Subgroups — #30874654

Order 17152.

838 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Dic₁₃₄	C₂	1	1	yes	maximal, characteristic
2	#22563121	C₂	1	2	yes	maximal
3	#22560970	C₂	1	2	yes	maximal
4	#22560969	C₂	1	4	yes	maximal
5	#22560364	C₂	1	2	yes	maximal
6	#106×Dic₆₇	C₂	1	2	yes	maximal
7	C₄×#14523975	C₂	1	1	yes	maximal, characteristic
8	C₆₇×#1844	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Dic₁₃₄	C₂²	1	1	yes	characteristic
10	C₂×#14524008	—	2	2	no	—
11	C₈×Dic₁₃₄	C₄	1	8	yes	complemented
12	C₂×#14523975	C₂²	1	1	yes	characteristic
13	C₂×#14523975	C₂²	1	1	yes	characteristic
14	C₂×#14523974	C₂²	1	4	yes	—
15	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
16	C₂×Dic₆₇⋊C₈	C₂²	1	2	yes	—
17	C₂×C₈×Dic₆₇	C₂²	1	2	yes	—
18	#16863292	—	2	2	no	—
19	#16863291	—	2	4	no	—
20	#16863265	C₂²	1	4	yes	—
21	C₄⋊C₄×Dic₆₇	C₂²	1	4	yes	—
22	C₄×C₄⋊Dic₆₇	C₂²	1	2	yes	—
23	#16863159	C₂²	1	2	yes	—
24	#16863158	C₂²	1	4	yes	—
25	C₂×#106×C₆₇	C₂²	1	2	yes	—
26	C₂×#106×C₆₇	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
28	#384×C₆₇	C₂²	1	2	yes	—
29	C₂²×Dic₁₃₄	D₄	1	1	yes	characteristic
30	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
31	C₄×Dic₁₃₄	C₂×C₄	1	4	yes	—
32	C₄×Dic₁₃₄	Q₈	1	4	yes	—
33	#14524008	—	4	8	no	—
34	C₂×C₄⋊Dic₆₇	D₄	1	1	yes	characteristic
35	C₂×C₄⋊Dic₆₇	C₂³	1	2	yes	—
36	C₂×C₄⋊Dic₆₇	C₂³	1	2	yes	—
37	C₂×Dic₆₇⋊C₄	—	2	4	no	—
38	C₂×Dic₆₇⋊C₄	—	2	4	no	—
39	C₂×Dic₆₇⋊C₄	D₄	1	2	yes	—
40	C₂×C₄×Dic₆₇	—	2	4	no	—
41	C₂×C₄×Dic₆₇	—	2	2	no	—
42	C₂×C₄×Dic₆₇	C₂³	1	2	yes	—
43	#14523975	—	2	2	no	—
44	#14523975	Q₈	1	4	yes	complemented
45	#14523974	—	4	4	no	—
46	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	4	yes	—
47	Dic₆₇⋊C₈	C₂×C₄	1	8	yes	—
48	C₈×Dic₆₇	C₂×C₄	1	8	yes	—
49	C₆₇⋊#95	—	2	4	no	—
50	C₆₇⋊#95	—	2	2	no	—
51	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
54	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
55	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
56	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
57	#106×C₆₇	—	2	4	no	—
58	#106×C₆₇	—	4	2	no	—
59	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	—
60	C₂×Dic₁₃₄	C₄⋊C₄	1	4	yes	—
61	C₂×Dic₁₃₄	SD₁₆	1	8	yes	—
62	C₂²×Dic₆₇	—	2	2	no	—
63	C₂²×Dic₆₇	—	2	2	no	—
64	C₂²×Dic₆₇	C₂×D₄	1	2	yes	—
65	C₂²×Dic₆₇	C₂×D₄	1	2	yes	—
66	C₂×C₆₇⋊C₈	—	2	2	no	—
67	C₂×C₆₇⋊C₈	—	2	2	no	—
68	C₂×C₆₇⋊C₈	C₂²×C₄	1	4	yes	—
69	C₄⋊Dic₆₇	—	2	2	no	—
70	C₄⋊Dic₆₇	—	2	2	no	—
71	C₄⋊Dic₆₇	—	2	2	no	—
72	C₄⋊Dic₆₇	C₄⋊C₄	1	4	yes	—
73	C₄⋊Dic₆₇	C₂×Q₈	1	4	yes	—
74	Dic₆₇⋊C₄	—	4	8	no	—
75	Dic₆₇⋊C₄	—	8	4	no	—
76	Dic₆₇⋊C₄	C₄⋊C₄	1	8	yes	—
77	C₄×Dic₆₇	—	4	4	no	—
78	C₄×Dic₆₇	—	4	8	no	—
79	C₄×Dic₆₇	C₂²×C₄	1	4	yes	—
80	C₄×Dic₆₇	C₂×Q₈	1	4	yes	—
81	C₂²×C₄×C₆₇	—	2	4	no	—
82	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
85	C₂×C₈×C₆₇	—	2	1	no	—
86	C₂×C₈×C₆₇	—	2	1	no	—
87	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
88	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
89	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
90	C₂×C₈×C₆₇	#41	1	2	yes	—
91	C₄⋊C₄×C₆₇	—	2	4	no	—
92	C₄⋊C₄×C₆₇	—	2	4	no	—
93	C₄⋊C₄×C₆₇	—	4	4	no	—
94	C₄⋊C₄×C₆₇	—	8	2	no	—
95	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
96	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
97	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
98	C₂×C₄×C₆₇	—	2	2	no	—
99	C₂×C₄×C₆₇	—	4	4	no	—
100	C₂×C₄×C₆₇	—	4	4	no	—
101	C₂×C₄×C₆₇	—	4	4	no	—
102	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
103	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
104	C₂×C₄×C₆₇	C₂×SD₁₆	1	4	yes	—
105	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
106	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
107	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
108	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
109	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
110	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
111	C₂×Dic₆₇	—	2	4	no	—
112	C₂×Dic₆₇	—	2	4	no	—
113	C₂×Dic₆₇	—	4	2	no	—
114	C₂×Dic₆₇	—	4	2	no	—
115	C₂×Dic₆₇	—	4	2	no	—
116	C₂×Dic₆₇	—	4	2	no	—
117	C₂×Dic₆₇	—	4	2	no	—
118	C₂×Dic₆₇	C₂×C₄⋊C₄	1	4	yes	—
119	C₂×Dic₆₇	C₂×C₄⋊C₄	1	4	yes	—
120	C₂×Dic₆₇	C₂×SD₁₆	1	8	yes	—
121	Dic₁₃₄	#106	1	16	yes	complemented
122	C₈×C₆₇	—	2	2	no	—
123	C₈×C₆₇	C₄×Q₈	1	4	yes	complemented
124	C₆₇⋊C₈	—	2	4	no	—
125	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
126	C₂²×C₆₇	D₄×Q₈	1	1	yes	characteristic
127	C₂²×C₆₇	#550	1	1	yes	characteristic
128	C₂²×C₆₇	C₂²×SD₁₆	1	2	yes	—
129	C₂²×C₆₇	#577	1	2	yes	—
130	C₄×C₆₇	—	2	2	no	—
131	C₄×C₆₇	—	8	4	no	—
132	C₄×C₆₇	C₂×#106	1	4	yes	—
133	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
134	C₄×C₆₇	#520	1	1	yes	characteristic
135	C₄×C₆₇	#557	1	1	yes	characteristic
136	C₄×C₆₇	Q₈²	1	1	yes	characteristic
137	Dic₆₇	—	2	4	no	—
138	Dic₆₇	—	4	4	no	—
139	Dic₆₇	—	8	2	no	—
140	Dic₆₇	C₂×#106	1	8	yes	—
141	Q₈×#106	—	67	1	no	maximal, Sylow
142	C₂×C₆₇	Q₈×C₄⋊C₄	1	1	yes	characteristic
143	C₂×C₆₇	C₂²×#106	1	1	yes	characteristic
144	C₂×C₆₇	#2912	1	1	yes	characteristic
145	C₂×C₆₇	Q₈×SD₁₆	1	2	yes	—
146	C₂×C₆₇	#3392	1	2	yes	—
147	C₂×C₈×Q₈	—	67	1	no	—
148	Q₈×C₄⋊C₄	—	67	2	no	—
149	#1948	—	67	4	no	—
150	#1948	—	67	2	no	—
151	#1844	—	67	2	no	—
152	#1844	D₆₇	1	1	yes	characteristic
153	C₄×#106	—	67	2	no	—
154	C₄×#106	—	67	1	no	—
155	C₆₇	Q₈×#106	1	1	yes	characteristic, complemented, Sylow
156	C₂×C₄×Q₈	—	67	1	no	—
157	C₂×C₄×Q₈	—	134	2	no	—
158	C₈×Q₈	—	67	8	no	—
159	C₂×#106	—	67	4	no	—
160	C₂×#106	—	67	1	no	—
161	C₂×#106	—	67	1	no	—
162	C₂×#106	C₂×D₆₇	1	2	yes	—
163	C₂×#106	C₂×D₆₇	1	2	yes	—
164	C₂×C₄⋊C₈	—	67	2	no	—
165	C₂×C₄⋊C₈	—	67	1	no	—
166	C₂×C₄×C₈	—	67	2	no	—
167	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
168	#389	—	134	4	no	—
169	#389	—	134	2	no	—
170	#384	—	67	4	no	—
171	#384	C₂×D₆₇	1	2	yes	—
172	C₄×C₄⋊C₄	—	67	4	no	—
173	C₄×C₄⋊C₄	—	67	2	no	—
174	#340	—	67	2	no	—
175	#340	—	67	4	no	—
176	C₂²×Q₈	—	67	1	no	—
177	C₂²×C₈	—	67	2	no	—
178	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
179	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
180	C₄×Q₈	—	268	8	no	—
181	C₄×Q₈	—	67	4	no	—
182	C₄×Q₈	—	67	4	no	—
183	C₂×C₄⋊C₄	—	2	2	no	—
184	C₂×C₄⋊C₄	—	2	2	no	—
185	C₂×C₄⋊C₄	—	67	2	no	—
186	C₂×C₄⋊C₄	—	67	1	no	—
187	C₂×C₄⋊C₄	—	67	2	no	—
188	C₂×C₄⋊C₄	—	67	2	no	—
189	C₂×C₄⋊C₄	—	134	4	no	—
190	C₂×C₄⋊C₄	—	134	4	no	—
191	C₂×C₄⋊C₄	C₂²×D₆₇	1	4	yes	—
192	C₂×C₄²	—	67	2	no	—
193	C₂×C₄²	—	134	4	no	—
194	C₂×C₄²	—	134	2	no	—
195	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
196	#106	—	2	4	no	—
197	#106	—	4	2	no	—
198	#106	—	268	4	no	—
199	#106	—	67	4	no	—
200	#106	—	134	2	no	—
201	C₄⋊C₈	—	67	8	no	—
202	C₄⋊C₈	—	67	4	no	—
203	C₄×C₈	—	67	8	no	—
204	C₄×C₈	C₄×D₆₇	1	4	yes	—
205	#95	—	134	4	no	—
206	#95	—	134	2	no	—
207	C₂×Q₈	—	67	4	no	—
208	C₂×Q₈	—	67	8	no	—
209	C₂²×C₄	—	2	4	no	—
210	C₂²×C₄	—	67	2	no	—
211	C₂²×C₄	—	67	2	no	—
212	C₂²×C₄	—	134	2	no	—
213	C₂²×C₄	—	134	2	no	—
214	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
215	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
216	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
217	C₂×C₈	—	2	1	no	—
218	C₂×C₈	—	2	1	no	—
219	C₂×C₈	—	67	4	no	—
220	C₂×C₈	—	134	2	no	—
221	C₂×C₈	—	134	2	no	—
222	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
223	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
224	C₂×C₈	Q₈×D₆₇	1	2	yes	—
225	C₂×C₈	#11761627	1	2	yes	—
226	C₄⋊C₄	—	2	4	no	—
227	C₄⋊C₄	—	2	4	no	—
228	C₄⋊C₄	—	4	4	no	—
229	C₄⋊C₄	—	8	2	no	—
230	C₄⋊C₄	—	268	8	no	—
231	C₄⋊C₄	—	536	4	no	—
232	C₄⋊C₄	—	67	8	no	—
233	C₄⋊C₄	—	67	4	no	—
234	C₄⋊C₄	—	67	4	no	—
235	C₄⋊C₄	—	134	2	no	—
236	C₄⋊C₄	—	134	2	no	—
237	C₄⋊C₄	—	134	2	no	—
238	C₄²	—	268	8	no	—
239	C₄²	—	268	4	no	—
240	C₄²	—	67	4	no	—
241	C₄²	—	67	4	no	—
242	C₄²	C₂×C₄×D₆₇	1	2	yes	—
243	C₄²	Q₈×D₆₇	1	2	yes	—
244	C₂³	C₂×D₄×D₆₇	1	1	yes	characteristic
245	Q₈	—	67	16	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	4	4	no	—
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	—	268	2	no	—
251	C₂×C₄	—	268	2	no	—
252	C₂×C₄	—	268	2	no	—
253	C₂×C₄	—	268	2	no	—
254	C₂×C₄	—	268	2	no	—
255	C₂×C₄	—	67	4	no	—
256	C₂×C₄	—	67	8	no	—
257	C₂×C₄	—	67	4	no	—
258	C₂×C₄	—	134	4	no	—
259	C₂×C₄	—	134	4	no	—
260	C₂×C₄	C₄⋊C₄×D₆₇	1	2	yes	—
261	C₂×C₄	C₄⋊C₄×D₆₇	1	2	yes	—
262	C₂×C₄	SD₁₆×D₆₇	1	4	yes	—
263	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
264	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
265	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
266	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
267	C₂×C₄	#14524088	1	1	yes	characteristic
268	C₂×C₄	#14524091	1	1	yes	characteristic
269	C₈	—	2	2	no	—
270	C₈	—	134	4	no	—
271	C₈	#14524008	1	4	yes	complemented
272	C₂²	C₂×C₄⋊C₄×D₆₇	1	1	yes	characteristic
273	C₂²	#16864061	1	1	yes	characteristic
274	C₂²	#16864066	1	1	yes	characteristic
275	C₂²	C₂×SD₁₆×D₆₇	1	2	yes	—
276	C₂²	#16864202	1	2	yes	—
277	C₄	—	2	2	no	—
278	C₄	—	8	4	no	—
279	C₄	—	268	4	no	—
280	C₄	—	536	2	no	—
281	C₄	—	67	8	no	—
282	C₄	—	134	4	no	—
283	C₄	#106×D₆₇	1	4	yes	—
284	C₄	C₂×#14524008	1	1	yes	characteristic
285	C₄	#16863877	1	1	yes	characteristic
286	C₄	#16864121	1	1	yes	characteristic
287	C₄	#16864126	1	1	yes	characteristic
288	C₂	#22563121	1	1	yes	characteristic
289	C₂	C₂×#106×D₆₇	1	1	yes	characteristic
290	C₂	#22568800	1	1	yes	characteristic
291	C₂	#22573066	1	2	yes	—
292	C₂	#22573187	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#1844	C₂
socle, prime radical	C₂³×C₆₇	C₂²×D₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#1844	D₆₇
hypocenter	C₆₇	Q₈×#106
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

Complements

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

N	H ≅ G/N	type
11. C₈×Dic₁₃₄	C₄	semidirect
44. #14523975	Q₈	semidirect
121. Dic₁₃₄	#106	semidirect
123. C₈×C₆₇	C₄×Q₈	semidirect
155. C₆₇	Q₈×#106	semidirect
271. C₈	#14524008	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Dic₁₃₄	2
#22563121	2
#22560970	2
#22560969	2
#22560364	2
#106×Dic₆₇	2
C₄×#14523975	2
C₆₇×#1844	2
Q₈×#106	67