Subgroups — #30874767

Order 17152.

822 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₄⋊Dic₆₇	C₂	1	1	yes	maximal, characteristic
3	#22560972	C₂	1	3	yes	maximal
4	#22560970	C₂	1	3	yes	maximal
5	#22560336	C₂	1	3	yes	maximal
6	C₄×#14523974	C₂	1	3	yes	maximal
7	Q₈×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂×Q₈×Dic₆₇	—	2	1	no	—
9	Q₈×C₆₇⋊C₈	C₄	1	8	yes	complemented
10	C₂×#14523974	C₂²	1	3	yes	—
11	C₂×#14523974	C₂²	1	3	yes	—
12	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	3	yes	—
13	C₂×C₄×C₆₇⋊C₈	C₂²	1	3	yes	—
14	#16863295	—	2	3	no	—
15	#16863266	C₂²	1	3	yes	—
16	C₄×C₄⋊Dic₆₇	C₂²	1	3	yes	—
17	#16863158	C₂²	1	3	yes	—
18	#16863158	C₂²	1	6	yes	—
19	#16863158	C₂²	1	1	yes	characteristic
20	C₂×C₄×Q₈×C₆₇	—	2	1	no	—
21	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
22	#389×C₆₇	—	2	3	no	—
23	#384×C₆₇	C₂²	1	3	yes	—
24	C₄×C₄⋊C₄×C₆₇	C₂²	1	3	yes	—
25	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
26	C₂²×C₆₇⋊C₈	C₂³	1	3	yes	—
27	Q₈×Dic₆₇	—	4	4	no	—
28	C₂×C₄⋊Dic₆₇	—	2	3	no	—
29	C₂×C₄⋊Dic₆₇	—	2	3	no	—
30	C₂×C₄⋊Dic₆₇	C₂³	1	3	yes	—
31	C₂×C₄⋊Dic₆₇	C₂³	1	1	yes	characteristic
32	C₂×C₄×Dic₆₇	—	2	3	no	—
33	#14523974	—	2	6	no	—
34	#14523974	—	2	6	no	—
35	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	12	yes	—
36	C₄×C₆₇⋊C₈	C₂×C₄	1	12	yes	—
37	C₆₇⋊#95	—	2	3	no	—
38	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
39	C₄×Q₈×C₆₇	—	4	4	no	—
40	C₄×Q₈×C₆₇	C₂×C₄	1	4	yes	—
41	C₄×Q₈×C₆₇	Q₈	1	4	yes	—
42	C₂×C₄⋊C₄×C₆₇	—	2	3	no	—
43	C₂×C₄⋊C₄×C₆₇	—	2	3	no	—
44	C₂×C₄⋊C₄×C₆₇	D₄	1	3	yes	—
45	C₂×C₄⋊C₄×C₆₇	C₂³	1	3	yes	—
46	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₆₇	—	2	3	no	—
48	C₂×C₄²×C₆₇	C₂³	1	3	yes	—
49	#95×C₆₇	—	2	3	no	—
50	C₂²×Dic₆₇	—	2	1	no	—
51	C₂²×Dic₆₇	—	2	3	no	—
52	C₂×C₆₇⋊C₈	—	2	3	no	—
53	C₂×C₆₇⋊C₈	—	2	3	no	—
54	C₂×C₆₇⋊C₈	—	2	1	no	—
55	C₂×C₆₇⋊C₈	—	2	1	no	—
56	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
57	C₂×C₆₇⋊C₈	C₂²×C₄	1	6	yes	—
58	C₄⋊Dic₆₇	—	2	1	no	—
59	C₄⋊Dic₆₇	—	2	3	no	—
60	C₄⋊Dic₆₇	—	2	3	no	—
61	C₄⋊Dic₆₇	—	4	6	no	—
62	C₄⋊Dic₆₇	—	8	3	no	—
63	C₄⋊Dic₆₇	#41	1	2	yes	—
64	C₄×Dic₆₇	—	4	6	no	—
65	C₂×Q₈×C₆₇	C₄⋊C₄	1	4	yes	—
66	C₂×Q₈×C₆₇	SD₁₆	1	4	yes	—
67	C₂×Q₈×C₆₇	SD₁₆	1	4	yes	—
68	C₂²×C₄×C₆₇	—	2	1	no	—
69	C₂²×C₄×C₆₇	—	2	3	no	—
70	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
71	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
72	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
73	C₄⋊C₄×C₆₇	—	2	1	no	—
74	C₄⋊C₄×C₆₇	—	2	3	no	—
75	C₄⋊C₄×C₆₇	—	2	3	no	—
76	C₄⋊C₄×C₆₇	—	4	6	no	—
77	C₄⋊C₄×C₆₇	—	8	3	no	—
78	C₄⋊C₄×C₆₇	C₄⋊C₄	1	12	yes	—
79	C₄⋊C₄×C₆₇	#41	1	2	yes	—
80	C₄²×C₆₇	—	4	6	no	—
81	C₄²×C₆₇	C₂²×C₄	1	6	yes	—
82	C₄²×C₆₇	C₂×Q₈	1	6	yes	—
83	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
84	Q₈×C₆₇	#106	1	16	yes	complemented
85	C₂×C₄×C₆₇	—	2	3	no	—
86	C₂×C₄×C₆₇	—	2	3	no	—
87	C₂×C₄×C₆₇	—	2	2	no	—
88	C₂×C₄×C₆₇	—	4	1	no	—
89	C₂×C₄×C₆₇	—	4	1	no	—
90	C₂×C₄×C₆₇	—	4	3	no	—
91	C₂×C₄×C₆₇	—	4	3	no	—
92	C₂×C₄×C₆₇	—	4	3	no	—
93	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	6	yes	—
94	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	6	yes	—
95	C₂×C₄×C₆₇	C₂×SD₁₆	1	6	yes	—
96	C₂×C₄×C₆₇	C₂×SD₁₆	1	6	yes	—
97	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
98	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
99	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
100	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
101	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
102	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
103	C₂×Dic₆₇	—	2	2	no	—
104	C₂×Dic₆₇	—	4	1	no	—
105	C₂×Dic₆₇	—	4	1	no	—
106	C₂×Dic₆₇	—	4	3	no	—
107	C₂×Dic₆₇	—	4	3	no	—
108	C₂×Dic₆₇	—	4	3	no	—
109	C₆₇⋊C₈	—	2	6	no	—
110	C₆₇⋊C₈	—	2	2	no	—
111	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
112	C₂²×C₆₇	#548	1	1	yes	characteristic
113	C₂²×C₆₇	#548	1	1	yes	characteristic
114	C₂²×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
115	C₂²×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
116	C₂²×C₆₇	#576	1	1	yes	characteristic
117	C₂²×C₆₇	#578	1	1	yes	characteristic
118	C₄×C₆₇	—	2	6	no	—
119	C₄×C₆₇	—	4	2	no	—
120	C₄×C₆₇	—	8	3	no	—
121	C₄×C₆₇	C₂×#106	1	12	yes	—
122	C₄×C₆₇	C₄×#41	1	1	yes	characteristic
123	C₄×C₆₇	#519	1	1	yes	characteristic
124	C₄×C₆₇	#557	1	1	yes	characteristic
125	C₄×C₆₇	#557	1	1	yes	characteristic
126	Dic₆₇	—	4	2	no	—
127	Dic₆₇	—	8	3	no	—
128	#22738	—	67	1	no	maximal, Sylow
129	C₂×C₆₇	#2351	1	1	yes	characteristic
130	C₂×C₆₇	C₂²×#106	1	1	yes	characteristic
131	C₂×C₆₇	#2913	1	1	yes	characteristic
132	C₂×C₆₇	#3359	1	1	yes	characteristic
133	C₂×C₆₇	#3359	1	1	yes	characteristic
134	C₂×C₆₇	#3362	1	1	yes	characteristic
135	C₂×C₆₇	#3362	1	1	yes	characteristic
136	C₂×C₈×Q₈	—	67	1	no	—
137	Q₈×C₄⋊C₄	—	67	1	no	—
138	Q₈×C₄⋊C₄	D₆₇	1	1	yes	characteristic
139	#1948	—	67	3	no	—
140	#1948	—	67	3	no	—
141	#1834	—	67	3	no	—
142	C₄×#106	—	67	3	no	—
143	C₆₇	#22738	1	1	yes	characteristic, complemented, Sylow
144	C₂×C₄×Q₈	—	2	1	no	—
145	C₂×C₄×Q₈	—	134	1	no	—
146	C₂×C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic
147	C₈×Q₈	—	67	8	no	—
148	C₂×#106	—	67	3	no	—
149	C₂×#106	—	67	3	no	—
150	C₂×C₄⋊C₈	—	67	3	no	—
151	C₂×C₄×C₈	—	67	3	no	—
152	#389	—	2	3	no	—
153	#389	—	134	3	no	—
154	#384	—	67	3	no	—
155	#384	C₂×D₆₇	1	3	yes	—
156	C₄×C₄⋊C₄	—	67	3	no	—
157	C₄×C₄⋊C₄	C₂×D₆₇	1	3	yes	—
158	#340	—	67	3	no	—
159	#340	—	67	6	no	—
160	#340	—	67	1	no	—
161	C₂²×Q₈	C₂²⋊D₆₇	1	1	yes	characteristic
162	C₂²×C₈	—	67	1	no	—
163	C₂²×C₈	—	67	3	no	—
164	C₄×Q₈	—	4	4	no	—
165	C₄×Q₈	—	268	4	no	—
166	C₄×Q₈	Dic₁₃₄	1	4	yes	—
167	C₄×Q₈	C₄×D₆₇	1	4	yes	—
168	C₂×C₄⋊C₄	—	2	3	no	—
169	C₂×C₄⋊C₄	—	2	3	no	—
170	C₂×C₄⋊C₄	—	67	1	no	—
171	C₂×C₄⋊C₄	—	67	3	no	—
172	C₂×C₄⋊C₄	—	134	3	no	—
173	C₂×C₄⋊C₄	—	134	3	no	—
174	C₂×C₄⋊C₄	C₂²⋊D₆₇	1	3	yes	—
175	C₂×C₄⋊C₄	C₂²×D₆₇	1	3	yes	—
176	C₂×C₄⋊C₄	C₂²×D₆₇	1	1	yes	characteristic
177	C₂×C₄²	—	2	3	no	—
178	C₂×C₄²	—	134	3	no	—
179	C₂×C₄²	C₂²×D₆₇	1	3	yes	—
180	#106	—	134	6	no	—
181	#106	—	134	6	no	—
182	C₄⋊C₈	—	67	12	no	—
183	C₄×C₈	—	67	12	no	—
184	#95	—	2	3	no	—
185	#95	—	134	3	no	—
186	C₂×Q₈	Dic₆₇⋊C₄	1	4	yes	—
187	C₂×Q₈	#11761615	1	4	yes	—
188	C₂×Q₈	#11761616	1	4	yes	—
189	C₂²×C₄	—	2	1	no	—
190	C₂²×C₄	—	2	3	no	—
191	C₂²×C₄	—	134	3	no	—
192	C₂²×C₄	—	134	1	no	—
193	C₂²×C₄	C₂×C₂²⋊D₆₇	1	3	yes	—
194	C₂²×C₄	C₂×C₂²⋊D₆₇	1	3	yes	—
195	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
196	C₂×C₈	—	67	2	no	—
197	C₂×C₈	—	67	6	no	—
198	C₂×C₈	—	134	3	no	—
199	C₂×C₈	—	134	3	no	—
200	C₂×C₈	—	134	1	no	—
201	C₂×C₈	—	134	1	no	—
202	C₄⋊C₄	—	2	1	no	—
203	C₄⋊C₄	—	2	3	no	—
204	C₄⋊C₄	—	2	3	no	—
205	C₄⋊C₄	—	4	6	no	—
206	C₄⋊C₄	—	8	3	no	—
207	C₄⋊C₄	—	268	6	no	—
208	C₄⋊C₄	—	536	3	no	—
209	C₄⋊C₄	—	67	2	no	—
210	C₄⋊C₄	—	134	3	no	—
211	C₄⋊C₄	—	134	3	no	—
212	C₄⋊C₄	—	134	1	no	—
213	C₄⋊C₄	Dic₆₇⋊C₄	1	12	yes	—
214	C₄⋊C₄	#11761629	1	2	yes	—
215	C₄²	—	4	6	no	—
216	C₄²	—	268	6	no	—
217	C₄²	C₂×C₄×D₆₇	1	6	yes	—
218	C₄²	C₂×Dic₁₃₄	1	6	yes	—
219	C₂³	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
220	Q₈	#14523974	1	16	yes	complemented
221	C₂×C₄	—	2	3	no	—
222	C₂×C₄	—	2	3	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	4	1	no	—
225	C₂×C₄	—	4	1	no	—
226	C₂×C₄	—	4	3	no	—
227	C₂×C₄	—	4	3	no	—
228	C₂×C₄	—	4	3	no	—
229	C₂×C₄	—	268	1	no	—
230	C₂×C₄	—	268	1	no	—
231	C₂×C₄	—	268	3	no	—
232	C₂×C₄	—	268	3	no	—
233	C₂×C₄	—	268	3	no	—
234	C₂×C₄	—	134	2	no	—
235	C₂×C₄	C₂×Dic₆₇⋊C₄	1	6	yes	—
236	C₂×C₄	C₂×Dic₆₇⋊C₄	1	6	yes	—
237	C₂×C₄	C₂×#11761615	1	6	yes	—
238	C₂×C₄	C₂×#11761616	1	6	yes	—
239	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
240	C₂×C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
241	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
242	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
243	C₂×C₄	#14524090	1	1	yes	characteristic
244	C₂×C₄	#14524091	1	1	yes	characteristic
245	C₈	—	134	6	no	—
246	C₈	—	134	2	no	—
247	C₂²	C₂²×Dic₆₇⋊C₄	1	1	yes	characteristic
248	C₂²	#16864053	1	1	yes	characteristic
249	C₂²	#16864057	1	1	yes	characteristic
250	C₂²	C₂²×#11761615	1	1	yes	characteristic
251	C₂²	C₂²×#11761616	1	1	yes	characteristic
252	C₂²	#16864199	1	1	yes	characteristic
253	C₂²	#16864211	1	1	yes	characteristic
254	C₄	—	2	6	no	—
255	C₄	—	4	2	no	—
256	C₄	—	8	3	no	—
257	C₄	—	268	2	no	—
258	C₄	—	536	3	no	—
259	C₄	C₂×#14523974	1	12	yes	—
260	C₄	C₄×#11761629	1	1	yes	characteristic
261	C₄	#16863872	1	1	yes	characteristic
262	C₄	#16864122	1	1	yes	characteristic
263	C₄	#16864125	1	1	yes	characteristic
264	C₂	#22563127	1	1	yes	characteristic
265	C₂	C₂²×#14523974	1	1	yes	characteristic
266	C₂	#22568807	1	1	yes	characteristic
267	C₂	#22572847	1	1	yes	characteristic
268	C₂	#22572851	1	1	yes	characteristic
269	C₂	#22572876	1	1	yes	characteristic
270	C₂	#22572880	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	Q₈×C₄⋊C₄×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂²×D₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	Q₈×C₄⋊C₄	D₆₇
hypocenter	C₆₇	#22738
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ Q₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
9. Q₈×C₆₇⋊C₈	C₄	semidirect
84. Q₈×C₆₇	#106	semidirect
143. C₆₇	#22738	semidirect
220. Q₈	#14523974	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×C₆₇⋊C₈	2
Q₈×C₄⋊Dic₆₇	2
#22560972	2
#22560970	2
#22560336	2
C₄×#14523974	2
Q₈×C₄⋊C₄×C₆₇	2
#22738	67