Subgroups — #30874149

Order 17152.

914 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863503	C₂	1	2	yes	maximal
2	Q₈×C₄⋊Dic₆₇	C₂	1	2	yes	maximal
3	#22560948	C₂	1	2	yes	maximal
4	#22560479	C₂	1	4	yes	maximal
5	#22560366	C₂	1	1	yes	maximal, characteristic
6	#22560214	C₂	1	2	yes	maximal
7	#22560076	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Q₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
9	C₂×Q₈×Dic₆₇	C₂²	1	4	yes	—
10	#16863503	—	2	8	no	—
11	C₂×#14523975	C₂²	1	2	yes	—
12	C₂×#14523975	C₂²	1	2	yes	—
13	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
14	C₂×#14523967	C₂²	1	4	yes	—
15	C₂×#14523953	C₂²	1	2	yes	—
16	#16863295	—	2	4	no	—
17	#16863295	C₂²	1	4	yes	—
18	#16863266	—	2	2	no	—
19	#16863266	C₂²	1	2	yes	—
20	C₄×C₄⋊Dic₆₇	—	2	2	no	—
21	C₄×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
22	#16863158	C₂²	1	4	yes	—
23	#16863150	C₂²	1	2	yes	—
24	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
26	C₄×Q₁₆×C₆₇	C₄	1	8	yes	complemented
27	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
28	C₂×#103×C₆₇	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
30	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
31	Q₈×Dic₆₇	—	2	8	no	—
32	Q₈×Dic₆₇	—	4	4	no	—
33	C₂×C₄⋊Dic₆₇	—	2	4	no	—
34	C₂×C₄⋊Dic₆₇	—	2	4	no	—
35	C₂×C₄⋊Dic₆₇	—	2	2	no	—
36	C₂×C₄⋊Dic₆₇	—	2	2	no	—
37	C₂×C₄⋊Dic₆₇	D₄	1	2	yes	—
38	C₂×C₄⋊Dic₆₇	C₂³	1	4	yes	—
39	C₂×C₄×Dic₆₇	—	2	4	no	—
40	C₂×C₄×Dic₆₇	C₂³	1	2	yes	—
41	#14523975	—	2	4	no	—
42	#14523975	—	4	2	no	—
43	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
44	#14523967	—	2	8	no	—
45	#14523953	—	2	4	no	—
46	C₆₇⋊#95	—	2	4	no	—
47	C₆₇⋊#95	D₄	1	2	yes	—
48	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
49	C₂×Q₁₆×C₆₇	C₂×C₄	1	4	yes	—
50	C₂×Q₁₆×C₆₇	D₄	1	4	yes	—
51	C₂×Q₁₆×C₆₇	Q₈	1	4	yes	—
52	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
54	C₄×Q₈×C₆₇	—	2	4	no	—
55	C₄×Q₈×C₆₇	C₂×C₄	1	8	yes	—
56	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
57	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
58	C₂×C₄²×C₆₇	—	2	2	no	—
59	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
60	#107×C₆₇	C₂×C₄	1	4	yes	—
61	#103×C₆₇	C₂×C₄	1	8	yes	—
62	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	—
63	C₂²×Dic₆₇	—	2	4	no	—
64	C₂²×Dic₆₇	C₂×D₄	1	4	yes	—
65	C₂×C₆₇⋊C₈	—	2	2	no	—
66	C₂×C₆₇⋊C₈	—	2	2	no	—
67	C₂×C₆₇⋊C₈	—	2	2	no	—
68	C₄⋊Dic₆₇	—	2	4	no	—
69	C₄⋊Dic₆₇	—	2	4	no	—
70	C₄⋊Dic₆₇	—	2	4	no	—
71	C₄⋊Dic₆₇	—	4	8	no	—
72	C₄⋊Dic₆₇	—	4	4	no	—
73	C₄⋊Dic₆₇	—	8	4	no	—
74	C₄⋊Dic₆₇	—	8	2	no	—
75	C₄×Dic₆₇	—	2	2	no	—
76	C₄×Dic₆₇	—	2	2	no	—
77	C₄×Dic₆₇	—	4	8	no	—
78	C₂×Q₈×C₆₇	—	2	2	no	—
79	C₂×Q₈×C₆₇	—	2	2	no	—
80	C₂×Q₈×C₆₇	—	2	2	no	—
81	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
82	C₂×Q₈×C₆₇	C₂×D₄	1	4	yes	—
83	C₂×Q₈×C₆₇	C₂×Q₈	1	4	yes	—
84	C₂²×C₄×C₆₇	—	2	2	no	—
85	C₂²×C₄×C₆₇	—	2	2	no	—
86	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
89	Q₁₆×C₆₇	C₄⋊C₄	1	16	yes	complemented
90	C₂×C₈×C₆₇	—	2	1	no	—
91	C₂×C₈×C₆₇	—	2	1	no	—
92	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
93	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
94	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
95	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
96	C₄⋊C₄×C₆₇	—	2	8	no	—
97	C₄⋊C₄×C₆₇	—	2	2	no	—
98	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
99	C₄²×C₆₇	—	2	8	no	—
100	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
101	C₄²×C₆₇	#41	1	2	yes	—
102	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
103	Q₈×C₆₇	—	2	4	no	—
104	Q₈×C₆₇	—	2	4	no	—
105	Q₈×C₆₇	—	2	4	no	—
106	Q₈×C₆₇	C₂×C₄⋊C₄	1	8	yes	—
107	C₂×C₄×C₆₇	—	2	4	no	—
108	C₂×C₄×C₆₇	—	2	4	no	—
109	C₂×C₄×C₆₇	—	2	4	no	—
110	C₂×C₄×C₆₇	—	2	2	no	—
111	C₂×C₄×C₆₇	—	2	4	no	—
112	C₂×C₄×C₆₇	—	4	2	no	—
113	C₂×C₄×C₆₇	—	4	2	no	—
114	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
115	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₆₇	#137	1	4	yes	—
117	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
118	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
119	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
120	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
121	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
122	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
123	C₂×Dic₆₇	—	2	4	no	—
124	C₂×Dic₆₇	—	2	4	no	—
125	C₂×Dic₆₇	—	2	4	no	—
126	C₂×Dic₆₇	—	4	4	no	—
127	C₂×Dic₆₇	—	4	4	no	—
128	C₂×Dic₆₇	—	4	4	no	—
129	C₈×C₆₇	—	2	2	no	—
130	C₈×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
131	C₆₇⋊C₈	—	4	2	no	—
132	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
133	C₂²×C₆₇	D₄²	1	1	yes	characteristic
134	C₂²×C₆₇	D₄×Q₈	1	1	yes	characteristic
135	C₂²×C₆₇	C₂×#137	1	2	yes	—
136	C₂²×C₆₇	#577	1	2	yes	—
137	C₄×C₆₇	—	2	8	no	—
138	C₄×C₆₇	—	2	2	no	—
139	C₄×C₆₇	—	4	4	no	—
140	C₄×C₆₇	#441	1	4	yes	complemented
141	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
142	C₄×C₆₇	#520	1	1	yes	characteristic
143	C₄×C₆₇	#547	1	1	yes	characteristic
144	C₄×C₆₇	#554	1	1	yes	characteristic
145	Dic₆₇	—	4	4	no	—
146	Dic₆₇	—	8	4	no	—
147	#22674	—	67	1	no	maximal, Sylow
148	C₂×C₆₇	D₄×C₄⋊C₄	1	1	yes	characteristic
149	C₂×C₆₇	C₂×#441	1	1	yes	characteristic
150	C₂×C₆₇	#2952	1	1	yes	characteristic
151	C₂×C₆₇	#3278	1	2	yes	—
152	C₂×C₆₇	#3387	1	2	yes	—
153	C₂×#441	—	67	2	no	—
154	C₂×C₄×Q₁₆	D₆₇	1	1	yes	characteristic
155	Q₈×C₄⋊C₄	—	67	2	no	—
156	#1943	—	67	2	no	—
157	#1865	—	67	4	no	—
158	#1844	—	67	1	no	—
159	#1803	—	67	2	no	—
160	#1770	—	67	1	no	—
161	C₆₇	#22674	1	1	yes	characteristic, complemented, Sylow
162	C₂²×Q₁₆	C₂×D₆₇	1	1	yes	characteristic
163	C₂×C₄×Q₈	—	67	4	no	—
164	C₂×C₄×Q₈	C₂×D₆₇	1	2	yes	—
165	#441	—	134	8	no	—
166	C₄×Q₁₆	Dic₆₇	1	8	yes	complemented
167	C₂×#107	C₂×D₆₇	1	1	yes	characteristic
168	C₂×#106	—	67	2	no	—
169	C₂×#106	—	67	2	no	—
170	C₂×C₄⋊C₈	—	67	1	no	—
171	C₂×#103	—	67	4	no	—
172	C₂×#103	C₂×D₆₇	1	2	yes	—
173	C₂×#97	—	67	2	no	—
174	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
175	#389	—	67	4	no	—
176	#389	—	134	4	no	—
177	#384	—	67	2	no	—
178	#384	—	134	2	no	—
179	C₄×C₄⋊C₄	—	67	1	no	—
180	C₄×C₄⋊C₄	—	134	2	no	—
181	#340	—	67	4	no	—
182	#336	—	67	2	no	—
183	C₂²×Q₈	C₂²×D₆₇	1	2	yes	—
184	C₂×Q₁₆	Dic₁₃₄	1	4	yes	—
185	C₂×Q₁₆	D₂₆₈	1	4	yes	—
186	C₂×Q₁₆	C₂×Dic₆₇	1	4	yes	—
187	C₂²×C₈	—	67	2	no	—
188	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
189	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
190	C₄×Q₈	—	2	4	no	—
191	C₄×Q₈	—	268	4	no	—
192	C₄×Q₈	—	134	8	no	—
193	C₄×Q₈	C₂×Dic₆₇	1	8	yes	—
194	C₂×C₄⋊C₄	—	2	2	no	—
195	C₂×C₄⋊C₄	—	67	2	no	—
196	C₂×C₄⋊C₄	—	67	4	no	—
197	C₂×C₄⋊C₄	—	134	4	no	—
198	C₂×C₄⋊C₄	—	134	2	no	—
199	C₂×C₄⋊C₄	—	134	4	no	—
200	C₂×C₄⋊C₄	—	134	2	no	—
201	C₂×C₄⋊C₄	C₂²×D₆₇	1	2	yes	—
202	C₂×C₄²	—	2	2	no	—
203	C₂×C₄²	—	67	2	no	—
204	C₂×C₄²	—	134	4	no	—
205	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
206	#107	C₂×Dic₆₇	1	4	yes	—
207	#106	—	268	2	no	—
208	#106	—	134	4	no	—
209	C₄⋊C₈	—	134	2	no	—
210	#103	—	134	8	no	—
211	#103	C₂×Dic₆₇	1	8	yes	—
212	#97	—	134	4	no	—
213	C₄×C₈	C₂×Dic₆₇	1	4	yes	—
214	#95	—	67	2	no	—
215	#95	—	134	4	no	—
216	C₂×Q₈	—	2	2	no	—
217	C₂×Q₈	—	2	2	no	—
218	C₂×Q₈	—	2	2	no	—
219	C₂×Q₈	C₂²×Dic₆₇	1	4	yes	—
220	C₂×Q₈	C₂×D₂₆₈	1	4	yes	—
221	C₂×Q₈	C₂×Dic₁₃₄	1	4	yes	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	67	4	no	—
225	C₂²×C₄	—	134	4	no	—
226	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
227	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
228	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
229	Q₁₆	C₄⋊Dic₆₇	1	16	yes	complemented
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	—	2	1	no	—
232	C₂×C₈	—	134	2	no	—
233	C₂×C₈	—	134	2	no	—
234	C₂×C₈	—	134	2	no	—
235	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
236	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
237	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
238	C₂×C₈	C₂×Dic₁₃₄	1	2	yes	—
239	C₄⋊C₄	—	2	8	no	—
240	C₄⋊C₄	—	2	2	no	—
241	C₄⋊C₄	—	268	8	no	—
242	C₄⋊C₄	—	268	4	no	—
243	C₄⋊C₄	—	536	2	no	—
244	C₄⋊C₄	—	536	4	no	—
245	C₄⋊C₄	—	134	4	no	—
246	C₄⋊C₄	—	134	4	no	—
247	C₄⋊C₄	—	134	4	no	—
248	C₄⋊C₄	C₂²×Dic₆₇	1	4	yes	—
249	C₄²	—	2	8	no	—
250	C₄²	—	268	8	no	—
251	C₄²	—	134	2	no	—
252	C₄²	—	134	2	no	—
253	C₄²	C₂²×Dic₆₇	1	2	yes	—
254	C₄²	#11761627	1	2	yes	—
255	C₂³	C₂×D₄×D₆₇	1	1	yes	characteristic
256	Q₈	—	2	4	no	—
257	Q₈	—	2	4	no	—
258	Q₈	—	2	4	no	—
259	Q₈	C₂×C₄⋊Dic₆₇	1	8	yes	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	268	4	no	—
268	C₂×C₄	—	268	4	no	—
269	C₂×C₄	—	268	4	no	—
270	C₂×C₄	—	134	4	no	—
271	C₂×C₄	—	134	4	no	—
272	C₂×C₄	—	134	4	no	—
273	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
274	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
275	C₂×C₄	#14524075	1	4	yes	—
276	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
277	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
278	C₂×C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
279	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
280	C₂×C₄	#14524089	1	1	yes	characteristic
281	C₂×C₄	#14524090	1	1	yes	characteristic
282	C₈	—	2	2	no	—
283	C₈	—	268	2	no	—
284	C₈	C₂×C₄⋊Dic₆₇	1	4	yes	—
285	C₂²	C₂×D₄×Dic₆₇	1	1	yes	characteristic
286	C₂²	D₄×D₂₆₈	1	1	yes	characteristic
287	C₂²	D₄×Dic₁₃₄	1	1	yes	characteristic
288	C₂²	C₂×#14524075	1	2	yes	—
289	C₂²	#16864203	1	2	yes	—
290	C₄	—	2	8	no	—
291	C₄	—	2	2	no	—
292	C₄	—	4	4	no	—
293	C₄	—	268	4	no	—
294	C₄	—	536	4	no	—
295	C₄	#16863503	1	4	yes	complemented
296	C₄	C₂²×C₄⋊Dic₆₇	1	1	yes	characteristic
297	C₄	#16863879	1	1	yes	characteristic
298	C₄	#16864050	1	1	yes	characteristic
299	C₄	#16864103	1	1	yes	characteristic
300	C₂	D₄×C₄⋊Dic₆₇	1	1	yes	characteristic
301	C₂	C₂×#16863503	1	1	yes	characteristic
302	C₂	#22569093	1	1	yes	characteristic
303	C₂	#22571876	1	2	yes	—
304	C₂	#22573139	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂×C₄×Q₁₆×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂²×D₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂×C₄×Q₁₆	D₆₇
hypocenter	C₆₇	#22674
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
26. C₄×Q₁₆×C₆₇	C₄	semidirect
89. Q₁₆×C₆₇	C₄⋊C₄	semidirect
140. C₄×C₆₇	#441	semidirect
161. C₆₇	#22674	semidirect
166. C₄×Q₁₆	Dic₆₇	semidirect
229. Q₁₆	C₄⋊Dic₆₇	semidirect
295. C₄	#16863503	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863503	2
Q₈×C₄⋊Dic₆₇	2
#22560948	2
#22560479	2
#22560366	2
#22560214	2
#22560076	2
C₂×C₄×Q₁₆×C₆₇	2
#22674	67