Subgroups — C₂×C₆₇×#1759 #30699654

Order 17152.

1662 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#41×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×#104×C₆₇	C₂	1	2	yes	maximal
3	C₂×C₄×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#376×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1759	C₂	1	8	yes	maximal, direct product factor
6	C₂×#339×C₆₇	C₂	1	2	yes	maximal, complemented
7	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
8	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
9	C₄×#41×C₆₇	—	2	2	no	—
10	C₄×#41×C₆₇	—	2	2	no	—
11	C₄×#41×C₆₇	C₄	1	4	yes	complemented
12	C₄×#41×C₆₇	C₂²	1	4	yes	—
13	C₂×C₄×Q₈×C₆₇	C₄	1	2	yes	complemented
14	C₂×C₄×D₄×C₆₇	—	2	2	no	—
15	C₂×C₄×D₄×C₆₇	C₄	1	2	yes	complemented
16	C₂×#117×C₆₇	—	2	1	no	—
17	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
18	C₂²×C₄²×C₆₇	—	2	1	no	—
19	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
20	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×#104×C₆₇	C₄	1	16	yes	complemented
22	C₂×#104×C₆₇	C₂²	1	8	yes	—
23	C₄×M₁₆×C₆₇	—	2	2	no	—
24	C₄×M₁₆×C₆₇	C₂²	1	4	yes	complemented
25	C₂×#97×C₆₇	—	2	1	no	—
26	C₂×C₄×C₈×C₆₇	—	2	1	no	—
27	#376×C₆₇	—	2	2	no	—
28	#376×C₆₇	C₂²	1	4	yes	complemented
29	C₂×#95×C₆₇	—	2	2	no	—
30	#339×C₆₇	C₂²	1	8	yes	complemented
31	C₂×#41×C₆₇	—	2	1	no	—
32	C₂×#41×C₆₇	—	2	1	no	—
33	C₂×#41×C₆₇	—	2	4	no	—
34	C₂×#41×C₆₇	C₂×C₄	1	8	yes	—
35	C₂×#41×C₆₇	C₂×C₄	1	2	yes	—
36	C₂×#41×C₆₇	C₂³	1	2	yes	—
37	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
38	C₂²×D₄×C₆₇	—	2	1	no	—
39	C₂²×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
40	C₂³×C₄×C₆₇	—	2	1	no	—
41	C₂³×C₄×C₆₇	—	2	2	no	—
42	C₂³×C₄×C₆₇	—	2	2	no	—
43	C₂³×C₄×C₆₇	D₄	1	2	yes	—
44	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
45	C₂×M₁₆×C₆₇	—	2	2	no	—
46	C₂×M₁₆×C₆₇	—	2	4	no	—
47	C₂×M₁₆×C₆₇	C₂×C₄	1	8	yes	complemented
48	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
49	C₂²×C₈×C₆₇	—	2	2	no	—
50	C₄×Q₈×C₆₇	—	2	4	no	—
51	C₄×Q₈×C₆₇	C₂×C₄	1	8	yes	complemented
52	C₄×D₄×C₆₇	—	2	8	no	—
53	C₄×D₄×C₆₇	—	2	8	no	—
54	C₄×D₄×C₆₇	—	2	4	no	—
55	C₄×D₄×C₆₇	C₂×C₄	1	8	yes	complemented
56	#117×C₆₇	—	2	1	no	—
57	#117×C₆₇	—	2	1	no	—
58	#117×C₆₇	—	2	4	no	—
59	#117×C₆₇	—	2	4	no	—
60	#117×C₆₇	C₂×C₄	1	2	yes	—
61	#117×C₆₇	C₂³	1	2	yes	—
62	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
63	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
64	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
65	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
66	C₂×C₄²×C₆₇	—	2	2	no	—
67	C₂×C₄²×C₆₇	—	2	2	no	—
68	C₂×C₄²×C₆₇	—	2	1	no	—
69	C₂×C₄²×C₆₇	—	2	4	no	—
70	C₂×C₄²×C₆₇	—	2	4	no	—
71	C₂×C₄²×C₆₇	—	2	4	no	—
72	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
73	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
74	C₂×C₄²×C₆₇	C₂×C₄	1	8	yes	complemented
75	C₂×C₄²×C₆₇	D₄	1	2	yes	—
76	C₂×C₄²×C₆₇	D₄	1	2	yes	—
77	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
78	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
79	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
80	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
81	#104×C₆₇	C₂×C₄	1	32	yes	complemented
82	#97×C₆₇	—	2	4	no	—
83	C₄×C₈×C₆₇	—	2	4	no	—
84	#95×C₆₇	—	2	8	no	—
85	C₂⁴×C₆₇	—	2	1	no	—
86	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
87	#41×C₆₇	—	2	4	no	—
88	#41×C₆₇	—	2	8	no	—
89	#41×C₆₇	—	2	8	no	—
90	#41×C₆₇	—	2	4	no	—
91	#41×C₆₇	C₄²	1	8	yes	—
92	#41×C₆₇	C₂²×C₄	1	8	yes	—
93	C₂×Q₈×C₆₇	—	2	1	no	—
94	C₂×Q₈×C₆₇	—	2	2	no	—
95	C₂×Q₈×C₆₇	C₄²	1	4	yes	—
96	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
97	C₂×D₄×C₆₇	—	2	2	no	—
98	C₂×D₄×C₆₇	—	2	8	no	—
99	C₂×D₄×C₆₇	—	2	2	no	—
100	C₂×D₄×C₆₇	—	2	1	no	—
101	C₂×D₄×C₆₇	—	2	2	no	—
102	C₂×D₄×C₆₇	C₄²	1	4	yes	—
103	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
104	C₂²×C₄×C₆₇	—	2	1	no	—
105	C₂²×C₄×C₆₇	—	2	2	no	—
106	C₂²×C₄×C₆₇	—	2	8	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	—	2	4	no	—
111	C₂²×C₄×C₆₇	—	2	8	no	—
112	C₂²×C₄×C₆₇	—	4	4	no	—
113	C₂²×C₄×C₆₇	—	4	8	no	—
114	C₂²×C₄×C₆₇	—	4	8	no	—
115	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
116	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
117	C₂²×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
118	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
119	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
120	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
121	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
122	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
123	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
124	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
125	C₂²×C₄×C₆₇	#41	1	4	yes	—
126	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
127	M₁₆×C₆₇	—	2	8	no	—
128	M₁₆×C₆₇	—	2	4	no	—
129	M₁₆×C₆₇	C₂²×C₄	1	8	yes	complemented
130	C₂×C₈×C₆₇	—	2	8	no	—
131	C₂×C₈×C₆₇	—	2	4	no	—
132	C₄⋊C₄×C₆₇	—	2	8	no	—
133	C₄⋊C₄×C₆₇	—	2	2	no	—
134	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
135	C₂²⋊C₄×C₆₇	—	2	2	no	—
136	C₂²⋊C₄×C₆₇	—	2	2	no	—
137	C₂²⋊C₄×C₆₇	—	2	8	no	—
138	C₄²×C₆₇	—	2	2	no	—
139	C₄²×C₆₇	—	2	8	no	—
140	C₄²×C₆₇	—	2	8	no	—
141	C₄²×C₆₇	—	2	2	no	—
142	C₄²×C₆₇	—	2	4	no	—
143	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
144	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
145	C₄²×C₆₇	C₂²⋊C₄	1	4	yes	—
146	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
147	C₄²×C₆₇	C₂²×C₄	1	8	yes	complemented
148	C₄²×C₆₇	C₂×D₄	1	2	yes	—
149	C₄²×C₆₇	C₂×D₄	1	2	yes	—
150	C₂³×C₆₇	—	2	4	no	—
151	C₂³×C₆₇	—	2	2	no	—
152	C₂³×C₆₇	—	2	4	no	—
153	C₂³×C₆₇	—	4	4	no	—
154	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
155	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
156	C₂³×C₆₇	C₄×D₄	1	4	yes	—
157	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
158	Q₈×C₆₇	—	2	2	no	—
159	Q₈×C₆₇	—	2	4	no	—
160	Q₈×C₆₇	C₂×C₄²	1	4	yes	—
161	D₄×C₆₇	—	2	8	no	—
162	D₄×C₆₇	—	2	8	no	—
163	D₄×C₆₇	—	2	2	no	—
164	D₄×C₆₇	—	2	4	no	—
165	D₄×C₆₇	C₂×C₄²	1	4	yes	—
166	C₂×C₄×C₆₇	—	2	2	no	—
167	C₂×C₄×C₆₇	—	2	4	no	—
168	C₂×C₄×C₆₇	—	2	4	no	—
169	C₂×C₄×C₆₇	—	2	4	no	—
170	C₂×C₄×C₆₇	—	2	8	no	—
171	C₂×C₄×C₆₇	—	2	8	no	—
172	C₂×C₄×C₆₇	—	2	4	no	—
173	C₂×C₄×C₆₇	—	2	8	no	—
174	C₂×C₄×C₆₇	—	4	4	no	—
175	C₂×C₄×C₆₇	—	4	4	no	—
176	C₂×C₄×C₆₇	—	4	16	no	—
177	C₂×C₄×C₆₇	—	4	8	no	—
178	C₂×C₄×C₆₇	—	4	16	no	—
179	C₂×C₄×C₆₇	—	4	4	no	—
180	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
181	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
182	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
183	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
184	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
185	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
186	C₂×C₄×C₆₇	#117	1	2	yes	—
187	C₂×C₄×C₆₇	#117	1	2	yes	—
188	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
189	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
190	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
191	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
192	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
193	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
194	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
195	C₈×C₆₇	—	2	8	no	—
196	C₂²×C₆₇	—	2	2	no	—
197	C₂²×C₆₇	—	2	2	no	—
198	C₂²×C₆₇	—	2	8	no	—
199	C₂²×C₆₇	—	2	4	no	—
200	C₂²×C₆₇	—	4	4	no	—
201	C₂²×C₆₇	—	4	8	no	—
202	C₂²×C₆₇	C₄×C₂²⋊C₄	1	2	yes	—
203	C₂²×C₆₇	C₄×C₂²⋊C₄	1	2	yes	—
204	C₂²×C₆₇	#421	1	4	yes	—
205	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
206	C₂²×C₆₇	C₂×C₄×D₄	1	2	yes	—
207	C₄×C₆₇	—	2	4	no	—
208	C₄×C₆₇	—	2	4	no	—
209	C₄×C₆₇	—	2	4	no	—
210	C₄×C₆₇	—	4	8	no	—
211	C₄×C₆₇	—	4	4	no	—
212	C₄×C₆₇	C₄×C₂²⋊C₄	1	2	yes	—
213	C₄×C₆₇	C₄×C₂²⋊C₄	1	2	yes	—
214	C₄×C₆₇	C₂²×C₄²	1	1	yes	characteristic
215	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
216	C₄×C₆₇	C₂×C₄×D₄	1	2	yes	—
217	C₂×#1759	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
218	C₂×C₆₇	—	2	2	no	—
219	C₂×C₆₇	—	2	2	no	—
220	C₂×C₆₇	—	4	4	no	—
221	C₂×C₆₇	#1759	1	4	yes	direct product factor
222	C₂×C₆₇	C₂×C₄×C₂²⋊C₄	1	1	yes	characteristic
223	C₂×C₆₇	C₂×#421	1	2	yes	—
224	C₂×C₄×#41	C₂×C₆₇	1	1	yes	characteristic
225	C₂²×#104	C₂×C₆₇	1	2	yes	—
226	C₂×C₄×M₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
227	C₂×#376	C₂×C₆₇	1	1	yes	characteristic, complemented
228	#1759	C₂×C₆₇	1	8	yes	direct product factor
229	C₂×#339	C₂×C₆₇	1	2	yes	complemented
230	C₆₇	C₂×#1759	1	1	yes	characteristic, direct product factor, Sylow
231	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
232	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
233	C₄×#41	—	2	2	no	—
234	C₄×#41	—	2	2	no	—
235	C₄×#41	C₄×C₆₇	1	4	yes	complemented
236	C₄×#41	C₂²×C₆₇	1	4	yes	—
237	C₂×C₄×Q₈	C₄×C₆₇	1	2	yes	complemented
238	C₂×C₄×D₄	—	2	2	no	—
239	C₂×C₄×D₄	C₄×C₆₇	1	2	yes	complemented
240	C₂×#117	—	2	1	no	—
241	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
242	C₂²×C₄²	—	2	1	no	—
243	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
244	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
245	C₂×#104	C₄×C₆₇	1	16	yes	complemented
246	C₂×#104	C₂²×C₆₇	1	8	yes	—
247	C₄×M₁₆	—	2	2	no	—
248	C₄×M₁₆	C₂²×C₆₇	1	4	yes	complemented
249	C₂×#97	—	2	1	no	—
250	C₂×C₄×C₈	—	2	1	no	—
251	#376	—	2	2	no	—
252	#376	C₂²×C₆₇	1	4	yes	complemented
253	C₂×#95	—	2	2	no	—
254	#339	C₂²×C₆₇	1	8	yes	complemented
255	C₂×#41	—	2	1	no	—
256	C₂×#41	—	2	1	no	—
257	C₂×#41	—	2	4	no	—
258	C₂×#41	C₂×C₄×C₆₇	1	8	yes	—
259	C₂×#41	C₂×C₄×C₆₇	1	2	yes	—
260	C₂×#41	C₂³×C₆₇	1	2	yes	—
261	C₂²×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
262	C₂²×D₄	—	2	1	no	—
263	C₂²×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
264	C₂³×C₄	—	2	1	no	—
265	C₂³×C₄	—	2	2	no	—
266	C₂³×C₄	—	2	2	no	—
267	C₂³×C₄	D₄×C₆₇	1	2	yes	—
268	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
269	C₂×M₁₆	—	2	2	no	—
270	C₂×M₁₆	—	2	4	no	—
271	C₂×M₁₆	C₂×C₄×C₆₇	1	8	yes	complemented
272	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
273	C₂²×C₈	—	2	2	no	—
274	C₄×Q₈	—	2	4	no	—
275	C₄×Q₈	C₂×C₄×C₆₇	1	8	yes	complemented
276	C₄×D₄	—	2	8	no	—
277	C₄×D₄	—	2	8	no	—
278	C₄×D₄	—	2	4	no	—
279	C₄×D₄	C₂×C₄×C₆₇	1	8	yes	complemented
280	#117	—	2	1	no	—
281	#117	—	2	1	no	—
282	#117	—	2	4	no	—
283	#117	—	2	4	no	—
284	#117	C₂×C₄×C₆₇	1	2	yes	—
285	#117	C₂³×C₆₇	1	2	yes	—
286	C₂×C₄⋊C₄	—	2	2	no	—
287	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
288	C₂×C₂²⋊C₄	—	2	1	no	—
289	C₂×C₂²⋊C₄	—	2	2	no	—
290	C₂×C₄²	—	2	2	no	—
291	C₂×C₄²	—	2	1	no	—
292	C₂×C₄²	—	2	2	no	—
293	C₂×C₄²	—	2	4	no	—
294	C₂×C₄²	—	2	4	no	—
295	C₂×C₄²	—	2	4	no	—
296	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
297	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
298	C₂×C₄²	C₂×C₄×C₆₇	1	8	yes	complemented
299	C₂×C₄²	D₄×C₆₇	1	2	yes	—
300	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
301	C₂×C₄²	D₄×C₆₇	1	2	yes	—
302	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
303	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
304	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
305	#104	C₂×C₄×C₆₇	1	32	yes	complemented
306	#97	—	2	4	no	—
307	C₄×C₈	—	2	4	no	—
308	#95	—	2	8	no	—
309	C₂⁴	—	2	1	no	—
310	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
311	#41	—	2	4	no	—
312	#41	—	2	8	no	—
313	#41	—	2	8	no	—
314	#41	—	2	4	no	—
315	#41	C₄²×C₆₇	1	8	yes	—
316	#41	C₂²×C₄×C₆₇	1	8	yes	—
317	C₂×Q₈	—	2	1	no	—
318	C₂×Q₈	—	2	2	no	—
319	C₂×Q₈	C₄²×C₆₇	1	4	yes	—
320	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
321	C₂×D₄	—	2	2	no	—
322	C₂×D₄	—	2	8	no	—
323	C₂×D₄	—	2	2	no	—
324	C₂×D₄	—	2	1	no	—
325	C₂×D₄	—	2	2	no	—
326	C₂×D₄	C₄²×C₆₇	1	4	yes	—
327	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
328	C₂²×C₄	—	2	1	no	—
329	C₂²×C₄	—	2	2	no	—
330	C₂²×C₄	—	2	8	no	—
331	C₂²×C₄	—	2	2	no	—
332	C₂²×C₄	—	2	2	no	—
333	C₂²×C₄	—	2	8	no	—
334	C₂²×C₄	—	2	4	no	—
335	C₂²×C₄	—	2	8	no	—
336	C₂²×C₄	—	4	4	no	—
337	C₂²×C₄	—	4	8	no	—
338	C₂²×C₄	—	4	8	no	—
339	C₂²×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
340	C₂²×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
341	C₂²×C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
342	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
343	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
344	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
345	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
346	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
347	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
348	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
349	C₂²×C₄	#41×C₆₇	1	4	yes	—
350	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
351	M₁₆	—	2	8	no	—
352	M₁₆	—	2	4	no	—
353	M₁₆	C₂²×C₄×C₆₇	1	8	yes	complemented
354	C₂×C₈	—	2	8	no	—
355	C₂×C₈	—	2	4	no	—
356	C₄⋊C₄	—	2	8	no	—
357	C₄⋊C₄	—	2	2	no	—
358	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
359	C₂²⋊C₄	—	2	2	no	—
360	C₂²⋊C₄	—	2	2	no	—
361	C₂²⋊C₄	—	2	8	no	—
362	C₄²	—	2	2	no	—
363	C₄²	—	2	8	no	—
364	C₄²	—	2	8	no	—
365	C₄²	—	2	2	no	—
366	C₄²	—	2	4	no	—
367	C₄²	C₂²⋊C₄×C₆₇	1	2	yes	—
368	C₄²	C₂²⋊C₄×C₆₇	1	2	yes	—
369	C₄²	C₂²⋊C₄×C₆₇	1	4	yes	—
370	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
371	C₄²	C₂²×C₄×C₆₇	1	8	yes	complemented
372	C₄²	C₂×D₄×C₆₇	1	2	yes	—
373	C₄²	C₂×D₄×C₆₇	1	2	yes	—
374	C₂³	—	2	4	no	—
375	C₂³	—	2	2	no	—
376	C₂³	—	2	4	no	—
377	C₂³	—	4	4	no	—
378	C₂³	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
379	C₂³	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
380	C₂³	C₄×D₄×C₆₇	1	4	yes	—
381	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
382	Q₈	—	2	2	no	—
383	Q₈	—	2	4	no	—
384	Q₈	C₂×C₄²×C₆₇	1	4	yes	—
385	D₄	—	2	8	no	—
386	D₄	—	2	8	no	—
387	D₄	—	2	2	no	—
388	D₄	—	2	4	no	—
389	D₄	C₂×C₄²×C₆₇	1	4	yes	—
390	C₂×C₄	—	2	2	no	—
391	C₂×C₄	—	2	4	no	—
392	C₂×C₄	—	2	4	no	—
393	C₂×C₄	—	2	4	no	—
394	C₂×C₄	—	2	8	no	—
395	C₂×C₄	—	2	8	no	—
396	C₂×C₄	—	2	4	no	—
397	C₂×C₄	—	2	8	no	—
398	C₂×C₄	—	4	4	no	—
399	C₂×C₄	—	4	4	no	—
400	C₂×C₄	—	4	16	no	—
401	C₂×C₄	—	4	16	no	—
402	C₂×C₄	—	4	8	no	—
403	C₂×C₄	—	4	4	no	—
404	C₂×C₄	C₂×C₄²×C₆₇	1	2	yes	—
405	C₂×C₄	C₂×C₄²×C₆₇	1	2	yes	—
406	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
407	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
408	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
409	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
410	C₂×C₄	#117×C₆₇	1	2	yes	—
411	C₂×C₄	#117×C₆₇	1	2	yes	—
412	C₂×C₄	C₄×D₄×C₆₇	1	4	yes	—
413	C₂×C₄	C₄×D₄×C₆₇	1	4	yes	—
414	C₂×C₄	C₄×D₄×C₆₇	1	4	yes	—
415	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
416	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
417	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
418	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
419	C₈	—	2	8	no	—
420	C₂²	—	2	8	no	—
421	C₂²	—	2	2	no	—
422	C₂²	—	2	2	no	—
423	C₂²	—	2	4	no	—
424	C₂²	—	4	8	no	—
425	C₂²	—	4	4	no	—
426	C₂²	C₄×C₂²⋊C₄×C₆₇	1	2	yes	—
427	C₂²	C₄×C₂²⋊C₄×C₆₇	1	2	yes	—
428	C₂²	#421×C₆₇	1	4	yes	—
429	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
430	C₂²	C₂×C₄×D₄×C₆₇	1	2	yes	—
431	C₄	—	2	4	no	—
432	C₄	—	2	4	no	—
433	C₄	—	2	4	no	—
434	C₄	—	4	8	no	—
435	C₄	—	4	4	no	—
436	C₄	C₄×C₂²⋊C₄×C₆₇	1	2	yes	—
437	C₄	C₄×C₂²⋊C₄×C₆₇	1	2	yes	—
438	C₄	C₂²×C₄²×C₆₇	1	1	yes	characteristic
439	C₄	C₂×#117×C₆₇	1	1	yes	characteristic
440	C₄	C₂×C₄×D₄×C₆₇	1	2	yes	—
441	C₂	—	2	2	no	—
442	C₂	—	2	2	no	—
443	C₂	—	4	4	no	—
444	C₂	C₆₇×#1759	1	4	yes	direct product factor
445	C₂	C₂×C₄×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
446	C₂	C₂×#421×C₆₇	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂×D₄
commutator	C₄	C₂²×C₄²×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂³×C₆₇	C₂×C₂²⋊C₄
prime radical	C₂²×#41×C₆₇	C₂²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₄×M₁₆×C₆₇	C₂	semidirect
4. C₂×#376×C₆₇	C₂	semidirect
5. C₆₇×#1759	C₂	direct (N × H)
6. C₂×#339×C₆₇	C₂	semidirect
11. C₄×#41×C₆₇	C₄	semidirect
13. C₂×C₄×Q₈×C₆₇	C₄	semidirect
15. C₂×C₄×D₄×C₆₇	C₄	semidirect
21. C₂×#104×C₆₇	C₄	semidirect
24. C₄×M₁₆×C₆₇	C₂²	semidirect
28. #376×C₆₇	C₂²	semidirect
30. #339×C₆₇	C₂²	semidirect
47. C₂×M₁₆×C₆₇	C₂×C₄	semidirect
51. C₄×Q₈×C₆₇	C₂×C₄	semidirect
55. C₄×D₄×C₆₇	C₂×C₄	semidirect
74. C₂×C₄²×C₆₇	C₂×C₄	semidirect
81. #104×C₆₇	C₂×C₄	semidirect
129. M₁₆×C₆₇	C₂²×C₄	semidirect
147. C₄²×C₆₇	C₂²×C₄	semidirect
217. C₂×#1759	C₆₇	direct (N × H)
221. C₂×C₆₇	#1759	direct (N × H)
226. C₂×C₄×M₁₆	C₂×C₆₇	semidirect
227. C₂×#376	C₂×C₆₇	semidirect
228. #1759	C₂×C₆₇	direct (N × H)
229. C₂×#339	C₂×C₆₇	semidirect
230. C₆₇	C₂×#1759	direct (N × H)
235. C₄×#41	C₄×C₆₇	semidirect
237. C₂×C₄×Q₈	C₄×C₆₇	semidirect
239. C₂×C₄×D₄	C₄×C₆₇	semidirect
245. C₂×#104	C₄×C₆₇	semidirect
248. C₄×M₁₆	C₂²×C₆₇	semidirect
252. #376	C₂²×C₆₇	semidirect
254. #339	C₂²×C₆₇	semidirect
271. C₂×M₁₆	C₂×C₄×C₆₇	semidirect
275. C₄×Q₈	C₂×C₄×C₆₇	semidirect
279. C₄×D₄	C₂×C₄×C₆₇	semidirect
298. C₂×C₄²	C₂×C₄×C₆₇	semidirect
305. #104	C₂×C₄×C₆₇	semidirect
353. M₁₆	C₂²×C₄×C₆₇	semidirect
371. C₄²	C₂²×C₄×C₆₇	semidirect
444. C₂	C₆₇×#1759	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×C₄×#41×C₆₇	2
C₂²×#104×C₆₇	2
C₂×C₄×M₁₆×C₆₇	2
C₂×#376×C₆₇	2
C₆₇×#1759	2
C₂×#339×C₆₇	2
C₂×#1759	67

Subgroups — C2×C67×#1759 #30699654

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#1759 #30699654