Subgroups — C₂×C₆₇×#1553 #30696962

Order 17152.

1254 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#532×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#432×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#431×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1553	C₂	1	8	yes	maximal, direct product factor
5	C₂×#332×C₆₇	C₂	1	2	yes	maximal, complemented
6	C₂×#331×C₆₇	C₂	1	2	yes	maximal, complemented
7	C₂²×#41×C₆₇	—	2	1	no	—
8	#532×C₆₇	—	2	2	no	—
9	#532×C₆₇	—	2	2	no	—
10	#532×C₆₇	C₄	1	4	yes	—
11	#532×C₆₇	C₂²	1	4	yes	—
12	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×#124×C₆₇	—	2	1	no	—
15	C₂×#121×C₆₇	—	2	1	no	—
16	C₂×#121×C₆₇	C₄	1	2	yes	—
17	C₂×C₄×D₄×C₆₇	—	2	1	no	—
18	C₂×C₄×D₄×C₆₇	C₄	1	2	yes	—
19	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
20	#432×C₆₇	—	2	2	no	—
21	#432×C₆₇	C₂²	1	4	yes	complemented
22	#431×C₆₇	—	2	2	no	—
23	#431×C₆₇	C₂²	1	4	yes	complemented
24	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
25	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
26	C₂×C₂²⋊C₈×C₆₇	—	2	1	no	—
27	C₂×C₂²⋊C₈×C₆₇	—	2	1	no	—
28	C₂×#97×C₆₇	—	2	1	no	—
29	C₂×C₄×C₈×C₆₇	—	2	1	no	—
30	#332×C₆₇	C₂²	1	8	yes	complemented
31	#331×C₆₇	C₂²	1	8	yes	complemented
32	C₂×#41×C₆₇	—	2	4	no	—
33	C₂×#41×C₆₇	—	2	4	no	—
34	C₂×#41×C₆₇	—	4	8	no	—
35	C₂²×Q₈×C₆₇	—	2	1	no	—
36	C₂²×D₄×C₆₇	—	2	2	no	—
37	C₂²×D₄×C₆₇	—	2	1	no	—
38	C₂²×D₄×C₆₇	—	2	1	no	—
39	C₂²×D₄×C₆₇	C₂×C₄	1	2	yes	—
40	C₂³×C₄×C₆₇	—	2	2	no	—
41	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
42	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
43	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₆₇	—	2	2	no	—
45	C₂²×C₈×C₆₇	—	2	2	no	—
46	C₄⋊Q₈×C₆₇	—	2	1	no	—
47	C₄⋊Q₈×C₆₇	—	2	1	no	—
48	C₄⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
49	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
50	#127×C₆₇	—	2	1	no	—
51	#127×C₆₇	—	2	1	no	—
52	#127×C₆₇	C₂×C₄	1	2	yes	—
53	#127×C₆₇	C₂³	1	2	yes	—
54	#124×C₆₇	—	2	4	no	—
55	#124×C₆₇	—	2	4	no	—
56	#121×C₆₇	—	2	4	no	—
57	#121×C₆₇	—	2	4	no	—
58	#121×C₆₇	—	2	4	no	—
59	#121×C₆₇	C₂×C₄	1	8	yes	—
60	C₄×D₄×C₆₇	—	2	4	no	—
61	C₄×D₄×C₆₇	—	2	4	no	—
62	C₄×D₄×C₆₇	—	2	4	no	—
63	C₄×D₄×C₆₇	C₂×C₄	1	8	yes	—
64	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
65	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
66	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
67	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
68	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
70	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
71	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
72	C₂×C₄²×C₆₇	D₄	1	2	yes	—
73	C₂×C₄²×C₆₇	D₄	1	2	yes	—
74	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
75	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
76	C₄⋊C₈×C₆₇	—	2	2	no	—
77	C₄⋊C₈×C₆₇	—	2	2	no	—
78	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
79	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
80	C₂²⋊C₈×C₆₇	—	2	4	no	—
81	C₂²⋊C₈×C₆₇	—	2	4	no	—
82	#97×C₆₇	—	2	4	no	—
83	C₄×C₈×C₆₇	—	2	4	no	—
84	C₂⁴×C₆₇	—	2	2	no	—
85	C₂⁴×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
86	#41×C₆₇	—	4	16	no	—
87	#41×C₆₇	—	4	16	no	—
88	C₂×Q₈×C₆₇	—	2	2	no	—
89	C₂×Q₈×C₆₇	—	2	2	no	—
90	C₂×Q₈×C₆₇	—	4	4	no	—
91	C₂×D₄×C₆₇	—	2	8	no	—
92	C₂×D₄×C₆₇	—	2	2	no	—
93	C₂×D₄×C₆₇	—	2	2	no	—
94	C₂×D₄×C₆₇	—	2	2	no	—
95	C₂×D₄×C₆₇	—	2	2	no	—
96	C₂×D₄×C₆₇	—	2	2	no	—
97	C₂×D₄×C₆₇	—	4	4	no	—
98	C₂×D₄×C₆₇	—	4	8	no	—
99	C₂×D₄×C₆₇	—	4	4	no	—
100	C₂×D₄×C₆₇	—	4	4	no	—
101	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
102	C₂²×C₄×C₆₇	—	2	4	no	—
103	C₂²×C₄×C₆₇	—	2	4	no	—
104	C₂²×C₄×C₆₇	—	2	2	no	—
105	C₂²×C₄×C₆₇	—	2	8	no	—
106	C₂²×C₄×C₆₇	—	4	2	no	—
107	C₂²×C₄×C₆₇	—	4	8	no	—
108	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
109	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
110	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
111	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
112	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
113	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
114	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
115	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
116	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
117	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
118	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
119	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
120	C₂×C₈×C₆₇	—	2	4	no	—
121	C₂×C₈×C₆₇	—	2	4	no	—
122	C₂×C₈×C₆₇	—	4	4	no	—
123	C₂×C₈×C₆₇	—	4	4	no	—
124	C₄⋊C₄×C₆₇	—	2	2	no	—
125	C₄⋊C₄×C₆₇	—	2	2	no	—
126	C₄⋊C₄×C₆₇	—	2	2	no	—
127	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
128	C₂²⋊C₄×C₆₇	—	2	8	no	—
129	C₂²⋊C₄×C₆₇	—	2	8	no	—
130	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
131	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
132	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
133	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
134	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
135	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
136	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
137	C₄²×C₆₇	C₂×D₄	1	2	yes	—
138	C₄²×C₆₇	C₂×D₄	1	2	yes	—
139	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
140	C₂³×C₆₇	—	2	2	no	—
141	C₂³×C₆₇	—	2	2	no	—
142	C₂³×C₆₇	—	2	2	no	—
143	C₂³×C₆₇	—	2	4	no	—
144	C₂³×C₆₇	—	4	4	no	—
145	C₂³×C₆₇	—	4	4	no	—
146	C₂³×C₆₇	—	4	4	no	—
147	C₂³×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
148	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
149	Q₈×C₆₇	—	4	4	no	—
150	Q₈×C₆₇	—	4	4	no	—
151	D₄×C₆₇	—	4	4	no	—
152	D₄×C₆₇	—	4	4	no	—
153	D₄×C₆₇	—	4	16	no	—
154	D₄×C₆₇	—	4	4	no	—
155	D₄×C₆₇	—	4	4	no	—
156	D₄×C₆₇	—	4	4	no	—
157	D₄×C₆₇	—	4	4	no	—
158	C₂×C₄×C₆₇	—	2	2	no	—
159	C₂×C₄×C₆₇	—	2	2	no	—
160	C₂×C₄×C₆₇	—	2	2	no	—
161	C₂×C₄×C₆₇	—	2	2	no	—
162	C₂×C₄×C₆₇	—	2	4	no	—
163	C₂×C₄×C₆₇	—	2	4	no	—
164	C₂×C₄×C₆₇	—	2	4	no	—
165	C₂×C₄×C₆₇	—	2	4	no	—
166	C₂×C₄×C₆₇	—	2	4	no	—
167	C₂×C₄×C₆₇	—	4	2	no	—
168	C₂×C₄×C₆₇	—	4	2	no	—
169	C₂×C₄×C₆₇	—	4	16	no	—
170	C₂×C₄×C₆₇	—	4	4	no	—
171	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
172	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
173	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
174	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
175	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
176	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
177	C₂×C₄×C₆₇	#135	1	4	yes	—
178	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	4	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×D₄	1	1	yes	characteristic
182	C₈×C₆₇	—	4	4	no	—
183	C₈×C₆₇	—	4	4	no	—
184	C₂²×C₆₇	—	2	2	no	—
185	C₂²×C₆₇	—	2	2	no	—
186	C₂²×C₆₇	—	2	2	no	—
187	C₂²×C₆₇	—	4	4	no	—
188	C₂²×C₆₇	—	4	4	no	—
189	C₂²×C₆₇	—	4	4	no	—
190	C₂²×C₆₇	—	4	4	no	—
191	C₂²×C₆₇	—	8	8	no	—
192	C₂²×C₆₇	#413	1	2	yes	—
193	C₂²×C₆₇	#416	1	2	yes	—
194	C₂²×C₆₇	#418	1	2	yes	—
195	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
196	C₄×C₆₇	—	2	2	no	—
197	C₄×C₆₇	—	2	1	no	—
198	C₄×C₆₇	—	2	1	no	—
199	C₄×C₆₇	—	2	2	no	—
200	C₄×C₆₇	—	2	2	no	—
201	C₄×C₆₇	—	4	4	no	—
202	C₄×C₆₇	#416	1	2	yes	—
203	C₄×C₆₇	#418	1	2	yes	—
204	C₄×C₆₇	C₂×#135	1	2	yes	—
205	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
206	C₂×#1553	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
207	C₂×C₆₇	—	4	2	no	—
208	C₂×C₆₇	—	8	4	no	—
209	C₂×C₆₇	#1553	1	4	yes	direct product factor
210	C₂×C₆₇	C₂×#413	1	1	yes	characteristic
211	C₂×C₆₇	C₂×#416	1	1	yes	characteristic
212	C₂×C₆₇	C₂×#418	1	1	yes	characteristic
213	C₂×#532	C₂×C₆₇	1	1	yes	characteristic
214	C₂×#432	C₂×C₆₇	1	1	yes	characteristic, complemented
215	C₂×#431	C₂×C₆₇	1	1	yes	characteristic, complemented
216	#1553	C₂×C₆₇	1	8	yes	direct product factor
217	C₂×#332	C₂×C₆₇	1	2	yes	complemented
218	C₂×#331	C₂×C₆₇	1	2	yes	complemented
219	C₆₇	C₂×#1553	1	1	yes	characteristic, direct product factor, Sylow
220	C₂²×#41	—	2	1	no	—
221	#532	—	2	2	no	—
222	#532	—	2	2	no	—
223	#532	C₄×C₆₇	1	4	yes	—
224	#532	C₂²×C₆₇	1	4	yes	—
225	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
226	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
227	C₂×#124	—	2	1	no	—
228	C₂×#121	—	2	1	no	—
229	C₂×#121	C₄×C₆₇	1	2	yes	—
230	C₂×C₄×D₄	—	2	1	no	—
231	C₂×C₄×D₄	C₄×C₆₇	1	2	yes	—
232	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
233	#432	—	2	2	no	—
234	#432	C₂²×C₆₇	1	4	yes	complemented
235	#431	—	2	2	no	—
236	#431	C₂²×C₆₇	1	4	yes	complemented
237	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
238	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
239	C₂×C₂²⋊C₈	—	2	1	no	—
240	C₂×C₂²⋊C₈	—	2	1	no	—
241	C₂×#97	—	2	1	no	—
242	C₂×C₄×C₈	—	2	1	no	—
243	#332	C₂²×C₆₇	1	8	yes	complemented
244	#331	C₂²×C₆₇	1	8	yes	complemented
245	C₂×#41	—	2	4	no	—
246	C₂×#41	—	2	4	no	—
247	C₂×#41	—	4	8	no	—
248	C₂²×Q₈	—	2	1	no	—
249	C₂²×D₄	—	2	2	no	—
250	C₂²×D₄	—	2	1	no	—
251	C₂²×D₄	—	2	1	no	—
252	C₂²×D₄	C₂×C₄×C₆₇	1	2	yes	—
253	C₂³×C₄	—	2	2	no	—
254	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
255	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
256	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
257	C₂²×C₈	—	2	2	no	—
258	C₂²×C₈	—	2	2	no	—
259	C₄⋊Q₈	—	2	1	no	—
260	C₄⋊Q₈	—	2	1	no	—
261	C₄⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
262	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
263	#127	—	2	1	no	—
264	#127	—	2	1	no	—
265	#127	C₂×C₄×C₆₇	1	2	yes	—
266	#127	C₂³×C₆₇	1	2	yes	—
267	#124	—	2	4	no	—
268	#124	—	2	4	no	—
269	#121	—	2	4	no	—
270	#121	—	2	4	no	—
271	#121	—	2	4	no	—
272	#121	C₂×C₄×C₆₇	1	8	yes	—
273	C₄×D₄	—	2	4	no	—
274	C₄×D₄	—	2	4	no	—
275	C₄×D₄	—	2	4	no	—
276	C₄×D₄	C₂×C₄×C₆₇	1	8	yes	—
277	C₂×C₄⋊C₄	—	2	1	no	—
278	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
279	C₂×C₂²⋊C₄	—	2	2	no	—
280	C₂×C₂²⋊C₄	—	2	2	no	—
281	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
282	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
283	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
284	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
285	C₂×C₄²	D₄×C₆₇	1	2	yes	—
286	C₂×C₄²	D₄×C₆₇	1	2	yes	—
287	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
288	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
289	C₄⋊C₈	—	2	2	no	—
290	C₄⋊C₈	—	2	2	no	—
291	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
292	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
293	C₂²⋊C₈	—	2	4	no	—
294	C₂²⋊C₈	—	2	4	no	—
295	#97	—	2	4	no	—
296	C₄×C₈	—	2	4	no	—
297	C₂⁴	—	2	2	no	—
298	C₂⁴	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
299	#41	—	4	16	no	—
300	#41	—	4	16	no	—
301	C₂×Q₈	—	2	2	no	—
302	C₂×Q₈	—	2	2	no	—
303	C₂×Q₈	—	4	4	no	—
304	C₂×D₄	—	2	8	no	—
305	C₂×D₄	—	2	2	no	—
306	C₂×D₄	—	2	2	no	—
307	C₂×D₄	—	2	2	no	—
308	C₂×D₄	—	2	2	no	—
309	C₂×D₄	—	2	2	no	—
310	C₂×D₄	—	4	4	no	—
311	C₂×D₄	—	4	8	no	—
312	C₂×D₄	—	4	4	no	—
313	C₂×D₄	—	4	4	no	—
314	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
315	C₂²×C₄	—	2	4	no	—
316	C₂²×C₄	—	2	4	no	—
317	C₂²×C₄	—	2	2	no	—
318	C₂²×C₄	—	2	8	no	—
319	C₂²×C₄	—	4	2	no	—
320	C₂²×C₄	—	4	8	no	—
321	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
322	C₂²×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
323	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
324	C₂²×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
325	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
326	C₂²×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
327	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
328	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
329	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
330	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
331	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
332	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
333	C₂×C₈	—	2	4	no	—
334	C₂×C₈	—	2	4	no	—
335	C₂×C₈	—	4	4	no	—
336	C₂×C₈	—	4	4	no	—
337	C₄⋊C₄	—	2	2	no	—
338	C₄⋊C₄	—	2	2	no	—
339	C₄⋊C₄	—	2	2	no	—
340	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
341	C₂²⋊C₄	—	2	8	no	—
342	C₂²⋊C₄	—	2	8	no	—
343	C₄²	C₂²⋊C₄×C₆₇	1	2	yes	—
344	C₄²	C₂²⋊C₄×C₆₇	1	2	yes	—
345	C₄²	C₂²⋊C₄×C₆₇	1	2	yes	—
346	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
347	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
348	C₄²	C₂×D₄×C₆₇	1	2	yes	—
349	C₄²	C₂×D₄×C₆₇	1	2	yes	—
350	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
351	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
352	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
353	C₂³	—	2	2	no	—
354	C₂³	—	2	2	no	—
355	C₂³	—	2	2	no	—
356	C₂³	—	2	4	no	—
357	C₂³	—	4	4	no	—
358	C₂³	—	4	4	no	—
359	C₂³	—	4	4	no	—
360	C₂³	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
361	C₂³	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
362	Q₈	—	4	4	no	—
363	Q₈	—	4	4	no	—
364	D₄	—	4	4	no	—
365	D₄	—	4	4	no	—
366	D₄	—	4	16	no	—
367	D₄	—	4	4	no	—
368	D₄	—	4	4	no	—
369	D₄	—	4	4	no	—
370	D₄	—	4	4	no	—
371	C₂×C₄	—	2	2	no	—
372	C₂×C₄	—	2	2	no	—
373	C₂×C₄	—	2	2	no	—
374	C₂×C₄	—	2	2	no	—
375	C₂×C₄	—	2	4	no	—
376	C₂×C₄	—	2	4	no	—
377	C₂×C₄	—	2	4	no	—
378	C₂×C₄	—	2	4	no	—
379	C₂×C₄	—	2	4	no	—
380	C₂×C₄	—	4	2	no	—
381	C₂×C₄	—	4	2	no	—
382	C₂×C₄	—	4	16	no	—
383	C₂×C₄	—	4	4	no	—
384	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
385	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
386	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
387	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
388	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
389	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
390	C₂×C₄	#135×C₆₇	1	4	yes	—
391	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	4	yes	—
392	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
393	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
394	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
395	C₈	—	4	4	no	—
396	C₈	—	4	4	no	—
397	C₂²	—	2	2	no	—
398	C₂²	—	2	2	no	—
399	C₂²	—	2	2	no	—
400	C₂²	—	4	4	no	—
401	C₂²	—	4	4	no	—
402	C₂²	—	4	4	no	—
403	C₂²	—	4	4	no	—
404	C₂²	—	8	8	no	—
405	C₂²	#413×C₆₇	1	2	yes	—
406	C₂²	#416×C₆₇	1	2	yes	—
407	C₂²	#418×C₆₇	1	2	yes	—
408	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
409	C₄	—	2	1	no	—
410	C₄	—	2	2	no	—
411	C₄	—	2	1	no	—
412	C₄	—	2	2	no	—
413	C₄	—	2	2	no	—
414	C₄	—	4	4	no	—
415	C₄	#416×C₆₇	1	2	yes	—
416	C₄	#418×C₆₇	1	2	yes	—
417	C₄	C₂×#135×C₆₇	1	2	yes	—
418	C₄	C₂×AGL(1,ℤ/8)×C₆₇	1	2	yes	—
419	C₂	—	4	2	no	—
420	C₂	—	8	4	no	—
421	C₂	C₆₇×#1553	1	4	yes	direct product factor
422	C₂	C₂×#413×C₆₇	1	1	yes	characteristic
423	C₂	C₂×#416×C₆₇	1	1	yes	characteristic
424	C₂	C₂×#418×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₄²	C₂⁴×C₆₇
prime radical	C₂×#532×C₆₇	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ 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
2. C₂×#432×C₆₇	C₂	semidirect
3. C₂×#431×C₆₇	C₂	semidirect
4. C₆₇×#1553	C₂	direct (N × H)
5. C₂×#332×C₆₇	C₂	semidirect
6. C₂×#331×C₆₇	C₂	semidirect
21. #432×C₆₇	C₂²	semidirect
23. #431×C₆₇	C₂²	semidirect
30. #332×C₆₇	C₂²	semidirect
31. #331×C₆₇	C₂²	semidirect
206. C₂×#1553	C₆₇	direct (N × H)
209. C₂×C₆₇	#1553	direct (N × H)
214. C₂×#432	C₂×C₆₇	semidirect
215. C₂×#431	C₂×C₆₇	semidirect
216. #1553	C₂×C₆₇	direct (N × H)
217. C₂×#332	C₂×C₆₇	semidirect
218. C₂×#331	C₂×C₆₇	semidirect
219. C₆₇	C₂×#1553	direct (N × H)
234. #432	C₂²×C₆₇	semidirect
236. #431	C₂²×C₆₇	semidirect
243. #332	C₂²×C₆₇	semidirect
244. #331	C₂²×C₆₇	semidirect
421. C₂	C₆₇×#1553	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×#532×C₆₇	2
C₂×#432×C₆₇	2
C₂×#431×C₆₇	2
C₆₇×#1553	2
C₂×#332×C₆₇	2
C₂×#331×C₆₇	2
C₂×#1553	67

Subgroups — C2×C67×#1553 #30696962

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#1553 #30696962