Subgroups — #30761567

Order 17152.

718 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863161	C₂	1	2	yes	maximal
2	C₂×#16863161	C₂	1	2	yes	maximal
3	#22559903	C₂	1	2	yes	maximal
4	C₂²×C₄×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11761612	C₂²	1	4	yes	—
6	#16863161	—	2	4	no	—
7	#16863161	C₄	1	8	yes	complemented
8	#16863151	—	2	4	no	—
9	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
10	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
11	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₆₇	—	2	4	no	—
13	C₂×C₄×C₈×C₆₇	C₄	1	4	yes	—
14	C₂×#11761612	—	2	4	no	—
15	C₂×#11761612	—	2	4	no	—
16	C₂×#11761612	—	2	4	no	—
17	C₂×#11761612	—	2	4	no	—
18	C₂×#11761612	—	2	4	no	—
19	C₂×#11761612	C₂×C₄	1	8	yes	—
20	C₂²×C₆₇⋊C₈	—	2	4	no	—
21	C₂³×C₄×C₆₇	Q₈	1	2	yes	—
22	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
23	C₂²×C₈×C₆₇	—	2	2	no	—
24	C₂²×C₈×C₆₇	—	2	2	no	—
25	C₂²×C₈×C₆₇	—	2	2	no	—
26	C₂²×C₈×C₆₇	—	2	2	no	—
27	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
28	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₆₇	D₄	1	4	yes	—
31	C₂×C₄²×C₆₇	—	2	2	no	—
32	C₂×C₄²×C₆₇	—	2	2	no	—
33	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₆₇	D₄	1	2	yes	—
35	C₄×C₈×C₆₇	—	2	8	no	—
36	#11761612	—	2	8	no	—
37	#11761612	—	2	8	no	—
38	#11761612	—	4	4	no	—
39	#11761612	—	4	4	no	—
40	C₂×C₆₇⋊C₈	—	2	8	no	—
41	C₂×C₆₇⋊C₈	—	4	4	no	—
42	C₂×C₆₇⋊C₈	—	4	4	no	—
43	C₂⁴×C₆₇	C₂×Q₈	1	1	yes	characteristic
44	C₂²×C₄×C₆₇	—	2	4	no	—
45	C₂²×C₄×C₆₇	—	2	8	no	—
46	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
47	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
48	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
50	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₆₇	#41	1	2	yes	—
52	C₂²×C₄×C₆₇	#41	1	2	yes	—
53	C₂²×C₄×C₆₇	#41	1	2	yes	—
54	C₂²×C₄×C₆₇	#41	1	2	yes	—
55	C₂×C₈×C₆₇	—	2	2	no	—
56	C₂×C₈×C₆₇	—	2	2	no	—
57	C₂×C₈×C₆₇	—	2	2	no	—
58	C₂×C₈×C₆₇	—	2	2	no	—
59	C₂×C₈×C₆₇	—	2	8	no	—
60	C₂×C₈×C₆₇	—	2	8	no	—
61	C₂×C₈×C₆₇	C₄²	1	4	yes	—
62	C₂×C₈×C₆₇	C₂²⋊C₄	1	4	yes	—
63	C₄²×C₆₇	—	2	2	no	—
64	C₄²×C₆₇	—	2	2	no	—
65	C₄²×C₆₇	—	2	4	no	—
66	C₂³×C₆₇	—	2	4	no	—
67	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂³×C₆₇	C₄×Q₈	1	2	yes	—
69	C₂³×C₆₇	C₂²⋊Q₈	1	2	yes	—
70	C₂³×C₆₇	#125	1	2	yes	—
71	C₂×C₄×C₆₇	—	2	16	no	—
72	C₂×C₄×C₆₇	—	2	4	no	—
73	C₂×C₄×C₆₇	—	2	4	no	—
74	C₂×C₄×C₆₇	#108	1	8	yes	—
75	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
76	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₆₇	#117	1	4	yes	—
78	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
79	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
80	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
81	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
82	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
83	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
84	C₂×C₄×C₆₇	#121	1	2	yes	—
85	C₂×C₄×C₆₇	#123	1	4	yes	—
86	C₂×C₄×C₆₇	#124	1	2	yes	—
87	C₂×C₄×C₆₇	#126	1	4	yes	—
88	C₈×C₆₇	—	2	4	no	—
89	C₈×C₆₇	—	2	4	no	—
90	C₆₇⋊C₈	—	4	8	no	—
91	C₂²×C₆₇	—	2	4	no	—
92	C₂²×C₆₇	—	2	4	no	—
93	C₂²×C₆₇	—	2	4	no	—
94	C₂²×C₆₇	C₄×C₄⋊C₄	1	1	yes	characteristic
95	C₂²×C₆₇	#380	1	1	yes	characteristic
96	C₂²×C₆₇	#382	1	1	yes	characteristic
97	C₂²×C₆₇	#387	1	4	yes	—
98	C₂²×C₆₇	C₂×#108	1	4	yes	—
99	C₄×C₆₇	—	2	8	no	—
100	C₄×C₆₇	#376	1	1	yes	characteristic
101	C₄×C₆₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
102	C₄×C₆₇	#381	1	1	yes	characteristic
103	C₄×C₆₇	#383	1	1	yes	characteristic
104	C₄×C₆₇	#388	1	4	yes	—
105	#10822	—	67	1	no	maximal, Sylow
106	C₂×C₆₇	—	2	4	no	—
107	C₂×C₆₇	#1434	1	1	yes	characteristic
108	C₂×C₆₇	C₄×#108	1	2	yes	—
109	C₂×C₆₇	#1810	1	2	yes	—
110	C₂×C₆₇	#1837	1	2	yes	—
111	C₂²×C₄×C₈	D₆₇	1	1	yes	characteristic
112	C₂×#341	—	67	2	no	—
113	C₂×#341	—	67	2	no	—
114	#1728	—	67	2	no	—
115	C₆₇	#10822	1	1	yes	characteristic, complemented, Sylow
116	C₂²×M₁₆	—	67	4	no	—
117	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
118	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
119	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
120	C₂×C₄×C₈	—	2	4	no	—
121	C₂×C₄×C₈	Dic₆₇	1	4	yes	—
122	#341	—	67	8	no	—
123	#341	—	134	4	no	—
124	#336	—	134	4	no	—
125	C₂³×C₄	Dic₁₃₄	1	2	yes	—
126	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
127	C₂×M₁₆	—	67	8	no	—
128	C₂×M₁₆	—	134	4	no	—
129	C₂×M₁₆	—	134	4	no	—
130	C₂×M₁₆	—	134	4	no	—
131	C₂×M₁₆	—	134	4	no	—
132	C₂×M₁₆	—	134	4	no	—
133	C₂²×C₈	—	2	2	no	—
134	C₂²×C₈	—	2	2	no	—
135	C₂²×C₈	—	2	2	no	—
136	C₂²×C₈	—	2	2	no	—
137	C₂²×C₈	—	134	4	no	—
138	C₂²×C₈	C₄×D₆₇	1	4	yes	—
139	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
140	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
141	C₂²×C₈	C₂²⋊D₆₇	1	4	yes	—
142	C₂×C₄²	—	2	2	no	—
143	C₂×C₄²	—	2	2	no	—
144	C₂×C₄²	D₂₆₈	1	2	yes	—
145	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
146	C₄×C₈	—	2	8	no	—
147	C₂⁴	C₂×Dic₁₃₄	1	1	yes	characteristic
148	C₂²×C₄	—	2	4	no	—
149	C₂²×C₄	—	2	8	no	—
150	C₂²×C₄	C₄⋊Dic₆₇	1	4	yes	—
151	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
152	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
153	C₂²×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
154	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
155	C₂²×C₄	#11761628	1	2	yes	—
156	C₂²×C₄	#11761628	1	2	yes	—
157	C₂²×C₄	#11761628	1	2	yes	—
158	C₂²×C₄	#11761628	1	2	yes	—
159	M₁₆	—	268	4	no	—
160	M₁₆	—	268	4	no	—
161	M₁₆	—	134	8	no	—
162	M₁₆	—	134	8	no	—
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	2	8	no	—
168	C₂×C₈	—	2	8	no	—
169	C₂×C₈	—	268	4	no	—
170	C₂×C₈	—	268	4	no	—
171	C₂×C₈	—	134	8	no	—
172	C₂×C₈	C₄×Dic₆₇	1	4	yes	—
173	C₂×C₈	C₂²⋊Dic₆₇	1	4	yes	—
174	C₄²	—	2	2	no	—
175	C₄²	—	2	2	no	—
176	C₄²	—	2	4	no	—
177	C₂³	—	2	4	no	—
178	C₂³	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
179	C₂³	C₄×Dic₁₃₄	1	2	yes	—
180	C₂³	#14524022	1	2	yes	—
181	C₂³	#14524035	1	2	yes	—
182	C₂×C₄	—	2	16	no	—
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	#14523979	1	8	yes	—
186	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
187	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
188	C₂×C₄	#14524000	1	4	yes	—
189	C₂×C₄	#14524002	1	1	yes	characteristic
190	C₂×C₄	#14524002	1	1	yes	characteristic
191	C₂×C₄	#14524002	1	1	yes	characteristic
192	C₂×C₄	#14524002	1	1	yes	characteristic
193	C₂×C₄	C₄×C₂²⋊D₆₇	1	4	yes	—
194	C₂×C₄	C₄×D₂₆₈	1	2	yes	—
195	C₂×C₄	#14524017	1	2	yes	—
196	C₂×C₄	#14524026	1	4	yes	—
197	C₂×C₄	#14524031	1	2	yes	—
198	C₂×C₄	#14524037	1	4	yes	—
199	C₈	—	2	4	no	—
200	C₈	—	2	4	no	—
201	C₈	—	268	8	no	—
202	C₂²	—	2	4	no	—
203	C₂²	—	2	4	no	—
204	C₂²	—	2	4	no	—
205	C₂²	C₄×C₄⋊Dic₆₇	1	1	yes	characteristic
206	C₂²	#16863256	1	1	yes	characteristic
207	C₂²	#16863262	1	1	yes	characteristic
208	C₂²	#16863280	1	4	yes	—
209	C₂²	C₂×#14523979	1	4	yes	—
210	C₄	—	2	8	no	—
211	C₄	#16863242	1	1	yes	characteristic
212	C₄	C₄×C₂²⋊Dic₆₇	1	1	yes	characteristic
213	C₄	#16863259	1	1	yes	characteristic
214	C₄	#16863264	1	1	yes	characteristic
215	C₄	#16863287	1	4	yes	—
216	C₂	—	2	4	no	—
217	C₂	#22558436	1	1	yes	characteristic
218	C₂	C₄×#14523979	1	2	yes	—
219	C₂	#22560247	1	2	yes	—
220	C₂	#22560347	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₆₇
Fitting	C₂²×C₄×C₈×C₆₇	C₂
socle	C₂³×C₆₇	C₂×C₄⋊C₄
prime radical	C₂⁴×C₆₇	C₂×Q₈
cosocle	C₂³×C₄×C₆₇	C₂³
hypercenter	C₂²×C₄×C₈	D₆₇
hypocenter	C₆₇	#10822
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
7. #16863161	C₄	semidirect
115. C₆₇	#10822	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863161	2
#22559903	2
C₂²×C₄×C₈×C₆₇	2
#10822	67