Subgroups — C₄×Q₈²×C₆₇ #30713421

Order 17152.

2324 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈²×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2369	C₂	1	9	yes	maximal
3	Q₈×C₄⋊C₄×C₆₇	C₂	1	6	yes	maximal
4	C₄×C₄⋊Q₈×C₆₇	C₂	1	9	yes	maximal
5	C₄²×Q₈×C₆₇	C₂	1	6	yes	maximal
6	Q₈²×C₆₇	C₄	1	16	yes	direct product factor
7	C₂×C₄⋊Q₈×C₆₇	C₂²	1	9	yes	—
8	C₂×C₄×Q₈×C₆₇	C₂²	1	6	yes	—
9	C₂×C₄×Q₈×C₆₇	C₂²	1	6	yes	—
10	C₂×C₄×Q₈×C₆₇	C₂²	1	6	yes	—
11	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
12	#391×C₆₇	C₂²	1	18	yes	—
13	#389×C₆₇	C₂²	1	36	yes	—
14	#384×C₆₇	C₂²	1	9	yes	—
15	C₄×C₄⋊C₄×C₆₇	C₂²	1	18	yes	—
16	C₄×C₄⋊C₄×C₆₇	C₂²	1	18	yes	—
17	C₄×C₄⋊C₄×C₆₇	C₂²	1	18	yes	—
18	C₄³×C₆₇	C₂²	1	9	yes	—
19	C₂²×Q₈×C₆₇	C₂³	1	6	yes	—
20	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
21	C₄⋊Q₈×C₆₇	C₂×C₄	1	72	yes	—
22	C₄×Q₈×C₆₇	—	2	24	no	—
23	C₄×Q₈×C₆₇	—	4	12	no	—
24	C₄×Q₈×C₆₇	C₂×C₄	1	48	yes	—
25	C₄×Q₈×C₆₇	Q₈	1	16	yes	direct product factor
26	C₂×C₄⋊C₄×C₆₇	C₂³	1	18	yes	—
27	C₂×C₄⋊C₄×C₆₇	C₂³	1	18	yes	—
28	C₂×C₄⋊C₄×C₆₇	C₂³	1	18	yes	—
29	C₂×C₄⋊C₄×C₆₇	C₂³	1	6	yes	—
30	C₂×C₄²×C₆₇	C₂³	1	18	yes	—
31	C₂×C₄²×C₆₇	C₂³	1	9	yes	—
32	C₂×C₄²×C₆₇	C₂³	1	9	yes	—
33	C₂×C₄²×C₆₇	C₂³	1	9	yes	—
34	C₂×C₄²×C₆₇	C₂³	1	6	yes	—
35	#95×C₆₇	C₂³	1	36	yes	—
36	C₂×Q₈×C₆₇	—	4	6	no	—
37	C₂×Q₈×C₆₇	—	4	6	no	—
38	C₂×Q₈×C₆₇	C₂²×C₄	1	24	yes	—
39	C₂×Q₈×C₆₇	C₂²×C₄	1	8	yes	—
40	C₂×Q₈×C₆₇	C₂×Q₈	1	8	yes	—
41	C₂×Q₈×C₆₇	#41	1	8	yes	—
42	C₂²×C₄×C₆₇	C₂⁴	1	9	yes	—
43	C₂²×C₄×C₆₇	C₂⁴	1	9	yes	—
44	C₂²×C₄×C₆₇	C₂⁴	1	6	yes	—
45	C₂²×C₄×C₆₇	C₂⁴	1	6	yes	—
46	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
47	C₄⋊C₄×C₆₇	—	2	36	no	—
48	C₄⋊C₄×C₆₇	—	4	18	no	—
49	C₄⋊C₄×C₆₇	C₂²×C₄	1	72	yes	—
50	C₄⋊C₄×C₆₇	C₂×Q₈	1	24	yes	—
51	C₄²×C₆₇	—	2	36	no	—
52	C₄²×C₆₇	—	2	18	no	—
53	C₄²×C₆₇	—	4	9	no	—
54	C₄²×C₆₇	C₂²×C₄	1	36	yes	—
55	C₄²×C₆₇	C₂×Q₈	1	24	yes	—
56	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
57	Q₈×C₆₇	—	4	24	no	—
58	Q₈×C₆₇	C₄×Q₈	1	32	yes	direct product factor
59	C₂×C₄×C₆₇	—	2	6	no	—
60	C₂×C₄×C₆₇	—	2	9	no	—
61	C₂×C₄×C₆₇	—	2	6	no	—
62	C₂×C₄×C₆₇	—	2	9	no	—
63	C₂×C₄×C₆₇	—	2	18	no	—
64	C₂×C₄×C₆₇	—	2	9	no	—
65	C₂×C₄×C₆₇	C₂³×C₄	1	18	yes	—
66	C₂×C₄×C₆₇	C₂³×C₄	1	12	yes	—
67	C₂×C₄×C₆₇	C₂²×Q₈	1	12	yes	—
68	C₂×C₄×C₆₇	C₂²×Q₈	1	12	yes	—
69	C₂×C₄×C₆₇	C₂²×Q₈	1	4	yes	—
70	C₂×C₄×C₆₇	C₂×#41	1	12	yes	—
71	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
72	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
73	C₂²×C₆₇	C₂³×Q₈	1	2	yes	—
74	C₂²×C₆₇	C₂²×#41	1	2	yes	—
75	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
76	C₂²×C₆₇	#585	1	1	yes	characteristic
77	C₄×C₆₇	—	2	18	no	—
78	C₄×C₆₇	—	2	12	no	—
79	C₄×C₆₇	—	4	9	no	—
80	C₄×C₆₇	C₂×C₄×Q₈	1	24	yes	—
81	C₄×C₆₇	Q₈²	1	4	yes	direct product factor
82	C₄×Q₈²	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
83	C₂×C₆₇	C₂²×C₄×Q₈	1	2	yes	—
84	C₂×C₆₇	C₄×H(5,𝔽₂)	1	1	yes	characteristic
85	C₂×C₆₇	C₂×Q₈²	1	1	yes	characteristic
86	C₂×C₆₇	Q₈×#41	1	2	yes	—
87	C₂×C₆₇	#3483	1	1	yes	characteristic
88	C₂×Q₈²	C₂×C₆₇	1	1	yes	characteristic
89	#2369	C₂×C₆₇	1	9	yes	—
90	Q₈×C₄⋊C₄	C₂×C₆₇	1	6	yes	—
91	C₄×C₄⋊Q₈	C₂×C₆₇	1	9	yes	—
92	C₄²×Q₈	C₂×C₆₇	1	6	yes	—
93	C₆₇	C₄×Q₈²	1	1	yes	characteristic, direct product factor, Sylow
94	Q₈²	C₄×C₆₇	1	16	yes	direct product factor
95	C₂×C₄⋊Q₈	C₂²×C₆₇	1	9	yes	—
96	C₂×C₄×Q₈	C₂²×C₆₇	1	6	yes	—
97	C₂×C₄×Q₈	C₂²×C₆₇	1	6	yes	—
98	C₂×C₄×Q₈	C₂²×C₆₇	1	6	yes	—
99	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
100	#391	C₂²×C₆₇	1	18	yes	—
101	#389	C₂²×C₆₇	1	36	yes	—
102	#384	C₂²×C₆₇	1	9	yes	—
103	C₄×C₄⋊C₄	C₂²×C₆₇	1	18	yes	—
104	C₄×C₄⋊C₄	C₂²×C₆₇	1	18	yes	—
105	C₄×C₄⋊C₄	C₂²×C₆₇	1	18	yes	—
106	C₄³	C₂²×C₆₇	1	9	yes	—
107	C₂²×Q₈	C₂³×C₆₇	1	6	yes	—
108	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
109	C₄⋊Q₈	C₂×C₄×C₆₇	1	72	yes	—
110	C₄×Q₈	—	2	24	no	—
111	C₄×Q₈	—	4	12	no	—
112	C₄×Q₈	C₂×C₄×C₆₇	1	48	yes	—
113	C₄×Q₈	Q₈×C₆₇	1	16	yes	direct product factor
114	C₂×C₄⋊C₄	C₂³×C₆₇	1	18	yes	—
115	C₂×C₄⋊C₄	C₂³×C₆₇	1	18	yes	—
116	C₂×C₄⋊C₄	C₂³×C₆₇	1	18	yes	—
117	C₂×C₄⋊C₄	C₂³×C₆₇	1	6	yes	—
118	C₂×C₄²	C₂³×C₆₇	1	9	yes	—
119	C₂×C₄²	C₂³×C₆₇	1	18	yes	—
120	C₂×C₄²	C₂³×C₆₇	1	9	yes	—
121	C₂×C₄²	C₂³×C₆₇	1	9	yes	—
122	C₂×C₄²	C₂³×C₆₇	1	6	yes	—
123	#95	C₂³×C₆₇	1	36	yes	—
124	C₂×Q₈	—	4	6	no	—
125	C₂×Q₈	—	4	6	no	—
126	C₂×Q₈	C₂²×C₄×C₆₇	1	24	yes	—
127	C₂×Q₈	C₂²×C₄×C₆₇	1	8	yes	—
128	C₂×Q₈	C₂×Q₈×C₆₇	1	8	yes	—
129	C₂×Q₈	#41×C₆₇	1	8	yes	—
130	C₂²×C₄	C₂⁴×C₆₇	1	9	yes	—
131	C₂²×C₄	C₂⁴×C₆₇	1	9	yes	—
132	C₂²×C₄	C₂⁴×C₆₇	1	6	yes	—
133	C₂²×C₄	C₂⁴×C₆₇	1	6	yes	—
134	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
135	C₄⋊C₄	—	2	36	no	—
136	C₄⋊C₄	—	4	18	no	—
137	C₄⋊C₄	C₂²×C₄×C₆₇	1	72	yes	—
138	C₄⋊C₄	C₂×Q₈×C₆₇	1	24	yes	—
139	C₄²	—	2	36	no	—
140	C₄²	—	2	18	no	—
141	C₄²	—	4	9	no	—
142	C₄²	C₂²×C₄×C₆₇	1	36	yes	—
143	C₄²	C₂×Q₈×C₆₇	1	24	yes	—
144	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
145	Q₈	—	4	24	no	—
146	Q₈	C₄×Q₈×C₆₇	1	32	yes	direct product factor
147	C₂×C₄	—	2	6	no	—
148	C₂×C₄	—	2	9	no	—
149	C₂×C₄	—	2	6	no	—
150	C₂×C₄	—	2	9	no	—
151	C₂×C₄	—	2	18	no	—
152	C₂×C₄	—	2	9	no	—
153	C₂×C₄	C₂³×C₄×C₆₇	1	18	yes	—
154	C₂×C₄	C₂³×C₄×C₆₇	1	12	yes	—
155	C₂×C₄	C₂²×Q₈×C₆₇	1	12	yes	—
156	C₂×C₄	C₂²×Q₈×C₆₇	1	12	yes	—
157	C₂×C₄	C₂²×Q₈×C₆₇	1	4	yes	—
158	C₂×C₄	C₂×#41×C₆₇	1	12	yes	—
159	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
160	C₂²	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
161	C₂²	C₂³×Q₈×C₆₇	1	2	yes	—
162	C₂²	C₂²×#41×C₆₇	1	2	yes	—
163	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
164	C₂²	#585×C₆₇	1	1	yes	characteristic
165	C₄	—	2	18	no	—
166	C₄	—	2	12	no	—
167	C₄	—	4	9	no	—
168	C₄	C₂×C₄×Q₈×C₆₇	1	24	yes	—
169	C₄	Q₈²×C₆₇	1	4	yes	direct product factor
170	C₂	C₂²×C₄×Q₈×C₆₇	1	2	yes	—
171	C₂	C₄×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
172	C₂	C₂×Q₈²×C₆₇	1	1	yes	characteristic
173	C₂	Q₈×#41×C₆₇	1	2	yes	—
174	C₂	C₆₇×#3483	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂⁴
commutator	C₂²	C₂⁴×C₄×C₆₇
Frattini, cosocle	C₂³	C₂⁵×C₆₇
socle, prime radical	C₂³×C₆₇	C₂⁵
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
6. Q₈²×C₆₇	C₄	direct (N × H)
25. C₄×Q₈×C₆₇	Q₈	direct (N × H)
58. Q₈×C₆₇	C₄×Q₈	direct (N × H)
81. C₄×C₆₇	Q₈²	direct (N × H)
82. C₄×Q₈²	C₆₇	direct (N × H)
93. C₆₇	C₄×Q₈²	direct (N × H)
94. Q₈²	C₄×C₆₇	direct (N × H)
113. C₄×Q₈	Q₈×C₆₇	direct (N × H)
146. Q₈	C₄×Q₈×C₆₇	direct (N × H)
169. C₄	Q₈²×C₆₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×Q₈²×C₆₇	2
C₆₇×#2369	2
Q₈×C₄⋊C₄×C₆₇	2
C₄×C₄⋊Q₈×C₆₇	2
C₄²×Q₈×C₆₇	2
C₄×Q₈²	67

Subgroups — C4×Q82×C67 #30713421

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×Q₈²×C₆₇ #30713421