Subgroups — C₆₇×#33234 #30712485

Order 17152.

946 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#491×C₆₇	C₂	1	2	yes	maximal, complemented
2	C₂×#490×C₆₇	C₂	1	2	yes	maximal
3	C₆₇×#2859	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2858	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1959	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1958	C₂	1	4	yes	maximal, complemented
7	C₄×#97×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
9	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
10	C₂×#125×C₆₇	C₂²	1	2	yes	—
11	C₂×#124×C₆₇	C₂²	1	2	yes	—
12	C₂×#124×C₆₇	C₂²	1	2	yes	—
13	#491×C₆₇	—	2	8	no	—
14	#490×C₆₇	—	2	8	no	—
15	C₂×#103×C₆₇	C₂²	1	8	yes	—
16	C₂×#102×C₆₇	C₂²	1	8	yes	—
17	C₂×#97×C₆₇	C₂²	1	2	yes	—
18	C₂×#97×C₆₇	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
20	#390×C₆₇	—	2	4	no	—
21	#389×C₆₇	—	2	4	no	—
22	#384×C₆₇	C₂²	1	2	yes	—
23	#382×C₆₇	C₂²	1	2	yes	—
24	C₄³×C₆₇	C₂²	1	1	yes	characteristic
25	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
26	C₂²×D₄×C₆₇	—	2	2	no	—
27	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
28	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
29	C₄⋊Q₈×C₆₇	—	2	4	no	—
30	#127×C₆₇	—	2	4	no	—
31	#125×C₆₇	—	2	4	no	—
32	#125×C₆₇	—	2	4	no	—
33	#124×C₆₇	—	2	8	no	—
34	#124×C₆₇	—	2	8	no	—
35	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
36	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
37	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
38	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
39	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
40	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
41	C₂×C₄²×C₆₇	D₄	1	4	yes	—
42	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
45	#103×C₆₇	—	2	16	no	—
46	#102×C₆₇	—	2	16	no	—
47	#97×C₆₇	—	2	4	no	—
48	#97×C₆₇	D₄	1	8	yes	—
49	C₄×C₈×C₆₇	—	2	4	no	—
50	#95×C₆₇	—	2	4	no	—
51	C₂⁴×C₆₇	—	2	2	no	—
52	C₂×Q₈×C₆₇	—	2	4	no	—
53	C₂×Q₈×C₆₇	—	2	4	no	—
54	C₂×Q₈×C₆₇	—	2	4	no	—
55	C₂×D₄×C₆₇	—	2	4	no	—
56	C₂×D₄×C₆₇	—	2	4	no	—
57	C₂×D₄×C₆₇	—	2	4	no	—
58	C₂×D₄×C₆₇	—	4	8	no	—
59	C₂×D₄×C₆₇	—	4	4	no	—
60	C₂²×C₄×C₆₇	—	2	2	no	—
61	C₂²×C₄×C₆₇	—	2	4	no	—
62	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
66	C₂×C₈×C₆₇	—	2	4	no	—
67	C₂×C₈×C₆₇	—	2	4	no	—
68	C₂×C₈×C₆₇	C₂×D₄	1	8	yes	—
69	C₄⋊C₄×C₆₇	—	2	4	no	—
70	C₄⋊C₄×C₆₇	—	2	4	no	—
71	C₄⋊C₄×C₆₇	—	4	8	no	—
72	C₄⋊C₄×C₆₇	—	4	8	no	—
73	C₄⋊C₄×C₆₇	—	4	4	no	—
74	C₂²⋊C₄×C₆₇	—	4	8	no	—
75	C₂²⋊C₄×C₆₇	—	4	8	no	—
76	C₄²×C₆₇	—	4	4	no	—
77	C₄²×C₆₇	C₂×D₄	1	4	yes	—
78	C₄²×C₆₇	#41	1	4	yes	—
79	C₄²×C₆₇	#41	1	4	yes	—
80	C₂³×C₆₇	—	4	2	no	—
81	C₂³×C₆₇	—	4	2	no	—
82	C₂³×C₆₇	—	4	2	no	—
83	C₂³×C₆₇	—	4	8	no	—
84	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
85	Q₈×C₆₇	—	4	8	no	—
86	D₄×C₆₇	—	4	8	no	—
87	D₄×C₆₇	—	8	8	no	—
88	C₂×C₄×C₆₇	—	2	4	no	—
89	C₂×C₄×C₆₇	—	2	4	no	—
90	C₂×C₄×C₆₇	—	2	4	no	—
91	C₂×C₄×C₆₇	—	4	2	no	—
92	C₂×C₄×C₆₇	—	4	2	no	—
93	C₂×C₄×C₆₇	—	4	4	no	—
94	C₂×C₄×C₆₇	—	4	2	no	—
95	C₂×C₄×C₆₇	—	4	8	no	—
96	C₂×C₄×C₆₇	#124	1	4	yes	—
97	C₂×C₄×C₆₇	#124	1	4	yes	—
98	C₂×C₄×C₆₇	#127	1	4	yes	—
99	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
100	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
101	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
102	C₈×C₆₇	—	2	8	no	—
103	C₂²×C₆₇	—	8	2	no	—
104	C₂²×C₆₇	—	8	4	no	—
105	C₂²×C₆₇	—	8	4	no	—
106	C₂²×C₆₇	—	8	4	no	—
107	C₂²×C₆₇	C₂×#124	1	1	yes	characteristic
108	C₂²×C₆₇	C₂×#124	1	1	yes	characteristic
109	C₂²×C₆₇	C₂×#127	1	1	yes	characteristic
110	C₂²×C₆₇	#575	1	4	yes	—
111	C₄×C₆₇	—	2	4	no	—
112	C₄×C₆₇	—	2	4	no	—
113	C₄×C₆₇	—	2	4	no	—
114	C₄×C₆₇	—	8	2	no	—
115	C₄×C₆₇	—	8	4	no	—
116	C₄×C₆₇	#532	1	4	yes	—
117	#33234	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
118	C₂×C₆₇	—	16	2	no	—
119	C₂×C₆₇	#2858	1	1	yes	characteristic
120	C₂×C₆₇	#3135	1	2	yes	—
121	C₂×C₆₇	#3142	1	2	yes	—
122	C₂×C₆₇	#3153	1	2	yes	—
123	C₂×#491	C₂×C₆₇	1	2	yes	complemented
124	C₂×#490	C₂×C₆₇	1	2	yes	—
125	#2859	C₂×C₆₇	1	1	yes	characteristic, complemented
126	#2858	C₂×C₆₇	1	1	yes	characteristic
127	#1959	C₂×C₆₇	1	4	yes	complemented
128	#1958	C₂×C₆₇	1	4	yes	complemented
129	C₄×#97	C₂×C₆₇	1	1	yes	characteristic, complemented
130	C₆₇	#33234	1	1	yes	characteristic, direct product factor, Sylow
131	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
132	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
133	C₂×#125	C₂²×C₆₇	1	2	yes	—
134	C₂×#124	C₂²×C₆₇	1	2	yes	—
135	C₂×#124	C₂²×C₆₇	1	2	yes	—
136	#491	—	2	8	no	—
137	#490	—	2	8	no	—
138	C₂×#103	C₂²×C₆₇	1	8	yes	—
139	C₂×#102	C₂²×C₆₇	1	8	yes	—
140	C₂×#97	C₂²×C₆₇	1	2	yes	—
141	C₂×#97	C₂²×C₆₇	1	2	yes	—
142	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
143	#390	—	2	4	no	—
144	#389	—	2	4	no	—
145	#384	C₂²×C₆₇	1	2	yes	—
146	#382	C₂²×C₆₇	1	2	yes	—
147	C₄³	C₂²×C₆₇	1	1	yes	characteristic
148	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
149	C₂²×D₄	—	2	2	no	—
150	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
151	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
152	C₄⋊Q₈	—	2	4	no	—
153	#127	—	2	4	no	—
154	#125	—	2	4	no	—
155	#125	—	2	4	no	—
156	#124	—	2	8	no	—
157	#124	—	2	8	no	—
158	C₂×C₄⋊C₄	—	2	4	no	—
159	C₂×C₄⋊C₄	—	2	2	no	—
160	C₂×C₄⋊C₄	—	2	4	no	—
161	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
162	C₂×C₂²⋊C₄	—	2	4	no	—
163	C₂×C₂²⋊C₄	—	2	4	no	—
164	C₂×C₄²	D₄×C₆₇	1	4	yes	—
165	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
166	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
167	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
168	#103	—	2	16	no	—
169	#102	—	2	16	no	—
170	#97	—	2	4	no	—
171	#97	D₄×C₆₇	1	8	yes	—
172	C₄×C₈	—	2	4	no	—
173	#95	—	2	4	no	—
174	C₂⁴	—	2	2	no	—
175	C₂×Q₈	—	2	4	no	—
176	C₂×Q₈	—	2	4	no	—
177	C₂×Q₈	—	2	4	no	—
178	C₂×D₄	—	2	4	no	—
179	C₂×D₄	—	2	4	no	—
180	C₂×D₄	—	2	4	no	—
181	C₂×D₄	—	4	8	no	—
182	C₂×D₄	—	4	4	no	—
183	C₂²×C₄	—	2	4	no	—
184	C₂²×C₄	—	2	2	no	—
185	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
186	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
187	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
188	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
189	C₂×C₈	—	2	4	no	—
190	C₂×C₈	—	2	4	no	—
191	C₂×C₈	C₂×D₄×C₆₇	1	8	yes	—
192	C₄⋊C₄	—	2	4	no	—
193	C₄⋊C₄	—	2	4	no	—
194	C₄⋊C₄	—	4	8	no	—
195	C₄⋊C₄	—	4	8	no	—
196	C₄⋊C₄	—	4	4	no	—
197	C₂²⋊C₄	—	4	8	no	—
198	C₂²⋊C₄	—	4	8	no	—
199	C₄²	—	4	4	no	—
200	C₄²	C₂×D₄×C₆₇	1	4	yes	—
201	C₄²	#41×C₆₇	1	4	yes	—
202	C₄²	#41×C₆₇	1	4	yes	—
203	C₂³	—	4	2	no	—
204	C₂³	—	4	2	no	—
205	C₂³	—	4	2	no	—
206	C₂³	—	4	8	no	—
207	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
208	Q₈	—	4	8	no	—
209	D₄	—	4	8	no	—
210	D₄	—	8	8	no	—
211	C₂×C₄	—	2	4	no	—
212	C₂×C₄	—	2	4	no	—
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	—	4	2	no	—
216	C₂×C₄	—	4	4	no	—
217	C₂×C₄	—	4	2	no	—
218	C₂×C₄	—	4	8	no	—
219	C₂×C₄	#124×C₆₇	1	4	yes	—
220	C₂×C₄	#124×C₆₇	1	4	yes	—
221	C₂×C₄	#127×C₆₇	1	4	yes	—
222	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
223	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
224	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
225	C₈	—	2	8	no	—
226	C₂²	—	8	2	no	—
227	C₂²	—	8	4	no	—
228	C₂²	—	8	4	no	—
229	C₂²	—	8	4	no	—
230	C₂²	C₂×#124×C₆₇	1	1	yes	characteristic
231	C₂²	C₂×#124×C₆₇	1	1	yes	characteristic
232	C₂²	C₂×#127×C₆₇	1	1	yes	characteristic
233	C₂²	#575×C₆₇	1	4	yes	—
234	C₄	—	2	4	no	—
235	C₄	—	2	4	no	—
236	C₄	—	2	4	no	—
237	C₄	—	8	2	no	—
238	C₄	—	8	4	no	—
239	C₄	#532×C₆₇	1	4	yes	—
240	C₂	—	16	2	no	—
241	C₂	C₆₇×#2858	1	1	yes	characteristic
242	C₂	C₆₇×#3135	1	2	yes	—
243	C₂	C₆₇×#3142	1	2	yes	—
244	C₂	C₆₇×#3153	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂×#127×C₆₇	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₄³×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#491×C₆₇	C₂	semidirect
3. C₆₇×#2859	C₂	semidirect
5. C₆₇×#1959	C₂	semidirect
6. C₆₇×#1958	C₂	semidirect
7. C₄×#97×C₆₇	C₂	semidirect
117. #33234	C₆₇	direct (N × H)
123. C₂×#491	C₂×C₆₇	semidirect
125. #2859	C₂×C₆₇	semidirect
127. #1959	C₂×C₆₇	semidirect
128. #1958	C₂×C₆₇	semidirect
129. C₄×#97	C₂×C₆₇	semidirect
130. C₆₇	#33234	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×#491×C₆₇	2
C₂×#490×C₆₇	2
C₆₇×#2859	2
C₆₇×#2858	2
C₆₇×#1959	2
C₆₇×#1958	2
C₄×#97×C₆₇	2
#33234	67

Subgroups — C67×#33234 #30712485

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33234 #30712485