Subgroups — #30872639

Order 17152.

1086 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×#16864125	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₄⋊Dic₆₇	C₂	1	1	yes	maximal, characteristic
3	#22561131	C₂	1	3	yes	maximal
4	#22561011	C₂	1	3	yes	maximal
5	#22560215	C₂	1	3	yes	maximal
6	C₄×#14523968	C₂	1	3	yes	maximal
7	C₂×C₈×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
8	#16864125	—	2	4	no	—
9	#16864125	C₄	1	8	yes	complemented
10	C₂×#14524043	C₂²	1	3	yes	—
11	C₂×#14524034	—	2	3	no	—
12	C₂×Q₈×Dic₆₇	—	2	1	no	—
13	C₂×Q₈×Dic₆₇	—	2	1	no	—
14	C₂×C₄×Dic₁₃₄	C₂²	1	3	yes	—
15	C₂×#14523968	C₂²	1	3	yes	—
16	C₂×#14523968	C₂²	1	3	yes	—
17	C₂×#14523968	C₂²	1	6	yes	—
18	#16863295	—	2	3	no	—
19	#16863266	C₂²	1	3	yes	—
20	C₄×C₄⋊Dic₆₇	C₂²	1	3	yes	—
21	#16863159	C₂²	1	1	yes	characteristic
22	#16863159	C₂²	1	3	yes	—
23	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₈×Q₈×C₆₇	—	2	4	no	—
25	C₂×C₄⋊C₈×C₆₇	C₂²	1	3	yes	—
26	C₂×C₄×C₈×C₆₇	C₂²	1	3	yes	—
27	C₂²×Dic₁₃₄	C₂³	1	3	yes	—
28	#14524043	—	2	6	no	—
29	#14524043	C₂×C₄	1	12	yes	—
30	#14524034	—	2	24	no	—
31	Q₈×Dic₆₇	—	2	8	no	—
32	Q₈×Dic₆₇	—	4	4	no	—
33	C₄×Dic₁₃₄	—	2	6	no	—
34	C₄×Dic₁₃₄	C₂×C₄	1	12	yes	—
35	C₂×C₄⋊Dic₆₇	—	2	3	no	—
36	C₂×C₄⋊Dic₆₇	—	2	3	no	—
37	C₂×C₄⋊Dic₆₇	—	2	3	no	—
38	C₂×C₄⋊Dic₆₇	—	2	3	no	—
39	C₂×C₄⋊Dic₆₇	C₂³	1	3	yes	—
40	C₂×C₄⋊Dic₆₇	C₂³	1	1	yes	characteristic
41	C₂×C₄⋊Dic₆₇	C₂³	1	1	yes	characteristic
42	C₂×Dic₆₇⋊C₄	—	2	3	no	—
43	C₂×C₄×Dic₆₇	—	2	3	no	—
44	C₂×C₄×Dic₆₇	—	2	3	no	—
45	#14523968	—	2	6	no	—
46	#14523968	—	2	6	no	—
47	#14523968	—	4	6	no	—
48	C₆₇⋊#95	—	2	3	no	—
49	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
50	C₂²×C₈×C₆₇	C₂³	1	3	yes	—
51	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
52	C₄×Q₈×C₆₇	C₂×C₄	1	4	yes	—
53	C₄×Q₈×C₆₇	D₄	1	4	yes	—
54	C₂×C₄⋊C₄×C₆₇	D₄	1	3	yes	—
55	C₂×C₄²×C₆₇	C₂³	1	3	yes	—
56	C₄⋊C₈×C₆₇	—	2	6	no	—
57	C₄×C₈×C₆₇	—	2	6	no	—
58	C₂×Dic₁₃₄	—	2	3	no	—
59	C₂×Dic₁₃₄	—	2	3	no	—
60	C₂×Dic₁₃₄	—	2	3	no	—
61	C₂×Dic₁₃₄	—	2	3	no	—
62	C₂×Dic₁₃₄	—	2	3	no	—
63	C₂×Dic₁₃₄	C₂²×C₄	1	6	yes	—
64	C₂²×Dic₆₇	—	2	1	no	—
65	C₂²×Dic₆₇	—	2	1	no	—
66	C₂²×Dic₆₇	—	2	3	no	—
67	C₂²×Dic₆₇	—	2	3	no	—
68	C₄⋊Dic₆₇	—	2	12	no	—
69	C₄⋊Dic₆₇	—	2	12	no	—
70	C₄⋊Dic₆₇	—	2	1	no	—
71	C₄⋊Dic₆₇	—	2	1	no	—
72	C₄⋊Dic₆₇	—	2	3	no	—
73	C₄⋊Dic₆₇	—	2	3	no	—
74	C₄⋊Dic₆₇	—	4	6	no	—
75	C₄⋊Dic₆₇	—	8	3	no	—
76	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
77	C₄⋊Dic₆₇	#41	1	2	yes	—
78	Dic₆₇⋊C₄	—	2	12	no	—
79	C₄×Dic₆₇	—	2	12	no	—
80	C₄×Dic₆₇	—	4	6	no	—
81	C₂×Q₈×C₆₇	C₂²⋊C₄	1	4	yes	—
82	C₂×Q₈×C₆₇	SD₁₆	1	4	yes	—
83	C₂×Q₈×C₆₇	Q₁₆	1	4	yes	—
84	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
85	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
86	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
87	C₂×C₈×C₆₇	—	2	1	no	—
88	C₂×C₈×C₆₇	—	2	1	no	—
89	C₂×C₈×C₆₇	—	2	3	no	—
90	C₂×C₈×C₆₇	—	2	3	no	—
91	C₂×C₈×C₆₇	—	2	3	no	—
92	C₂×C₈×C₆₇	#41	1	2	yes	—
93	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	12	yes	—
94	C₄²×C₆₇	C₂²×C₄	1	6	yes	—
95	C₄²×C₆₇	C₂×D₄	1	6	yes	—
96	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
97	Q₈×C₆₇	#103	1	16	yes	complemented
98	C₂×C₄×C₆₇	—	2	3	no	—
99	C₂×C₄×C₆₇	—	2	3	no	—
100	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	6	yes	—
101	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	6	yes	—
102	C₂×C₄×C₆₇	C₂×SD₁₆	1	6	yes	—
103	C₂×C₄×C₆₇	C₂×Q₁₆	1	6	yes	—
104	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
105	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
106	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
107	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
108	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
109	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
110	C₂×Dic₆₇	—	2	2	no	—
111	C₂×Dic₆₇	—	2	6	no	—
112	C₂×Dic₆₇	—	2	2	no	—
113	C₂×Dic₆₇	—	4	3	no	—
114	C₂×Dic₆₇	—	4	3	no	—
115	C₂×Dic₆₇	—	4	1	no	—
116	C₂×Dic₆₇	—	4	1	no	—
117	C₂×Dic₆₇	—	4	1	no	—
118	C₂×Dic₆₇	—	4	1	no	—
119	C₂×Dic₆₇	—	4	3	no	—
120	C₂×Dic₆₇	—	4	3	no	—
121	C₂×Dic₆₇	—	4	3	no	—
122	Dic₁₃₄	—	2	6	no	—
123	Dic₁₃₄	—	2	6	no	—
124	Dic₁₃₄	—	4	3	no	—
125	Dic₁₃₄	—	4	3	no	—
126	C₈×C₆₇	—	2	2	no	—
127	C₈×C₆₇	—	4	3	no	—
128	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
129	C₂²×C₆₇	#548	1	1	yes	characteristic
130	C₂²×C₆₇	#550	1	1	yes	characteristic
131	C₂²×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
132	C₂²×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
133	C₂²×C₆₇	#577	1	1	yes	characteristic
134	C₂²×C₆₇	#578	1	1	yes	characteristic
135	C₄×C₆₇	—	2	6	no	—
136	C₄×C₆₇	C₂×#103	1	12	yes	—
137	C₄×C₆₇	C₄×#41	1	1	yes	characteristic
138	C₄×C₆₇	#520	1	1	yes	characteristic
139	C₄×C₆₇	#548	1	1	yes	characteristic
140	C₄×C₆₇	#550	1	1	yes	characteristic
141	Dic₆₇	—	4	2	no	—
142	Dic₆₇	—	4	6	no	—
143	Dic₆₇	—	4	2	no	—
144	Dic₆₇	—	8	3	no	—
145	#22542	—	67	1	no	maximal, Sylow
146	C₂×C₆₇	#2341	1	1	yes	characteristic
147	C₂×C₆₇	C₂²×#103	1	1	yes	characteristic
148	C₂×C₆₇	#2898	1	1	yes	characteristic
149	C₂×C₆₇	#3359	1	1	yes	characteristic
150	C₂×C₆₇	#3363	1	1	yes	characteristic
151	C₂×C₆₇	#3392	1	1	yes	characteristic
152	C₂×C₆₇	#3395	1	1	yes	characteristic
153	C₂×#557	—	67	1	no	—
154	C₂×C₈×Q₈	D₆₇	1	1	yes	characteristic
155	Q₈×C₄⋊C₄	—	67	1	no	—
156	#1981	—	67	3	no	—
157	#1956	—	67	3	no	—
158	#1803	—	67	3	no	—
159	C₄×#103	—	67	3	no	—
160	C₆₇	#22542	1	1	yes	characteristic, complemented, Sylow
161	#557	—	67	8	no	—
162	#557	—	134	4	no	—
163	C₂×C₄⋊Q₈	—	67	3	no	—
164	C₂×#125	—	134	3	no	—
165	C₂×C₄×Q₈	—	67	3	no	—
166	C₂×C₄×Q₈	—	134	1	no	—
167	C₂×C₄×Q₈	—	134	1	no	—
168	C₂×C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic
169	C₈×Q₈	—	2	4	no	—
170	C₂×C₄⋊C₈	C₂×D₆₇	1	3	yes	—
171	C₂×#103	—	67	6	no	—
172	C₂×#103	—	67	3	no	—
173	C₂×#103	—	67	3	no	—
174	C₂×C₄×C₈	C₂×D₆₇	1	3	yes	—
175	#389	—	134	3	no	—
176	#384	—	67	3	no	—
177	C₄×C₄⋊C₄	—	67	3	no	—
178	#340	—	67	3	no	—
179	#340	—	67	1	no	—
180	C₂²×Q₈	—	67	3	no	—
181	C₂²×Q₈	D₂₆₈	1	1	yes	characteristic
182	C₂²×C₈	C₂²×D₆₇	1	3	yes	—
183	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
184	C₄⋊Q₈	—	67	12	no	—
185	C₄⋊Q₈	—	134	6	no	—
186	#125	—	134	24	no	—
187	C₄×Q₈	—	268	4	no	—
188	C₄×Q₈	—	67	12	no	—
189	C₄×Q₈	—	134	8	no	—
190	C₄×Q₈	—	134	6	no	—
191	C₄×Q₈	C₄×D₆₇	1	4	yes	—
192	C₄×Q₈	C₂²⋊D₆₇	1	4	yes	—
193	C₂×C₄⋊C₄	—	67	1	no	—
194	C₂×C₄⋊C₄	—	67	1	no	—
195	C₂×C₄⋊C₄	—	67	3	no	—
196	C₂×C₄⋊C₄	—	134	3	no	—
197	C₂×C₄⋊C₄	—	134	3	no	—
198	C₂×C₄⋊C₄	—	134	3	no	—
199	C₂×C₄⋊C₄	—	134	3	no	—
200	C₂×C₄⋊C₄	—	134	3	no	—
201	C₂×C₄⋊C₄	D₂₆₈	1	3	yes	—
202	C₂×C₄²	—	134	3	no	—
203	C₂×C₄²	—	134	3	no	—
204	C₂×C₄²	C₂²×D₆₇	1	3	yes	—
205	C₄⋊C₈	—	2	6	no	—
206	#103	—	268	6	no	—
207	#103	—	134	6	no	—
208	#103	—	134	6	no	—
209	C₄×C₈	—	2	6	no	—
210	#95	—	134	3	no	—
211	C₂×Q₈	—	67	6	no	—
212	C₂×Q₈	—	134	3	no	—
213	C₂×Q₈	—	134	3	no	—
214	C₂×Q₈	—	134	3	no	—
215	C₂×Q₈	—	134	3	no	—
216	C₂×Q₈	—	134	3	no	—
217	C₂×Q₈	(C₂×D₆₇)⋊C₄	1	4	yes	—
218	C₂×Q₈	#11761617	1	4	yes	—
219	C₂×Q₈	Dic₂₆₈	1	4	yes	—
220	C₂²×C₄	—	134	3	no	—
221	C₂²×C₄	—	134	1	no	—
222	C₂²×C₄	—	134	1	no	—
223	C₂²×C₄	—	134	3	no	—
224	C₂²×C₄	C₂×D₂₆₈	1	3	yes	—
225	C₂²×C₄	C₂×D₂₆₈	1	3	yes	—
226	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	2	1	no	—
229	C₂×C₈	—	2	3	no	—
230	C₂×C₈	—	2	3	no	—
231	C₂×C₈	—	2	3	no	—
232	C₂×C₈	#11761629	1	2	yes	—
233	C₄⋊C₄	—	268	6	no	—
234	C₄⋊C₄	—	536	3	no	—
235	C₄⋊C₄	—	67	2	no	—
236	C₄⋊C₄	—	67	2	no	—
237	C₄⋊C₄	—	134	3	no	—
238	C₄⋊C₄	—	134	3	no	—
239	C₄⋊C₄	—	134	12	no	—
240	C₄⋊C₄	—	134	12	no	—
241	C₄⋊C₄	—	134	12	no	—
242	C₄⋊C₄	—	134	1	no	—
243	C₄⋊C₄	—	134	1	no	—
244	C₄⋊C₄	(C₂×D₆₇)⋊C₄	1	12	yes	—
245	C₄²	—	268	6	no	—
246	C₄²	—	134	12	no	—
247	C₄²	C₂×C₄×D₆₇	1	6	yes	—
248	C₄²	C₂×C₂²⋊D₆₇	1	6	yes	—
249	C₂³	C₂²×D₂₆₈	1	1	yes	characteristic
250	Q₈	—	268	3	no	—
251	Q₈	—	268	3	no	—
252	Q₈	—	134	6	no	—
253	Q₈	—	134	6	no	—
254	Q₈	#14523968	1	16	yes	complemented
255	C₂×C₄	—	2	3	no	—
256	C₂×C₄	—	2	3	no	—
257	C₂×C₄	—	268	3	no	—
258	C₂×C₄	—	268	3	no	—
259	C₂×C₄	—	268	1	no	—
260	C₂×C₄	—	268	1	no	—
261	C₂×C₄	—	268	1	no	—
262	C₂×C₄	—	268	1	no	—
263	C₂×C₄	—	268	3	no	—
264	C₂×C₄	—	268	3	no	—
265	C₂×C₄	—	268	3	no	—
266	C₂×C₄	—	134	2	no	—
267	C₂×C₄	—	134	2	no	—
268	C₂×C₄	—	134	6	no	—
269	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	6	yes	—
270	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	6	yes	—
271	C₂×C₄	C₂×#11761617	1	6	yes	—
272	C₂×C₄	C₂×Dic₂₆₈	1	6	yes	—
273	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
274	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
275	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
276	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
277	C₂×C₄	#14524089	1	1	yes	characteristic
278	C₂×C₄	#14524091	1	1	yes	characteristic
279	C₈	—	2	2	no	—
280	C₈	—	4	3	no	—
281	C₂²	C₂²×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
282	C₂²	#16864058	1	1	yes	characteristic
283	C₂²	#16864068	1	1	yes	characteristic
284	C₂²	C₂²×#11761617	1	1	yes	characteristic
285	C₂²	C₂²×Dic₂₆₈	1	1	yes	characteristic
286	C₂²	#16864208	1	1	yes	characteristic
287	C₂²	#16864213	1	1	yes	characteristic
288	C₄	—	2	6	no	—
289	C₄	—	268	2	no	—
290	C₄	—	268	2	no	—
291	C₄	—	268	6	no	—
292	C₄	—	536	3	no	—
293	C₄	C₂×#14523968	1	12	yes	—
294	C₄	C₄×#11761629	1	1	yes	characteristic
295	C₄	#16863878	1	1	yes	characteristic
296	C₄	#16864057	1	1	yes	characteristic
297	C₄	#16864065	1	1	yes	characteristic
298	C₂	#22563047	1	1	yes	characteristic
299	C₂	C₂²×#14523968	1	1	yes	characteristic
300	C₂	#22568743	1	1	yes	characteristic
301	C₂	#22572852	1	1	yes	characteristic
302	C₂	#22572888	1	1	yes	characteristic
303	C₂	#22573191	1	1	yes	characteristic
304	C₂	#22573215	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂×C₈×Q₈×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂²×D₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂×C₈×Q₈	D₆₇
hypocenter	C₆₇	#22542
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
9. #16864125	C₄	semidirect
97. Q₈×C₆₇	#103	semidirect
160. C₆₇	#22542	semidirect
254. Q₈	#14523968	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16864125	2
Q₈×C₄⋊Dic₆₇	2
#22561131	2
#22561011	2
#22560215	2
C₄×#14523968	2
C₂×C₈×Q₈×C₆₇	2
#22542	67