Subgroups — C₂²×#396×C₆₇ #30713787

Order 17152.

5838 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	C₂³×C₄⋊C₄×C₆₇	C₂	1	3	yes	maximal, complemented
2	C₂³×C₂²⋊C₄×C₆₇	C₂	1	3	yes	maximal
3	C₂×#396×C₆₇	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#95×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂⁶×C₆₇	C₂²	1	1	yes	characteristic
6	C₂⁴×C₄×C₆₇	C₂²	1	3	yes	—
7	C₂⁴×C₄×C₆₇	C₂²	1	3	yes	—
8	C₂²×C₄⋊C₄×C₆₇	—	2	24	no	—
9	C₂²×C₄⋊C₄×C₆₇	C₂²	1	36	yes	complemented
10	C₂²×C₂²⋊C₄×C₆₇	—	2	24	no	—
11	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	36	yes	—
12	#396×C₆₇	C₂²	1	64	yes	direct product factor
13	C₂×#95×C₆₇	C₂²	1	12	yes	complemented
14	C₂⁵×C₆₇	—	2	12	no	—
15	C₂⁵×C₆₇	—	2	12	no	—
16	C₂⁵×C₆₇	Q₈	1	8	yes	—
17	C₂⁵×C₆₇	C₂³	1	6	yes	—
18	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
19	C₂³×C₄×C₆₇	—	2	12	no	—
20	C₂³×C₄×C₆₇	—	2	12	no	—
21	C₂³×C₄×C₆₇	—	2	12	no	—
22	C₂³×C₄×C₆₇	—	2	12	no	—
23	C₂³×C₄×C₆₇	—	2	12	no	—
24	C₂³×C₄×C₆₇	D₄	1	24	yes	—
25	C₂³×C₄×C₆₇	C₂³	1	18	yes	—
26	C₂³×C₄×C₆₇	C₂³	1	18	yes	—
27	C₂×C₄⋊C₄×C₆₇	—	2	144	no	—
28	C₂×C₄⋊C₄×C₆₇	C₂³	1	48	yes	complemented
29	C₂×C₂²⋊C₄×C₆₇	—	2	144	no	—
30	C₂×C₂²⋊C₄×C₆₇	C₂³	1	48	yes	—
31	#95×C₆₇	C₂³	1	16	yes	complemented
32	C₂⁴×C₆₇	—	2	96	no	—
33	C₂⁴×C₆₇	—	2	36	no	—
34	C₂⁴×C₆₇	—	2	36	no	—
35	C₂⁴×C₆₇	—	4	48	no	—
36	C₂⁴×C₆₇	—	4	16	no	—
37	C₂⁴×C₆₇	C₂×D₄	1	12	yes	—
38	C₂⁴×C₆₇	C₂×Q₈	1	24	yes	—
39	C₂⁴×C₆₇	C₂×Q₈	1	4	yes	—
40	C₂⁴×C₆₇	#41	1	12	yes	—
41	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
42	C₂⁴×C₆₇	C₂⁴	1	3	yes	—
43	C₂²×C₄×C₆₇	—	2	96	no	—
44	C₂²×C₄×C₆₇	—	2	36	no	—
45	C₂²×C₄×C₆₇	—	2	36	no	—
46	C₂²×C₄×C₆₇	—	2	36	no	—
47	C₂²×C₄×C₆₇	—	2	36	no	—
48	C₂²×C₄×C₆₇	—	2	36	no	—
49	C₂²×C₄×C₆₇	—	4	48	no	—
50	C₂²×C₄×C₆₇	C₂×D₄	1	72	yes	—
51	C₂²×C₄×C₆₇	C₂⁴	1	12	yes	—
52	C₂²×C₄×C₆₇	C₂⁴	1	12	yes	—
53	C₄⋊C₄×C₆₇	—	2	96	no	—
54	C₂²⋊C₄×C₆₇	—	2	96	no	—
55	C₂³×C₆₇	—	2	144	no	—
56	C₂³×C₆₇	—	2	12	no	—
57	C₂³×C₆₇	—	2	12	no	—
58	C₂³×C₆₇	—	4	72	no	—
59	C₂³×C₆₇	—	4	24	no	—
60	C₂³×C₆₇	—	4	128	no	—
61	C₂³×C₆₇	#121	1	48	yes	—
62	C₂³×C₆₇	C₂²⋊Q₈	1	48	yes	—
63	C₂³×C₆₇	#126	1	16	yes	—
64	C₂³×C₆₇	C₂²×D₄	1	18	yes	—
65	C₂³×C₆₇	C₂²×Q₈	1	8	yes	—
66	C₂³×C₆₇	C₂²×Q₈	1	6	yes	—
67	C₂³×C₆₇	C₂×#41	1	18	yes	—
68	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
69	C₂×C₄×C₆₇	—	2	144	no	—
70	C₂×C₄×C₆₇	—	2	12	no	—
71	C₂×C₄×C₆₇	—	2	12	no	—
72	C₂×C₄×C₆₇	—	2	12	no	—
73	C₂×C₄×C₆₇	—	2	12	no	—
74	C₂×C₄×C₆₇	—	2	12	no	—
75	C₂×C₄×C₆₇	—	4	72	no	—
76	C₂×C₄×C₆₇	C₂²×D₄	1	24	yes	—
77	C₂²×C₆₇	—	2	24	no	—
78	C₂²×C₆₇	—	4	96	no	—
79	C₂²×C₆₇	—	4	12	no	—
80	C₂²×C₆₇	—	4	4	no	—
81	C₂²×C₆₇	#396	1	64	yes	direct product factor
82	C₂²×C₆₇	C₂×#121	1	36	yes	—
83	C₂²×C₆₇	C₂×C₂²⋊Q₈	1	36	yes	—
84	C₂²×C₆₇	C₂×#126	1	12	yes	—
85	C₂²×C₆₇	C₂³×D₄	1	3	yes	—
86	C₂²×C₆₇	C₂³×Q₈	1	1	yes	characteristic
87	C₂²×C₆₇	C₂²×#41	1	3	yes	—
88	C₄×C₆₇	—	2	24	no	—
89	C₄×C₆₇	—	4	12	no	—
90	C₂²×#396	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₂×C₆₇	—	4	8	no	—
92	C₂×C₆₇	C₂×#396	1	24	yes	direct product factor
93	C₂×C₆₇	C₂²×#121	1	3	yes	—
94	C₂×C₆₇	C₂²×C₂²⋊Q₈	1	3	yes	—
95	C₂×C₆₇	C₂²×#126	1	1	yes	characteristic
96	C₂³×C₄⋊C₄	C₂×C₆₇	1	3	yes	complemented
97	C₂³×C₂²⋊C₄	C₂×C₆₇	1	3	yes	—
98	C₂×#396	C₂×C₆₇	1	24	yes	direct product factor
99	C₂²×#95	C₂×C₆₇	1	1	yes	characteristic, complemented
100	C₆₇	C₂²×#396	1	1	yes	characteristic, direct product factor, Sylow
101	C₂⁶	C₂²×C₆₇	1	1	yes	characteristic
102	C₂⁴×C₄	C₂²×C₆₇	1	3	yes	—
103	C₂⁴×C₄	C₂²×C₆₇	1	3	yes	—
104	C₂²×C₄⋊C₄	—	2	24	no	—
105	C₂²×C₄⋊C₄	C₂²×C₆₇	1	36	yes	complemented
106	C₂²×C₂²⋊C₄	—	2	24	no	—
107	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	36	yes	—
108	#396	C₂²×C₆₇	1	64	yes	direct product factor
109	C₂×#95	C₂²×C₆₇	1	12	yes	complemented
110	C₂⁵	—	2	12	no	—
111	C₂⁵	—	2	12	no	—
112	C₂⁵	Q₈×C₆₇	1	8	yes	—
113	C₂⁵	C₂³×C₆₇	1	6	yes	—
114	C₂⁵	C₂³×C₆₇	1	1	yes	characteristic
115	C₂³×C₄	—	2	12	no	—
116	C₂³×C₄	—	2	12	no	—
117	C₂³×C₄	—	2	12	no	—
118	C₂³×C₄	—	2	12	no	—
119	C₂³×C₄	—	2	12	no	—
120	C₂³×C₄	D₄×C₆₇	1	24	yes	—
121	C₂³×C₄	C₂³×C₆₇	1	18	yes	—
122	C₂³×C₄	C₂³×C₆₇	1	18	yes	—
123	C₂×C₄⋊C₄	—	2	144	no	—
124	C₂×C₄⋊C₄	C₂³×C₆₇	1	48	yes	complemented
125	C₂×C₂²⋊C₄	—	2	144	no	—
126	C₂×C₂²⋊C₄	C₂³×C₆₇	1	48	yes	—
127	#95	C₂³×C₆₇	1	16	yes	complemented
128	C₂⁴	—	2	96	no	—
129	C₂⁴	—	2	36	no	—
130	C₂⁴	—	2	36	no	—
131	C₂⁴	—	4	48	no	—
132	C₂⁴	—	4	16	no	—
133	C₂⁴	C₂×D₄×C₆₇	1	12	yes	—
134	C₂⁴	C₂×Q₈×C₆₇	1	24	yes	—
135	C₂⁴	C₂×Q₈×C₆₇	1	4	yes	—
136	C₂⁴	#41×C₆₇	1	12	yes	—
137	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
138	C₂⁴	C₂⁴×C₆₇	1	3	yes	—
139	C₂²×C₄	—	2	96	no	—
140	C₂²×C₄	—	2	36	no	—
141	C₂²×C₄	—	2	36	no	—
142	C₂²×C₄	—	2	36	no	—
143	C₂²×C₄	—	2	36	no	—
144	C₂²×C₄	—	2	36	no	—
145	C₂²×C₄	—	4	48	no	—
146	C₂²×C₄	C₂×D₄×C₆₇	1	72	yes	—
147	C₂²×C₄	C₂⁴×C₆₇	1	12	yes	—
148	C₂²×C₄	C₂⁴×C₆₇	1	12	yes	—
149	C₄⋊C₄	—	2	96	no	—
150	C₂²⋊C₄	—	2	96	no	—
151	C₂³	—	2	144	no	—
152	C₂³	—	2	12	no	—
153	C₂³	—	2	12	no	—
154	C₂³	—	4	72	no	—
155	C₂³	—	4	24	no	—
156	C₂³	—	4	128	no	—
157	C₂³	#121×C₆₇	1	48	yes	—
158	C₂³	C₂²⋊Q₈×C₆₇	1	48	yes	—
159	C₂³	#126×C₆₇	1	16	yes	—
160	C₂³	C₂²×D₄×C₆₇	1	18	yes	—
161	C₂³	C₂²×Q₈×C₆₇	1	8	yes	—
162	C₂³	C₂²×Q₈×C₆₇	1	6	yes	—
163	C₂³	C₂×#41×C₆₇	1	18	yes	—
164	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
165	C₂×C₄	—	2	144	no	—
166	C₂×C₄	—	2	12	no	—
167	C₂×C₄	—	2	12	no	—
168	C₂×C₄	—	2	12	no	—
169	C₂×C₄	—	2	12	no	—
170	C₂×C₄	—	2	12	no	—
171	C₂×C₄	—	4	72	no	—
172	C₂×C₄	C₂²×D₄×C₆₇	1	24	yes	—
173	C₂²	—	2	24	no	—
174	C₂²	—	4	96	no	—
175	C₂²	—	4	12	no	—
176	C₂²	—	4	4	no	—
177	C₂²	#396×C₆₇	1	64	yes	direct product factor
178	C₂²	C₂×#121×C₆₇	1	36	yes	—
179	C₂²	C₂×C₂²⋊Q₈×C₆₇	1	36	yes	—
180	C₂²	C₂×#126×C₆₇	1	12	yes	—
181	C₂²	C₂³×D₄×C₆₇	1	3	yes	—
182	C₂²	C₂³×Q₈×C₆₇	1	1	yes	characteristic
183	C₂²	C₂²×#41×C₆₇	1	3	yes	—
184	C₄	—	2	24	no	—
185	C₄	—	4	12	no	—
186	C₂	—	4	8	no	—
187	C₂	C₂×#396×C₆₇	1	24	yes	direct product factor
188	C₂	C₂²×#121×C₆₇	1	3	yes	—
189	C₂	C₂²×C₂²⋊Q₈×C₆₇	1	3	yes	—
190	C₂	C₂²×#126×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁵×C₆₇	C₂³
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
prime radical	C₂⁶×C₆₇	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×C₄⋊C₄×C₆₇	C₂	semidirect
3. C₂×#396×C₆₇	C₂	direct (N × H)
4. C₂²×#95×C₆₇	C₂	semidirect
9. C₂²×C₄⋊C₄×C₆₇	C₂²	semidirect
12. #396×C₆₇	C₂²	direct (N × H)
13. C₂×#95×C₆₇	C₂²	semidirect
28. C₂×C₄⋊C₄×C₆₇	C₂³	semidirect
31. #95×C₆₇	C₂³	semidirect
81. C₂²×C₆₇	#396	direct (N × H)
90. C₂²×#396	C₆₇	direct (N × H)
92. C₂×C₆₇	C₂×#396	direct (N × H)
96. C₂³×C₄⋊C₄	C₂×C₆₇	semidirect
98. C₂×#396	C₂×C₆₇	direct (N × H)
99. C₂²×#95	C₂×C₆₇	semidirect
100. C₆₇	C₂²×#396	direct (N × H)
105. C₂²×C₄⋊C₄	C₂²×C₆₇	semidirect
108. #396	C₂²×C₆₇	direct (N × H)
109. C₂×#95	C₂²×C₆₇	semidirect
124. C₂×C₄⋊C₄	C₂³×C₆₇	semidirect
127. #95	C₂³×C₆₇	semidirect
177. C₂²	#396×C₆₇	direct (N × H)
187. C₂	C₂×#396×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
C₂³×C₄⋊C₄×C₆₇	2
C₂³×C₂²⋊C₄×C₆₇	2
C₂×#396×C₆₇	2
C₂²×#95×C₆₇	2
C₂²×#396	67

Subgroups — C22×#396×C67 #30713787

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#396×C₆₇ #30713787