Subgroups — C₆₇×#22277 #30701528

Order 17152.

826 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#494×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2327	C₂	1	2	yes	maximal
3	C₆₇×#1960	C₂	1	2	yes	maximal
4	C₆₇×#1878	C₂	1	8	yes	maximal
5	C₆₇×#1835	C₂	1	1	yes	maximal, characteristic
6	C₆₇×#1746	C₂	1	1	yes	maximal, characteristic
7	C₂²×Q₁₆×C₆₇	C₂²	1	4	yes	—
8	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
9	C₂×C₄×Q₈×C₆₇	C₂²	1	4	yes	—
10	#494×C₆₇	C₄	1	8	yes	complemented
11	C₂×#103×C₆₇	C₂²	1	8	yes	—
12	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	#391×C₆₇	C₂²	1	4	yes	—
15	#387×C₆₇	—	2	8	no	—
16	#382×C₆₇	C₂²	1	2	yes	—
17	#376×C₆₇	C₂²	1	1	yes	characteristic
18	#340×C₆₇	C₂²	1	4	yes	—
19	#336×C₆₇	C₂²	1	4	yes	—
20	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
21	C₂×Q₁₆×C₆₇	—	4	8	no	—
22	C₂×Q₁₆×C₆₇	C₂×C₄	1	16	yes	—
23	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
24	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
25	C₄⋊Q₈×C₆₇	—	2	4	no	—
26	C₄⋊Q₈×C₆₇	C₂×C₄	1	8	yes	—
27	C₄×Q₈×C₆₇	—	4	8	no	—
28	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
29	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
30	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
31	C₂×C₄²×C₆₇	—	2	4	no	—
32	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
33	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
34	#103×C₆₇	—	4	8	no	—
35	#97×C₆₇	—	2	2	no	—
36	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	—
37	#95×C₆₇	—	2	4	no	—
38	#95×C₆₇	—	2	8	no	—
39	#95×C₆₇	D₄	1	4	yes	—
40	C₂×Q₈×C₆₇	—	2	4	no	—
41	C₂×Q₈×C₆₇	—	2	8	no	—
42	C₂×Q₈×C₆₇	—	2	8	no	—
43	C₂×Q₈×C₆₇	C₂²×C₄	1	8	yes	—
44	C₂²×C₄×C₆₇	—	2	4	no	—
45	C₂²×C₄×C₆₇	—	2	4	no	—
46	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
47	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
48	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
49	Q₁₆×C₆₇	—	4	16	no	—
50	C₂×C₈×C₆₇	—	2	4	no	—
51	C₂×C₈×C₆₇	—	2	2	no	—
52	C₂×C₈×C₆₇	—	2	4	no	—
53	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
54	C₄⋊C₄×C₆₇	—	2	16	no	—
55	C₄⋊C₄×C₆₇	—	2	8	no	—
56	C₄⋊C₄×C₆₇	—	4	8	no	—
57	C₄²×C₆₇	—	2	4	no	—
58	C₄²×C₆₇	—	4	8	no	—
59	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
60	C₄²×C₆₇	#41	1	2	yes	—
61	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
62	Q₈×C₆₇	—	2	16	no	—
63	Q₈×C₆₇	—	4	8	no	—
64	C₂×C₄×C₆₇	—	2	4	no	—
65	C₂×C₄×C₆₇	—	2	8	no	—
66	C₂×C₄×C₆₇	—	2	4	no	—
67	C₂×C₄×C₆₇	—	2	8	no	—
68	C₂×C₄×C₆₇	—	4	4	no	—
69	C₂×C₄×C₆₇	—	4	4	no	—
70	C₂×C₄×C₆₇	—	4	4	no	—
71	C₂×C₄×C₆₇	—	4	8	no	—
72	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
73	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
74	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
75	C₂×C₄×C₆₇	H(5,𝔽₂)	1	4	yes	—
76	C₈×C₆₇	—	2	4	no	—
77	C₈×C₆₇	—	4	2	no	—
78	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
79	C₂²×C₆₇	#535	1	2	yes	—
80	C₂²×C₆₇	#578	1	4	yes	—
81	C₄×C₆₇	—	2	4	no	—
82	C₄×C₆₇	—	4	8	no	—
83	C₄×C₆₇	—	4	4	no	—
84	C₄×C₆₇	—	8	4	no	—
85	C₄×C₆₇	#518	1	2	yes	—
86	C₄×C₆₇	#540	1	2	yes	—
87	#22277	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
88	C₂×C₆₇	#2328	1	1	yes	characteristic
89	C₂×C₆₇	#2950	1	2	yes	—
90	C₂×C₆₇	#3223	1	4	yes	—
91	C₂×#494	C₂×C₆₇	1	1	yes	characteristic
92	#2327	C₂×C₆₇	1	2	yes	—
93	#1960	C₂×C₆₇	1	2	yes	—
94	#1878	C₂×C₆₇	1	8	yes	—
95	#1835	C₂×C₆₇	1	1	yes	characteristic
96	#1746	C₂×C₆₇	1	1	yes	characteristic
97	C₆₇	#22277	1	1	yes	characteristic, direct product factor, Sylow
98	C₂²×Q₁₆	C₂²×C₆₇	1	4	yes	—
99	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
100	C₂×C₄×Q₈	C₂²×C₆₇	1	4	yes	—
101	#494	C₄×C₆₇	1	8	yes	complemented
102	C₂×#103	C₂²×C₆₇	1	8	yes	—
103	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
104	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
105	#391	C₂²×C₆₇	1	4	yes	—
106	#387	—	2	8	no	—
107	#382	C₂²×C₆₇	1	2	yes	—
108	#376	C₂²×C₆₇	1	1	yes	characteristic
109	#340	C₂²×C₆₇	1	4	yes	—
110	#336	C₂²×C₆₇	1	4	yes	—
111	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
112	C₂×Q₁₆	—	4	8	no	—
113	C₂×Q₁₆	C₂×C₄×C₆₇	1	16	yes	—
114	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
115	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
116	C₄⋊Q₈	—	2	4	no	—
117	C₄⋊Q₈	C₂×C₄×C₆₇	1	8	yes	—
118	C₄×Q₈	—	4	8	no	—
119	C₂×C₄⋊C₄	—	2	4	no	—
120	C₂×C₄⋊C₄	—	2	4	no	—
121	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
122	C₂×C₄²	—	2	4	no	—
123	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
124	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
125	#103	—	4	8	no	—
126	#97	—	2	2	no	—
127	C₄×C₈	C₂×C₄×C₆₇	1	4	yes	—
128	#95	—	2	4	no	—
129	#95	—	2	8	no	—
130	#95	D₄×C₆₇	1	4	yes	—
131	C₂×Q₈	—	2	4	no	—
132	C₂×Q₈	—	2	8	no	—
133	C₂×Q₈	—	2	8	no	—
134	C₂×Q₈	C₂²×C₄×C₆₇	1	8	yes	—
135	C₂²×C₄	—	2	4	no	—
136	C₂²×C₄	—	2	4	no	—
137	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
138	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
139	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
140	Q₁₆	—	4	16	no	—
141	C₂×C₈	—	2	4	no	—
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	2	4	no	—
144	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
145	C₄⋊C₄	—	2	16	no	—
146	C₄⋊C₄	—	2	8	no	—
147	C₄⋊C₄	—	4	8	no	—
148	C₄²	—	2	4	no	—
149	C₄²	—	4	8	no	—
150	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
151	C₄²	#41×C₆₇	1	2	yes	—
152	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
153	Q₈	—	2	16	no	—
154	Q₈	—	4	8	no	—
155	C₂×C₄	—	2	4	no	—
156	C₂×C₄	—	2	8	no	—
157	C₂×C₄	—	2	4	no	—
158	C₂×C₄	—	2	8	no	—
159	C₂×C₄	—	4	4	no	—
160	C₂×C₄	—	4	4	no	—
161	C₂×C₄	—	4	4	no	—
162	C₂×C₄	—	4	8	no	—
163	C₂×C₄	C₄×D₄×C₆₇	1	4	yes	—
164	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
165	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
166	C₂×C₄	H(5,𝔽₂)×C₆₇	1	4	yes	—
167	C₈	—	2	4	no	—
168	C₈	—	4	2	no	—
169	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
170	C₂²	#535×C₆₇	1	2	yes	—
171	C₂²	#578×C₆₇	1	4	yes	—
172	C₄	—	2	4	no	—
173	C₄	—	4	8	no	—
174	C₄	—	4	4	no	—
175	C₄	—	8	4	no	—
176	C₄	#518×C₆₇	1	2	yes	—
177	C₄	#540×C₆₇	1	2	yes	—
178	C₂	C₆₇×#2328	1	1	yes	characteristic
179	C₂	C₆₇×#2950	1	2	yes	—
180	C₂	C₆₇×#3223	1	4	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ #376×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
10. #494×C₆₇	C₄	semidirect
87. #22277	C₆₇	direct (N × H)
97. C₆₇	#22277	direct (N × H)
101. #494	C₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#494×C₆₇	2
C₆₇×#2327	2
C₆₇×#1960	2
C₆₇×#1878	2
C₆₇×#1835	2
C₆₇×#1746	2
#22277	67

Subgroups — C67×#22277 #30701528

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22277 #30701528