Subgroups — #30746524

Order 17152.

412 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22568854	C₂	1	1	yes	maximal, characteristic
2	#22558592	C₂	1	1	yes	maximal, characteristic
3	#22558589	C₂	1	1	yes	maximal, characteristic
4	#22558502	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16863118	C₂	1	1	yes	maximal, characteristic, complemented
6	#22558047	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1278	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#11761628	C₂²	1	1	yes	characteristic
9	#16863458	—	2	2	no	—
10	#16863449	C₂²	1	1	yes	characteristic
11	#16863447	—	2	1	no	—
12	#16863447	—	2	1	no	—
13	#16863424	C₄	1	2	yes	—
14	#16863364	C₄	1	2	yes	—
15	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
17	#16863128	C₄	1	2	yes	—
18	#16863126	C₄	1	2	yes	—
19	#16863125	C₄	1	2	yes	—
20	#16863123	C₄	1	2	yes	—
21	#16863118	—	2	2	no	—
22	#16863118	—	2	1	no	—
23	#16863117	—	2	1	no	—
24	#431×C₆₇	C₂²	1	1	yes	characteristic
25	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×#11761628	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₆₇⋊C₈	—	2	1	no	—
29	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
30	C₄×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
31	C₄×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
32	C₄×D₂₆₈	C₂×C₄	1	1	yes	characteristic
33	C₄×D₂₆₈	C₂×C₄	1	1	yes	characteristic
34	C₄×C₂²⋊D₆₇	—	2	1	no	—
35	C₄×C₂²⋊D₆₇	—	2	1	no	—
36	#14524002	C₂×C₄	1	1	yes	characteristic
37	#14524000	—	2	1	no	—
38	C₄²×D₆₇	—	2	1	no	—
39	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
40	Dic₆₇⋊C₈	—	2	2	no	—
41	Dic₆₇⋊C₈	—	2	2	no	—
42	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
43	(C₂×D₆₇)⋊C₈	—	2	2	no	—
44	(C₂×D₆₇)⋊C₈	—	2	2	no	—
45	C₄×C₆₇⋊C₈	—	2	1	no	—
46	C₄×C₆₇⋊C₈	—	2	1	no	—
47	C₄×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
48	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
49	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
50	C₄×C₆₇⋊C₈	D₄	1	2	yes	—
51	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
52	C₂²×C₈×C₆₇	—	2	1	no	—
53	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
54	C₄⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
55	C₂²⋊C₈×C₆₇	—	2	1	no	—
56	#97×C₆₇	—	2	1	no	—
57	C₄×C₈×C₆₇	—	2	1	no	—
58	C₄×C₈×C₆₇	—	2	1	no	—
59	C₄×C₈×C₆₇	—	2	1	no	—
60	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
61	#11761628	—	2	2	no	—
62	#11761628	—	2	2	no	—
63	C₂×Dic₁₃₄	C₄²	1	1	yes	characteristic
64	C₂×D₂₆₈	C₄²	1	1	yes	characteristic
65	C₂×C₂²⋊D₆₇	—	2	1	no	—
66	C₂×C₄×D₆₇	—	2	1	no	—
67	C₂×C₄×D₆₇	—	2	1	no	—
68	C₂×C₄×D₆₇	—	2	1	no	—
69	C₂×C₆₇⋊C₈	—	2	2	no	—
70	C₂×C₆₇⋊C₈	—	2	2	no	—
71	C₂×C₆₇⋊C₈	—	2	2	no	—
72	C₂×C₆₇⋊C₈	—	2	2	no	—
73	C₂×C₆₇⋊C₈	—	4	2	no	—
74	C₂×C₆₇⋊C₈	C₂²⋊C₄	1	2	yes	—
75	C₂×C₆₇⋊C₈	C₂²⋊C₄	1	2	yes	—
76	C₄⋊Dic₆₇	C₄²	1	1	yes	characteristic
77	C₄⋊Dic₆₇	C₄²	1	1	yes	characteristic
78	Dic₆₇⋊C₄	—	2	1	no	—
79	Dic₆₇⋊C₄	—	2	1	no	—
80	C₂²⋊Dic₆₇	—	2	1	no	—
81	(C₂×D₆₇)⋊C₄	—	2	1	no	—
82	(C₂×D₆₇)⋊C₄	—	2	1	no	—
83	C₄×Dic₆₇	—	2	1	no	—
84	C₄×Dic₆₇	—	2	1	no	—
85	C₄×Dic₆₇	—	2	1	no	—
86	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
88	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
89	M₁₆×C₆₇	—	2	2	no	—
90	M₁₆×C₆₇	—	2	2	no	—
91	C₂×C₈×C₆₇	—	2	2	no	—
92	C₂×C₈×C₆₇	—	2	2	no	—
93	C₂×C₈×C₆₇	—	2	2	no	—
94	C₂×C₈×C₆₇	—	4	2	no	—
95	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
97	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
99	C₂³×C₆₇	#117	1	1	yes	characteristic
100	C₂²×D₆₇	—	2	1	no	—
101	C₂×C₄×C₆₇	—	2	2	no	—
102	C₂×C₄×C₆₇	—	2	2	no	—
103	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
104	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
108	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₆₇	#131	1	2	yes	—
110	C₂×C₄×C₆₇	#131	1	2	yes	—
111	C₂²⋊D₆₇	—	4	2	no	—
112	C₂×Dic₆₇	—	2	1	no	—
113	C₂×Dic₆₇	—	2	1	no	—
114	C₂×Dic₆₇	—	2	1	no	—
115	D₂₆₈	—	2	1	no	—
116	D₂₆₈	—	2	1	no	—
117	C₄×D₆₇	—	4	2	no	—
118	C₄×D₆₇	—	4	2	no	—
119	C₄×D₆₇	—	4	2	no	—
120	Dic₁₃₄	—	2	1	no	—
121	Dic₁₃₄	—	2	1	no	—
122	C₈×C₆₇	—	4	2	no	—
123	C₈×C₆₇	—	4	2	no	—
124	C₈×C₆₇	—	4	2	no	—
125	C₆₇⋊C₈	—	2	2	no	—
126	C₆₇⋊C₈	—	2	2	no	—
127	C₆₇⋊C₈	—	4	2	no	—
128	C₆₇⋊C₈	—	4	2	no	—
129	C₂²×C₆₇	—	2	1	no	—
130	C₂²×C₆₇	—	2	1	no	—
131	C₂²×C₆₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
132	C₂²×C₆₇	#433	1	2	yes	—
133	C₂×D₆₇	—	4	1	no	—
134	C₂×D₆₇	—	4	1	no	—
135	C₂×D₆₇	—	4	1	no	—
136	C₄×C₆₇	—	2	1	no	—
137	C₄×C₆₇	—	2	1	no	—
138	C₄×C₆₇	#405	1	1	yes	characteristic
139	C₄×C₆₇	#405	1	1	yes	characteristic
140	C₄×C₆₇	#408	1	2	yes	—
141	C₄×C₆₇	#444	1	2	yes	—
142	C₄×C₆₇	#444	1	2	yes	—
143	Dic₆₇	—	4	1	no	—
144	Dic₆₇	—	4	1	no	—
145	Dic₆₇	—	4	1	no	—
146	#8188	—	67	1	no	maximal, Sylow
147	C₂×C₆₇	—	2	2	no	—
148	C₂×C₆₇	#1750	1	1	yes	characteristic
149	C₂×C₆₇	#1765	1	1	yes	characteristic
150	C₂×C₆₇	#1765	1	1	yes	characteristic
151	D₆₇	—	8	1	no	—
152	#2922	—	67	1	no	—
153	#1476	—	67	1	no	—
154	#1476	—	67	1	no	—
155	#1452	—	67	1	no	—
156	C₂×#322	—	67	1	no	—
157	#1278	—	67	1	no	—
158	#1278	D₆₇	1	1	yes	characteristic, complemented
159	C₆₇	#8188	1	1	yes	characteristic, complemented, Sylow
160	C₄×#41	—	67	1	no	—
161	#433	—	134	2	no	—
162	#431	—	67	1	no	—
163	#431	—	134	1	no	—
164	#431	—	134	1	no	—
165	#431	C₂×D₆₇	1	1	yes	characteristic
166	#424	—	67	2	no	—
167	#408	—	67	2	no	—
168	C₄×M₁₆	C₂×D₆₇	1	1	yes	characteristic
169	C₂×C₄×C₈	—	67	1	no	—
170	C₂×C₄×C₈	—	67	1	no	—
171	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
172	Q₈⋊C₈	—	67	2	no	—
173	Q₈⋊C₈	—	67	2	no	—
174	D₄⋊C₈	—	67	2	no	—
175	D₄⋊C₈	—	67	2	no	—
176	#322	—	134	1	no	—
177	#322	—	134	2	no	—
178	#322	—	134	1	no	—
179	C₂×#41	—	67	1	no	—
180	C₂×M₁₆	C₄×D₆₇	1	2	yes	—
181	C₂²×C₈	—	2	1	no	—
182	C₂²×C₈	—	67	2	no	—
183	C₂²×C₈	—	134	1	no	—
184	C₄×Q₈	—	67	1	no	—
185	C₄×Q₈	—	67	1	no	—
186	C₄×D₄	—	67	1	no	—
187	C₄×D₄	—	67	1	no	—
188	C₄×D₄	—	134	1	no	—
189	C₄×D₄	—	134	1	no	—
190	#117	—	67	1	no	—
191	#117	—	134	1	no	—
192	C₂×C₄²	—	134	1	no	—
193	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
194	C₄⋊C₈	—	67	2	no	—
195	C₄⋊C₈	—	134	2	no	—
196	C₄⋊C₈	—	134	2	no	—
197	C₄⋊C₈	C₄×D₆₇	1	2	yes	—
198	C₂²⋊C₈	—	2	1	no	—
199	C₂²⋊C₈	—	134	2	no	—
200	C₂²⋊C₈	—	134	2	no	—
201	C₂²⋊C₈	—	134	1	no	—
202	#97	—	2	1	no	—
203	C₄×C₈	—	2	1	no	—
204	C₄×C₈	—	2	1	no	—
205	C₄×C₈	—	2	1	no	—
206	C₄×C₈	—	67	2	no	—
207	C₄×C₈	—	67	2	no	—
208	C₄×C₈	—	67	1	no	—
209	C₄×C₈	—	67	1	no	—
210	C₄×C₈	—	134	1	no	—
211	C₄×C₈	—	134	1	no	—
212	C₄×C₈	C₄×D₆₇	1	2	yes	—
213	#41	—	134	2	no	—
214	#41	—	134	2	no	—
215	C₂×Q₈	—	67	1	no	—
216	C₂×D₄	—	67	1	no	—
217	C₂×D₄	—	134	1	no	—
218	C₂²×C₄	—	134	1	no	—
219	C₂²×C₄	—	134	1	no	—
220	C₂²×C₄	—	134	1	no	—
221	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
222	C₂²×C₄	#11761627	1	1	yes	characteristic
223	C₂²×C₄	#11761629	1	1	yes	characteristic
224	M₁₆	—	2	2	no	—
225	M₁₆	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	4	2	no	—
230	C₂×C₈	—	268	2	no	—
231	C₂×C₈	—	67	2	no	—
232	C₂×C₈	—	67	2	no	—
233	C₂×C₈	—	134	2	no	—
234	C₂×C₈	—	134	2	no	—
235	C₂×C₈	—	134	2	no	—
236	C₂×C₈	—	134	2	no	—
237	C₄⋊C₄	—	67	1	no	—
238	C₄⋊C₄	—	67	1	no	—
239	C₄⋊C₄	—	134	1	no	—
240	C₄⋊C₄	—	134	1	no	—
241	C₂²⋊C₄	—	134	1	no	—
242	C₂²⋊C₄	—	134	1	no	—
243	C₂²⋊C₄	—	134	1	no	—
244	C₄²	—	134	1	no	—
245	C₄²	—	134	1	no	—
246	C₄²	—	134	1	no	—
247	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
248	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
249	C₄²	D₄×D₆₇	1	1	yes	characteristic
250	C₄²	D₄×D₆₇	1	1	yes	characteristic
251	C₂³	—	134	1	no	—
252	C₂³	#14524001	1	1	yes	characteristic
253	Q₈	—	134	1	no	—
254	Q₈	—	134	1	no	—
255	D₄	—	268	2	no	—
256	D₄	—	134	1	no	—
257	D₄	—	134	1	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	268	2	no	—
261	C₂×C₄	—	268	2	no	—
262	C₂×C₄	—	268	2	no	—
263	C₂×C₄	—	134	1	no	—
264	C₂×C₄	—	134	1	no	—
265	C₂×C₄	—	134	1	no	—
266	C₂×C₄	C₄²×D₆₇	1	1	yes	characteristic
267	C₂×C₄	C₂²⋊C₄×D₆₇	1	1	yes	characteristic
268	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
269	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
270	C₂×C₄	#14524005	1	1	yes	characteristic
271	C₂×C₄	#14524005	1	1	yes	characteristic
272	C₂×C₄	#14524049	1	2	yes	—
273	C₂×C₄	#14524051	1	2	yes	—
274	C₈	—	4	2	no	—
275	C₈	—	4	2	no	—
276	C₈	—	4	2	no	—
277	C₈	—	268	2	no	—
278	C₈	—	268	2	no	—
279	C₈	—	134	2	no	—
280	C₈	—	134	2	no	—
281	C₂²	—	2	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	268	1	no	—
284	C₂²	—	268	1	no	—
285	C₂²	—	268	1	no	—
286	C₂²	#16863243	1	1	yes	characteristic
287	C₂²	#16863456	1	2	yes	—
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	268	1	no	—
291	C₄	—	268	1	no	—
292	C₄	—	268	1	no	—
293	C₄	#16863350	1	1	yes	characteristic
294	C₄	#16863353	1	1	yes	characteristic
295	C₄	#16863361	1	2	yes	—
296	C₄	#16863515	1	2	yes	—
297	C₄	#16863517	1	2	yes	—
298	C₂	—	2	2	no	—
299	C₂	—	536	1	no	—
300	C₂	#22559974	1	1	yes	characteristic
301	C₂	#22560050	1	1	yes	characteristic
302	C₂	#22560052	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1278	C₂
socle	C₂²×C₆₇	C₄×C₂²⋊C₄
prime radical	C₂×#11761628	C₂×C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1278	D₆₇
hypocenter	C₆₇	#8188
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ #1278

Complements

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

N	H ≅ G/N	type
4. #22558502	C₂	semidirect
5. C₂×#16863118	C₂	semidirect
6. #22558047	C₂	semidirect
7. C₆₇×#1278	C₂	semidirect
158. #1278	D₆₇	semidirect
159. C₆₇	#8188	semidirect

Maximal subgroups

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

subgroup	index
#22568854	2
#22558592	2
#22558589	2
#22558502	2
C₂×#16863118	2
#22558047	2
C₆₇×#1278	2
#8188	67