Subgroups — #30761663

Order 17152.

774 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863420	C₂	1	2	yes	maximal
2	C₂×#16863161	C₂	1	4	yes	maximal
3	C₂²×#97×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11761612	C₂²	1	4	yes	—
5	#16863420	—	2	4	no	—
6	#16863420	C₄	1	8	yes	—
7	C₂×C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
8	#16863161	—	2	8	no	—
9	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
10	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
11	C₂×#97×C₆₇	—	2	4	no	—
12	C₂×#97×C₆₇	C₄	1	4	yes	—
13	C₂×#11761612	—	2	4	no	—
14	C₂×#11761612	—	2	8	no	—
15	C₂×#11761612	—	2	8	no	—
16	C₂×#11761612	C₂×C₄	1	8	yes	—
17	C₂²×C₆₇⋊C₈	—	2	4	no	—
18	C₄⋊(C₆₇⋊C₈)	—	2	16	no	—
19	C₂³×C₄×C₆₇	Q₈	1	2	yes	—
20	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
21	C₂²×C₈×C₆₇	—	2	4	no	—
22	C₂²×C₈×C₆₇	—	2	4	no	—
23	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
24	C₂²×C₈×C₆₇	D₄	1	4	yes	—
25	C₂²×C₈×C₆₇	Q₈	1	4	yes	—
26	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
27	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₆₇	D₄	1	2	yes	—
29	C₂×C₄²×C₆₇	D₄	1	4	yes	—
30	#97×C₆₇	—	2	8	no	—
31	#11761612	—	2	16	no	—
32	#11761612	—	4	8	no	—
33	C₂×C₆₇⋊C₈	—	2	8	no	—
34	C₂×C₆₇⋊C₈	—	4	4	no	—
35	C₂×C₆₇⋊C₈	—	4	4	no	—
36	C₂⁴×C₆₇	C₂×Q₈	1	1	yes	characteristic
37	C₂²×C₄×C₆₇	—	4	4	no	—
38	C₂²×C₄×C₆₇	C₄⋊C₄	1	8	yes	—
39	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
40	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
41	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
43	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
44	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
46	C₂²×C₄×C₆₇	#41	1	4	yes	—
47	C₂×C₈×C₆₇	—	2	4	no	—
48	C₂×C₈×C₆₇	—	2	4	no	—
49	C₂×C₈×C₆₇	—	2	8	no	—
50	C₂×C₈×C₆₇	—	2	8	no	—
51	C₂×C₈×C₆₇	C₄⋊C₄	1	8	yes	—
52	C₄²×C₆₇	C₄²	1	4	yes	—
53	C₄²×C₆₇	C₂²⋊C₄	1	8	yes	—
54	C₄²×C₆₇	C₂²⋊C₄	1	4	yes	—
55	C₂³×C₆₇	—	2	4	no	—
56	C₂³×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
57	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂³×C₆₇	#125	1	2	yes	—
59	C₂³×C₆₇	C₄⋊Q₈	1	2	yes	—
60	C₂×C₄×C₆₇	—	2	8	no	—
61	C₂×C₄×C₆₇	—	4	8	no	—
62	C₂×C₄×C₆₇	#95	1	8	yes	—
63	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
64	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
65	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
67	C₂×C₄×C₆₇	#117	1	2	yes	—
68	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
69	C₂×C₄×C₆₇	C₄×Q₈	1	4	yes	—
70	C₂×C₄×C₆₇	#121	1	4	yes	—
71	C₂×C₄×C₆₇	C₂²⋊Q₈	1	4	yes	—
72	C₂×C₄×C₆₇	#124	1	2	yes	—
73	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
74	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
75	C₈×C₆₇	—	2	8	no	—
76	C₆₇⋊C₈	—	4	8	no	—
77	C₂²×C₆₇	—	2	4	no	—
78	C₂²×C₆₇	—	2	4	no	—
79	C₂²×C₆₇	—	2	4	no	—
80	C₂²×C₆₇	C₂×#95	1	1	yes	characteristic
81	C₂²×C₆₇	#382	1	1	yes	characteristic
82	C₂²×C₆₇	#384	1	1	yes	characteristic
83	C₂²×C₆₇	#389	1	4	yes	—
84	C₂²×C₆₇	#430	1	4	yes	—
85	C₄×C₆₇	—	2	8	no	—
86	C₄×C₆₇	C₄×C₄⋊C₄	1	2	yes	—
87	C₄×C₆₇	#380	1	2	yes	—
88	C₄×C₆₇	#390	1	2	yes	—
89	C₄×C₆₇	#391	1	2	yes	—
90	#10845	—	67	1	no	maximal, Sylow
91	C₂×C₆₇	—	2	4	no	—
92	C₂×C₆₇	C₄⋊#95	1	1	yes	characteristic
93	C₂×C₆₇	#1738	1	2	yes	—
94	C₂×C₆₇	#1838	1	2	yes	—
95	C₂×C₆₇	#1849	1	2	yes	—
96	C₂×#423	—	67	2	no	—
97	C₂²×#97	D₆₇	1	1	yes	characteristic
98	C₂×#341	—	67	4	no	—
99	C₆₇	#10845	1	1	yes	characteristic, complemented, Sylow
100	C₂²×M₁₆	—	67	4	no	—
101	C₂³×C₈	C₂×D₆₇	1	2	yes	—
102	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
103	#423	—	67	8	no	—
104	#423	—	134	4	no	—
105	C₂×C₄⋊C₈	—	134	4	no	—
106	C₂×#97	—	2	4	no	—
107	C₂×#97	Dic₆₇	1	4	yes	—
108	#341	—	134	8	no	—
109	C₂³×C₄	Dic₁₃₄	1	2	yes	—
110	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
111	C₂×M₁₆	—	67	8	no	—
112	C₂×M₁₆	—	134	8	no	—
113	C₂×M₁₆	—	134	8	no	—
114	C₂×M₁₆	—	134	4	no	—
115	C₂²×C₈	—	2	4	no	—
116	C₂²×C₈	—	2	4	no	—
117	C₂²×C₈	—	134	4	no	—
118	C₂²×C₈	Dic₁₃₄	1	4	yes	—
119	C₂²×C₈	D₂₆₈	1	4	yes	—
120	C₂²×C₈	C₂×Dic₆₇	1	4	yes	—
121	C₂×C₄²	C₄×D₆₇	1	4	yes	—
122	C₂×C₄²	D₂₆₈	1	2	yes	—
123	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
124	C₂×C₄²	C₂²⋊D₆₇	1	4	yes	—
125	C₄⋊C₈	—	134	16	no	—
126	#97	—	2	8	no	—
127	C₂⁴	C₂×Dic₁₃₄	1	1	yes	characteristic
128	C₂²×C₄	—	4	4	no	—
129	C₂²×C₄	Dic₆₇⋊C₄	1	8	yes	—
130	C₂²×C₄	C₄⋊Dic₆₇	1	4	yes	—
131	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
132	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
133	C₂²×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
134	C₂²×C₄	C₂×D₂₆₈	1	2	yes	—
135	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
136	C₂²×C₄	C₂×Dic₁₃₄	1	2	yes	—
137	C₂²×C₄	#11761628	1	4	yes	—
138	M₁₆	—	268	8	no	—
139	M₁₆	—	134	16	no	—
140	C₂×C₈	—	2	4	no	—
141	C₂×C₈	—	2	4	no	—
142	C₂×C₈	—	2	8	no	—
143	C₂×C₈	—	2	8	no	—
144	C₂×C₈	—	268	4	no	—
145	C₂×C₈	—	268	4	no	—
146	C₂×C₈	—	134	8	no	—
147	C₂×C₈	C₄⋊Dic₆₇	1	8	yes	—
148	C₄²	C₄×Dic₆₇	1	4	yes	—
149	C₄²	(C₂×D₆₇)⋊C₄	1	8	yes	—
150	C₄²	C₂²⋊Dic₆₇	1	4	yes	—
151	C₂³	—	2	4	no	—
152	C₂³	C₂×Dic₆₇⋊C₄	1	2	yes	—
153	C₂³	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
154	C₂³	#14524035	1	2	yes	—
155	C₂³	#14524043	1	2	yes	—
156	C₂×C₄	—	2	8	no	—
157	C₂×C₄	—	4	8	no	—
158	C₂×C₄	C₆₇⋊#95	1	8	yes	—
159	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
160	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	2	yes	—
161	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
162	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
163	C₂×C₄	#14524002	1	2	yes	—
164	C₂×C₄	C₄×D₂₆₈	1	4	yes	—
165	C₂×C₄	C₄×Dic₁₃₄	1	4	yes	—
166	C₂×C₄	#14524017	1	4	yes	—
167	C₂×C₄	#14524022	1	4	yes	—
168	C₂×C₄	#14524031	1	2	yes	—
169	C₂×C₄	#14524040	1	1	yes	characteristic
170	C₂×C₄	#14524043	1	1	yes	characteristic
171	C₈	—	2	8	no	—
172	C₈	—	268	8	no	—
173	C₂²	—	2	4	no	—
174	C₂²	—	2	4	no	—
175	C₂²	—	2	4	no	—
176	C₂²	C₂×C₆₇⋊#95	1	1	yes	characteristic
177	C₂²	#16863262	1	1	yes	characteristic
178	C₂²	#16863266	1	1	yes	characteristic
179	C₂²	#16863294	1	4	yes	—
180	C₂²	#16863445	1	4	yes	—
181	C₄	—	2	8	no	—
182	C₄	C₄×C₄⋊Dic₆₇	1	2	yes	—
183	C₄	#16863256	1	2	yes	—
184	C₄	#16863298	1	2	yes	—
185	C₄	#16863302	1	2	yes	—
186	C₂	—	2	4	no	—
187	C₂	#22558439	1	1	yes	characteristic
188	C₂	#22559936	1	2	yes	—
189	C₂	#22560350	1	2	yes	—
190	C₂	#22560384	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₆₇
Fitting	C₂²×#97×C₆₇	C₂
socle	C₂³×C₆₇	C₂×C₄⋊C₄
prime radical	C₂⁴×C₆₇	C₂×Q₈
cosocle	C₂³×C₄×C₆₇	C₂³
hypercenter	C₂²×#97	D₆₇
hypocenter	C₆₇	#10845
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×#97

Complements

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

N	H ≅ G/N	type
99. C₆₇	#10845	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863420	2
C₂×#16863161	2
C₂²×#97×C₆₇	2
#10845	67