Subgroups — C₆₇×#21170 #30700421

Order 17152.

1082 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#428×C₆₇	C₂	1	2	yes	maximal
2	C₂×#423×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2285	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1949	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1948	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1947	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1736	C₂	1	2	yes	maximal
9	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
10	C₂×#117×C₆₇	C₂²	1	2	yes	—
11	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
12	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
13	C₂²×C₄²×C₆₇	—	2	1	no	—
14	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
15	#428×C₆₇	—	2	8	no	—
16	C₂×#107×C₆₇	C₂²	1	4	yes	—
17	C₂×#106×C₆₇	C₂²	1	4	yes	—
18	#423×C₆₇	C₄	1	8	yes	complemented
19	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
20	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
21	#384×C₆₇	C₂²	1	1	yes	characteristic
22	#384×C₆₇	C₂²	1	1	yes	characteristic
23	#382×C₆₇	C₂²	1	2	yes	—
24	#380×C₆₇	—	2	1	no	—
25	#380×C₆₇	—	2	1	no	—
26	C₄×C₄⋊C₄×C₆₇	—	2	1	no	—
27	C₄×C₄⋊C₄×C₆₇	—	2	1	no	—
28	C₄×C₄⋊C₄×C₆₇	—	2	1	no	—
29	C₄×C₄⋊C₄×C₆₇	—	2	1	no	—
30	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
31	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
32	C₂×#95×C₆₇	—	2	1	no	—
33	#340×C₆₇	C₂²	1	4	yes	—
34	#340×C₆₇	C₂²	1	4	yes	—
35	C₂³×C₄×C₆₇	—	2	1	no	—
36	C₂³×C₄×C₆₇	—	2	1	no	—
37	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
38	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
39	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
40	C₂×M₁₆×C₆₇	—	2	4	no	—
41	C₂×M₁₆×C₆₇	—	2	4	no	—
42	C₂×M₁₆×C₆₇	C₂×C₄	1	8	yes	—
43	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
44	#117×C₆₇	—	2	4	no	—
45	#117×C₆₇	—	2	4	no	—
46	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
47	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
48	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
49	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
50	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
51	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
52	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
53	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
54	C₂×C₄⋊C₄×C₆₇	D₄	1	4	yes	—
55	C₂×C₄⋊C₄×C₆₇	Q₈	1	4	yes	—
56	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
57	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
58	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
59	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
60	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
61	C₂×C₄²×C₆₇	—	2	2	no	—
62	C₂×C₄²×C₆₇	—	2	2	no	—
63	C₂×C₄²×C₆₇	—	2	2	no	—
64	C₂×C₄²×C₆₇	—	4	4	no	—
65	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
66	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
67	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
68	C₂×C₄²×C₆₇	Q₈	1	4	yes	—
69	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
70	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
71	#107×C₆₇	—	2	8	no	—
72	#106×C₆₇	—	2	8	no	—
73	C₄⋊C₈×C₆₇	C₂×C₄	1	8	yes	complemented
74	C₄⋊C₈×C₆₇	C₂×C₄	1	8	yes	complemented
75	#95×C₆₇	—	2	2	no	—
76	#95×C₆₇	—	2	2	no	—
77	#95×C₆₇	—	2	2	no	—
78	#95×C₆₇	—	2	2	no	—
79	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₆₇	—	2	2	no	—
81	C₂²×C₄×C₆₇	—	2	2	no	—
82	C₂²×C₄×C₆₇	—	2	4	no	—
83	C₂²×C₄×C₆₇	—	2	4	no	—
84	C₂²×C₄×C₆₇	—	2	2	no	—
85	C₂²×C₄×C₆₇	—	2	2	no	—
86	C₂²×C₄×C₆₇	—	2	2	no	—
87	C₂²×C₄×C₆₇	—	2	2	no	—
88	C₂²×C₄×C₆₇	—	4	1	no	—
89	C₂²×C₄×C₆₇	—	4	1	no	—
90	C₂²×C₄×C₆₇	—	4	2	no	—
91	C₂²×C₄×C₆₇	—	4	2	no	—
92	C₂²×C₄×C₆₇	—	4	2	no	—
93	C₂²×C₄×C₆₇	—	4	2	no	—
94	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
95	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
96	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
97	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
99	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
103	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
104	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
105	C₂²×C₄×C₆₇	#41	1	2	yes	—
106	C₂²×C₄×C₆₇	#41	1	2	yes	—
107	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
108	M₁₆×C₆₇	—	2	16	no	—
109	C₂×C₈×C₆₇	—	2	4	no	—
110	C₂×C₈×C₆₇	—	2	4	no	—
111	C₂×C₈×C₆₇	C₂²×C₄	1	8	yes	—
112	C₄⋊C₄×C₆₇	—	2	4	no	—
113	C₄⋊C₄×C₆₇	—	2	4	no	—
114	C₄⋊C₄×C₆₇	—	2	2	no	—
115	C₄⋊C₄×C₆₇	—	2	2	no	—
116	C₄⋊C₄×C₆₇	—	2	2	no	—
117	C₄⋊C₄×C₆₇	—	2	2	no	—
118	C₄⋊C₄×C₆₇	—	2	2	no	—
119	C₄⋊C₄×C₆₇	—	2	2	no	—
120	C₄⋊C₄×C₆₇	—	4	2	no	—
121	C₄⋊C₄×C₆₇	—	4	2	no	—
122	C₄⋊C₄×C₆₇	—	4	2	no	—
123	C₄⋊C₄×C₆₇	—	4	2	no	—
124	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
125	C₄⋊C₄×C₆₇	C₂×Q₈	1	4	yes	—
126	C₂²⋊C₄×C₆₇	—	4	4	no	—
127	C₄²×C₆₇	—	4	4	no	—
128	C₄²×C₆₇	—	4	4	no	—
129	C₄²×C₆₇	—	4	4	no	—
130	C₄²×C₆₇	C₄⋊C₄	1	4	yes	—
131	C₄²×C₆₇	C₄⋊C₄	1	4	yes	—
132	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
133	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
134	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
135	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
136	C₂³×C₆₇	—	2	1	no	—
137	C₂³×C₆₇	—	2	1	no	—
138	C₂³×C₆₇	—	2	1	no	—
139	C₂³×C₆₇	—	2	1	no	—
140	C₂³×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
141	C₂³×C₆₇	#121	1	2	yes	—
142	C₂³×C₆₇	C₂²⋊Q₈	1	2	yes	—
143	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₄×C₆₇	—	2	1	no	—
145	C₂×C₄×C₆₇	—	2	1	no	—
146	C₂×C₄×C₆₇	—	2	1	no	—
147	C₂×C₄×C₆₇	—	2	1	no	—
148	C₂×C₄×C₆₇	—	2	1	no	—
149	C₂×C₄×C₆₇	—	2	1	no	—
150	C₂×C₄×C₆₇	—	2	8	no	—
151	C₂×C₄×C₆₇	—	2	4	no	—
152	C₂×C₄×C₆₇	—	2	4	no	—
153	C₂×C₄×C₆₇	—	4	4	no	—
154	C₂×C₄×C₆₇	—	4	8	no	—
155	C₂×C₄×C₆₇	—	4	8	no	—
156	C₂×C₄×C₆₇	—	4	2	no	—
157	C₂×C₄×C₆₇	—	4	2	no	—
158	C₂×C₄×C₆₇	—	4	2	no	—
159	C₂×C₄×C₆₇	—	4	2	no	—
160	C₂×C₄×C₆₇	—	4	2	no	—
161	C₂×C₄×C₆₇	—	4	2	no	—
162	C₂×C₄×C₆₇	—	4	2	no	—
163	C₂×C₄×C₆₇	—	4	2	no	—
164	C₂×C₄×C₆₇	—	4	2	no	—
165	C₂×C₄×C₆₇	—	4	2	no	—
166	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
167	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
168	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
169	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
170	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
171	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
172	C₂×C₄×C₆₇	C₄×Q₈	1	4	yes	—
173	C₂×C₄×C₆₇	#121	1	2	yes	—
174	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
175	C₂×C₄×C₆₇	#125	1	4	yes	—
176	C₂×C₄×C₆₇	C₄⋊Q₈	1	4	yes	—
177	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
178	C₂×C₄×C₆₇	#137	1	2	yes	—
179	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
180	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
181	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
182	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
183	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
184	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
185	C₈×C₆₇	—	2	8	no	—
186	C₂²×C₆₇	—	2	2	no	—
187	C₂²×C₆₇	—	2	2	no	—
188	C₂²×C₆₇	—	2	2	no	—
189	C₂²×C₆₇	—	2	2	no	—
190	C₂²×C₆₇	—	2	2	no	—
191	C₂²×C₆₇	—	2	2	no	—
192	C₂²×C₆₇	#389	1	4	yes	—
193	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
194	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
195	C₂²×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
196	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
197	C₂²×C₆₇	C₂×#137	1	1	yes	characteristic
198	C₂²×C₆₇	#575	1	1	yes	characteristic
199	C₂²×C₆₇	#575	1	1	yes	characteristic
200	C₄×C₆₇	—	2	2	no	—
201	C₄×C₆₇	—	2	2	no	—
202	C₄×C₆₇	—	2	2	no	—
203	C₄×C₆₇	—	4	4	no	—
204	C₄×C₆₇	—	4	4	no	—
205	C₄×C₆₇	—	8	2	no	—
206	C₄×C₆₇	—	8	2	no	—
207	C₄×C₆₇	#389	1	4	yes	—
208	C₄×C₆₇	#428	1	4	yes	complemented
209	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
210	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
211	C₄×C₆₇	C₂×#125	1	1	yes	characteristic
212	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
213	#21170	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
214	C₂×C₆₇	—	2	4	no	—
215	C₂×C₆₇	C₂×#389	1	1	yes	characteristic
216	C₂×C₆₇	C₂×#428	1	1	yes	characteristic
217	C₂×C₆₇	#2911	1	1	yes	characteristic
218	C₂×C₆₇	#3036	1	1	yes	characteristic
219	C₂×C₆₇	#3037	1	1	yes	characteristic
220	C₂×C₆₇	#3077	1	1	yes	characteristic
221	C₂×C₆₇	#3078	1	1	yes	characteristic
222	C₂×#428	C₂×C₆₇	1	2	yes	—
223	C₂×#423	C₂×C₆₇	1	1	yes	characteristic
224	#2285	C₂×C₆₇	1	1	yes	characteristic
225	C₂×C₄×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic
226	#1949	C₂×C₆₇	1	4	yes	complemented
227	#1948	C₂×C₆₇	1	2	yes	complemented
228	#1947	C₂×C₆₇	1	2	yes	complemented
229	#1736	C₂×C₆₇	1	2	yes	—
230	C₆₇	#21170	1	1	yes	characteristic, direct product factor, Sylow
231	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
232	C₂×#117	C₂²×C₆₇	1	2	yes	—
233	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
234	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
235	C₂²×C₄²	—	2	1	no	—
236	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
237	#428	—	2	8	no	—
238	C₂×#107	C₂²×C₆₇	1	4	yes	—
239	C₂×#106	C₂²×C₆₇	1	4	yes	—
240	#423	C₄×C₆₇	1	8	yes	complemented
241	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
242	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
243	#384	C₂²×C₆₇	1	1	yes	characteristic
244	#384	C₂²×C₆₇	1	1	yes	characteristic
245	#382	C₂²×C₆₇	1	2	yes	—
246	#380	—	2	1	no	—
247	#380	—	2	1	no	—
248	C₄×C₄⋊C₄	—	2	1	no	—
249	C₄×C₄⋊C₄	—	2	1	no	—
250	C₄×C₄⋊C₄	—	2	1	no	—
251	C₄×C₄⋊C₄	—	2	1	no	—
252	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
253	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
254	C₂×#95	—	2	1	no	—
255	#340	C₂²×C₆₇	1	4	yes	—
256	#340	C₂²×C₆₇	1	4	yes	—
257	C₂³×C₄	—	2	1	no	—
258	C₂³×C₄	—	2	1	no	—
259	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
260	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
261	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
262	C₂×M₁₆	—	2	4	no	—
263	C₂×M₁₆	—	2	4	no	—
264	C₂×M₁₆	C₂×C₄×C₆₇	1	8	yes	—
265	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
266	#117	—	2	4	no	—
267	#117	—	2	4	no	—
268	C₂×C₄⋊C₄	—	2	1	no	—
269	C₂×C₄⋊C₄	—	2	1	no	—
270	C₂×C₄⋊C₄	—	2	2	no	—
271	C₂×C₄⋊C₄	—	2	2	no	—
272	C₂×C₄⋊C₄	—	2	1	no	—
273	C₂×C₄⋊C₄	—	2	1	no	—
274	C₂×C₄⋊C₄	—	2	1	no	—
275	C₂×C₄⋊C₄	—	2	1	no	—
276	C₂×C₄⋊C₄	D₄×C₆₇	1	4	yes	—
277	C₂×C₄⋊C₄	Q₈×C₆₇	1	4	yes	—
278	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
279	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
280	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
281	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
282	C₂×C₂²⋊C₄	—	2	2	no	—
283	C₂×C₄²	—	2	2	no	—
284	C₂×C₄²	—	2	2	no	—
285	C₂×C₄²	—	2	2	no	—
286	C₂×C₄²	—	4	4	no	—
287	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
288	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
289	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
290	C₂×C₄²	Q₈×C₆₇	1	4	yes	—
291	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
292	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
293	#107	—	2	8	no	—
294	#106	—	2	8	no	—
295	C₄⋊C₈	C₂×C₄×C₆₇	1	8	yes	complemented
296	C₄⋊C₈	C₂×C₄×C₆₇	1	8	yes	complemented
297	#95	—	2	2	no	—
298	#95	—	2	2	no	—
299	#95	—	2	2	no	—
300	#95	—	2	2	no	—
301	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
302	C₂²×C₄	—	2	2	no	—
303	C₂²×C₄	—	2	2	no	—
304	C₂²×C₄	—	2	4	no	—
305	C₂²×C₄	—	2	4	no	—
306	C₂²×C₄	—	2	2	no	—
307	C₂²×C₄	—	2	2	no	—
308	C₂²×C₄	—	2	2	no	—
309	C₂²×C₄	—	2	2	no	—
310	C₂²×C₄	—	4	1	no	—
311	C₂²×C₄	—	4	1	no	—
312	C₂²×C₄	—	4	2	no	—
313	C₂²×C₄	—	4	2	no	—
314	C₂²×C₄	—	4	2	no	—
315	C₂²×C₄	—	4	2	no	—
316	C₂²×C₄	C₄⋊C₄×C₆₇	1	4	yes	—
317	C₂²×C₄	C₄⋊C₄×C₆₇	1	4	yes	—
318	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
319	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
320	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
321	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
322	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
323	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
324	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
325	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
326	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
327	C₂²×C₄	#41×C₆₇	1	2	yes	—
328	C₂²×C₄	#41×C₆₇	1	2	yes	—
329	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
330	M₁₆	—	2	16	no	—
331	C₂×C₈	—	2	4	no	—
332	C₂×C₈	—	2	4	no	—
333	C₂×C₈	C₂²×C₄×C₆₇	1	8	yes	—
334	C₄⋊C₄	—	2	4	no	—
335	C₄⋊C₄	—	2	4	no	—
336	C₄⋊C₄	—	2	2	no	—
337	C₄⋊C₄	—	2	2	no	—
338	C₄⋊C₄	—	2	2	no	—
339	C₄⋊C₄	—	2	2	no	—
340	C₄⋊C₄	—	2	2	no	—
341	C₄⋊C₄	—	2	2	no	—
342	C₄⋊C₄	—	4	2	no	—
343	C₄⋊C₄	—	4	2	no	—
344	C₄⋊C₄	—	4	2	no	—
345	C₄⋊C₄	—	4	2	no	—
346	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
347	C₄⋊C₄	C₂×Q₈×C₆₇	1	4	yes	—
348	C₂²⋊C₄	—	4	4	no	—
349	C₄²	—	4	4	no	—
350	C₄²	—	4	4	no	—
351	C₄²	—	4	4	no	—
352	C₄²	C₄⋊C₄×C₆₇	1	4	yes	—
353	C₄²	C₄⋊C₄×C₆₇	1	4	yes	—
354	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
355	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
356	C₄²	C₂×Q₈×C₆₇	1	2	yes	—
357	C₄²	C₂×Q₈×C₆₇	1	2	yes	—
358	C₂³	—	2	1	no	—
359	C₂³	—	2	1	no	—
360	C₂³	—	2	1	no	—
361	C₂³	—	2	1	no	—
362	C₂³	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
363	C₂³	#121×C₆₇	1	2	yes	—
364	C₂³	C₂²⋊Q₈×C₆₇	1	2	yes	—
365	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
366	C₂×C₄	—	2	1	no	—
367	C₂×C₄	—	2	1	no	—
368	C₂×C₄	—	2	1	no	—
369	C₂×C₄	—	2	1	no	—
370	C₂×C₄	—	2	1	no	—
371	C₂×C₄	—	2	1	no	—
372	C₂×C₄	—	2	8	no	—
373	C₂×C₄	—	2	4	no	—
374	C₂×C₄	—	2	4	no	—
375	C₂×C₄	—	4	4	no	—
376	C₂×C₄	—	4	8	no	—
377	C₂×C₄	—	4	8	no	—
378	C₂×C₄	—	4	2	no	—
379	C₂×C₄	—	4	2	no	—
380	C₂×C₄	—	4	2	no	—
381	C₂×C₄	—	4	2	no	—
382	C₂×C₄	—	4	2	no	—
383	C₂×C₄	—	4	2	no	—
384	C₂×C₄	—	4	2	no	—
385	C₂×C₄	—	4	2	no	—
386	C₂×C₄	—	4	2	no	—
387	C₂×C₄	—	4	2	no	—
388	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
389	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
390	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
391	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
392	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
393	C₂×C₄	C₄×D₄×C₆₇	1	4	yes	—
394	C₂×C₄	C₄×Q₈×C₆₇	1	4	yes	—
395	C₂×C₄	#121×C₆₇	1	2	yes	—
396	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
397	C₂×C₄	#125×C₆₇	1	4	yes	—
398	C₂×C₄	C₄⋊Q₈×C₆₇	1	4	yes	—
399	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
400	C₂×C₄	#137×C₆₇	1	2	yes	—
401	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
402	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
403	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
404	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
405	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
406	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
407	C₈	—	2	8	no	—
408	C₂²	—	2	2	no	—
409	C₂²	—	2	2	no	—
410	C₂²	—	2	2	no	—
411	C₂²	—	2	2	no	—
412	C₂²	—	2	2	no	—
413	C₂²	—	2	2	no	—
414	C₂²	#389×C₆₇	1	4	yes	—
415	C₂²	C₂²×C₄⋊C₄×C₆₇	1	1	yes	characteristic
416	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
417	C₂²	C₂×C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
418	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
419	C₂²	C₂×#137×C₆₇	1	1	yes	characteristic
420	C₂²	#575×C₆₇	1	1	yes	characteristic
421	C₂²	#575×C₆₇	1	1	yes	characteristic
422	C₄	—	2	2	no	—
423	C₄	—	2	2	no	—
424	C₄	—	2	2	no	—
425	C₄	—	4	4	no	—
426	C₄	—	4	4	no	—
427	C₄	—	8	2	no	—
428	C₄	—	8	2	no	—
429	C₄	#389×C₆₇	1	4	yes	—
430	C₄	#428×C₆₇	1	4	yes	complemented
431	C₄	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
432	C₄	C₂×C₄×Q₈×C₆₇	1	1	yes	characteristic
433	C₄	C₂×#125×C₆₇	1	1	yes	characteristic
434	C₄	C₂×C₄⋊Q₈×C₆₇	1	1	yes	characteristic
435	C₂	—	2	4	no	—
436	C₂	C₂×#389×C₆₇	1	1	yes	characteristic
437	C₂	C₂×#428×C₆₇	1	1	yes	characteristic
438	C₂	C₆₇×#2911	1	1	yes	characteristic
439	C₂	C₆₇×#3036	1	1	yes	characteristic
440	C₂	C₆₇×#3037	1	1	yes	characteristic
441	C₂	C₆₇×#3077	1	1	yes	characteristic
442	C₂	C₆₇×#3078	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂⁴×C₆₇	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₆₇×#1949	C₂	semidirect
6. C₆₇×#1948	C₂	semidirect
7. C₆₇×#1947	C₂	semidirect
18. #423×C₆₇	C₄	semidirect
73. C₄⋊C₈×C₆₇	C₂×C₄	semidirect
74. C₄⋊C₈×C₆₇	C₂×C₄	semidirect
208. C₄×C₆₇	#428	semidirect
213. #21170	C₆₇	direct (N × H)
226. #1949	C₂×C₆₇	semidirect
227. #1948	C₂×C₆₇	semidirect
228. #1947	C₂×C₆₇	semidirect
230. C₆₇	#21170	direct (N × H)
240. #423	C₄×C₆₇	semidirect
295. C₄⋊C₈	C₂×C₄×C₆₇	semidirect
296. C₄⋊C₈	C₂×C₄×C₆₇	semidirect
430. C₄	#428×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#428×C₆₇	2
C₂×#423×C₆₇	2
C₆₇×#2285	2
C₂×C₄×C₄⋊C₄×C₆₇	2
C₆₇×#1949	2
C₆₇×#1948	2
C₆₇×#1947	2
C₆₇×#1736	2
#21170	67

Subgroups — C67×#21170 #30700421

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21170 #30700421