Subgroups — C₂×C₆₇×#1551 #30696960

Order 17152.

1310 conjugacy classes of subgroups.

The subgroup lattice has 442 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	C₂²×C₄⋊C₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1551	C₂	1	8	yes	maximal, direct product factor
5	C₂×#333×C₆₇	C₂	1	2	yes	maximal, complemented
6	C₂×#331×C₆₇	C₂	1	2	yes	maximal, complemented
7	C₂²×#41×C₆₇	C₄	1	2	yes	—
8	C₂³×C₈×C₆₇	—	2	1	no	—
9	#532×C₆₇	—	2	2	no	—
10	#532×C₆₇	—	2	2	no	—
11	#532×C₆₇	C₄	1	4	yes	—
12	#532×C₆₇	C₂²	1	4	yes	—
13	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×#124×C₆₇	C₄	1	2	yes	—
16	C₂×#121×C₆₇	—	2	1	no	—
17	C₂×#121×C₆₇	—	2	1	no	—
18	C₂×C₄×D₄×C₆₇	—	2	1	no	—
19	C₂×C₄×D₄×C₆₇	—	2	1	no	—
20	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
21	#432×C₆₇	—	2	2	no	—
22	#432×C₆₇	C₂²	1	4	yes	complemented
23	C₂×C₄⋊C₈×C₆₇	—	2	2	no	—
24	C₂×C₄⋊C₈×C₆₇	—	2	1	no	—
25	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
26	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
27	C₂×C₄⋊C₈×C₆₇	C₂²	1	4	yes	complemented
28	C₂×C₂²⋊C₈×C₆₇	—	2	1	no	—
29	C₂×#97×C₆₇	—	2	1	no	—
30	#333×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	—	4	8	no	—
34	C₂×#41×C₆₇	C₂×C₄	1	8	yes	—
35	C₂²×Q₈×C₆₇	C₂×C₄	1	2	yes	—
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	4	no	—
46	C₂²×C₈×C₆₇	—	2	2	no	—
47	C₂²×C₈×C₆₇	—	4	4	no	—
48	C₄⋊Q₈×C₆₇	—	2	1	no	—
49	C₄⋊Q₈×C₆₇	—	2	1	no	—
50	C₄⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
51	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
52	#127×C₆₇	—	2	1	no	—
53	#127×C₆₇	—	2	1	no	—
54	#127×C₆₇	C₂×C₄	1	2	yes	—
55	#127×C₆₇	C₂³	1	2	yes	—
56	#124×C₆₇	—	2	4	no	—
57	#124×C₆₇	C₂×C₄	1	8	yes	—
58	#121×C₆₇	—	2	4	no	—
59	#121×C₆₇	—	2	4	no	—
60	#121×C₆₇	—	2	4	no	—
61	#121×C₆₇	—	2	4	no	—
62	C₄×D₄×C₆₇	—	2	4	no	—
63	C₄×D₄×C₆₇	—	2	4	no	—
64	C₄×D₄×C₆₇	—	2	4	no	—
65	C₄×D₄×C₆₇	—	2	4	no	—
66	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
67	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
68	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
69	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
70	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
71	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
72	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
73	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
74	C₂×C₄²×C₆₇	D₄	1	2	yes	—
75	C₂×C₄²×C₆₇	D₄	1	2	yes	—
76	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
77	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
78	C₄⋊C₈×C₆₇	—	2	2	no	—
79	C₄⋊C₈×C₆₇	—	2	4	no	—
80	C₄⋊C₈×C₆₇	—	2	2	no	—
81	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
82	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
83	C₂²⋊C₈×C₆₇	—	2	4	no	—
84	#97×C₆₇	—	2	4	no	—
85	C₂⁴×C₆₇	—	2	2	no	—
86	C₂⁴×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
87	#41×C₆₇	—	4	16	no	—
88	#41×C₆₇	—	4	16	no	—
89	C₂×Q₈×C₆₇	—	2	2	no	—
90	C₂×Q₈×C₆₇	—	4	4	no	—
91	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
92	C₂×D₄×C₆₇	—	2	8	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	—	2	2	no	—
98	C₂×D₄×C₆₇	—	4	4	no	—
99	C₂×D₄×C₆₇	—	4	8	no	—
100	C₂×D₄×C₆₇	—	4	4	no	—
101	C₂×D₄×C₆₇	—	4	2	no	—
102	C₂×D₄×C₆₇	—	4	2	no	—
103	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
104	C₂²×C₄×C₆₇	—	2	4	no	—
105	C₂²×C₄×C₆₇	—	2	4	no	—
106	C₂²×C₄×C₆₇	—	2	2	no	—
107	C₂²×C₄×C₆₇	—	2	2	no	—
108	C₂²×C₄×C₆₇	—	2	2	no	—
109	C₂²×C₄×C₆₇	—	2	8	no	—
110	C₂²×C₄×C₆₇	—	4	8	no	—
111	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
112	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
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	1	yes	characteristic
116	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
117	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
118	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
119	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
120	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
121	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
122	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
123	C₂×C₈×C₆₇	—	2	4	no	—
124	C₂×C₈×C₆₇	—	2	4	no	—
125	C₂×C₈×C₆₇	—	4	8	no	—
126	C₂×C₈×C₆₇	—	4	4	no	—
127	C₂×C₈×C₆₇	—	4	4	no	—
128	C₄⋊C₄×C₆₇	—	2	2	no	—
129	C₄⋊C₄×C₆₇	—	2	2	no	—
130	C₄⋊C₄×C₆₇	—	2	2	no	—
131	C₄⋊C₄×C₆₇	—	2	2	no	—
132	C₂²⋊C₄×C₆₇	—	2	8	no	—
133	C₂²⋊C₄×C₆₇	—	2	8	no	—
134	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
135	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
136	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
137	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
138	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
139	C₄²×C₆₇	C₂×D₄	1	2	yes	—
140	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
141	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
142	C₄²×C₆₇	C₂×D₄	1	2	yes	—
143	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
144	C₂³×C₆₇	—	2	2	no	—
145	C₂³×C₆₇	—	2	2	no	—
146	C₂³×C₆₇	—	2	4	no	—
147	C₂³×C₆₇	—	4	4	no	—
148	C₂³×C₆₇	—	4	4	no	—
149	C₂³×C₆₇	—	4	4	no	—
150	C₂³×C₆₇	#100	1	4	yes	—
151	C₂³×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
152	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
153	Q₈×C₆₇	—	4	4	no	—
154	Q₈×C₆₇	—	4	4	no	—
155	D₄×C₆₇	—	4	4	no	—
156	D₄×C₆₇	—	4	4	no	—
157	D₄×C₆₇	—	4	16	no	—
158	D₄×C₆₇	—	4	4	no	—
159	D₄×C₆₇	—	4	2	no	—
160	D₄×C₆₇	—	4	2	no	—
161	D₄×C₆₇	—	4	4	no	—
162	D₄×C₆₇	—	4	4	no	—
163	C₂×C₄×C₆₇	—	2	2	no	—
164	C₂×C₄×C₆₇	—	2	4	no	—
165	C₂×C₄×C₆₇	—	2	2	no	—
166	C₂×C₄×C₆₇	—	2	4	no	—
167	C₂×C₄×C₆₇	—	2	2	no	—
168	C₂×C₄×C₆₇	—	2	2	no	—
169	C₂×C₄×C₆₇	—	2	4	no	—
170	C₂×C₄×C₆₇	—	2	2	no	—
171	C₂×C₄×C₆₇	—	2	2	no	—
172	C₂×C₄×C₆₇	—	2	4	no	—
173	C₂×C₄×C₆₇	—	2	4	no	—
174	C₂×C₄×C₆₇	—	2	4	no	—
175	C₂×C₄×C₆₇	—	4	16	no	—
176	C₂×C₄×C₆₇	—	4	4	no	—
177	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
178	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
179	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
180	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
181	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
182	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
183	C₂×C₄×C₆₇	#135	1	4	yes	—
184	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
185	C₂×C₄×C₆₇	#137	1	2	yes	—
186	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
187	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
188	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
189	C₈×C₆₇	—	4	4	no	—
190	C₈×C₆₇	—	4	4	no	—
191	C₂²×C₆₇	—	2	8	no	—
192	C₂²×C₆₇	—	2	2	no	—
193	C₂²×C₆₇	—	2	2	no	—
194	C₂²×C₆₇	—	4	4	no	—
195	C₂²×C₆₇	—	4	4	no	—
196	C₂²×C₆₇	—	4	4	no	—
197	C₂²×C₆₇	—	8	8	no	—
198	C₂²×C₆₇	C₂×#100	1	4	yes	—
199	C₂²×C₆₇	C₂×#100	1	2	yes	—
200	C₂²×C₆₇	#416	1	2	yes	—
201	C₂²×C₆₇	#417	1	2	yes	—
202	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
203	C₄×C₆₇	—	2	2	no	—
204	C₄×C₆₇	—	2	1	no	—
205	C₄×C₆₇	—	2	1	no	—
206	C₄×C₆₇	—	2	2	no	—
207	C₄×C₆₇	—	2	2	no	—
208	C₄×C₆₇	—	4	4	no	—
209	C₄×C₆₇	#416	1	2	yes	—
210	C₄×C₆₇	#417	1	2	yes	—
211	C₄×C₆₇	C₂×#135	1	2	yes	—
212	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
213	C₄×C₆₇	C₂×#137	1	1	yes	characteristic
214	C₂×#1551	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
215	C₂×C₆₇	—	2	4	no	—
216	C₂×C₆₇	—	8	4	no	—
217	C₂×C₆₇	#1551	1	4	yes	direct product factor
218	C₂×C₆₇	C₂²×#100	1	1	yes	characteristic
219	C₂×C₆₇	C₂×#416	1	1	yes	characteristic
220	C₂×C₆₇	C₂×#417	1	1	yes	characteristic
221	C₂×#532	C₂×C₆₇	1	1	yes	characteristic
222	C₂×#432	C₂×C₆₇	1	1	yes	characteristic, complemented
223	C₂²×C₄⋊C₈	C₂×C₆₇	1	1	yes	characteristic, complemented
224	#1551	C₂×C₆₇	1	8	yes	direct product factor
225	C₂×#333	C₂×C₆₇	1	2	yes	complemented
226	C₂×#331	C₂×C₆₇	1	2	yes	complemented
227	C₆₇	C₂×#1551	1	1	yes	characteristic, direct product factor, Sylow
228	C₂²×#41	C₄×C₆₇	1	2	yes	—
229	C₂³×C₈	—	2	1	no	—
230	#532	—	2	2	no	—
231	#532	—	2	2	no	—
232	#532	C₄×C₆₇	1	4	yes	—
233	#532	C₂²×C₆₇	1	4	yes	—
234	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
235	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
236	C₂×#124	C₄×C₆₇	1	2	yes	—
237	C₂×#121	—	2	1	no	—
238	C₂×#121	—	2	1	no	—
239	C₂×C₄×D₄	—	2	1	no	—
240	C₂×C₄×D₄	—	2	1	no	—
241	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
242	#432	—	2	2	no	—
243	#432	C₂²×C₆₇	1	4	yes	complemented
244	C₂×C₄⋊C₈	—	2	2	no	—
245	C₂×C₄⋊C₈	—	2	1	no	—
246	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
247	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
248	C₂×C₄⋊C₈	C₂²×C₆₇	1	4	yes	complemented
249	C₂×C₂²⋊C₈	—	2	1	no	—
250	C₂×#97	—	2	1	no	—
251	#333	C₂²×C₆₇	1	8	yes	complemented
252	#331	C₂²×C₆₇	1	8	yes	complemented
253	C₂×#41	—	2	4	no	—
254	C₂×#41	—	4	8	no	—
255	C₂×#41	C₂×C₄×C₆₇	1	8	yes	—
256	C₂²×Q₈	C₂×C₄×C₆₇	1	2	yes	—
257	C₂²×D₄	—	2	2	no	—
258	C₂²×D₄	—	2	1	no	—
259	C₂²×D₄	—	2	1	no	—
260	C₂²×D₄	C₂×C₄×C₆₇	1	2	yes	—
261	C₂³×C₄	—	2	2	no	—
262	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
263	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
264	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
265	C₂²×C₈	—	2	2	no	—
266	C₂²×C₈	—	2	4	no	—
267	C₂²×C₈	—	2	2	no	—
268	C₂²×C₈	—	4	4	no	—
269	C₄⋊Q₈	—	2	1	no	—
270	C₄⋊Q₈	—	2	1	no	—
271	C₄⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
272	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
273	#127	—	2	1	no	—
274	#127	—	2	1	no	—
275	#127	C₂×C₄×C₆₇	1	2	yes	—
276	#127	C₂³×C₆₇	1	2	yes	—
277	#124	—	2	4	no	—
278	#124	C₂×C₄×C₆₇	1	8	yes	—
279	#121	—	2	4	no	—
280	#121	—	2	4	no	—
281	#121	—	2	4	no	—
282	#121	—	2	4	no	—
283	C₄×D₄	—	2	4	no	—
284	C₄×D₄	—	2	4	no	—
285	C₄×D₄	—	2	4	no	—
286	C₄×D₄	—	2	4	no	—
287	C₂×C₄⋊C₄	—	2	1	no	—
288	C₂×C₄⋊C₄	—	2	1	no	—
289	C₂×C₂²⋊C₄	—	2	2	no	—
290	C₂×C₂²⋊C₄	—	2	2	no	—
291	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
292	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
293	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
294	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
295	C₂×C₄²	D₄×C₆₇	1	2	yes	—
296	C₂×C₄²	D₄×C₆₇	1	2	yes	—
297	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
298	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
299	C₄⋊C₈	—	2	2	no	—
300	C₄⋊C₈	—	2	4	no	—
301	C₄⋊C₈	—	2	2	no	—
302	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
303	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
304	C₂²⋊C₈	—	2	4	no	—
305	#97	—	2	4	no	—
306	C₂⁴	—	2	2	no	—
307	C₂⁴	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
308	#41	—	4	16	no	—
309	#41	—	4	16	no	—
310	C₂×Q₈	—	2	2	no	—
311	C₂×Q₈	—	4	4	no	—
312	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
313	C₂×D₄	—	2	8	no	—
314	C₂×D₄	—	2	2	no	—
315	C₂×D₄	—	2	2	no	—
316	C₂×D₄	—	2	2	no	—
317	C₂×D₄	—	2	2	no	—
318	C₂×D₄	—	2	2	no	—
319	C₂×D₄	—	4	4	no	—
320	C₂×D₄	—	4	8	no	—
321	C₂×D₄	—	4	4	no	—
322	C₂×D₄	—	4	2	no	—
323	C₂×D₄	—	4	2	no	—
324	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
325	C₂²×C₄	—	2	4	no	—
326	C₂²×C₄	—	2	4	no	—
327	C₂²×C₄	—	2	2	no	—
328	C₂²×C₄	—	2	2	no	—
329	C₂²×C₄	—	2	2	no	—
330	C₂²×C₄	—	2	8	no	—
331	C₂²×C₄	—	4	8	no	—
332	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
333	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
334	C₂²×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
335	C₂²×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
336	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
337	C₂²×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
338	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
339	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
340	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
341	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
342	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
343	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
344	C₂×C₈	—	2	4	no	—
345	C₂×C₈	—	2	4	no	—
346	C₂×C₈	—	4	8	no	—
347	C₂×C₈	—	4	4	no	—
348	C₂×C₈	—	4	4	no	—
349	C₄⋊C₄	—	2	2	no	—
350	C₄⋊C₄	—	2	2	no	—
351	C₄⋊C₄	—	2	2	no	—
352	C₄⋊C₄	—	2	2	no	—
353	C₂²⋊C₄	—	2	8	no	—
354	C₂²⋊C₄	—	2	8	no	—
355	C₄²	C₂²⋊C₄×C₆₇	1	2	yes	—
356	C₄²	C₂²⋊C₄×C₆₇	1	2	yes	—
357	C₄²	C₂²⋊C₄×C₆₇	1	2	yes	—
358	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
359	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
360	C₄²	C₂×D₄×C₆₇	1	2	yes	—
361	C₄²	C₂×D₄×C₆₇	1	2	yes	—
362	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
363	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
364	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
365	C₂³	—	2	2	no	—
366	C₂³	—	2	2	no	—
367	C₂³	—	2	4	no	—
368	C₂³	—	4	4	no	—
369	C₂³	—	4	4	no	—
370	C₂³	—	4	4	no	—
371	C₂³	#100×C₆₇	1	4	yes	—
372	C₂³	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
373	C₂³	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
374	Q₈	—	4	4	no	—
375	Q₈	—	4	4	no	—
376	D₄	—	4	4	no	—
377	D₄	—	4	4	no	—
378	D₄	—	4	16	no	—
379	D₄	—	4	4	no	—
380	D₄	—	4	2	no	—
381	D₄	—	4	2	no	—
382	D₄	—	4	4	no	—
383	D₄	—	4	4	no	—
384	C₂×C₄	—	2	2	no	—
385	C₂×C₄	—	2	4	no	—
386	C₂×C₄	—	2	2	no	—
387	C₂×C₄	—	2	4	no	—
388	C₂×C₄	—	2	2	no	—
389	C₂×C₄	—	2	2	no	—
390	C₂×C₄	—	2	4	no	—
391	C₂×C₄	—	2	2	no	—
392	C₂×C₄	—	2	2	no	—
393	C₂×C₄	—	2	4	no	—
394	C₂×C₄	—	2	4	no	—
395	C₂×C₄	—	2	4	no	—
396	C₂×C₄	—	4	16	no	—
397	C₂×C₄	—	4	4	no	—
398	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
399	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
400	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
401	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
402	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
403	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
404	C₂×C₄	#135×C₆₇	1	4	yes	—
405	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
406	C₂×C₄	#137×C₆₇	1	2	yes	—
407	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
408	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
409	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
410	C₈	—	4	4	no	—
411	C₈	—	4	4	no	—
412	C₂²	—	2	8	no	—
413	C₂²	—	2	2	no	—
414	C₂²	—	2	2	no	—
415	C₂²	—	4	4	no	—
416	C₂²	—	4	4	no	—
417	C₂²	—	4	4	no	—
418	C₂²	—	8	8	no	—
419	C₂²	C₂×#100×C₆₇	1	4	yes	—
420	C₂²	C₂×#100×C₆₇	1	2	yes	—
421	C₂²	#416×C₆₇	1	2	yes	—
422	C₂²	#417×C₆₇	1	2	yes	—
423	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
424	C₄	—	2	2	no	—
425	C₄	—	2	1	no	—
426	C₄	—	2	1	no	—
427	C₄	—	2	2	no	—
428	C₄	—	2	2	no	—
429	C₄	—	4	4	no	—
430	C₄	#416×C₆₇	1	2	yes	—
431	C₄	#417×C₆₇	1	2	yes	—
432	C₄	C₂×#135×C₆₇	1	2	yes	—
433	C₄	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
434	C₄	C₂×#137×C₆₇	1	1	yes	characteristic
435	C₂	—	2	4	no	—
436	C₂	—	8	4	no	—
437	C₂	C₆₇×#1551	1	4	yes	direct product factor
438	C₂	C₂²×#100×C₆₇	1	1	yes	characteristic
439	C₂	C₂×#416×C₆₇	1	1	yes	characteristic
440	C₂	C₂×#417×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. C₂²×C₄⋊C₈×C₆₇	C₂	semidirect
4. C₆₇×#1551	C₂	direct (N × H)
5. C₂×#333×C₆₇	C₂	semidirect
6. C₂×#331×C₆₇	C₂	semidirect
22. #432×C₆₇	C₂²	semidirect
27. C₂×C₄⋊C₈×C₆₇	C₂²	semidirect
30. #333×C₆₇	C₂²	semidirect
31. #331×C₆₇	C₂²	semidirect
214. C₂×#1551	C₆₇	direct (N × H)
217. C₂×C₆₇	#1551	direct (N × H)
222. C₂×#432	C₂×C₆₇	semidirect
223. C₂²×C₄⋊C₈	C₂×C₆₇	semidirect
224. #1551	C₂×C₆₇	direct (N × H)
225. C₂×#333	C₂×C₆₇	semidirect
226. C₂×#331	C₂×C₆₇	semidirect
227. C₆₇	C₂×#1551	direct (N × H)
243. #432	C₂²×C₆₇	semidirect
248. C₂×C₄⋊C₈	C₂²×C₆₇	semidirect
251. #333	C₂²×C₆₇	semidirect
252. #331	C₂²×C₆₇	semidirect
437. C₂	C₆₇×#1551	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
C₂²×C₄⋊C₈×C₆₇	2
C₆₇×#1551	2
C₂×#333×C₆₇	2
C₂×#331×C₆₇	2
C₂×#1551	67

Subgroups — C2×C67×#1551 #30696960

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#1551 #30696960