Subgroups — C₆₇×#21231 #30700482

Order 17152.

1630 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×SD₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#427×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#419×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#418×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#405×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#2310	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#2309	C₂	1	1	yes	maximal, characteristic
8	C₆₇×#1973	C₂	1	4	yes	maximal, complemented
9	C₆₇×#1968	C₂	1	4	yes	maximal, complemented
10	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
11	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
12	C₂²×SD₁₆×C₆₇	—	2	4	no	—
13	C₂²×SD₁₆×C₆₇	—	2	4	no	—
14	C₂²×SD₁₆×C₆₇	C₄	1	8	yes	complemented
15	C₂²×SD₁₆×C₆₇	C₂²	1	2	yes	—
16	C₂²×SD₁₆×C₆₇	C₂²	1	2	yes	—
17	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
18	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
22	#427×C₆₇	—	2	4	no	—
23	C₂×#107×C₆₇	C₂²	1	2	yes	complemented
24	C₂×#106×C₆₇	C₂²	1	2	yes	complemented
25	#419×C₆₇	—	2	4	no	—
26	#418×C₆₇	—	2	4	no	—
27	C₂×#103×C₆₇	C₂²	1	4	yes	complemented
28	C₂×#102×C₆₇	C₂²	1	4	yes	—
29	#405×C₆₇	—	2	4	no	—
30	C₂×#97×C₆₇	C₂²	1	2	yes	complemented
31	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	complemented
32	#391×C₆₇	—	2	2	no	—
33	#391×C₆₇	C₂²	1	4	yes	complemented
34	#390×C₆₇	—	2	2	no	—
35	#390×C₆₇	C₂²	1	4	yes	—
36	#381×C₆₇	—	2	2	no	—
37	#379×C₆₇	—	2	2	no	—
38	C₂⁵×C₆₇	—	2	1	no	—
39	C₂²×Q₈×C₆₇	—	2	1	no	—
40	C₂²×Q₈×C₆₇	—	2	2	no	—
41	C₂²×Q₈×C₆₇	—	2	2	no	—
42	C₂²×Q₈×C₆₇	—	2	2	no	—
43	C₂²×Q₈×C₆₇	—	2	2	no	—
44	C₂²×Q₈×C₆₇	—	2	2	no	—
45	C₂²×Q₈×C₆₇	C₂×C₄	1	4	yes	—
46	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
47	C₂²×D₄×C₆₇	—	2	2	no	—
48	C₂²×D₄×C₆₇	—	2	2	no	—
49	C₂²×D₄×C₆₇	—	2	2	no	—
50	C₂²×D₄×C₆₇	—	2	1	no	—
51	C₂²×D₄×C₆₇	—	2	2	no	—
52	C₂²×D₄×C₆₇	—	2	2	no	—
53	C₂²×D₄×C₆₇	C₂×C₄	1	4	yes	—
54	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
55	C₂³×C₄×C₆₇	—	2	1	no	—
56	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
57	C₂×SD₁₆×C₆₇	—	2	32	no	—
58	C₂×SD₁₆×C₆₇	—	2	4	no	—
59	C₂×SD₁₆×C₆₇	—	2	4	no	—
60	C₂×SD₁₆×C₆₇	—	2	4	no	—
61	C₂×SD₁₆×C₆₇	—	2	4	no	—
62	C₂×SD₁₆×C₆₇	C₂×C₄	1	8	yes	—
63	C₂×SD₁₆×C₆₇	C₂×C₄	1	8	yes	—
64	C₂×M₁₆×C₆₇	—	2	2	no	—
65	C₂×M₁₆×C₆₇	—	2	2	no	—
66	C₂×M₁₆×C₆₇	D₄	1	4	yes	—
67	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
68	C₂²×C₈×C₆₇	D₄	1	4	yes	—
69	C₂²×C₈×C₆₇	D₄	1	4	yes	—
70	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
71	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
72	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
73	#117×C₆₇	—	2	2	no	—
74	#117×C₆₇	—	2	2	no	—
75	#117×C₆₇	—	2	2	no	—
76	#117×C₆₇	—	2	2	no	—
77	#117×C₆₇	—	2	2	no	—
78	#117×C₆₇	—	2	2	no	—
79	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
80	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
81	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
82	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
83	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
84	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
85	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
86	C₂×C₄²×C₆₇	—	2	1	no	—
87	C₂×C₄²×C₆₇	—	2	1	no	—
88	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
89	#107×C₆₇	—	2	4	no	—
90	#106×C₆₇	—	2	4	no	—
91	#103×C₆₇	—	2	8	no	—
92	#102×C₆₇	—	2	8	no	—
93	#97×C₆₇	—	2	4	no	—
94	C₄×C₈×C₆₇	—	2	4	no	—
95	#95×C₆₇	—	2	8	no	—
96	C₂⁴×C₆₇	—	2	2	no	—
97	C₂⁴×C₆₇	—	2	4	no	—
98	C₂⁴×C₆₇	—	4	2	no	—
99	C₂⁴×C₆₇	—	4	2	no	—
100	C₂⁴×C₆₇	—	8	4	no	—
101	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
102	C₂×Q₈×C₆₇	—	2	4	no	—
103	C₂×Q₈×C₆₇	—	2	2	no	—
104	C₂×Q₈×C₆₇	—	2	8	no	—
105	C₂×Q₈×C₆₇	—	2	8	no	—
106	C₂×Q₈×C₆₇	—	2	2	no	—
107	C₂×Q₈×C₆₇	—	2	2	no	—
108	C₂×Q₈×C₆₇	—	2	2	no	—
109	C₂×Q₈×C₆₇	—	2	2	no	—
110	C₂×Q₈×C₆₇	—	4	2	no	—
111	C₂×Q₈×C₆₇	—	4	4	no	—
112	C₂×Q₈×C₆₇	—	4	4	no	—
113	C₂×Q₈×C₆₇	—	4	2	no	—
114	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
115	C₂×D₄×C₆₇	—	2	8	no	—
116	C₂×D₄×C₆₇	—	2	8	no	—
117	C₂×D₄×C₆₇	—	2	4	no	—
118	C₂×D₄×C₆₇	—	2	2	no	—
119	C₂×D₄×C₆₇	—	2	2	no	—
120	C₂×D₄×C₆₇	—	2	2	no	—
121	C₂×D₄×C₆₇	—	2	2	no	—
122	C₂×D₄×C₆₇	—	2	2	no	—
123	C₂×D₄×C₆₇	—	4	4	no	—
124	C₂×D₄×C₆₇	—	4	2	no	—
125	C₂×D₄×C₆₇	—	4	4	no	—
126	C₂×D₄×C₆₇	—	4	2	no	—
127	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
128	C₂²×C₄×C₆₇	—	2	4	no	—
129	C₂²×C₄×C₆₇	—	2	2	no	—
130	C₂²×C₄×C₆₇	—	2	2	no	—
131	C₂²×C₄×C₆₇	—	2	2	no	—
132	C₂²×C₄×C₆₇	—	4	2	no	—
133	C₂²×C₄×C₆₇	—	4	2	no	—
134	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
135	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
136	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
137	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
138	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
139	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
140	C₂²×C₄×C₆₇	#41	1	2	yes	—
141	C₂²×C₄×C₆₇	#41	1	2	yes	—
142	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
143	SD₁₆×C₆₇	—	2	16	no	—
144	SD₁₆×C₆₇	—	2	16	no	—
145	M₁₆×C₆₇	—	2	8	no	—
146	C₂×C₈×C₆₇	—	2	2	no	—
147	C₂×C₈×C₆₇	—	2	2	no	—
148	C₂×C₈×C₆₇	C₂²⋊C₄	1	16	yes	complemented
149	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
150	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
151	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
152	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
153	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
154	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
155	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
156	C₄⋊C₄×C₆₇	—	2	4	no	—
157	C₄⋊C₄×C₆₇	—	2	2	no	—
158	C₄⋊C₄×C₆₇	—	2	2	no	—
159	C₄⋊C₄×C₆₇	—	2	2	no	—
160	C₄⋊C₄×C₆₇	—	2	2	no	—
161	C₂²⋊C₄×C₆₇	—	2	4	no	—
162	C₂²⋊C₄×C₆₇	—	4	2	no	—
163	C₂²⋊C₄×C₆₇	—	4	2	no	—
164	C₂²⋊C₄×C₆₇	—	8	8	no	—
165	C₄²×C₆₇	—	2	2	no	—
166	C₄²×C₆₇	—	2	2	no	—
167	C₄²×C₆₇	—	4	2	no	—
168	C₄²×C₆₇	—	4	2	no	—
169	C₂³×C₆₇	—	2	4	no	—
170	C₂³×C₆₇	—	2	4	no	—
171	C₂³×C₆₇	—	4	2	no	—
172	C₂³×C₆₇	—	4	2	no	—
173	C₂³×C₆₇	—	4	8	no	—
174	C₂³×C₆₇	—	4	8	no	—
175	C₂³×C₆₇	—	8	8	no	—
176	C₂³×C₆₇	—	8	8	no	—
177	C₂³×C₆₇	—	8	8	no	—
178	C₂³×C₆₇	C₄×D₄	1	2	yes	—
179	C₂³×C₆₇	#121	1	2	yes	—
180	C₂³×C₆₇	#127	1	2	yes	—
181	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
182	Q₈×C₆₇	—	2	4	no	—
183	Q₈×C₆₇	—	2	4	no	—
184	Q₈×C₆₇	—	4	2	no	—
185	Q₈×C₆₇	—	4	2	no	—
186	Q₈×C₆₇	—	4	8	no	—
187	D₄×C₆₇	—	2	4	no	—
188	D₄×C₆₇	—	2	4	no	—
189	D₄×C₆₇	—	4	2	no	—
190	D₄×C₆₇	—	4	2	no	—
191	D₄×C₆₇	—	4	8	no	—
192	C₂×C₄×C₆₇	—	2	4	no	—
193	C₂×C₄×C₆₇	—	2	4	no	—
194	C₂×C₄×C₆₇	—	2	8	no	—
195	C₂×C₄×C₆₇	—	4	2	no	—
196	C₂×C₄×C₆₇	—	4	2	no	—
197	C₂×C₄×C₆₇	—	4	8	no	—
198	C₂×C₄×C₆₇	—	4	2	no	—
199	C₂×C₄×C₆₇	—	4	2	no	—
200	C₂×C₄×C₆₇	—	4	4	no	—
201	C₂×C₄×C₆₇	—	4	4	no	—
202	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
203	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
204	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
205	C₂×C₄×C₆₇	#121	1	4	yes	—
206	C₂×C₄×C₆₇	#121	1	2	yes	—
207	C₂×C₄×C₆₇	#124	1	4	yes	—
208	C₂×C₄×C₆₇	#127	1	2	yes	—
209	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
210	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
211	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
212	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
213	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
214	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
215	C₈×C₆₇	—	2	4	no	—
216	C₈×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
217	C₈×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
218	C₂²×C₆₇	—	2	4	no	—
219	C₂²×C₆₇	—	2	4	no	—
220	C₂²×C₆₇	—	2	4	no	—
221	C₂²×C₆₇	—	4	8	no	—
222	C₂²×C₆₇	—	4	4	no	—
223	C₂²×C₆₇	—	8	16	no	—
224	C₂²×C₆₇	—	8	4	no	—
225	C₂²×C₆₇	—	8	4	no	—
226	C₂²×C₆₇	#390	1	4	yes	—
227	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
228	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
229	C₂²×C₆₇	C₂×#127	1	1	yes	characteristic
230	C₂²×C₆₇	#577	1	4	yes	—
231	C₄×C₆₇	—	4	4	no	—
232	C₄×C₆₇	—	4	4	no	—
233	C₄×C₆₇	—	8	2	no	—
234	C₄×C₆₇	—	8	2	no	—
235	C₄×C₆₇	#390	1	4	yes	—
236	C₄×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
237	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
238	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
239	C₄×C₆₇	C₂×#124	1	1	yes	characteristic
240	#21231	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
241	C₂×C₆₇	—	2	4	no	—
242	C₂×C₆₇	—	8	4	no	—
243	C₂×C₆₇	C₂×#390	1	1	yes	characteristic
244	C₂×C₆₇	#2949	1	2	yes	—
245	C₂×C₆₇	#3057	1	2	yes	—
246	C₂×C₆₇	#3154	1	2	yes	—
247	C₂³×SD₁₆	C₂×C₆₇	1	1	yes	characteristic
248	C₂×#427	C₂×C₆₇	1	1	yes	characteristic, complemented
249	C₂×#419	C₂×C₆₇	1	1	yes	characteristic, complemented
250	C₂×#418	C₂×C₆₇	1	1	yes	characteristic
251	C₂×#405	C₂×C₆₇	1	1	yes	characteristic, complemented
252	#2310	C₂×C₆₇	1	1	yes	characteristic, complemented
253	#2309	C₂×C₆₇	1	1	yes	characteristic
254	#1973	C₂×C₆₇	1	4	yes	complemented
255	#1968	C₂×C₆₇	1	4	yes	complemented
256	C₆₇	#21231	1	1	yes	characteristic, direct product factor, Sylow
257	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
258	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
259	C₂²×SD₁₆	—	2	4	no	—
260	C₂²×SD₁₆	—	2	4	no	—
261	C₂²×SD₁₆	C₄×C₆₇	1	8	yes	complemented
262	C₂²×SD₁₆	C₂²×C₆₇	1	2	yes	—
263	C₂²×SD₁₆	C₂²×C₆₇	1	2	yes	—
264	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
265	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
266	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
267	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
268	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
269	#427	—	2	4	no	—
270	C₂×#107	C₂²×C₆₇	1	2	yes	complemented
271	C₂×#106	C₂²×C₆₇	1	2	yes	complemented
272	#419	—	2	4	no	—
273	#418	—	2	4	no	—
274	C₂×#103	C₂²×C₆₇	1	4	yes	complemented
275	C₂×#102	C₂²×C₆₇	1	4	yes	—
276	#405	—	2	4	no	—
277	C₂×#97	C₂²×C₆₇	1	2	yes	complemented
278	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	complemented
279	#391	—	2	2	no	—
280	#391	C₂²×C₆₇	1	4	yes	complemented
281	#390	—	2	2	no	—
282	#390	C₂²×C₆₇	1	4	yes	—
283	#381	—	2	2	no	—
284	#379	—	2	2	no	—
285	C₂⁵	—	2	1	no	—
286	C₂²×Q₈	—	2	1	no	—
287	C₂²×Q₈	—	2	2	no	—
288	C₂²×Q₈	—	2	2	no	—
289	C₂²×Q₈	—	2	2	no	—
290	C₂²×Q₈	—	2	2	no	—
291	C₂²×Q₈	—	2	2	no	—
292	C₂²×Q₈	C₂×C₄×C₆₇	1	4	yes	—
293	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
294	C₂²×D₄	—	2	2	no	—
295	C₂²×D₄	—	2	2	no	—
296	C₂²×D₄	—	2	2	no	—
297	C₂²×D₄	—	2	1	no	—
298	C₂²×D₄	—	2	2	no	—
299	C₂²×D₄	—	2	2	no	—
300	C₂²×D₄	C₂×C₄×C₆₇	1	4	yes	—
301	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
302	C₂³×C₄	—	2	1	no	—
303	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
304	C₂×SD₁₆	—	2	32	no	—
305	C₂×SD₁₆	—	2	4	no	—
306	C₂×SD₁₆	—	2	4	no	—
307	C₂×SD₁₆	—	2	4	no	—
308	C₂×SD₁₆	—	2	4	no	—
309	C₂×SD₁₆	C₂×C₄×C₆₇	1	8	yes	—
310	C₂×SD₁₆	C₂×C₄×C₆₇	1	8	yes	—
311	C₂×M₁₆	—	2	2	no	—
312	C₂×M₁₆	—	2	2	no	—
313	C₂×M₁₆	D₄×C₆₇	1	4	yes	—
314	C₂²×C₈	C₂×C₄×C₆₇	1	4	yes	—
315	C₂²×C₈	D₄×C₆₇	1	4	yes	—
316	C₂²×C₈	D₄×C₆₇	1	4	yes	—
317	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
318	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
319	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
320	#117	—	2	2	no	—
321	#117	—	2	2	no	—
322	#117	—	2	2	no	—
323	#117	—	2	2	no	—
324	#117	—	2	2	no	—
325	#117	—	2	2	no	—
326	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
327	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
328	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
329	C₂×C₂²⋊C₄	—	2	1	no	—
330	C₂×C₂²⋊C₄	—	2	1	no	—
331	C₂×C₂²⋊C₄	—	2	8	no	—
332	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
333	C₂×C₄²	—	2	1	no	—
334	C₂×C₄²	—	2	1	no	—
335	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
336	#107	—	2	4	no	—
337	#106	—	2	4	no	—
338	#103	—	2	8	no	—
339	#102	—	2	8	no	—
340	#97	—	2	4	no	—
341	C₄×C₈	—	2	4	no	—
342	#95	—	2	8	no	—
343	C₂⁴	—	2	2	no	—
344	C₂⁴	—	2	4	no	—
345	C₂⁴	—	4	2	no	—
346	C₂⁴	—	4	2	no	—
347	C₂⁴	—	8	4	no	—
348	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
349	C₂×Q₈	—	2	4	no	—
350	C₂×Q₈	—	2	2	no	—
351	C₂×Q₈	—	2	8	no	—
352	C₂×Q₈	—	2	8	no	—
353	C₂×Q₈	—	2	2	no	—
354	C₂×Q₈	—	2	2	no	—
355	C₂×Q₈	—	2	2	no	—
356	C₂×Q₈	—	2	2	no	—
357	C₂×Q₈	—	4	2	no	—
358	C₂×Q₈	—	4	4	no	—
359	C₂×Q₈	—	4	4	no	—
360	C₂×Q₈	—	4	2	no	—
361	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
362	C₂×D₄	—	2	8	no	—
363	C₂×D₄	—	2	8	no	—
364	C₂×D₄	—	2	4	no	—
365	C₂×D₄	—	2	2	no	—
366	C₂×D₄	—	2	2	no	—
367	C₂×D₄	—	2	2	no	—
368	C₂×D₄	—	2	2	no	—
369	C₂×D₄	—	2	2	no	—
370	C₂×D₄	—	4	4	no	—
371	C₂×D₄	—	4	2	no	—
372	C₂×D₄	—	4	4	no	—
373	C₂×D₄	—	4	2	no	—
374	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
375	C₂²×C₄	—	2	4	no	—
376	C₂²×C₄	—	2	2	no	—
377	C₂²×C₄	—	2	2	no	—
378	C₂²×C₄	—	2	2	no	—
379	C₂²×C₄	—	4	2	no	—
380	C₂²×C₄	—	4	2	no	—
381	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
382	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
383	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
384	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
385	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
386	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
387	C₂²×C₄	#41×C₆₇	1	2	yes	—
388	C₂²×C₄	#41×C₆₇	1	2	yes	—
389	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
390	SD₁₆	—	2	16	no	—
391	SD₁₆	—	2	16	no	—
392	M₁₆	—	2	8	no	—
393	C₂×C₈	—	2	2	no	—
394	C₂×C₈	—	2	2	no	—
395	C₂×C₈	C₂²⋊C₄×C₆₇	1	16	yes	complemented
396	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
397	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
398	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
399	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
400	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
401	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
402	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
403	C₄⋊C₄	—	2	4	no	—
404	C₄⋊C₄	—	2	2	no	—
405	C₄⋊C₄	—	2	2	no	—
406	C₄⋊C₄	—	2	2	no	—
407	C₄⋊C₄	—	2	2	no	—
408	C₂²⋊C₄	—	2	4	no	—
409	C₂²⋊C₄	—	4	2	no	—
410	C₂²⋊C₄	—	4	2	no	—
411	C₂²⋊C₄	—	8	8	no	—
412	C₄²	—	2	2	no	—
413	C₄²	—	2	2	no	—
414	C₄²	—	4	2	no	—
415	C₄²	—	4	2	no	—
416	C₂³	—	2	4	no	—
417	C₂³	—	2	4	no	—
418	C₂³	—	4	2	no	—
419	C₂³	—	4	2	no	—
420	C₂³	—	4	8	no	—
421	C₂³	—	4	8	no	—
422	C₂³	—	8	8	no	—
423	C₂³	—	8	8	no	—
424	C₂³	—	8	8	no	—
425	C₂³	C₄×D₄×C₆₇	1	2	yes	—
426	C₂³	#121×C₆₇	1	2	yes	—
427	C₂³	#127×C₆₇	1	2	yes	—
428	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
429	Q₈	—	2	4	no	—
430	Q₈	—	2	4	no	—
431	Q₈	—	4	2	no	—
432	Q₈	—	4	2	no	—
433	Q₈	—	4	8	no	—
434	D₄	—	2	4	no	—
435	D₄	—	2	4	no	—
436	D₄	—	4	2	no	—
437	D₄	—	4	2	no	—
438	D₄	—	4	8	no	—
439	C₂×C₄	—	2	4	no	—
440	C₂×C₄	—	2	4	no	—
441	C₂×C₄	—	2	8	no	—
442	C₂×C₄	—	4	2	no	—
443	C₂×C₄	—	4	2	no	—
444	C₂×C₄	—	4	8	no	—
445	C₂×C₄	—	4	2	no	—
446	C₂×C₄	—	4	2	no	—
447	C₂×C₄	—	4	4	no	—
448	C₂×C₄	—	4	4	no	—
449	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
450	C₂×C₄	C₄×D₄×C₆₇	1	4	yes	—
451	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
452	C₂×C₄	#121×C₆₇	1	4	yes	—
453	C₂×C₄	#121×C₆₇	1	2	yes	—
454	C₂×C₄	#124×C₆₇	1	4	yes	—
455	C₂×C₄	#127×C₆₇	1	2	yes	—
456	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
457	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
458	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
459	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
460	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
461	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
462	C₈	—	2	4	no	—
463	C₈	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
464	C₈	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
465	C₂²	—	2	4	no	—
466	C₂²	—	2	4	no	—
467	C₂²	—	2	4	no	—
468	C₂²	—	4	4	no	—
469	C₂²	—	4	8	no	—
470	C₂²	—	8	16	no	—
471	C₂²	—	8	4	no	—
472	C₂²	—	8	4	no	—
473	C₂²	#390×C₆₇	1	4	yes	—
474	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
475	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
476	C₂²	C₂×#127×C₆₇	1	1	yes	characteristic
477	C₂²	#577×C₆₇	1	4	yes	—
478	C₄	—	4	4	no	—
479	C₄	—	4	4	no	—
480	C₄	—	8	2	no	—
481	C₄	—	8	2	no	—
482	C₄	#390×C₆₇	1	4	yes	—
483	C₄	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
484	C₄	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
485	C₄	C₂×#121×C₆₇	1	1	yes	characteristic
486	C₄	C₂×#124×C₆₇	1	1	yes	characteristic
487	C₂	—	2	4	no	—
488	C₂	—	8	4	no	—
489	C₂	C₂×#390×C₆₇	1	1	yes	characteristic
490	C₂	C₆₇×#2949	1	2	yes	—
491	C₂	C₆₇×#3057	1	2	yes	—
492	C₂	C₆₇×#3154	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂³×D₄×C₆₇	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×#117×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#427×C₆₇	C₂	semidirect
3. C₂×#419×C₆₇	C₂	semidirect
5. C₂×#405×C₆₇	C₂	semidirect
6. C₆₇×#2310	C₂	semidirect
8. C₆₇×#1973	C₂	semidirect
9. C₆₇×#1968	C₂	semidirect
14. C₂²×SD₁₆×C₆₇	C₄	semidirect
23. C₂×#107×C₆₇	C₂²	semidirect
24. C₂×#106×C₆₇	C₂²	semidirect
27. C₂×#103×C₆₇	C₂²	semidirect
30. C₂×#97×C₆₇	C₂²	semidirect
31. C₂×C₄×C₈×C₆₇	C₂²	semidirect
33. #391×C₆₇	C₂²	semidirect
148. C₂×C₈×C₆₇	C₂²⋊C₄	semidirect
240. #21231	C₆₇	direct (N × H)
248. C₂×#427	C₂×C₆₇	semidirect
249. C₂×#419	C₂×C₆₇	semidirect
251. C₂×#405	C₂×C₆₇	semidirect
252. #2310	C₂×C₆₇	semidirect
254. #1973	C₂×C₆₇	semidirect
255. #1968	C₂×C₆₇	semidirect
256. C₆₇	#21231	direct (N × H)
261. C₂²×SD₁₆	C₄×C₆₇	semidirect
270. C₂×#107	C₂²×C₆₇	semidirect
271. C₂×#106	C₂²×C₆₇	semidirect
274. C₂×#103	C₂²×C₆₇	semidirect
277. C₂×#97	C₂²×C₆₇	semidirect
278. C₂×C₄×C₈	C₂²×C₆₇	semidirect
280. #391	C₂²×C₆₇	semidirect
395. C₂×C₈	C₂²⋊C₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂³×SD₁₆×C₆₇	2
C₂×#427×C₆₇	2
C₂×#419×C₆₇	2
C₂×#418×C₆₇	2
C₂×#405×C₆₇	2
C₆₇×#2310	2
C₆₇×#2309	2
C₆₇×#1973	2
C₆₇×#1968	2
#21231	67

Subgroups — C67×#21231 #30700482

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21231 #30700482