no	—
141	C₂×C₆₇	—	2	1	no	—
142	C₂×C₆₇	—	2	1	no	—
143	C₂×C₆₇	#1752	1	1	yes	characteristic
144	D₆₇	—	4	2	no	—
145	C₂×#504	—	67	1	no	—
146	#2115	—	67	1	no	—
147	#2110	—	67	1	no	—
148	#2110	—	67	1	no	—
149	#1757	—	67	1	no	—
150	#1379	—	67	1	no	—
151	#1379	D₆₇	1	1	yes	characteristic, complemented
152	C₆₇	#12617	1	1	yes	characteristic, complemented, Sylow
153	C₂×#131	—	67	1	no	—
154	#504	—	134	2	no	—
155	#504	—	134	2	no	—
156	C₂×M₃₂	—	67	1	no	—
157	C₂×M₃₂	—	67	2	no	—
158	C₂×M₃₂	—	67	1	no	—
159	C₂×M₃₂	C₂×D₆₇	1	1	yes	characteristic
160	C₂²×C₁₆	—	67	2	no	—
161	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
162	#413	—	67	1	no	—
163	#410	—	67	1	no	—
164	#405	—	67	1	no	—
165	#405	C₂×D₆₇	1	1	yes	characteristic
166	#349	—	67	4	no	—
167	#347	—	134	2	no	—
168	#347	—	134	2	no	—
169	#344	—	67	1	no	—
170	#344	—	67	1	no	—
171	C₂×#41	—	67	1	no	—
172	#131	—	134	1	no	—
173	#131	—	134	1	no	—
174	#131	—	134	1	no	—
175	#131	—	134	1	no	—
176	C₂×M₁₆	—	67	1	no	—
177	C₂×M₁₆	—	67	2	no	—
178	C₂×M₁₆	—	67	1	no	—
179	C₂×M₁₆	C₄×D₆₇	1	1	yes	characteristic
180	C₂²×C₈	—	67	2	no	—
181	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
182	#117	—	67	1	no	—
183	#117	C₄×D₆₇	1	1	yes	characteristic
184	M₃₂	—	2	1	no	—
185	M₃₂	—	67	2	no	—
186	M₃₂	—	67	2	no	—
187	M₃₂	—	67	2	no	—
188	M₃₂	—	134	4	no	—
189	M₃₂	—	134	1	no	—
190	M₃₂	C₄×D₆₇	1	2	yes	—
191	C₂×C₁₆	—	2	1	no	—
192	C₂×C₁₆	—	2	1	no	—
193	C₂×C₁₆	—	2	1	no	—
194	C₂×C₁₆	—	67	2	no	—
195	C₂×C₁₆	—	134	4	no	—
196	C₂×C₁₆	—	134	1	no	—
197	C₂×C₁₆	C₄×D₆₇	1	1	yes	characteristic
198	C₂×C₁₆	C₄×D₆₇	1	1	yes	characteristic
199	#101	—	134	1	no	—
200	#100	—	134	1	no	—
201	#99	—	134	2	no	—
202	#97	—	2	1	no	—
203	#97	—	134	1	no	—
204	C₄×C₈	—	2	1	no	—
205	C₄×C₈	—	134	1	no	—
206	#41	—	134	1	no	—
207	#41	—	134	1	no	—
208	#41	—	134	1	no	—
209	#41	—	134	1	no	—
210	C₂×Q₈	—	67	1	no	—
211	C₂×D₄	—	67	2	no	—
212	C₂×D₄	—	67	1	no	—
213	C₂²×C₄	—	67	2	no	—
214	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
215	M₁₆	—	2	1	no	—
216	M₁₆	—	2	1	no	—
217	M₁₆	—	134	1	no	—
218	M₁₆	—	134	1	no	—
219	M₁₆	—	134	1	no	—
220	M₁₆	—	134	1	no	—
221	M₁₆	—	134	2	no	—
222	M₁₆	—	134	2	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	67	2	no	—
225	C₂×C₈	—	134	1	no	—
226	C₂×C₈	—	134	2	no	—
227	C₂×C₈	—	134	2	no	—
228	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
229	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
230	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
231	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
232	C₂×C₈	#11761627	1	1	yes	characteristic
233	C₂×C₈	#11761629	1	1	yes	characteristic
234	C₄⋊C₄	—	2	1	no	—
235	C₄⋊C₄	—	134	1	no	—
236	C₂²⋊C₄	—	2	1	no	—
237	C₂²⋊C₄	—	134	1	no	—
238	C₄²	—	2	1	no	—
239	C₄²	—	134	1	no	—
240	C₁₆	—	2	1	no	—
241	C₁₆	—	2	1	no	—
242	C₁₆	—	2	2	no	—
243	C₁₆	—	67	4	no	—
244	C₁₆	—	134	2	no	—
245	C₂³	—	67	2	no	—
246	C₂³	M₁₆×D₆₇	1	1	yes	characteristic
247	Q₈	—	134	1	no	—
248	Q₈	—	134	1	no	—
249	D₄	—	134	2	no	—
250	D₄	—	134	2	no	—
251	D₄	—	134	1	no	—
252	D₄	—	134	1	no	—
253	C₂×C₄	—	4	1	no	—
254	C₂×C₄	—	268	1	no	—
255	C₂×C₄	—	67	2	no	—
256	C₂×C₄	—	134	2	no	—
257	C₂×C₄	—	134	2	no	—
258	C₂×C₄	C₄²×D₆₇	1	1	yes	characteristic
259	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
260	C₂×C₄	#14524005	1	1	yes	characteristic
261	C₂×C₄	M₁₆×D₆₇	1	1	yes	characteristic
262	C₂×C₄	#14524049	1	1	yes	characteristic
263	C₂×C₄	#14524049	1	1	yes	characteristic
264	C₈	—	4	1	no	—
265	C₈	—	268	1	no	—
266	C₈	—	134	2	no	—
267	C₈	C₂²⋊C₄×D₆₇	1	1	yes	characteristic
268	C₈	#14524001	1	1	yes	characteristic
269	C₈	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
270	C₈	#14524005	1	1	yes	characteristic
271	C₂²	—	4	1	no	—
272	C₂²	—	268	2	no	—
273	C₂²	—	134	2	no	—
274	C₂²	#97×D₆₇	1	1	yes	characteristic
275	C₂²	#16863466	1	1	yes	characteristic
276	C₂²	#16863470	1	1	yes	characteristic
277	C₄	—	8	1	no	—
278	C₄	—	536	1	no	—
279	C₄	—	134	2	no	—
280	C₄	#16863243	1	1	yes	characteristic
281	C₄	#16863350	1	1	yes	characteristic
282	C₄	#16863466	1	1	yes	characteristic
283	C₄	#16863470	1	1	yes	characteristic
284	C₂	—	2	1	no	—
285	C₂	—	2	1	no	—
286	C₂	—	2	1	no	—
287	C₂	—	268	2	no	—
288	C₂	#22559984	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#14524001
commutator	C₂×C₄×C₆₇	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#1379	C₂
socle	C₂×C₆₇	#1752
prime radical	C₂×#11761628	C₂×C₄
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#1379	D₆₇
hypocenter	C₆₇	#12617
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #1379

Complements

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

N	H ≅ G/N	type
3. #22561814	C₂	semidirect
4. #22561813	C₂	semidirect
6. #22558297	C₂	semidirect
7. C₆₇×#1379	C₂	semidirect
13. C₂×#14523982	C₄	semidirect
14. C₂×C₁₆×D₆₇	C₄	semidirect
18. #16863177	C₄	semidirect
151. #1379	D₆₇	semidirect
152. C₆₇	#12617	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863812	2
#22561829	2
#22561814	2
#22561813	2
#22560010	2
#22558297	2
C₆₇×#1379	2
#12617	67